Quantum physics has come a long way

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1 Vol 453 Issue no June 28 QUANTUM COHERENCE Cover illustrtion The entngling of toms through spin coupling in doule-well potentil (Courtesy of I. Bloch) Editor, Nture Philip Cmpell Insights Pulisher Steven Inchcoome Insights Editor Krl Ziemelis Production Editor Dvin Ddley-Moore Senior Art Editor Mrtin Hrrison Art Editor Nik Spencer Sponsorship Amélie Pequignot Production Jocelyn Hilton Mrketing Kty Dunninghm Elen Woodstock Editoril Assistnt Alison McGill Quntum physics hs come long wy since its theoreticl eginnings in the erly twentieth century. Techniques to mnipulte light nd mtter hve ecome incresingly sophisticted, fcilitting fundmentl studies of quntum effects nd inspiring new technologies. From tomic networks to semiconductor spintronics, seemingly disprte res of reserch re eing driven y shred gol to hrness nd exploit quntum coherence nd entnglement. Inevitly, these lortory endevours hve necessitted new theoreticl toolox. The imge of pir of photons zooming off in opposite directions, ech sensitive to the other through their quntum entnglement, is conceptully tidy. But wht hppens when descriing the quntum properties of more complex systems? This Insight on quntum coherence nd entnglement strts with Progress rticle tht ddresses the prolem of thinking ig : how cn entnglement e quntified or mesured in system tht comprises mny prticles nd degrees of freedom? The reviews in this Insight highlight the exciting experimentl progress in such systems, covering wide rnge of physicl settings. They descrie oth ottom-up pproches, in which reserchers strive to chieve incresingly complex systems strting from very smll numer of prticles nd degrees of freedom, nd top-down pproches, in which the individul nd collective degrees of freedom in lrger systems re controlled. Ultimtely, the gol is to control mny-prticle systems t the quntum limit, n ttrctive prospect for quntum simultion nd informtion pplictions. As such, this Insight rings together vried reserch. We trust, however, tht you will find coherence in this diversity. PROGRESS 14 Quntifying entnglement in mcroscopic systems V. Vedrl REVIEWS 18 Entngled sttes of trpped tomic ions R. Bltt & D. Winelnd 116 Quntum coherence nd entnglement with ultrcold toms in opticl lttices I. Bloch 123 The quntum internet H. J. Kimle 131 Superconducting quntum its J. Clrke & F. K. Wilhelm 143 Coherent mnipultion of single spins in semiconductors R. Hnson & D. D. Awschlom Kren Southwell, Senior Editor 13

2 INSIGHT PROGRESS NATURE Vol June 28 doi:1.138/nture7124 Quntifying entnglement in mcroscopic systems Vltko Vedrl 1,2,3 Trditionlly, entnglement ws considered to e quirk of microscopic ojects tht defied common-sense explntion. Now, however, entnglement is recognized to e uiquitous nd roust. With the reliztion tht entnglement cn occur in mcroscopic systems nd with the development of experiments imed t exploiting this fct new tools re required to define nd quntify entnglement eyond the originl microscopic frmework. In the pst decde, there hs een n explosion of interest in entnglement in mcroscopic (mny ody) physicl systems 1. The trnsformtion in how entnglement is perceived hs een remrkle. In less thn century, reserchers hve moved from distrusting entnglement ecuse of its spooky ction t distnce to strting to regrd it s n essentil property of the mcroscopic world. There re three sic motivtions for studying entnglement in the mcroscopic world. The first motivtion is fundmentl. Reserchers wnt to know whether lrge ojects cn support entnglement. The conventionl wisdom is tht system tht consists of lrge numer of susystems (for exmple, 1 26 of them, similr to the numer of toms in living room) immersed in n environment t high temperture (room temperture, for exmple) ought to ehve fully clssiclly. Studying mcroscopic entnglement is thus wy of proing the quntum-to-clssicl trnsition. The second motivtion is physicl nd reltes to the different phses of mtter. Trditionlly, the ide of n order prmeter is used to quntify phse trnsitions. For exmple, non-mgnetic system (in the dis ordered phse ) cn e mgnetized (or ecome ordered) in certin cond itions, nd this trnsition is indicted y n rupt chnge in the order prmeter of the system. In this cse, the mgnetiztion itself is relevnt order prmeter, ut the interesting question is whether entngle ment is useful order prmeter for other phse trnsitions 2,3. The third motivtion comes from technology. If the power of entnglement is to e hrnessed through quntum computing, then entngled systems of incresingly lrge sizes need to e hndled, which is itself chllenge. It is cler tht the modern perspective on entnglement differs gretly from the initil ides out its seemingly prdoxicl nture. Reserchers re now relizing how generl nd roust entnglement is. Lrger nd lrger entngled systems re eing mnipulted coherently in different physicl implementtions. And it is not s surprising s it once ws to find tht entnglement contriutes to some phenomen. Not ll of the mystery hs vnished, however. As is common in scientific reserch, nswering one question genertes mny new ones, in this cse relted to the type of entnglement tht is useful for studies motivted y ech of the three resons ove. These questions ring reserchers closer to the hert of the current understnding of entnglement. Here I first exmine wht entnglement is nd how it is quntified in physicl systems. Different clsses of entnglement re then discussed, nd I conclude y considering the possiilities of chieving nd exploiting lrge-scle entnglement in the lortory. Wht is entnglement? The first chpter of lmost ny elementry quntum-mechnics textook usully sttes tht quntum ehviour is not relevnt for systems with physicl size much lrger thn their de Broglie wvelength. The de Broglie wvelength, which cn intuitively e thought of s the quntum extent of the system, scles inversely s (the squre root of) mss Lser light HV + VH Figure 1 A wy of generting entngled photons y using down conversion. The input lser light is shone onto nonliner crystl (green ox). The nonlinerity of the crystl mens tht there is non-zero proility tht two photons will e emitted from the crystl. The cones represent the regions where ech of the two photons is emitted. Owing to energy conservtion, the frequencies of the photons need to dd up to the originl frequency. Their moment lso must cncel in the perpendiculr direction nd dd up to the originl momentum in the forwrd direction. One of the photons is horizontlly polrized (H), nd one is verticlly polrized (V). However, in the regions where the two cones overlp, the stte of the photons will e HV + VH. It is round these points tht entngled photons re generted. 1 School of Physics nd Astronomy, University of Leeds, Leeds LS2 9JT, UK. 2 Centre for Quntum Technologies, Ntionl University of Singpore, 3 Science Drive 2, Singpore Deprtment of Physics, Ntionl University of Singpore, 2 Science Drive 3, Singpore

3 NATURE Vol June 28 INSIGHT PROGRESS times temperture. From this, it cn e concluded tht mssive nd hot systems which could lmost e considered s synonymous with mcroscopic systems should not ehve quntum mechniclly. As I show in the next section, however, de-broglie-type rguments re too simplistic. First, entnglement cn e found in mcroscopic systems 4 (including t high tempertures 5 ). And, second, entnglement turns out to e crucil for explining the ehviour of lrge systems 6. For exmple, the low vlues of mgnetic susceptiilities in some mgnetic systems cn e explined only y using entngled sttes of those systems. Now, wht exctly is entnglement? After ll is sid nd done, it tkes (t lest) two to tngle 7, lthough these two need not e prticles. To study entnglement, two or more susystems need to e identified, together with the pproprite degrees of freedom tht might e entngled. The susystems re techniclly known s modes, nd the possily entngled degrees of freedom re clled oservles. Most formlly, entnglement is the degree of correltion etween oservles pertining to different modes tht exceeds ny correltion llowed y the lws of clssicl physics. I now descrie severl exmples of entngled systems. Two photons tht hve een generted y, for exmple, prmetric down conversion 8 re in the overll polrized stte HV + VH (where H is horizontl polriztion nd V is verticl polriztion) nd re entngled s fr s their polriztion is concerned (Fig. 1). A photon is n excittion of the electromgnetic field, nd its polriztion denotes the direction of the electric field. Ech of the two entngled photons represents susystem, nd the relevnt oservles re the polriztions in different directions. (Two electrons could lso e entngled in terms of their spin vlue in n nlogous wy.) When two susystems in pure sttes ecome entngled, the overll stte cn no longer e written s product of the individul sttes (for exmple, HV ). A pure stte mens tht the informtion out how the stte ws prepred is complete. A stte is clled mixed if some knowledge is lcking out the detils of system preprtion. For exmple, if the pprtus prepres either the ground stte or the first excited stte 1 in rndom mnner, with respective proilities p nd 1 p, then the overll stte will need to e descried s the mixture p + (1 p) 1 1. In this cse, the proilities need to e used to descrie the stte ecuse of the lck of knowledge. Consequently, quntifying entnglement for mixed sttes is complex. Systems cn lso e entngled in terms of their externl degrees of freedom (such s in sptil prmeters). For exmple, two prticles could hve their positions nd moment entngled. This ws the originl mening of entnglement, s defined y Alert Einstein, Boris Podolsky nd Nthn Rosen 9. When the susystems hve een identified, sttes re referred to s entngled when they re not of the disentngled (or seprle) form 1 : ρ sep = Σ i p i ρ i 1 ρ i 2 ρ i n, where Σ i p i = 1 is proility distriution nd ρ i 1, ρ i 2,, ρ i n re the sttes (generlly mixed) of susystem 1, 2,, n, respectively. On the one hnd, two susystems descried y the density mtrix ρ 12 = ½( ) re one such exmple of seprle stte. The stte of three susystems, + 111, on the other hnd, cn esily e confirmed to e not seprle nd therefore (y definition) entngled. This simple mthemticl definition hides gret del of physicl sutlety. For exmple, Bose Einstein condenstes re creted when ll prticles in system go into the sme ground stte. It seems tht the overll stte is just the product of the individul prticle sttes nd is therefore (y definition) disentngled. However, in this cse, entnglement lies in the correltions etween prticle numers in different sptil modes. Systems cn lso seem to e entngled ut, on closer inspection, re not (Fig. 2). Witnesses nd mesures of entnglement In this section, I present two surprising results from recent studies of mny-ody entnglement: first, entnglement cn e witnessed y mcroscopic oservles 11,12 (see the susection Witnessing entnglement ); nd second, entnglement cn persist in the thermodynmic limit t ritrrily high tempertures 13. The first sttement is surprising ecuse oservles represent verges over ll susystems, so it is expected tht entnglement disppers s result of this verging. The effect of temperture is similr. Incresing the temperture mens tht n incresing mount of noise is dded to the entnglement, so the second finding tht entnglement cn persist t high tempertures is lso surprising. Before these findings re descried in more detil, simple oservtion cn e mde. The entnglement of two susystems in pure stte is very esy to quntify. This is ecuse the more entngled the stte, the more mixed the suset of the system. This property of quntum sttes nmely tht lthough exct informtion out the overll stte is ville, informtion out prts of the system cn e incomplete ws first emphsized y Erwin Schrödinger 14, in the fmous pper in which he descried the Schrödinger s ct thought experiment. This logic fils for mixed sttes, however. For exmple, n equl mixture of nd 11 lso results in mximlly mixed sttes for ech quntum it (quit), ut the overll stte is not entngled. It lso fils for quntifying quntum correltions etween more thn two components. In fct, in this lst cse, it is even difficult to determine whether stte of mny susystems is entngled in the first plce. This leds on to the concept of witnessing entnglement. Witnessing entnglement Entnglement witnesses 15 re oservles whose expecttion vlue cn indicte something out the entnglement in given stte. Suppose tht there is n oservle W, which hs the property tht for ll dis entngled sttes, the verge vlue is ounded y some numer, Figure 2 Seprle sttes. Two exmples of disentngled systems re shown., Two electrons re shown confined to two sptil regions nd with their internl spins pointing up. In this cse, their spin sttes re oth in the sme upwrds direction. Becuse electrons re fermions, the overll stte of this system must e ntisymmetricl. The internl stte is symmetricl (ecuse the electrons re pointing up), nd so their specil wvefunctions must e ntisymmetrized, Ψ 1 Ψ 2 Ψ 2 Ψ 1. The sptil prt of the electronic stte, therefore, seems to e entngled ut this only seems to e the cse. The electrons in question re fully distinguishle (ecuse they re fr prt), so ny experiment on one of them is not correlted to ny experiment on the other one. Therefore, these electrons cnnot e entngled., A doule-well potentil, with ech well contining five prticles, is shown. Experiments tht trp toms in this wy re now routine. It is cler tht there is no entnglement etween the two wells, ecuse ech well contins fixed, clerly defined numer of prticles, lthough there could e some entnglement within ech well, depending on the exct circumstnces. 15

4 INSIGHT PROGRESS NATURE Vol June 28 χ Mgnetic susceptiility (cgs mol 1 ) Temperture (K) Figure 3 Susceptiility s mcroscopic witness of entnglement., The typicl ehviour of mgnetic susceptiility χ versus temperture for mgnetic system is depicted (lck). The ehviour of the entnglement witness is lso depicted (red). Vlues of mgnetic susceptiility elow the red line re entngled, nd the dshed line indictes the trnsitionl point., One of the erliest experimentl confirmtions of entnglement 6 involved copper nitrte, Cu(NO 3 ) 2 2.5H 2 O, with entnglement existing t less thn ~5 K. A moleculr imge of copper nitrte is depicted, with copper in red, nitrogen in lue, oxygen in lck nd hydrogen in green. The one-dimensionl chin (red), which consists of intercting copper toms, is the physiclly relevnt chin in terms of the mgnetic properties of the compound nd cn e thought of s collection of dimers (shown seprted y dshed red lines). W. Suppose, furthermore, tht resercher is given physicl stte nd experimentlly shows tht W >, then the only explntion is tht the stte is entngled. Imgine two spins ( dimer ) coupled through Heisenerg interction 4 : H = Jσ σ, where H is the hmiltonin, σ denotes Puli spin mtrix nd J is the strength of coupling. I now use the hmiltonin s n entnglement witness. It is esy to see tht the verge vlue of H with respect to disentngled (seprle) sttes cnnot exceed the vlue J: H = Tr(ρ sep H) = J σ σ J. However, if the expected vlues re computed for the singlet stte (which is the ground stte of H), then the following is otined: Tr(ρ sin H) = 3J, where ρ sin is the density mtrix of the singlet stte. This vlue is clerly outside the rnge of seprle sttes. The singlet is therefore entngled. This logic generlizes to more complex hmiltonins (with ritrrily mny prticles), nd it cn e shown tht oservles other thn energy (for exmple, mgnetic susceptiility) cn e good witnesses of entnglement 1 (Fig. 3). In fct, y using this method, ground sttes of ntiferromgnets, s well s other intercting systems, cn generlly e shown to e entngled t low tempertures (k B T J, where k B is the Boltzmnn constnt nd T is temperture; this seems to e universl temperture ound for the existence of entnglement 16. Mesuring entnglement Mesuring entnglement is complex, nd there re mny pproches 17. Here I discuss two mesures of entnglement: overll entnglement nd connectivity. Further mesures re descried in ref. 17. The first mesure is overll entnglement, lso known s the reltive entropy of entnglement 18, which is mesure of the difference etween given quntum stte nd ny clssiclly correlted stte. It turns out tht the est pproximtion to the Greenerger Horne Zeilinger (GHZ) 19 stte, + 111, is mixture of the form For W sttes 2 (y which I men ny symmetricl superposition of zeros nd ones, such s ), the est clssicl pproximtion is slightly more elorte mixture 5. On the sis of overll entnglement, W sttes re more entngled thn GHZ sttes. Wht is the most entngled stte of N quits ccording to the overll entnglement? The nswer is tht the mximum possile overll entnglement is N/2, nd one such stte tht chieves this (y no mens the only one) is collection of dimers (tht is, mximlly entngled pirs of quits). This is esy to understnd when considering tht ech dimer hs one unit of entnglement nd tht there re N/2 dimers in totl. The second mesure (originlly termed disconnectivity) is referred to here s connectivity 21. Mesuring connectivity is designed to ddress the question of how fr correltions stretch. Tke GHZ stte of N quits, It is cler tht the first two quits re s correlted s the first nd the third nd, in fct, s the first nd the lst. Correltions of GHZ sttes therefore hve long rnge. GHZ sttes hve connectivity equl to N. The W stte, y contrst, cn e well descried y nerest-neighour correltions. The W stte contining N/2 zeros nd N/2 ones cn e well pproximted y the sttes etween nerest neighours. Therefore, correltions do not stretch fr, nd the connectivity is only equl to 2. The ove considertions of how to quntify entnglement re generl nd pply to ll discrete (spin) systems, s well s to continuous systems (such s hrmonic chins 22 nd quntum fields 23 ), lthough continuous systems need to e treted with extr cre ecuse of their infinite dimensionlity. Although the discussed witnesses nd mesures cn e pplied to mixed sttes, I now focus on pure sttes for simplicity. Different types of mcroscopic entnglement There re mny types of entnglement. Here I discuss the four types tht cover ll three motivtions mentioned erlier: GHZ, W, reson ting vlence ond (RVB) 24 nd cluster 25. GHZ sttes re typiclly used in testing the non-loclity of quntum mechnics, ecuse they hve high vlue of connectivity. W nd RVB sttes nturlly occur for rnge of physicl systems. For exmple, oth Bose Einstein condenstes (such s superfluid nd superconducting mterils) nd ferromgnets hve W sttes s ground sttes 5. RVB sttes re uilt from singlet sttes etween pirs of spins. It is cler tht connectivity of RVBs is only 2, ut the sttes themselves hve high overll degree of entnglement, N/2 (ref. 26). It is intriguing tht nturl sttes hve low connectivity ut high overll entnglement tht scles s log N or even N/2, wheres GHZ sttes, which do not occur nturlly, hve high connectivity of the order of N ut very low overll entnglement (Box 1). Are there sttes tht hve oth connectivity nd overll entnglement tht scle s the numer of susystems? The nswer is, surprisingly, yes. Even more interestingly, these sttes, which re known s cluster sttes, re importnt for quntum computing 25. Cluster sttes re highly entngled rrys of quits, nd this entnglement is used to crry out quntum computing through single quit mesurements. Entnglement drives the dynmics of these computers 27, which is why high overll entnglement is needed. But the type of entnglement is lso responsile for the implementtion of vrious gtes during the opertion of these computers, which is why high connectivity is needed. Experimentl considertions nd eyond There re mny pths to prepring nd experimenting with lrger collections of entngled systems. As I hve descried, nturl entnglement is not strong in generl nd is fr from eing mximl with respect to overll entnglement or connectivity. To crete high overll entnglement nd connectivity invrily involves gret del of effort. There re two sic pproches to generting lrge-scle entnglement: ottom up nd top down. The first pproch, the ottom-up pproch, relies on gining precise control of single system first nd then extending this control to two systems nd scling it up further. So fr, ottom-up experiments hve otined up to eight entngled ions in n ion trp (in W stte) 28 nd six entngled photons 29. During nucler mgnetic resonnce spectroscopy, 13 nuclei cn e pseudoentngled 3. Lrger systems, however, re exceedingly difficult to control in this wy. The second pproch is the top-down pproch. As descried erlier, mny nturl systems, with mny degrees of freedom (1 million toms, for exmple), cn ecome entngled without the need for difficult mnipultions (for exmple, the only requirement might e to decrese the temperture to less thn 5 K, which is physiclly possile). Moreover, in mny systems, certin types of entnglement re present in therml equilirium nd even ove room temperture, without the need for ny mnipultion. 16

5 NATURE Vol June 28 INSIGHT PROGRESS Box 1 Comprison of four types of entngled quit stte 3. Osorne, T. J. & Nielsen, M. A. Entnglement in simple quntum phse trnsition. Phys. Rev. A 66, 3211 (22). Quit stte Overll entnglement Connectivity GHZ 1 N W Log N 2 RVB N/2 2 Cluster N/2 N The nturlly occurring sttes, W nd RVB, hve much smller connectivity thn the sttes used for testing non-loclity (GHZ) nd for crrying out universl computing (cluster). In contrst to connectivity, the overll entnglement shows different scling. The importnt point is tht the overll entnglement nd connectivity cpture mrkedly different spects of the quntumness of mcroscopic sttes. Both of these mesures cn e thought of in terms of frgility of the entngled stte, ut they descrie different types of frgility. The connectivity is relted to the frgility of the stte under dephsing: tht is, the loss of phses etween vrious components in the superposition. The overll entnglement, y contrst, is relted to the frgility of the stte under the full removl of quits from the stte. For exmple, if one quit is removed from the GHZ stte, then the remining quits utomticlly ecome disentngled, which is why the overll entnglement of the GHZ stte is equl to 1. If ech quit dephses t the rte r, then N quits in GHZ sttes will dephse t the rte Nr, which is why the connectivity of the GHZ stte is N. By contrst, for RVB sttes, there re N/2 singlets, so hlf of the quits need to e removed to destroy entnglement. Similrly, this stte is not mrkedly susceptile to dephsing, indicting low vlue of connectivity. Given tht mcroscopic entnglement exists, n importnt technologicl question is how esy this entnglement would e to extrct nd use. Suppose tht two neutron ems re imed t mgnetic sustnce, ech t different section 31. It is fruitful leit not entirely mthemticlly precise to think of this interction s stte swp of the spins of the neutrons nd the spins of the toms in the solid. If the toms in the solid re themselves entngled, then this entnglement is trnsferred to ech of the scttered spins. This trnsferrl could then presumly e used for further informtion processing. Similrly, schemes cn e designed to extrct entnglement from Bose Einstein condenstes 32,33 nd superconductors 34 (which cn e thought of s Bose Einstein condenstes of Cooper pirs of electrons), lthough none of these extrction schemes hs een implemented s yet. There re mny open questions regrding entnglement. Here I hve stted tht, in theory, entnglement cn exist in ritrrily lrge nd hot systems. But how true is this in prctice? Another question is whether the entnglement of mssless odies fundmentlly differs from tht of mssive ones 35. Furthermore, does mcroscopic entnglement lso occur in living systems nd, if so, is it used y these systems? Some of the open questions might never e nswered. Some might turn out to e uninteresting or irrelevnt. One thing is certin though: current experimentl progress is so rpid tht future findings will surprise reserchers nd will tke the current knowledge of entnglement to nother level. 1. Amico, L., Fzio, R., Osterloh, A. & Vedrl, V. Mny-ody entnglement. Rev. Mod. Phys. 8, (28). 2. Osterloh, A. et l. Scling of entnglement close to quntum phse trnsition. Nture 416, (22). 4. Arnesen, M. C., Bose, S. & Vedrl, V. Nturl therml nd mgnetic entnglement in 1D Heisenerg model. Phys. Rev. Lett. 87, 1791 (21). 5. Vedrl, V. High temperture mcroscopic entnglement. New J. Phys. 6, 12 (24). 6. Brukner, C., Vedrl, V. & Zeilinger, A. Crucil role of entnglement in ulk properties of solids. Phys. Rev. A 73, 1211 (26). 7. Vedrl, V. A etter thn perfect mtch. Nture 439, 397 (26). 8. Zeilinger, A., Weihs, G., Jennewein, T. & Aspelmeyer, M. Hppy centenry, photon. Nture 433, (25). 9. Einstein, A., Podolsky, B. & Rosen, N. Cn quntum-mechnicl description of physicl relity e considered complete? Phys. Rev. 47, (1935). 1. Werner, R. F. Quntum sttes with Einstein Podolsky Rosen correltions dmitting hidden-vrile model. Phys. Rev. A 4, (1989). 11. Brukner, C. & Vedrl, V. Mcroscopic thermodynmicl witnesses of quntum entnglement. Preprint t < (24). 12. Toth, G. & Guhne, O. Detecting genuine multiprtite entnglement with two locl mesurements. Phys. Rev. Lett. 94, 651 (24). 13. Nrnhoffer, H. Seprility for lttice systems t high temperture. Phys. Rev. A 71, (25). 14. Schrödinger, E. Die gegenwärtige Sitution in der Quntenmechnik. Nturwissenschften 23, ; ; (1935). 15. Horodecki, M., Horodecki, P. & Horodecki, R. Seprility of mixed sttes: necessry nd sufficient conditions. Phys. Lett. A 223, 1 8 (1996). 16. Anders J. & Vedrl, V. Mcroscopic entnglement nd phse trnsitions. Open Sys. Inform. Dyn. 14, 1 16 (27). 17. Horodecki, M. Entnglement mesures. Qunt. Inform. Comput. 1, 3 26 (21). 18. Vedrl V. et l. Quntifying entnglement. Phys. Rev. Lett. 78, (1997). 19. Greenerger, D., Horne, M. A. & Zeilinger, A. in Bell s Theorem, Quntum Theory, nd Conceptions of the Universe (ed. Kftos, M.) (Kluwer Acdemic, Dordrecht, 1989). 2. Dur, W., Vidl, G. & Circ, J. I. Three quits cn e entngled in two inequivlent wys. Phys. Rev. A 62, (2). 21. Leggett, A. J. Mcroscopic quntum systems nd the quntum theory of mesurement. Prog. Theor. Phys. Suppl. 69, 8 1 (198). 22. Anders, J. & Winter, A. Entnglement nd seprility of quntum hrmonic oscilltor systems t finite temperture. Qunt. Inform. Comput. 8, (28). 23. Vedrl, V. Entnglement in the second quntistion formlism. Cent. Eur. J. Phys. 2, (23). 24. Anderson, P. W. Resonting vlence onds: new kind of insultor? Mter. Res. Bull. 81, 53 6 (1973). 25. Russendorf, R. & Briegel, H. J. A one-wy quntum computer. Phys. Rev. Lett. 86, (21). 26. Chndrn, A., Kszlikowski, D., Sen De, A., Sen, U. & Vedrl, V. Regionl versus glol entnglement in resonting-vlence-ond sttes. Phys. Rev. Lett. 99, 1752 (27). 27. Pge, D. N. & Wootters, W. K. Evolution without evolution: dynmics descried y sttionry oservles. Phys. Rev. D 27, (1983). 28. Hffner, H. et l. Sclle multiprticle entnglement of trpped ions. Nture 438, (25). 29. Lu, C.-Y. et l. Experimentl entnglement of six photons in grph sttes. Nture Phys. 3, (27). 3. Bugh, J. et l. Quntum informtion processing using nucler nd electron mgnetic resonnce: review nd prospects. Preprint t < (27). 31. de Chir, G. et l. A scheme for entnglement extrction from solid. New J. Phys. 8, 95 (26). 32. Toth, G. Entnglement detection in opticl lttices of osonic toms with collective mesurements. Phys. Rev. A 69, (24). 33. Heney, L., Anders, J., Kszlikowski, D. & Vedrl, V. Sptil entnglement from off-digonl long-rnge order in Bose Einstein condenste. Phys. Rev. A 76, 5365 (27). 34. Recher, P. & Loss, D. Superconductor coupled to two Luttinger liquids s n entngler for spin electrons. Phys. Rev. B 65, (22). 35. Verstrete, F. & Circ, J. I. Quntum nonloclity in the presence of superselection rules nd dt hiding protocols. Phys. Rev. Lett. 91, 144 (23). Acknowledgements I m grteful for funding from the Engineering nd Physicl Sciences Reserch Council, the Wolfson Foundtion, the Royl Society nd the Europen Union. My work is lso supported y the Ntionl Reserch Foundtion (Singpore) nd the Ministry of Eduction (Singpore). I thnk J. A. Dunninghm, A. J. Leggett, D. Mrkhm, E. Rieper, W. Son nd M. Willimson for discussions of this nd relted sujects. W. Son s help with illustrtions is lso grtefully cknowledged. Author Informtion Reprints nd permissions informtion is ville t npg.nture.com/reprints. The uthor declres no competing finncil interests. Correspondence should e ddressed to the uthor (vltko.vedrl@quntuminfo.org). 17

6 INSIGHT REVIEW NATURE Vol June 28 doi:1.138/nture7125 Entngled sttes of trpped tomic ions Riner Bltt 1,2 & Dvid Winelnd 3 To process informtion using quntum-mechnicl principles, the sttes of individul prticles need to e entngled nd mnipulted. One wy to do this is to use trpped, lser-cooled tomic ions. Attining generl-purpose quntum computer is, however, distnt gol, ut recent experiments show tht just few entngled trpped ions cn e used to improve the precision of mesurements. If the entnglement in such systems cn e scled up to lrger numers of ions, simultions tht re intrctle on clssicl computer might ecome possile. For more thn five decdes, quntum superposition sttes tht re coherent hve een studied nd used in pplictions such s photon interferometry nd Rmsey spectroscopy 1. However, entngled sttes, prticulrly those tht hve een engineered or creted for specific tsks, hve ecome routinely ville only in the pst two decdes (see pge 14). The initil experiments with pirs of entngled photons 2,3, strting in the 197s, were importnt ecuse they provided tests of non-loclity in quntum mechnics 4. Then, in the erly to mid-198s, Richrd Feynmn nd Dvid Deutsch proposed tht it might e possile wy to crry out certin computtions or quntum simultions efficiently y using quntum systems 5,6. This ide ws, however, lrgely considered curiosity until the mid-199s, when Peter Shor devised n lgorithm 7 tht could fctor lrge numers very efficiently with quntum computer. This mrked the eginning of widespred interest in quntum informtion processing 8 nd stimulted severl pro posls for the implementtion of quntum computer. Among these proposls, the use of trpped ions 9 hs proved to e one of the most successful wys of deterministiclly creting entngled sttes, nd for mnipulting, chrcterizing nd using these sttes for mesurement. At present, out 25 lortories worldwide re studying spects of quntum informtion processing with trpped ions. Ions provide reltively clen system, ecuse they cn e confined for long durtions while experiencing only smll perturtions from the environment, nd cn e coherently mnipulted. Although trpping ions in this wy involves severl technicl processes, the system is n ccessile one in which to test concepts tht might e pplicle to other systems, such s those involving neutrl trpped toms, quntum dots, nucler spins, Josephson junctions or photons. In this review, we highlight recent progress in creting nd mnipulting entngled sttes of ions, nd we descrie how these dvnces could help to generte quntum gtes for quntum informtion processing nd improve tools for high-precision mesurement. For review of erlier progress in quntum informtion processing with toms, including tomic ions, nd photons, see ref. 1. Trpped nd lser-cooled ions To study entnglement, it is desirle to hve collection of quntum systems tht cn e individully mnipulted, their sttes entngled, nd their coherences mintined for long durtions, while suppressing the detrimentl effects of unwnted couplings to the environment. This cn e relized y confining nd lser cooling group of tomic ions in prticulr rrngement of electric nd/or mgnetic fields 11,12. With such trps, tomic ions cn e stored nerly indefinitely nd cn e loclized in spce to within few nnometres. Coherence times of s long s ten minutes hve een oserved for superpositions of two hyperfine tomic sttes of lser-cooled, trpped tomic ions 13,14. In the context of quntum informtion processing, typicl experiment involves trpping few ions y using comintion of sttic nd sinusoidlly oscillting electric potentils tht re pplied etween the electrodes of liner qudrupole, n rrngement known s Pul trp 12 (Fig. 1). When the trpped ions re lser cooled, they form liner string, in which the spcings re determined y lnce etween the horizontl (xil) confining fields nd mutul Coulom repulsion. Scttered fluorescence, induced y lser em, cn e imged with cmer (Fig. 1). The use of tightly focused lser ems llows the mnipultion of individul ions. For simplicity, in this review, we focus on two specific internl sttes of ech ion, which we refer to s the ground nd excited sttes ( g nd e, respectively). This quntum it (quit) structure is dressed y the oscilltor sttes n of frequency ω m of prticulr mode (Fig. 1). We denote the internl sttes s spin sttes, in nlogy to the two sttes of spin ½ prticle. If the energy etween internl sttes corresponds to n opticl frequency ω eg, this tomic trnsition cn e driven y lser rdition t frequency ω eg, which couples sttes g, n e, n, where g, n denotes the comined stte g n. Spin nd motionl degrees of freedom cn e coupled y tuning the lser to sidend frequencies ω eg ± ω m, which drives trnsitions g, n e, n + Δn (refs 15 18), with Δn = ±1. In this cse, stte evolution cn e descried s rottion R Δn (θ, ϕ) of the stte vector on the Bloch sphere 8,18 nd is defined here s R Δn (θ, ϕ) g, n cos θ g, n + ie iϕ sin θ e, n + Δn 2 2 R θ θ Δn (θ, ϕ) e, n + Δn ie iϕ sin g, n + cos e, n + Δn (1) 2 2 where θ depends on the strength nd the durtion of the pplied lser pulse, ϕ is the lser em phse t the ion s position nd i = 1. For Δn = ±1, entnglement is generted etween the spin nd motionl degrees of freedom. Higher-order couplings ( Δn > 1) re suppressed for lser-cooled ions, the sptil extent of which is much smller thn the lser wvelength, which is known s the Lm Dicke regime. In this regime, sidend lser cooling works y tuning the lser to induce sorption on the lower sidend frequency (Δn = 1), followed y spontneous emission decy, which occurs minly t the crrier 1 Institut für Experimentlphysik, Universität Innsruck, Technikerstrsse 25, A-62 Innsruck, Austri. 2 Institut für Quntenoptik und Qunteninformtion, Österreichische Akdemie der Wissenschften, Otto-Hittmir-Pltz 1, A-62 Innsruck, Austri. 3 Ntionl Institute of Stndrds nd Technology, 325 Brodwy, Boulder, Colordo 835, USA. 18

7 NATURE Vol June 28 INSIGHT REVIEW trnsition frequency (Δn = ). With repeted sorption emission cycles, the ions re opticlly pumped to the comined spin nd motion ground stte g, n = (ref. 19). If the spin energy levels correspond to microwve or lower frequencies (s occurs in hyperfine tomic sttes nd Zeemn sttes), the sme processes cn e relized y replcing single-photon opticl trnsitions with two-photon stimulted-rmn trn sitions nd y replcing spontneous emission with spontneous Rmn scttering It should e noted tht there re similrities etween the coupling of n ion s internl sttes to the hrmonic oscilltor ssocited with mode of motion nd the cse of cvity quntum electrodynmics, in which n tom s internl sttes re coupled to the hrmonic oscilltor ssocited with single electromgnetic mode of the cvity (see pge 123). The quit stte of n ion cn e detected with more thn 99% efficiency y mesuring resonnce fluorescence from n uxiliry stte tht is strongly coupled (y monitoring excittion) to one of the quit sttes ( g or e ) nd decys ck only to tht sme stte, known s cycling trnsition. This is usully clled quntum non-demolition (QND) detection ecuse when the ion hs een projected into prticulr spin stte, it will remin in tht stte throughout repeted excittion emission cycles. Therefore, cycle cn e repeted mny times, nd it is not necessry to detect every emitted photon to otin high overll detection efficiency. If the quit is projected to, or shelved in, the stte tht is not coupled to the fluorescing trnsition, then no photons re oserved, nd this stte cn therefore e distinguished from the fluorescing stte 2. Spin-entngled sttes In 1995, Igncio Circ nd Peter Zoller suggested how to use trppedion system to implement quntum computer 9. For universl quntum computing nd for the genertion of ritrry entn gled quit sttes, two sic gte opertions re required: first, individul quit rottions s descried y eqution (1); nd, second, two-quit-entngling opertion tht is the quntum counterprt to the clssicl opertion with the XOR logic gte, the controlled-not (CNOT)-gte opertion. The CNOT gte flips the stte of trget quit depending on the stte of control quit. And, importntly, when pplied to superposition sttes, it genertes entnglement. The CNOT opertion (Fig. 2) is chieved with sequence of crrier pulses (R (θ, ϕ)) nd red sidend pulses (R 1 (θ, ϕ)). The centrl prt of this sequence involves phse gte tht pplies phse shift e iπ = 1 to the g, n = 1 component of the trget ion s wvefunction. This is implemented y pplying coherent R 1 (2π, ϕ) pulse etween the g, 1 stte nd n uxiliry stte ux,. Becuse the pplied rdition cnnot excite the sttes g,, e, or e, 1, they re unf fected. This opertion is sndwiched etween rottions tht trnsfer phse chnges into stte chnges, s occurs in Rmsey spectroscopy. By using single ion, Christopher Monroe et l. 21 relized the CNOT opertion etween motion nd spin for 9 Be + ions. Susequently, Ferdinnd Schmidt-Kler et l. 22,23 nd lter Mrk Riee et l. 24 relized the complete CNOT opertion etween two individully ddressed 4 C + ions. Entngling gtes hve lso een relized y irrditing ions simultneously (Fig. 3). Although such gtes cn e implemented in single step, they still involve trnsitory entnglement with motionl mode, which effectively couples the spin quits. Ions hve lso een entngled with ech other in proilistic wy medited y entnglement with scttered photons 25 (Fig. 4). By sequentilly comining single-quit nd multiquit opertions, vrious entngled sttes of ions hve een creted deterministiclly or on demnd. A reserch group from the Ntionl Institute of Stndrds nd Technology (NIST), in Boulder, Colordo, creted 26 the stte Ψ e (ϕ) = 3 5 ge e iϕ 4 5 eg, where ϕ is controllle phse fctor nd ge denotes the comined stte g 1 e 2 for ions 1 nd 2. More generlly, y using entngling opertions nd single-quit rottions with djustle phses, ll Bell sttes Ψ ± = 1 2 ( ge ± eg ), Φ ± = 1 2 ( gg ± ee ) nd ritrry superpositions cn e generted 27,28. The qulity or fidelity of quntum sttes is usully chrcterized y the degree with which they gree with the desired (or idel) stte, which is expressed s F = Ψ idel ρ exp Ψ idel (2) where ρ exp is the experimentlly chieved density mtrix, which chrcterizes oth pure nd non-pure sttes. In current experiments, fidelities F >.95 re chieved. In some cses, complete knowledge of the density mtrix is not required. For exmple, the fidelity of stte reltive to Φ + cn e derived from just three mtrix elements, F = 1 2(ρ gg,gg + ρ ee,ee ) + Reρ ee,gg, where ρ ee,gg ee ρ exp gg nd so on nd Re denotes the rel prt of the expression tht follows. The mtrix elements ρ gg,gg nd ρ ee,ee re otined from the mesured popultions of the respective sttes. The mtrix element ρ ee,gg Two-level ion Hrmonic trp x y z e g W g ]w m c Coupled system e, n 1 e, n e, n + 1 g, n 1 g, n g, n + 1 Figure 1 Ions confined in trp., A liner qudrupole ion trp (known s Pul trp; eige) contining individully ddressed 4 C + ions (lue) is depicted. After cooling y lser ems (red), the trpped ions form string nd re then imged y using chrge-coupled device (CCD). In the CCD imge shown, the spcing of the two centre ions is ~8 µm. The electrode rrngement in the Pul trp provides n lmost hrmonic three-dimensionl well. For single ion, this is chrcterized y three frequencies 17 : ω x, ω y nd ω z, where x, y nd z denote the confining potentil xes. In this cse, z points long the trp xis nd x, y in the trnsverse directions. Owing to the Coulom coupling tht occurs etween ions, the motion is est descried in terms of norml modes; string of ions cn therefore e viewed s pseudo-molecule. In generl, the normlmode frequencies ω m differ from ech other, nd prticulr mode cn e ccessed y spectrl selection., The energy levels of two-level ion (left) nd one mode of the ion s motion (right) re shown. On the left is depicted the ion s ground stte g nd excited stte e, intercting with rdition chrcterized y the Ri frequency Ω nd decying with the rte γ. On the right is depicted the hrmonic oscilltor potentil nd eqully spced energy levels for one mode of motion. Both the two-level system nd the hrmonic oscilltor cn e descried jointly in quntummechnicl wy, indicted y the direct product, resulting in mnifold of two-level systems seprted y the mode frequency ω m (s shown in c). c, The level structure of the coupled ion hrmonic-oscilltor system is shown, with sttes jointly descried y the spin ( g nd e ) nd motionl (, 1,..., n ) degrees of freedom, where g n = g, n nd e n = e, n. Arrows indicte the trnsitions tht re possile when ppropritely tuned rdition is pplied; dshed lines indicted connections to levels not shown. 19

8 INSIGHT REVIEW NATURE Vol June 28 cn e otined y pplying rottion R (π/2, ϕ) to oth ions nd mesuring the prity P gg gg + ee ee ge ge eg eg of the resultnt stte s function of ϕ. The only component of the prity tht oscilltes sinusoidlly with frequency 2ϕ is proportionl to ρ ee,gg, which llows this element to e extrcted 29. As shown y eqution (2), the fidelity cn e otined y mesuring the full density mtrix. To do this, the quntum stte in question must e prepred mny times; in this wy, with the pproprite single-quit rottions pplied efore the quit mesurements, ll expecttion vlues of the density mtrix re otined. Such procedure is known s quntum-stte tomogrphy 28. When this procedure is pplied to Bell sttes, the density mtrix cn e completely chrcterized (Fig. 5). From the density mtrices, ll mesures cn su sequently e clculted. For exmple, in the cse of Bell s Ion 1 g, e Motion Ion 2 g, e, 1 g, e R (π/2, π/2) 1, 3 2 R 1 (π, f) R 1 (π, f + π) e, g, 1 3 CNOT 2 ux R 1 (2π, f ux ) gg ge eg ee gg ge ee eg Truth tle of CNOT R (π/2, π/2) e, 1 e, 1 e, g, 1 ux, 1 ux, g, g, 1 g, g, e CNOT ( g, e ) g, 1 Figure 2 A CNOT-gte opertion with two trpped ions., Consider two ions in the sme trp tht re initilly prepred in their motionl ground stte. In step 1, lower-sidend pulse R 1 (π, ϕ) is pplied to the first ion (ion 1; the control quit) nd mps the excited-stte mplitude of this ion to the first excited stte of selected motionl mode ( process known s SWAP opertion). Importntly, this motionl excittion is lso shred with the second ion (ion 2; the trget quit). In step 2, CNOT-gte opertion is implemented etween the motion quit (which is shred y oth spin quits) nd the spin stte of ion 2. Finlly, in step 3, the first step is reversed, therey restoring the initil spin stte of ion 1 nd returning the motion to its ground stte. The pulse sequence of the CNOT-gte opertion is lso shown (lower prt of )., On the left is CCD imge of two ions. The rrows indicte lser rdition tht is pplied to the ions in the order of the indicted numers (which correspond to the three steps in ). First, lser pulse is pplied to the upper ion (1), then the CNOT sequence is pplied to the lower ion (2). Finlly, lser pulse is pplied to the upper ion gin (3). On the right is the resultnt truth tle of the CNOT-gte opertion, with the first nd second symols denoting the stte of the control quit (ion 1) nd the trget quit (ion 2), respectively. inequlities, it is possile to determine the expecttion vlue of the opertor 3 A = σ (1) x σ (2) x z + σ (1) x σ (2) x+z + σ (1) z σ (2) x z σ (1) z σ (2) x+z, where σ x±z = (σ x ± σ z )/ 2 nd σ is Puli opertor nd the superscripts refer to the first nd second quits. For locl relistic theories, mesurements of A re predicted to e less thn 2, nd vlues of 2 < A < 2 2 re expected for sttes tht cn e descried only y quntum theory. With trpped ions, experiments yielded A = 2.25(3) t NIST 27, A = 2.52(6) t the Institute for Experimentl Physics, University of Innsruck (Innsruck, Austri) 28, nd A = 2.2(3) t the FOCUS Center nd Deprtment of Physics, University of Michign (Ann Aror, Michign) 31, where the numer in prentheses denotes the uncertinty in the lst digit, clerly corroorting quntum theory (Fig. 5). Moreover, ech time n experiment ws run, result ws recorded. This closed the detection loophole, which would provide wy to violte Bell s inequlities within locl relistic theories. The opertions outlined ove cn e generlized to entngle more thn two prticles. Among such sttes, the ct sttes, nmed fter Schrödinger s ct 32, re of prticulr interest. Ct sttes re usully defined s superpositions of two prticulr mximlly different sttes, such s Ψ ct = α ggg g + β eee e, nd they hve n importnt role in quntum informtion science. For three quits, ct sttes re lso known s GHZ sttes, which were nmed fter Dniel Greenerger, Michel Horne nd Anton Zeilinger, who showed tht these sttes could provide prticulrly cler contrdiction with locl relistic theories 33. They re fundmentl resource in fult-tolernt quntum computing, for error correction 34,35 nd for quntum communiction. In ddition, ecuse of their sensitivity to the interferometric phse ϕ, they cn lso improve signl-to-noise rtios in interferometry 36 (descried lter). With trpped ions, ct sttes with α = β hve een generted y using two pproches. At NIST, glol entngling opertions were used to demonstrte ct stte of four ions 29, GHZ stte with F =.89 (ref. 37), nd ct sttes of up to six ions 38. Using individully ddressed ions nd CNOTgte opertion, the reserch group t Innsruck produced GHZ sttes in n lgorithmic wy nd nlysed the sttes y using tomogrphic mesurements 39. In similr wy, the Innsruck group lso produced W sttes for N-ion quits, Ψ W = N 1 ( g gge + g geg + + eg g ), which elong to different clss of entngled sttes. Such clsses re distinct ecuse sttes of dif ferent clsses cnnot e trnsformed into ech other y locl opertions nd clssicl communiction 4. Nevertheless, oth ct nd W sttes cn violte Bell-type inequlities. In contrst to ct sttes, W sttes re remrkly roust in the fce of vriety of decoherence processes: for W sttes, even the loss of quits does not destroy entnglement completely. The Innsruck group deterministiclly prepred n eight-quit W stte 41 y using individul ion ddressing. Both the NIST nd Innsruck groups verified multiprtite entnglement y using n entnglement witness, n opertor constructed so tht its expecttion vlue must exceed (or e less thn) certin vlue to verify N-prticle entnglement 38,41. Demonstrting quntum-informtion-processing lgorithms Algorithms re lists of instructions for completing tsk 8. As is the cse in clssicl computtion, quntum lgorithms cn sometimes e viewed s suroutines in lrger computtion. A quntum-informtion-processing lgorithm generlly involves single-quit gtes nd multi quit gtes, s well s mesurements nd mesurement-dependent opertions. The result of such sequence of opertions could e deterministiclly prepred quntum stte (such s Bell, GHZ or W stte), conditioned stte (such s n error-corrected stte) or stte tht is susequently inferred from mesurement of the quntum register nd is then ville s clssicl informtion. In contrst to clssicl informtion processing, quntum informtion processing llows tests to e crried out using superpositions. A simple exmple showing the gin in efficiency tht is possile with quntum lgorithm ws proposed y Deutsch nd Richrd Jozs 42. The Deutsch Jozs lgorithm ws first demonstrted with two quits in nucler mgnetic resonnce spectroscopy 43, nd it ws demonstrted more recently with trpped ion 44, with the motionl nd spin properties of the ion quit serving s the two quits. 11

9 NATURE Vol June 28 INSIGHT REVIEW Another exmple lgorithm is teleporttion of the stte of one quit to nother quit, n importnt protocol for the trnsfer of quntum informtion 1,45. In this lgorithm, Alice wnts to send quit stte (which, in generl, is unknown) to Bo. To do this, Bell pir is generted, nd one quit from this pir is given to the sender, Alice, nd the other quit to the receiver, Bo. When the unknown stte is redy to e teleported, it is entngled with Alice s quit of the Bell pir. A su sequent mesurement of oth quits y Alice yields two its of clssicl informtion tht she sends to Bo. With this informtion, Bo knows which of four possile rottions to pply to his quit to otin Alice s originl unknown stte. Deterministic quntum teleporttion hs een demonstrted y the NIST 46 nd Innsruck 47 groups. The Innsruck group used individul lser-em ddressing of three quits; therefore, the stte ws teleported from one end of the ion string to the other end, distnce of ~1 µm. The NIST group used multizone liner-trp rry. By pplying control potentils to electrode segments, the ions could e seprted nd moved in nd out of one zone in which the lser ems were present. In this cse, the stte ws teleported cross few hundred micrometres. Teleporttion is n importnt uilding lock for quntum informtion processing nd cn re duce the computtionl resource requirements. Furthermore, it is the sic procedure for quntum communiction protocols, such s for implementing quntum repeters. Other lgorithms such s entnglement purifiction 48, quntum error correction 49, the quntum Fourier trnsform 5 nd deterministic entngle ment swpping (M. Riee, T. Monz, K. Kim, A. S. Villr, P. Schindler, M. Chwll, M. Hennrich nd R. Bltt, unpulished oservtions) hve lso een demonstrted with ion quits. These experiments demonstrte the sic fetures of quntum lgorithms, ut for the concten tion of processes nd repeted computtions, improved opertion fidelities will e required. In pr ticulr, full nd repetitive implementtion of quntum error correction, which could keep quit superposition live while sujected to decoherence, remins mjor chllenge in quntum in formtion processing. Applictions In the mid-199s, wve of interest in potentil pplictions for quntum informtion processing ws generted y Shor s period-finding lgorithm for fctoring lrge numers 7. Another noteworthy potentil ppliction is the implementtion of unstructured serches 51. However, to e of prcticl use, these pplictions re quire sustntil resources in terms of the numer of quits nd the numer of opertions, fr eyond the cpilities of current implementtions. Despite this, some elements of quntum informtion processing nd entnglement with smll numers of quits re eginning to find pplictions in metrology. Mny physicists lso expect tht useful quntum simultions will e crried out on reltively smll numer of quits, perhps up to 1, in the next decde. One ppliction in metrology is to improve interferometry. As n exmple, we discuss how entnglement cn e pplied to Rmsey spectroscopy 52, ut this scheme hs direct nlogue in electron, tom nd photon Mch Zehnder interferometry. Rmsey spectroscopy on the g e trnsi tion proceeds s follows. The tom is first prepred in the stte Ψ initil = g. Rdition t frequency ω ner ω eg is pplied in fst pulse to produce the stte R (π/2, π/2) g = 2 1 ( g + e ). The tom is now llowed to evolve for durtion T so tht the tom s upper stte ccumultes phse ϕ R = (ω ω ge )T reltive to the lower stte (when the prolem is viewed in frme tht rottes t frequency ω). Finlly, gin, rottion R (π/2, π/2) is pplied nd leves the tom in the stte (up to glol phse fctor) Ψ finl = sin(ϕ R /2) g + i cos(ϕ R /2) e. Therefore, the proility of finding the tom in the stte e is p e = 1 2(1 + cos [(ω ω ge )T]). For n ensemle of N toms, the detected signl will e Np e. In precision spectroscopy, the ide is to detect chnges in ω ω ge or ϕ R, s oserved through chnges in p e. Therefore, the N-ion signl cn e defined s S = d(np e )/dϕ R = N/2 sin(ϕ R ). The fundmentl noise in the signl is given y the projection noise : tht is, the fluctution in the numer of toms, from experiment to experiment, tht is mesured to e in the stte e (ref. 53). The vrince of this noise is given y V N = Np e (1 p e ), so the mgnitude of the signl-to-noise rtio is equl to S/ V N = N, essentilly the shot noise corresponding to the numer of toms. Now, suppose tht the first R (π/2, π/2) pulse cn e replced with n entngling π/2 pulse 37,38, which cretes the ct stte g 1 g 2 g N 1 2( g 1 g 2 g N + e 1 e 2 e N ) 1 2( g N + e N ) (3) After dely T, the e N stte ccumultes phse Nϕ R reltive to the g N stte. A finl entn gling π/2 pulse leves the toms in super position stte sin(nϕ R /2) g N + i cos(nϕ R /2) e N ; therefore, the proility of detecting the toms in the e N stte is p Ne = 1 2(1 + cos [N(ω ω ge )T]). It is s though spectroscopy hs een crried out on single supertom composed of sttes e N nd g N. The super-tom hs resonnt frequency tht is N times higher thn tht of single tom, s well s phse sen sitivity (to the Nth hrmonic of the pplied rdition) tht is N times higher. The resultnt gin in interferometric sensitivity must, however, e offset y the fct tht only single-prticle two-stte system ( e N nd g N ) is eing mesured. Nevertheless, fter sttisticlly significnt numer of repeted mesurements, the sensitivity is 2F Lser em 1 F Lser em 2 F F Trp (z) xis Figure 3 A two-quit phse gte. A phse gte with two ions (lue) is depicted. The opertion of such phse gtes relies on the fct tht when selected mode of the ions motion is displced in phse spce out closed pth, the ions wvefunction picks up phse tht is proportionl to the enclosed re. If this displcement depends on the ions quit sttes, then entnglement is generted This stte-dependent displcement cn e implemented y pplying opticl dipole forces (F) y using lserem intensity grdients. In this exmple, n intensity stnding wve is creted with two lser ems, nd the horizontl spcing of the ions is mde to e n integrl numer of wvelengths of the intensity pttern. The pttern sweeps cross the ions t the difference etween the frequencies of the ems, chosen to e ner the stretch-mode frequency. If the ions quit sttes g nd e feel different dipole forces, then only the g e nd e g components of the ions wvefunction re displced in phse spce. By mking the trjectories closed nd y choosing the size of the displcements ppropritely, the wvefunction is unchnged except for n e iπ/2 phse shift on the g e nd e g sttes, the desired phse gte. Such gte opertions hve een implemented with trpped 9 Be + ions 29,95 nd in similr wy with 111 Cd + ions 96 nd 4 C + ions 63,

10 INSIGHT REVIEW NATURE Vol June 28 Figure 4 Entnglement produced y conditionl mesurements. Entnglement cn e creted etween two seprted prticles y n interference effect nd stte projection ccompnying mesurement. In this exmple 25, it is ssumed tht the quits of two ions (lue) re encoded in hyperfine levels of the electronic ground sttes. These quits re first prepred in superposition sttes 1 2( g + e ). When excited with lser pulses tht re short enough tht oth quits simultneously undergo (single-photon) scttering, the frequencies (denoted red nd lue ) of the emitted photons long prticulr direction re correlted with the quit sttes, s indicted for entngled sttes Ψ 1 nd Ψ 2. These photons cn e simultneously sent into 5/5 em splitter nd then detected. In the cses when photons re simultneously detected t detector A nd detector B, the ions re projected into the Bell stte Ψ finl, even though the toms hve not directly intercted. For mny such experiments, photons do not rech either detector; however, when photons re simultneously detected, this herlds the formtion of the entngled stte Ψ finl, which cn then e sved nd used lter, such s in Bell s inequlity mesurements of remotely locted ions 98. One potentil use of this scheme is for entnglement-ssisted communiction etween ion loctions 1 nd 2. Simultneous excittion pulses Ion 1 Ion 2 Y 1 = 1 ( g 1 lue + e 1 red ) 2 Y2 = 1 ( g 2 lue + e 2 red ) 2 Detector A Detector B 5/5 Bem splitter On simultneous detection Yfinl = 1 ( g 1 e 2 e 1 g 2 ) 2 incresed y fctor of N y using entngling π/2 pulses compred with the cse of N unentn gled toms Becuse of technicl noise in the experiments, this theoreticl improvement is not fully relized; however, gin in sensitivity compred with the cse of unentngled toms hs een relized for up to six entngled ions 38,54,55. These rguments ssume tht noise results only from stte projection. In experiments, if there is correlted decoherence of quit phses, then ny gin in sensitivity my e lost s result of the fster decoherence of the ct sttes 56 or s result of noise in the oscilltor tht produces the rdition 18,57. If these sources of noise cn e suppressed, entngled sttes should e le to improve the signl-to-noise rtio in future spectroscopy experiments. Another ppliction of quntum-informtion-processing techniques is incresed fidelity of detection 58. This cn e useful if the quit does not hve cycling trnsition or if the QND spect of the shelving detection is not well stisfied. A simple implementtion is to ssume tht there re two quits, lelled q nd d, stored in the sme trp. The gol is to detect the stte of informtion quit q, y using detection quit d. Before ny mesurements re tken, quit q will generlly e in super position stte α g q + β e q. Using the SWAP opertions of the Circ Zoller gte, this superposition is first trnsferred to the quit composed of the nd 1 sttes of selected motionl mode, nd is then mpped to quit d. Then, quit d is mesured, therey in effect mesuring quit q. This protocol cn e crried out without disturing the initil proilities α 2 nd β 2 for quit q, even if the mpping steps re imperfect. Therefore, it is QND mesurement nd cn e repeted to increse detection efficiency. This scheme ws demonstrted in n experiment 59 in which quit q ws sed on n opticl trnsition in 27 Al + ion nd quit d ws sed on hyperfine trnsition in 9 Be + ion. In tht experiment, single round of detection hd fidelity of only F =.85; however, y repeting the mesurement, nd y using rel-time yesin nlysis, the fidelity ws improved to F = It should e noted tht this strtegy cn lso e used to prepre n eigenstte of quit q with high fidelity. In ddition to this demonstrtion, this protocol is now used in high-ccurcy opticl clock sed on single 27 Al + ions 6. This technique hs lso een extended so tht single detection quit cn e used to mesure the sttes of multiple ions 59, similr to the mesurement of the Fock sttes of photons y using multiple proe toms 61. Finlly, entnglement cn e used in metrology to crete sttes tht llow the mesurement of certin prmeters while suppressing sensitivity to others. This strtegy hs een used, for exmple, to mke precise mesurement of the qudrupole moment of 4 C + ion y crrying out spectroscopy on n entngled stte of two ions tht depended on the qudrupole moment ut ws insensitive to fluctutions in the mgnetic field r rel.5 gg ge eg.5 ee eg gg ge ee r rel Y + = 1 ( ge + eg ) 2.5 gg ge eg.5 ee eg gg ge ee Y = 1 ( ge eg ) 2 r imginry gg ge eg ee eg gg ge ee r imginry gg ge eg ee eg gg ge ee r rel.5 gg ge eg.5 ee eg gg ge ee r rel f + = 1 ( gg + ee ) 2.5 gg ge eg.5 ee eg gg ge ee f = 1 ( gg ee ) 2 r imginry gg ge eg ee eg gg ge ee r imginry gg ge eg ee eg gg ge ee Figure 5 Mesured density mtrices of Bell sttes. The rel (left) nd imginry (right) prts of the density mtrices otined for the Bell sttes Ψ + (upper left), Ψ (lower left), ϕ + (upper right) nd ϕ (lower right) prepred deterministiclly with two trpped 4 C + ions re shown. The sttes were nlysed y using quntum-stte tomogrphy, technique tht provides ll of the necessry informtion to reconstruct the corresponding density mtrix 8. More specificlly, the density mtrix for single quit cn e represented y ρ = 1 2(I + Σ i σ i σ i ), where σ i is Puli mtrix, i = x, y, z nd I is the identity mtrix. Mesurements project quit onto its energy eigensttes, which is equivlent to mesuring σ z. To determine σ x,y, n dditionl rottion of the Bloch sphere is pplied efore the mesurement. The tomogrphy procedure cn e extended to N quits, requiring of the order of 4 N expecttion vlues to e mesured. Owing to sttisticl errors, the experimentlly mesured expecttion vlues cn result in unphysicl elements in the density mtrix (with negtive eigenvlues). This outcome is voided y fitting the mesured expecttion vlues y using mximum-likelihood method nd then finding the most likely density mtrix tht descries the stte

11 NATURE Vol June 28 INSIGHT REVIEW Prospects Although the sic elements of quntum computtion hve een demonstrted with tomic ions, opertion errors must e significntly reduced nd the numer of ion quits must e sustntilly incresed if quntum computtion is to e prcticl. Nevertheless, efore fidelities nd quit numers rech those required for useful fctoring mchine, worthwhile quntum simultions might e relized. More ion quits nd etter fidelity To crete mny-ion entngled sttes, there re two importnt gols: improving gte fidelity, nd overcoming the dditionl prolems tht re ssocited with lrge numers of ions. For fult-tolernt opertion, resonle guideline is to ssume tht the proility of n error occurring during single gte opertion should e of the order of 1 4 or lower. An importnt enchmrk is the fidelity of two-quit gtes. The est error proility chieved so fr is pproximtely 1 2, which ws inferred from the fidelity of Bell-stte genertion 63. In generl, it seems tht gte fidelities re compromised y limited control of clssicl components (such s fluctutions in the lser-em intensity t the positions of the ions) nd y quntum limittions (such s decoherence cused y spontneous emission) 64. These re dunting technicl prolems; however, eventully, with sufficient cre nd engineering expertise, these fctors re likely to e suppressed. The multiquit opertions discussed in this review rely on the ility to isolte spectrlly single mode of the motion of n ion. Becuse there re 3N modes of motion for N trpped ions, s N ecomes lrge, the mode spectrum ecomes so dense tht the gte speeds must e significntly reduced to void off-resonnce coupling to other modes. Severl proposls hve een put forwrd to circumvent this prolem 65,66. Alterntively, wy to solve this prolem with gtes tht hve een demonstrted involves dis triuting the ions in n rry of multiple trp zones 18,67 69 (Fig. 6). In this rchitecture, multiquit gte opertions could e crried out on reltively smll numer of ions in mul tiple processing zones. Entnglement could e distriuted etween these zones y physiclly moving the ions 18,68,69 or y opticl mens 25,67,7 72. For quntum communiction over lrge distnces, opticl distriution seems to e the only prcticl choice; for experiments in which locl entnglement is desirle, moving ions is lso n option. Exmples of trps tht could e used for scling up the numer of ions used in n lgorithm re shown in Fig. 6. Ions cn e moved e tween zones y pplying pproprite control electric potentils to the vrious electrode segments 46, Individul ions hve een moved ~1 mm in ~5 µs without loss of coherence; the excittion of the ion s motion (in its locl well) ws less thn one quntum 73. Multiple ions present in single zone cn e seprted 46,73 y inserting n electric potentil wedge etween the ions. In the tele porttion experiment y the NIST group 46, two ions could e seprted from third in ~2 µs, with negligile excittion of the motionl mode used for susequent entngling opertions etween the two ions. This sence of motionl excittion ment tht n dditionl entngling-gte opertion on the sep rted ions could e implemented with resonle fidelity. For lgorithms tht operte over long time periods, the ions motion will eventully ecome excited s result of trnsporttion nd ckground noise from electric fields. To counterct this prolem, dditionl lser-cooled ions could e used to cool the quits symptheticlly (Fig. 6). These refrigertor ions could e identicl to the quit ions 76, of different isotope 77 or of different species 6,78. They could lso id in detection nd stte preprtion (descried erlier). For ll multiquit gtes tht hve een implemented so fr, the speeds re proportionl to the frequen cies of the modes of the ions, which scle s 1/d 2 qe, where d qe is the distnce of the ion to the nerest electrode. Therefore, it would e vlule to mke trps s smll s possile. Mny groups hve endevoured to chieve this, ut they hve ll oserved significnt heting of the ions, compromising gte fidelity. The heting is nomlously lrge compred with tht expected to result from therml noise, which rises from resistnce in, or coupled to, the trp electrodes 18, It scles pproximtely s 1/d 4 qe (refs 18, 79 83), which is consistent with the presence of independently fluctuting potentils on electrode ptches, the extent of which is smll compred with d qe (ref. 79). The source of the heting hs yet to e understood, ut recent experiments 8,82 indicte tht it is thermlly ctivted nd cn e significntly suppressed y operting t low temperture. For lrge trp rrys, roust mens of friction will e required, s well s mens of independently controlling very lrge numer of electrodes. Microelectromechnicl systems (MEMS) friction technologies cn e used for monolithic construction 83,84, nd trp structures cn e further simplified y plcing ll electrodes in plne 84,85. To mitigte the prolem of controlling mny electrodes, it might e possile to incorporte on-ord electronics close to individul trp zones 86. Lser ems must lso e pplied in severl loctions simultneously, ecuse it will e essentil to crry out prllel opertions when implementing complex lgorithms. The recycling of lser ems cn e used 86,87, ut the overll lser power requirements will still increse. If gtes re implemented y using stimulted-rmn trnsitions, then Refrigertor em Gte em(s) Quit memory zone To dditionl zones Figure 6 Multizone trp rrys., A schemtic representtion of multizone trp rry is shown. Ech control electrode is depicted s rectngle. Ions (lue circles) cn e seprted nd moved to specific zones, including memory zone, y pplying pproprite electricl potentils. Becuse the ions motion will ecome excited s result of trnsport (idirectionl rrow) nd noisy mient electric fields, refrigertor ions (red; which re cooled y the red lser em) re used to cool the ions efore gte opertions, which re implemented with the lue lser em., Exmples of the electrode configurtions of trp rrys re shown. In the upper left is two-lyer, six-zone liner trp in which entngled ions cn e seprted nd used for lgorithm demonstrtions, including teleporttion 46 (width of nrrow slot (where the ions re locted) = 2 µm). In the upper right is three-lyer, two-dimensionl multizone trp tht cn e used to switch ion positions 99 (width of slot = 2 µm). In the lower left is singlezone trp in which ll of the electrodes lie in single lyer; this design considerly simplifies friction 85. In the lower right is single-lyer, liner multizone trp fricted on silicon (width of open slot for loding ions 95 µm), which cn enle electronics to e fricted on the sme sustrte tht contins the trp electrodes. (Imge courtesy of R. Slusher, Georgi Tech Reserch Institute, Atlnt). 113

12 INSIGHT REVIEW NATURE Vol June 28 high lser-em intensity will lso e needed to suppress spontneous emission decoherence to fult-tolernt levels 64. Detection will lso need to e implemented simultneously in severl loctions. This issue might e resolved y coupling on-ord detectors or other forms of miniture integrted optics to opticl fires. Future pplictions In the erly 198s, Feynmn suggested tht one quntum system could perhps e used to simulte nother 5. This simultion could e ccomplished efficiently with lrge-scle quntum computer. But efore this gol is reched, it might e possile to tke dvntge of the fct tht current logic gtes re implemented y hmiltonins tht cn e used to simulte interctions in other systems. A simple exmple ws mentioned erlier in the dis cussion of spectroscopy with ct sttes; these experiments simulte the ction of electron, photon nd tom Mch Zehnder interferometers tht incorporte entngling em splitters 55. A more interesting prospect is tht the gte hmiltonins might e pplied in strtegic wy to sim ulte specific mny-ody hmiltonins. The sic ide cn e considered y noting tht the two-ion phse gte (Fig. 3) cn e written in the form R Z1 R Z2 e iξσz1σz2, where R Z1 nd R Z2 re rottions out the z xis nd ξ is the strength of coupling. Therefore, up to n overll rottion on the quits, the gte implements the hmiltonin H = κσ z1 σ z2, spin spin interction etween the two ddressed spins, where κ is the strength of the interction nd is h/2π (nd h is Plnck s constnt). By extending these cou plings to mny ion quits in n ensemle, Ising-type spin hmiltonins could, for exmple, e implemented The interctions etween ion pirs could e pplied in stepwise mnner ut might lso e implemented simultneously, therey incresing efficiency. Although simulting specific mny-ody hmiltonins is chl lenge given current experimentl cpilities, even with reltively smll numer of ions, interesting phenomen such s quntum phse trnsitions might e oservle. Conclusion As reserchers progress towrds generting lrge-scle quntuminformtion-processing device, it might e possile to shed light on more fundmentl issues of decoherence nd why mny-prticle sttes with the quntum ttriutes of Schrödinger s ct re not oserved. If it is possile to continue scling up such devices to lrge size, the issue of the sence of ct sttes ecomes more pressing. For exmple, suppose tht, in the future, lrge-n-quit ct sttes in the form of eqution (3) cn e mde. Then, this ct stte for N quits cn e rewritten s Ψ = 2( g 1 j π k j g N k + e j π k j e N k ), where the jth quit hs een (ritrrily) singled out nd k represents the other quits. For lrge N, this wvefunction hs the ttriutes of Schrödinger s ct in the sense tht the sttes of single two-level quntum system (the jth quit) re correlted with sttes tht hve mcroscopiclly distinct polriztions. If generting such sttes is successful, then the existence of, in essence, Schrödinger s cts will hve een shown. Such sttes re, however, more sensitive to the effects of phse decoherence 56, ut this seems to e technicl, not fundmentl, prolem. Therefore, if it ecomes impossile to mke such sttes or to uild lrge-scle quntum computer for non-technicl resons, this filure might indicte some new physics. 1. Rmsey, N. F. Moleculr Bems (Clrendon, London, 1956). 2. Freedmn, S. F. & Cluser, J. F. Experimentl test of locl hidden-vrile theories. Phys. Rev. Lett. 28, (1972). 3. Aspect, A., Grngier, P. & Roger, G. Experimentl tests of relistic locl theories vi Bell s theorem. Phys. Rev. Lett. 47, (1981). 4. Bell, J. S. Spekle nd Unspekle in Quntum Mechnics (Cmridge Univ. Press, Cmridge, UK, 1987). 5. Feynmn, R. P. Simulting physics with computers. Int. J. Theoret. Phys. 21, (1982). 6. Deutsch, D. Quntum theory, the Church Turing principle nd the universl quntum computer. Proc. R. Soc. Lond. A 4, (1985). 7. Shor, P. W. Algorithms for quntum computtion: discrete logrithms nd fctoring. In Proc. Annu. Symp. Found. Comput. Sci (1994). 8. Nielsen, M. A. & Chung, I. L. Quntum Computtion nd Quntum Informtion (Cmridge Univ. Press, Cmridge, UK, 2). 9. Circ, J. I. & Zoller, P. Quntum computtions with cold trpped ions. Phys. Rev. Lett. 74, (1995). 1. Monroe, C. Quntum informtion processing with toms nd photons. Nture 416, (22). 11. Dehmelt, H. Experiments with n isolted sutomic prticle t rest. Rev. Mod. Phys. 62, (199). 12. Pul, W. Electromgnetic trps for chrged nd neutrl prticles. Rev. Mod. Phys. 62, (199). 13. Bollinger, J. J., Heinzen, D. J., Itno, W. M., Gilert, S. L. & Winelnd, D. J. A 33-MHz frequency stndrd sed on trpped 9 Be + ions. IEEE Trns. Instrum. Mes. 4, (1991). 14. Fisk, P. T. H. et l. Very high q microwve spectroscopy on trpped 171 Y + ions: ppliction s frequencystndrd. IEEE Trns. Instrum. Mes. 44, (1995). 15. Bltt, R., Häffner, H., Roos, C., Becher, C. & Schmidt-Kler, F. in Quntum Entnglement nd Informtion Processing: Les Houches Session LXXIX (eds Estève, D. Rimond, J.-M. & Dlird, J.) (Elsevier, Amsterdm, 24). 16. Winelnd, D. J. in Quntum Entnglement nd Informtion Processing: Les Houches Session LXXIX (eds Estève, D. Rimond, J.-M. & Dlird, J.) (Elsevier, Amsterdm, 24). 17. Leifried, D., Bltt, R., Monroe, C. & Winelnd, D. Quntum dynmics of single trpped ions. Rev. Mod. Phys. 75, (23). 18. Winelnd, D. J. et l. Experimentl issues in coherent quntum-stte mnipultion of trpped tomic ions. J. Res. Ntl Inst. Stnd. Technol. 13, (1998). 19. Diedrich, F., Bergquist, J. C., Itno, W. M. & Winelnd, D. J. Lser cooling to the zero-point energy of motion. Phys. Rev. Lett. 62, (1989). 2. Dehmelt, H. G. Mono-ion oscilltor s potentil ultimte lser frequency stndrd. IEEE Trns. Instrum. Mes. 31, (1982). 21. Monroe, C., Meekhof, D. M., King, B. E., Itno, W. M. & Winelnd, D. J. Demonstrtion of fundmentl quntum logic gte. Phys. Rev. Lett. 75, (1995). 22. Schmidt-Kler, F. et l. Reliztion of the Circ Zoller controlled-not quntum gte. Nture 422, (23). 23. Schmidt-Kler, F. et l. How to relize universl quntum gte with trpped ions. Appl. Phys. B 77, (23). 24. Riee, M. et l. Process tomogrphy of ion trp quntum gtes. Phys. Rev. Lett. 97, 2247 (26). 25. Moehring, D. L. et l. Entnglement of single-tom quntum its t distnce. Nture 449, (27). 26. Turchette, Q. A. et l. Deterministic entnglement of two trpped ions. Phys. Rev. Lett. 81, (1998). 27. Rowe, M. A. et l. Experimentl violtion of Bell s inequlity with efficient detection. Nture 49, (21). 28. Roos, C. F. et l. Bell sttes of toms with ultrlong life times nd their tomogrphic stte nlysis. Phys. Rev. Lett. 92, 2242 (24). 29. Sckett, C. A. et l. Experimentl entnglement of four prticles. Nture 44, (2). 3. Cluser, J. F., Horne, M. A., Shimony, A. & Holt, R. A. Proposed experiment to test locl hidden-vrile theories. Phys. Rev. Lett. 23, (1969). 31. Moehring, D. L., Mdsen, M. J., Blinov, B. B. & Monroe, C. Experimentl Bell inequlity violtion with n tom nd photon. Phys. Rev. Lett. 93, 941 (24). 32. Schrödinger, E. Die gegenwärtige Sitution in der Quntenmechnik. Nturwissenschften 23, (1935). 33. Greenerger, D. M., Horne, M. A. & Zeilinger, A. in Going Beyond Bell s Theorem (ed. Kftos, M.) (Kluwer Acdemic, Dordrecht, 1989). 34. DiVincenzo, D. P. & Shor, P. W. Fult-tolernt error correction withefficient quntum codes. Phys. Rev. Lett. 77, (1996). 35. Stene, A. M. Error correcting codes in quntum theory. Phys. Rev. Lett. 77, (1996). 36. Bollinger, J. J., Itno, W. M., Winelnd, D. J. & Heinzen, D. J. Optiml frequency mesurements with mximlly correlted sttes. Phys. Rev. A 54, R4649 R4652 (1996). 37. Leifried, D. et l. Towrd Heisenerg-limited spectroscopy with multiprticle entngled sttes. Science 34, (24). 38. Leifried, D. et l. Cretion of six-tom Schrödinger ct stte. Nture 438, (25). 39. Roos, C. F. et l. Control nd mesurement of three-quit entngled sttes. Science 34, (24). 4. Dür, W., Vidl, G. & Circ, J. I. Three quits cn e entngled in two inequivlent wys. Phys. Rev. A 62, (2). 41. Häffner, H. et l. Sclle multiprticle entnglement of trpped ions. Nture 438, (25). 42. Deutsch, D. & Jozs, R. Rpid solution of prolems y quntum computtion. Proc. R. Soc. Lond. A 439, (1992). 43. Chung, I. L. et l. Experimentl reliztion of quntum lgorithm. Nture 393, (1998). 44. Gulde, S. et l. Implementtion of the Deutsch Jozs lgorithm on n ion-trp quntum computer. Nture 421, 48 5 (23). 45. Bennett, C. H. et l. Teleporting n unknown quntum stte vi dul clssicl nd Einstein Podolsky-Rosen chnnels. Phys. Rev. Lett. 7, (1993). 46. Brrett, M. D. et l. Deterministic quntum teleporttion of tomic quits. Nture 429, (24). 47. Riee, M. et l. Deterministic quntum teleporttion with toms. Nture 429, (24). 48. Reichle, R. et l. Experimentl purifiction of two-tom entnglement. Nture 443, (26). 49. Chiverini, J. et l. Reliztion of quntum error correction. Nture 432, (24). 5. Chiverini, J. et l. Implementtion of the semiclssicl quntum Fourier trnsform in sclle system. Science 38, (25). 51. Grover, L. K. Quntum mechnics helps in serching for needle in hystck. Phys. Rev. Lett. 79, (1997). 52. Winelnd, D. J., Bollinger, J. J., Itno, W. M., Moore, F. L. & Heinzen, D. J. Spin squeezing nd reduced quntum noise in spectroscopy. Phys. Rev. A 46, R6797 R68 (1992). 114

13 NATURE Vol June 28 INSIGHT REVIEW 53. Itno, W. M. et l. Quntum projection noise: popultion fluctutions in two-level systems. Phys. Rev. A 47, (1993). 54. Meyer, V. et l. Experimentl demonstrtion of entnglement-enhnced rottion ngle estimtion using trpped ions. Phys. Rev. Lett. 86, (21). 55. Leifried, D. et l. Trpped-ion quntum simultor: experimentl ppliction to nonliner interferometers. Phys. Rev. Lett. 89, (22). 56. Huelg, S. F. et l. Improvement of frequency stndrds with quntum entnglement. Phys. Rev. Lett. 79, (1997). 57. André, A., Sørensen, A. S. & Lukin, M. D. Stility of tomic clocks sed on entngled toms. Phys. Rev. Lett. 92, 2381 (24). 58. Schetz, T. et l. Enhnced quntum stte detection efficiency through quntum informtion processing. Phys. Rev. Lett. 94, 151 (25). 59. Hume, D. B., Rosennd, T. & Winelnd, D. J. High-fidelity dptive quit detection through repetitive quntum nondemolition mesurements. Phys. Rev. Lett. 99, 1252 (27). 6. Rosennd, T. et l. Frequency rtio of Al + nd Hg + single-ion opticl clocks; metrology t the 17th deciml plce. Science 319, (28). 61. Guerlin, C. et l. Progressive field-stte collpse nd quntum non-demolition photon counting. Nture 448, (27). 62. Roos, C. F., Chwll, M., Kim, K., Riee, M. & Bltt, R. Designer toms for quntum metrology. Nture 443, (26). 63. Benhelm, J., Kirchmir, G., Roos, C. F. & Bltt, R. Towrds fult-tolernt quntum computing with trpped ions. Nture Phys. 4, (28). 64. Ozeri, R. et l. Errors in trpped-ion quntumgtes due to spontneous photon scttering. Phys. Rev. A 75, (27). 65. Zhu, S.-L., Monroe, C. & Dun, L.-M. Aritrry-speed quntum gtes within lrge ion crystls through miminum control of lser ems. Europhys. Lett. 73, (26). 66. Dun, L.-M. Scling ion trp quntum computtion through fst quntum gtes. Phys. Rev. Lett. 93, 152 (24). 67. DeVoe, R. G. Ellipticl ion trps nd trp rrys for quntum computtion. Phys. Rev. A 58, (1998). 68. Circ, J. I. & Zoller, P. A sclle quntum computer with ions in n rry of microtrps. Nture 44, (2). 69. Kielpinski, D., Monroe, C. & Winelnd, D. J. Architecture for lrge-scle ion-trp quntum computer. Nture 417, (22). 7. Circ, I., Zoller, P., Kimle, J. & Muchi, H. Quntum stte trnsfer nd entnglement distriution mong distnt nodes in quntum network. Phys. Rev. Lett. 78, (1997). 71. Dun, L.-M. & Kimle, H. J. Sclle photonic quntum computtion through cvityssisted interctions. Phys. Rev. Lett. 92, (24). 72. Dun, L.-M. et l. Proilistic quntum gtes etween remote toms through interference of opticl frequency quits. Phys. Rev. A 73, (26). 73. Rowe, M. et l. Trnsport of quntum sttes nd seprtion of ions in dul rf ion trp. Quntum Inform. Comput. 2, (22). 74. Hucul, D. et l. On the trnsport of tomic ions in liner nd multidimensionl trp rrys. Preprint t < (27). 75. Huer, G. et l. Trnsport of ions in segmented liner Pul trp in printed-circuit-ord technology. New J. Phys. 1, 134 (28). 76. Rohde, H. et l. Sympthetic ground-stte cooling nd coherent mnipultion with two-ion crystls. J. Opt. Soc. Am. B 3, S34 S41 (21). 77. Blinov, B. B. et l. Sympthetic cooling of trpped Cd + isotopes. Phys. Rev. A 65, 434 (22). 78. Brrett, M. D. et l. Sympthetic cooling of 9 Be + nd 24 Mg + for quntum logic. Phys. Rev. A 68, 4232 (23). 79. Turchette, Q. A. et l. Heting of trpped ions from the quntum ground stte. Phys. Rev. A 61, (2). 8. Desluriers, L. et l. Scling nd suppression of nomlous heting in ion trps. Phys. Rev. Lett. 97, 137 (26). 81. Leirndt, D., Yurke, B. & Slusher, R. Modeling ion trp therml noise decoherence. Qunt. Inform. Comput. 7, (27). 82. Lziewicz, J. et l. Suppression of heting rtes in cryogenic surfce-electrode ion trps. Phys. Rev. Lett. 1, 131 (28). 83. Stick, D. et l. Ion trp in semiconductor chip. Nture Phys. 2, (26). 84. Chiverini, J. et l. Surfce-electrode rchitecture for ion-trp quntum informtion processing. Quntum Inform. Comput. 5, (25). 85. Seidelin, S. et l. Microfricted surfce-electrode ion trp for sclle quntum informtion processing. Phys. Rev. Lett. 96, 2533 (26). 86. Kim, J. et l. System design for lrge-scle ion trp quntum informtion processor. Qunt. Inform. Comput. 5, (25). 87. Leifried, D., Knill, E., Ospelkus, C. & Winelnd, D. J. Trnsport quntum logic gtes for trpped ions. Phys. Rev. A 76, (27). 88. Wunderlich, C. & Blzer, C. Quntum mesurements nd new concepts for experiments with trpped ions. Adv. At. Mol. Opt. Phys. 49, (23). 89. Porrs, D. & Circ, J. I. Quntum mnipultion of trpped ions in two dimensionl Coulom crystls. Phys. Rev. Lett. 96, 2551 (26). 9. Tylor, J. M. & Clrco, T. Wigner crystls of ions s quntum hrd drives. Preprint t < (27). 91. Chiverini, J. & Lyrger Jr, W. E. Lserless trpped-ion quntum simultions without spontneous scttering using microtrp rrys. Phys. Rev. A 77, (28). 92. Mølmer, K. & Sørensen, A. Multiprticle entnglement of hot trpped ions. Phys. Rev. Lett. 82, (1999). 93. Milurn, G. J., Schneider, S. & Jmes, D. F. Ion trp quntum computing with wrm ions. Fortschr. Physik 48, (2). 94. Solno, E., de Mtos Filho, R. L. & Zgury, N. Mesoscopic superpositions of vironic collective sttes of N trpped ions. Phys. Rev. Lett. 87, 642 (21). 95. Leifried, D. et l. Experimentl demonstrtion of roust, high-fidelity geometric two ionquit phse gte. Nture 422, (23). 96. Hljn, P. C. et l. Entnglement of trpped-ion clock sttes. Phys. Rev. A 72, (25). 97. Home, J. P. et l. Deterministic entnglement nd tomogrphy of ion spin quits. New J. Phys. 8, 188 (26). 98. Mtsukevich, D. N., Munz, P., Moehring, D. L., Olmschenk, S. & Monroe, C. Bell inequlity violtion with two remote tomic quits. Phys. Rev. Lett. 1, 1544 (28). 99. Hensinger, W. K. et l. T-junction ion trp rry for two-dimensionl ion shuttling storge, nd mnipultion. Appl. Phys. Lett. 88, 3411 (26). Acknowledgements We thnk H. Häffner, J. Home, E. Knill, D. Leifried, C. Roos nd P. Schmidt for comments on the mnuscript. Author Informtion Reprints nd permissions informtion is ville t The uthors declre no competing finncil interests. Correspondence should e ddressed to R.B. (Riner.Bltt@uik.c.t). 115

14 INSIGHT REVIEW NATURE Vol June 28 doi:1.138/nture7126 Quntum coherence nd entnglement with ultrcold toms in opticl lttices Immnuel Bloch1 At nnokelvin tempertures, ultrcold quntum gses cn e stored in opticl lttices, which re rrys of microscopic trpping potentils formed y lser light. Such lrge rrys of toms provide opportunities for investigting quntum coherence nd generting lrge-scle entnglement, ultimtely leding to quntum informtion processing in these rtificil crystl structures. These rrys cn lso function s verstile model systems for the study of strongly intercting mny-ody systems on lttice. Recent dvnces in the lser cooling of neutrl (unchrged) toms nd the cretion of ultrcold quntum gses1 hve opened up intriguing possiilities for the quntum mnipultion of rrys of neutrl toms. Around 15 2 yers go, spectculr progress ws mde on the trpping nd spectroscopy of single prticles, nd reserchers concentrted on studying such single prticles with ever-incresing precision. Now, reserchers re uilding on these exquisite mnipultion nd trpping techniques to extend this control over lrger rrys of prticles. Not only cn neutrl toms e trpped in microscopic potentils engineered y lser light2 4, ut the interctions etween these prticles cn e controlled with incresing precision. Given this success, the cretion of lrge-scle entnglement nd the use of ultrcold toms s interfces etween different quntum technologies hve come to the forefront of reserch, nd ultrcold toms re mong the hot cndidtes for quntum informtion processing, quntum simultions nd quntum communiction. Two complementry lines of reserch using ultrcold toms re dominting this field. In ottom-up pproch, rrys of toms cn e uilt up one y one. By contrst, top-down pproch uses the reliztion of degenerte ultrcold osonic5 7 nd fermionic8 1 quntum gses s n lterntive wy of estlishing lrge-scle rrys of ultrcold toms; this pproch llows the cretion of lrge numers of neutrl toms, with lmost perfect control over the motionl nd electronic degrees of freedom of millions of toms with tempertures in the nnokelvin rnge. When such ultrcold toms re loded into three-dimensionl rrys of microscopic trpping potentils, known s opticl lttices, the toms re sorted in such wy tht every lttice site is occupied y single tom, for exmple, y strong repulsive interctions in the cse of osons or y Puli locking in the cse of fermions. For osons, this corresponds to Mott insulting stte11 15, wheres for fermions nd insulting stte is creted16, oth of which form highly regulr, ordered, quntum register t close to zero kelvin. After initiliztion, the interctions nd the sttes of the toms re controlled to cox them into the correct possily entngled mcroscopic (mny ody) stte to e used in quntum informtion processing, for exmple, or metrology t the quntum limit. Ultrcold toms cnnot yet rivl the pristine control chieved using ion-trp experiments (see pge 18), ut some key fetures nevertheless render them highly ttrctive. First, neutrl toms couple only wekly to the environment, llowing long storge nd coherence times, even in the proximity of ulk mterils; this feture hs mde them highly successful in the field of cvity quntum electrodynmics (see pge 123). Second, 1 Institut für Physik, Johnnes Gutenerg-Universität Minz, 5599 Minz, Germny. 116 ultrcold toms in opticl lttices form the only system so fr in which lrge numer (up to millions) of prticles cn e initilized simultneously. Eventully, ny system proposed for quntum informtion processing will hve to del with such lrge rrys, nd mny of the perspectives (nd difficulties) ssocited with these cn lredy e tested using ultrcold toms tody. Ultrcold toms hve therefore lso ecome promising cndidtes in relted line of reserch quntum simultions4,17 19 in which highly controllle quntum mtter is used to unrvel some of the most intriguing questions in modern condensedmtter physics involving strongly correlted mny-ody quntum systems. In this review, I descrie sic spects of opticl trpping nd opticl lttices. I then discuss novel stte mnipultion nd entnglement schemes in opticl lttices, nd how these might e used to implement mesurement-sed quntum computing. X :fc[ Xkfd`Z ^Xj Y Figure 1 Formtion of opticl lttices., An opticl stnding wve is generted y superimposing two lser ems. The ntinodes (or nodes) of the stnding wve ct s perfectly periodic rry of microscopic lser trps for the toms. The crystl of light in which the cold toms cn move nd re stored is clled n opticl lttice., If severl stnding wves re overlpped, higher-dimensionl lttice structures cn e formed, such s the twodimensionl opticl lttice shown here.

15 NATURE Vol June 28 INSIGHT REVIEW c y x 2 VDT 1 3 HDT d 4 Finl distnce (µm) Initil distnce (µm) Figure 2 Atom sorting in n opticl lttice., Strings of toms cn e rerrnged y using two crossed stnding wves. Atoms cn e moved independently in the horizontl or verticl direction y tuning the frequency difference of the counterpropgting lser ems, forming single one-dimensionl opticl lttice. HDT, horizontl dipole trp; VDT, verticl dipole trp., Fluorescence imge of the initil tom distriution on the lttice. Scle r, 15 μm. c, Applying distnce-control opertions on six of the seven toms cretes string of toms with equidistnt seprtion. This is crried out y moving the two stnding wves through severl sequences (for exmple, 1, 2, nd then 3) s shown in. The toms follow the movement of the nodes of the lttices nd cn therey e repositioned. Scle r, 15 μm. d, For initil distnces of the toms lrger thn 1 µm, the toms cn e sorted to controlled seprtions of 15 µm. (Reproduced, with permission, from ref. 31.) Opticl trpping nd opticl lttices Neutrl toms cn e efficiently trpped y lser light thnks to the opticl dipole force. This technique in which cells cn e mnipulted with opticl tweezers, without touching them is widely used in iophysics. The sic principle is tht prticle with n electric dipole moment d plced in n externl electric field E experiences potentil energy: V dip = d E. In the cse of n oscillting electric field, n oscillting electric dipole moment is induced, for exmple when lser light intercts with n tom. Such n induced dipole moment is proportionl to the pplied electric field strength nd results in n opticl potentil tht is generlly proportionl to the intensity of the pplied light field. The opticl potentil cn either e ttrctive or repulsive, depending on whether the frequency of the pplied lser field is smller or lrger thn the tomic resonnce frequency 2. Periodic potentils cn e formed out of such opticl potentils y interfering lser ems propgting long different directions. The resultnt periodic pttern of right nd drk fringes is experienced y the toms s perfect rry of potentil mxim nd minim in which they move. In the simplest cse of two counterpropgting lser ems long the z xis, periodic potentil of the form V lt = V sin 2 (2πz/λ) is creted, with periodicity of λ/2, where λ is the wvelength of the light field nd V is the potentil depth of the lttice (Fig. 1). By superimposing severl of these stnding-wve lser fields long different directions, it is possile to crete lttice structures, in which toms cn e trpped, in one, two or three dimensions (Fig. 1). For three-dimensionl lttice, ech trpping site cn e viewed s n lmost perfect hrmonic oscilltor, with virtionl frequencies in the rnge of tens to hundreds of kilohertz. Such opticl-lttice potentils offer huge flexiility in their design. For exmple, the potentil depth cn e chnged long different directions independently, nd the generl lttice geometry cn e controlled, for exmple y interfering lser ems t different ngles. It hs recently ecome possile to engineer spin-dependent lttice potentils, where different tomic spin sttes experience different periodic potentils 2,21, or superlttice structures composed of rrys of doule wells When ech of these doule wells is filled with two toms, they cn mimic the ehviour of electronic doule-quntum-dot systems 25 27, nd similr strtegies cn e used to crete protected nd long-lived quits nd roust quntum gtes. The dditionl strength of opticl-lttice-sed systems, however, lies in the fct tht thousnds of potentil wells re present in prllel, ech of which cn e efficiently coupled with the neighouring well to crete mssively prllel cting quntum gtes. Atom trnsport nd stte mnipultion One importnt chllenge when deling with ultrcold toms is keeping to minimum ny possile heting, ecuse this could ffect the motionl or spin degrees of freedom. At the sme time, toms my need to e moved close together to initite quntum gtes etween ritrry pirs of toms in the rry. There hs recently een n impressive dvnce in the control nd movement of single toms. A French reserch tem hs shown how single tom, trpped in dipole trp, cn e moved in two-dimensionl plne in highly controlled wy with su-micrometre sptil resolution 28. The reserchers lso showed tht toms cn e moved without detectle perturtion even if they re prepred in coherent superposition of two internl spin sttes nd when trnsferred from one dipole trp to nother. In nother pproch, tem from the University of Bonn, Germny, used n tomic conveyer elt to move nd position toms trpped in the nodes of one-dimensionl stnding-wve light field 29. By slightly tuning the frequency difference etween the two counterpropgting lser fields, the stnding wve cn e turned into wlking wve, the motion of which the toms closely trck. By crossing two such conveyer elts long orthogonl directions, toms cn e ctively sorted in n rry. Such n tom sorting mchine hs een used to sort lttice rndomly filled with seven toms into perfectly ordered string of equidistnt single toms 3,31 (Fig. 2). These impressive fets oth contin crucil components for the controlled entnglement of tom pirs or strings of toms in the lttice (discussed in the next section). The control nd imging of single toms in n opticl lttice remins huge chllenge, ut Dvid Weiss nd co-workers hve recently shown how such imging cn work in three-dimensionl rry of toms 32. By using high-resolution opticl lens, the reserchers were le to imge two-dimensionl plnes in three-dimensionl opticl lttice x z y x z y x z y Figure 3 Imging of single toms in three-dimensionl opticl lttice. Up to 25 toms re loded from mgneto-opticl trp into threedimensionl opticl lttice with spcing of 4.9 µm. (Scle r, µm.) The toms cn e imged y collecting their fluorescence light through high-resolution ojective lens. Different plnes of the rry cn e trgeted y focusing the imging plne to different lttice plnes (left to right). The sme rry of toms cn e imged repetedly while only minimlly ffecting the tom distriution in the lttice. Imging ws crried out long the z xis, t time t = () nd t = 3 s (). (Reproduced, with permission, from ref. 32.) 117

16 INSIGHT REVIEW NATURE Vol June 28 Quit sis m F = 1 m F = 1 RF m F = m F = 1 m F = m F = 1 m F = m F = 1 Momentum ( K R / 2) 4 m F = m F = e g e e g e e 1 g e g Figure 4 Demonstrtion of SWAP opertion using exchnge interctions., Using rdio-frequency (RF) wves, two toms in doule-well potentil re rought into different spin sttes, denoted (red) nd 1 (lue), on different sides of the opticl doule-well potentil. The two logicl quits, nd 1, re encoded in the electronic hyperfine sttes with ngulr momentum m F = nd m F = 1 of the toms, respectively. When merging these into single well, quntum-mechnicl exchnge interctions induce n oscilltion etween the spin popultions in the lower nd upper virtionl level., This oscilltion cn e reveled y using n ditic nd mpping technique in which the popultion of different virtionl sttes is mpped onto different Brillouin zones fter slowly turning off the lttice potentil. (The Brillouin zones re given in units of K R 2, where K R is the recoil momentum of the lttice photons. The colours reflect the numer of toms with this momentum, incresing from lue to red.) This technique llows the popultion of the virtionl sttes of single lttice site to e mesured in spin-resolved wy (upper imges re exmples of experimentl results for the nd mpping for three distinct times during the exchnge oscilltion cycle, with the times denoted y dshed lines through the lower imges nd imges in c), reveling the exchnge-induced spin dynmics (lower imges, which show nd mpping results tken t different times in the exchnge oscilltion cycle). The lue oxes indicte the moment to which the different virtionl sttes, g nd e, re mpped. c, Multiple SWAP cycles re oserved, y mesuring the popultion in the excited virtionl stte e over time (red, toms in spin stte ; nd lue, toms in spin stte 1 ). These show negligile decy during the oscilltions, indicting the roust implementtion of the two-quit interction. For hlf of SWAP cycle, denoted s S W AP opertion, two toms cn e entngled to form Bell pir. (Reproduced, with permission, from ref. 23.) c.8 Frction in stte e Time (ms) filled with up to 25 toms loded from lser-cooled cloud of toms (Fig. 3). To chieve such single-site nd single-tom resolution, the tem used wider-spced opticl lttice with periodicity of 4.9 µm, nd the shllow depth of field of the opticl detection llowed them to select single lttice plne. Severl groups re lredy trying to chieve such single-site nd single-tom resolution for tightly spced lttices formed y counterpropgting lser ems in the opticl regime, with site spcing of only few hundred nnometres. When such rrys re loded from degenerte osonic or fermionic quntum gs, the lttice would e filled with hundreds of thousnds of toms, with ech plne contining n rry of typiclly 1, toms tht could e imged nd mnipulted simultneously. Entngling neutrl toms Storing, sorting nd controlling toms in lrge-scle rry of prticles is only one prt of the chllenge; the other consists of entngling the prticles to implement quntum gtes or to generte multiprticle entngled resource sttes for quntum informtion processing. This requires precise control over the internl-stte-dependent interctions etween the prticles in lttice. Idelly, the interctions etween ny pir of toms in the lttice should e controllle such tht they could e coxed into ny desired quntum-mechnicl superposition stte. One pproch is to use single-tom red-nd-write hed, moving toms in opticl tweezers to the desired loction to interct with other toms. However, the trnsport tkes precious time, during which hrmful decoherence processes could destroy the frgile quntum coherence stored in the register. Another possiility might e etter dpted to the lttice system nd tkes dvntge of the mssive prllelism with which opertions cn e crried out. The interctions etween neutrl toms re typiclly very short-rnged they re known s contct interctions nd only occur when two prticles re rought together t single lttice site, where they cn directly interct. But when ech tom is rought into contct with ech of its neighours, the collisions etween the prticles 118

17 NATURE Vol June 28 INSIGHT REVIEW cn crete highly entngled multiprticle stte 2,21, known s cluster stte 36, which cn e used s resource stte for quntum informtion processing. The superposition principle of quntum mechnics llows this to e chieved in highly prllel wy, using stte-dependent opticl lttice, in which different tomic spin sttes experience different periodic potentils 2,21. Strting from lttice where ech site is filled with single tom, the toms re first rought into superposition of two internl spin sttes. The spin-dependent lttice is then moved in such wy tht n tom in two different spin sttes splits up nd moves to the left nd right simultneously so tht it collides with its two neighours. In single opertion, whole string of toms cn therey e entngled. However, if the initil string of toms contined defects, n tom moving to the side my hve no prtner to collide with, so the length of the entngled cluster would e limited to the verge length etween two defects. The sorted rrys of toms produced y n tomic sorting mchine could prove to e n idel strting point for such collisionl quntum gtes, s the initil rrys re defect free. In ddition, defects could e efficiently removed y further ctive cooling of the quntum gses in the lttice. Indeed, such cooling is necessry to enhnce the regulrity of the filling chieved with the current lrge-scle ensemles. Severl concepts relted to drk stte cooling methods from quntum optics nd lser cooling could help in this cse. The toms could e ctively cooled into the desired mny-ody quntum stte, which is tilored to e non-intercting (tht is, drk) with the pplied cooling lser field 37,38. When constructing such entngled sttes, the prticles mny degrees of freedom cn couple to the environment, leding to decoherence, which will destroy the complex quntum superpositions of the toms. To void such decoherence processes, which ffect the system more the lrger it ecomes, it is desirle to construct mny-prticle sttes, which re highly insensitive to externl perturtions. Unfortuntely, when using the outlined controlled-collisions scheme to crete n tomic cluster stte, the tomic quits must e encoded in sttes tht undergo mximl de coherence with respect to mgnetic field fluctutions. Two recent experiments hve shown how decoherence could e voided, y imp lementing controlled exchnge interctions etween toms 23,39 ; this could led to new wys of creting roust entngled sttes (discussed in the next section). Another wy to void the prolem of decoherence is to pply fster quntum gtes, so more gte opertions could e crried out within fixed decoherence time. For the toms of ultrcold gses in opticl lttices, Feshch resonnces 4,41 cn e used to increse the collisionl interctions nd therey speed up gte opertions. However, the unitrity limit in scttering theory does not llow the collisionl interction energy to e incresed eyond the on-site virtionl oscilltion frequency, so the lower timescle for gte opertion is typiclly few tens of microseconds. Much lrger interction energies, nd hence fster gte times, could e chieved y using the electric dipole dipole interctions etween polr molecules 42, for exmple, or Ryderg toms 43,44 ; in the ltter cse, gte times well elow the microsecond rnge re possile. For Ryderg toms, phse gte etween two toms could e implemented y dipole-lockde mechnism, which inhiits the simultneous excittion of two toms nd therey induces phse shift in the two-prticle stte only when oth toms re initilly plced in the sme quntum stte. The first signs of such Ryderg dipole-lockde mechnism hve een oserved in mesoscopic cold nd ultrcold tom clouds 45 48, ut it remins to e seen how well they cn e used to implement quntum gtes etween two individul toms. Ryderg toms offer n importnt dvntge for the entnglement of neutrl toms: they cn interct over longer distnces, nd ddressing single toms in the lttice to turn the interctions etween these two toms on nd off voids the need for the toms to move. In ddition, the lttice does not hve to e perfectly filled for two toms to e entngled if their initil position is known efore pplying the Ryderg interction. Novel quntum gtes vi exchnge interctions Entngling neutrl toms requires stte-dependent interctions. A nturl wy to chieve this is to tune the collisionl interctions etween toms to different strengths for different spin sttes, or to llow explicitly only specific spin sttes into contct for controlled collisions. Another possiility is to exploit the symmetry of the underlying two-prticle wvefunctions to crete the desired gte opertions, even in the cse of completely spin-independent interctions etween toms. This principle lies t the hert of two experiments to control the spin spin interctions etween two prticles using exchnge symmetry 23,39,49, nd uilds on originl ides nd experiments involving doule quntumdot systems 25,26. Reserch tems t the Ntionl Institute of Stndrds nd Technology (NIST) t Githersurg, Mrylnd, nd the University of Minz, Germny, hve demonstrted such interctions for two toms in doule-well potentil. How do these exchnge interctions rise, nd how cn they e used to develop primitives (or uilding locks) for quntum informtion processing? As one of the fundmentl principles of quntum mechnics, the totl quntum stte of two prticles (used in two experiments) hs to e either symmetricl in the cse of osons or ntisymmetricl for fermions, with respect to exchnge of the two prticles. When trpped on single lttice site, two-prticle osonic wvefunction cn e fctored into sptil component, which descries the positions of the two prticles, nd spin component, which descries U Popultion nd spin imlnce J J ex Time (ms) Figure 5 Superexchnge coupling etween toms on neighouring lttice sites., Virtul hopping processes (left to right, nd right to left) medite n effective spin spin interction with strength J ex etween the toms, which cn e controlled in oth mgnitude nd sign y using potentil is etween the wells. U is the on-site interction energy etween the toms on single lttice site, nd J is the single-prticle tunnel coupling., The effective spin spin interction emerges when incresing the interction U etween the prticles reltive to their kinetic energy J (top to ottom). It cn e oserved in the time evolution of the mgnetiztion dynmics in the doule well. Blue circles indicte spin imlnce, nd rown circles indicte popultion imlnce. The curves denote fit to theoreticl model tking into ccount the full dynmics oserved within the Hurd model. (Reproduced, with permission, from ref. 39.) J 119

18 INSIGHT REVIEW NATURE Vol June 28 their spin orienttions. If the sptil wvefunction prt is sym metricl with respect to prticle exchnge, the spin prt must e sym metricl too, or they must oth e ntisymmetricl, so the totl wvefunction lwys retins the correct symmetry. The two comintions, however, hve different interction energies: in the cse of symmetricl sptil wvefunction, oth prticles re more likely to e locted in the sme position, wheres for n ntisymmetricl one they re never found t the sme loction. The former leds to strong collisionl interctions etween the prticles, wheres the ltter leds to vnishing interction energy. It is this energy difference etween the singlet (ntisymmetricl) nd triplet (symmetricl) spin sttes tht gives rise to n effective spin spin interction etween the two prticles. When the NIST tem plced two toms onto lttice site, with the spin-up prticle in the virtionl ground stte, g nd the spindown prticle in the first excited virtionl stte, e, the effective spin interction led to exchnge oscilltions etween the quit sttes, g, e, g, e. In computer terminology this is clled SWAP Figure 6 Arry of entngled Bell pirs otined using opticl superlttices., Using exchnge-medited S W AP opertions, rrys of Bell pirs (yellow) consisting of two toms in different spin sttes (red nd lue) cn e creted in mssively prllel wy., These two-prticle entngled sttes cn e extended to lrger multiprticle entngled sttes, y using spin spin interctions to connect toms on the edges of Bell pir (mrked y dditionl yellow onds etween the edges of previously unconnected Bell pirs). Applying this opertion dditionlly long the orthogonl direction leds to the cretion of lrge two-dimensionl cluster sttes or other useful entngled resource sttes 54. opertion nd is one of the fundmentl primitives of quntum computing 25. In fct, the exchnge opertion llows for ny trnsform tions y n ngle θ of the form, = cos(θ), + isin(θ),, for ny spin stte, of the prticles. When the SWAP opertion is crried only hlfwy through, denoted y S W AP, the two prticles end up s n entngled Bell pir. The NIST reserchers oserved such SWAP opertions y first prepring L R stte configurtion in the doule-well potentil (where L is the left well nd R is the right well) nd then ctively deforming the doule well, so oth prticles ended up on the sme lttice site. Exchnge oscilltions then flipped the spin configurtions over time; these were oserved in the experiment over up to 12 SWAP cycles without ny noticele dmping of the exchnge oscilltion signl 23 (Fig. 4). In the NIST experiments, the toms hd to e rought onto the sme lttice site to initite exchnge interctions, ut virtul tunnelling processes 24 cn chieve this without moving the prticles. In these processes, toms constntly proe their neighouring lttice site, fter which either they or their neighouring prticle return to the originl lttice site. Such process cn either leve the initil position of the toms intct or swp them over, therey giving rise to n effective spin spin interction etween the two prticles of the form H eff = J ex S i S j, where S i nd S j re the spin opertors on neighouring lttice sites i nd j. Such super-exchnge interctions therefore do not require ny direct wvefunction overlp of the two prticles, s this overlp is estlished during the toms virtul hopping process. The strength nd the sign of the coupling constnt J ex cn e evluted through second-order perturtion theory, resulting in J ex = 4J 2 /U, where J is the single-prticle tunnelling coupling nd U is the spin-independent interction energy etween two prticles occupying the sme lttice site The Minz reserchers could directly oserve nd control such superexchnge spin couplings etween two neighouring toms in the doule-well potentil creted y n opticl superlttice (Fig. 5). These controllle superexchnge interctions form the sic uilding lock of quntum mgnetism in strongly correlted electronic medi nd give rise, for exmple, to the ntiferromgnetic ordering of two-component Fermi gs on lttice 5. For quntum informtion processing, they too cn e used to implement SWAP opertions, ut their control over the spin sttes etween pirs of toms could find other uses s well. For exmple, y first creting n rry of Bell pirs in opticl superlttices using exchnge interctions or spin-chnging collisions 53, these Bell pirs could e connected to ech other using Isingtype superexchnge interctions to directly crete cluster sttes or other useful resource sttes 54 (Fig. 6). Compred with the controlled-collision pproches, however, these cluster sttes cn e encoded in susttes with vnishing totl mgnetiztion nd so could e more roust to glol field fluctutions leding to decoherence. Mesurement-sed quntum computing In the field of quntum computing, there re severl computtionl models, such s the quntum circuit model 55 57, ditic quntum computtion 58, the quntum Turing mchine 59,6, teleporttion-sed models nd the one-wy quntum computer 64,65, giving rise to lrge numer of possiilities for how to crry out quntum computtion. In the circuit model, for exmple, informtion is processed through series of unitry gte opertions, fter which the desired clcultion result is otined t the output. In the mesurement-sed one-wy quntum computer, informtion is processed through sequence of dptive mesurements on n initilly prepred, highly entngled resource stte. Mesurement-sed quntum computing (MBQC) lys out wholly new concept for the prcticl implementtion of quntum informtion processing tht is extremely well suited to lrge rrys of prticles, such s neutrl toms in opticl lttices. First, lrge, multiprticle, entngled resource stte, such s cluster stte, is creted y mens of controlled collisions or the methods outlined ove. A computtionl lgorithm is then implemented y crrying out sequence of dptive single-prticle mesurements, together with locl single-prticle unitry opertions (Fig. 7). The size of the initil entngled cluster is therey crucil, s it determines the length of the clcultion tht cn e crried out. Singlesite ddressing techniques tht re currently eing implemented in ls 12

19 NATURE Vol June 28 INSIGHT REVIEW Informtion flow Quntum gte In z direction In x direction or In x y plne Figure 7 Informtion processing in one-wy quntum computer. After initilly creting multiprticle entngled cluster stte, sequence of dptive single-prticle mesurements is crried out. In ech step of the computtion, the mesurement sis for the next quit depends on the specific progrm nd on the outcome of previous mesurement results. Finlly, fter ll the mesurements hve een crried out, the stte of the system is given y ξ {α} Ψ { α} out, where the mesured quits re given y the product stte ξ {α} nd the finl output stte is Ψ { α} out, which contins the computtion result up to unitry opertion tht depends on ll of the previous mesurement results, {α}. The short lck rrows in the figure denote the direction of the mesurement sis for the corresponding quit, nd the lrge rown rrows indicte the directions of informtion flow. When mesuring the quits etween two chins (lue rrows), quntum gte is relized. (Reproduced, with permission, from ref. 36.) could one dy led to cluster-stte computing in lttice-sed systems. Proof-of-principle demonstrtions hve lredy een crried out using photon-sed cluster sttes 66,67, nd the model could e implemented in ny system consisting of n rry of quits. So fr, MBQC hs lredy ecome mjor reserch field, currently minly driven y theory, with interdisciplinry connections to entnglement theory, grph theory, computtionl complexity, logic nd sttisticl physics. Severl fundmentl questions regrding MBQC hve now een nswered, such s, which multiprticle entngled sttes cn serve s universl resources Universlity in this context is defined s the ility to generte every possile quntum stte from the resource through single-quit opertions lone. Using this definition, it cn e shown tht the two-dimensionl cluster stte is universl resource stte, wheres the one-dimensionl cluster stte is not. Furthermore, universl resource stte must e mximlly entngled with respect to ll types of entnglement mesure. If this were not the cse, there could e stte with higher degree of entnglement tht could not e generted from the resource stte through single-quit opertions. Becuse singlequit opertions cnnot dd entnglement to the system, the initil stte could not hve een universl resource stte. For MBQC to e implemented in prctice, it is importnt to know how defects, such s missing toms or douly occupied sites, cn limit its computtionl power. Active cooling of the lttice gses could help to reduce such defects 37,38, lthough finite residul numer of defects will lwys e present. Astonishingly, the computtionl power degrdes shrply only when the numer of defects is incresed ove the percoltion threshold 72 of sttisticl physics. In ddition, cluster stte cn e universl resource even in the presence of defects, lthough the loction of the defects would need to e known in order to dpt mesurement sequence to them. In n effort to understnd the computtionl power of MBQC, severl tems hve lso shown how MBQC cn e connected to other mesurement-sed quntum computing schemes, such s teleporttion sed ones Any rel-world quntum computer will lso need to overcome the dverse effects of decoherence rising from interctions with the environment, which ffect the frgile quntum superpositions nd the entngled mny-ody sttes in the system nd result in errors in quntum computtion. In the drive to crete sclle quntum computer, quntum error correction hs crucil role in correcting such errors 77, while mintining the greter computtionl speed of quntum computer over clssicl computer. Quntum error correction llows n ritrrily long quntum computtion to e crried out with ritrry ccurcy, if the error level of the underlying opertions is elow threshold vlue By comining topologicl error-correction schemes originting from Alexei Kitev s toric code 81 nd mgic-stte distilltion into the one-wy quntum computer, it hs recently een shown tht n error threshold of up to cn e relized 82. For locl model in two dimensions, in which only nerest-neighour interctions etween the prticles re llowed, this is the highest threshold known, ut it is still eyond the rech of current experiments. Quntum simultions Ultrcold quntum gses in opticl lttices re lso eing used to simulte the ehviour of strongly intercting electronic systems 4,17,19, where they might e le to shed light on complex prolems emerging from condensed-mtter physics. A prominent exmple is the Hurd model, which forms simple theoreticl description of intercting fermions on lttice. Although the sic hmiltonin for such system cn e esily written down, solving it is one of the hrdest prolems in condensedmtter physics. One prolem tht ultrcold toms might help to nswer is whether high-temperture superconducting phse cn emerge from within the Hurd model 83. Such scenrio is widely thought to lie t the hert of the mystery of high-temperture superconductors 84. A strting point for such studies could e n ntiferromgneticlly ordered gs of fermions, which fter doping hs een proposed to trnsform into spin-liquid phse 84,85 tht cn support the formtion of hightemperture superconductor. Severl reserch groups re currently trying to estlish n ntiferromgneticlly ordered Mott insultor in fermionic tom clouds with two spin components. The temperture requirements to chieve this seem to e demnding 86, however, nd progress will gin depend on finding wys to cool the quntum gses within the lttice 37. Outlook From oth n experimentl nd theoreticl point of view, opticl lttices offer outstnding possiilities for implementing new designs for quntum informtion processing nd quntum simultions. Some of the mjor experimentl chllenges in the field re lowering the tempertures of the lttice-sed quntum gses nd chieving single-site ddressing, the ltter eing, for exmple, crucil requirement for the MBQC model. Although there might e specil situtions in which this cn e voided, such ddressility would provide fresh impetus for the field of quntum simultions. Imgine eing le to oserve nd control spin system in two dimensions with 1, prticles simultneously in view, ll with single-site nd single-tom resolution. Oserving dynmic evolutions in these systems, proing their sptil correltions nd finlly implementing quntum informtion processing in truly lrge-scle system would ll ecome possile. 1. Pitevskii, L. & Stringri, S. Bose Einstein Condenstion (Oxford Univ. Press, Oxford, 23). 2. Grimm, R., Weidemüller, M. & Ovchinnikov, Y. B. Opticl dipole trps for neutrl toms. Adv. At. Mol. Opt. Phys. 42, (2). 3. Jessen, P. S. & Deutsch, I. H. Opticl lttices. Adv. At. Mol. Opt. Phys. 37, (1996). 4. Bloch, I., Dlird, J. & Zwerger, W. Mny-ody physics with ultrcold gses. Preprint t (27). 5. Anderson, M. H., Ensher, J. R., Mtthews, M. R., Wiemn, C. E. & Cornell, E. A. Oservtion of Bose Einstein condenstion in dilute tomic vpor. Science 269, (1995). 6. Dvis, K. B. et l. Bose Einstein condenstion in gs of sodium toms. Phys. Rev. Lett. 75, (1995). 7. Brdley, C. C., Sckett, C. A., Tollett, J. J. & Hulet, R. G. Evidence of Bose Einstein condenstion in n tomic gs with ttrctive interctions. Phys. Rev. Lett. 75, (1995). 8. DeMrco, B. & Jin, D. D. Onset of Fermi degenercy in trpped tomic gs. Science 285, (1999). 9. Schreck, F. et l. Qusipure Bose Einstein condenste immersed in Fermi Se. Phys. Rev. Lett. 87, 843 (21). 1. Truscott, A. G., Strecker, K. E., McAlexnder, W. I., Prtridge, G. P. & Hulet, R.G. Oservtion of Fermi pressure in gs of trpped toms. Science 291, (21). 121

20 INSIGHT REVIEW NATURE Vol June Fisher, M. P. A., Weichmn, P. B., Grinstein, G. & Fisher, D. S. Boson locliztion nd the superfluid-insultor trnsition. Phys. Rev. B 4, (1989). 12. Jksch, D., Bruder, C., Circ, J. I., Grdiner, C. W. & Zoller, P. Cold osonic toms in opticl lttices. Phys. Rev. Lett. 81, (1998). 13. Greiner, M., Mndel, M. O., Esslinger, T., Hänsch, T. & Bloch, I. Quntum phse trnsition from superfluid to Mott insultor in gs of ultrcold toms. Nture 415, (22). 14. Stöferle, T., Moritz, H., Schori, C., Köhl, M. & Esslinger, T. Trnsition from strongly intercting 1D superfluid to Mott insultor. Phys. Rev. Lett. 92, 1343 (24). 15. Spielmn, I. B., Phillips, W. D. & Porto, J. V. The Mott insultor trnsition in two dimensions. Phys. Rev. Lett. 98, 844 (27). 16. Köhl, M., Moritz, H., Stöferle, T., Günter, K. & Esslinger, T. Fermionic toms in three dimensionl opticl lttice: oserving Fermi surfces, dynmics, nd interctions. Phys. Rev. Lett. 94, 843 (25). 17. Jksch, D. & Zoller, P. The cold toms Hurd toolox. Ann. Phys. (NY) 315, (25). 18. Bloch, I. Ultrcold quntum gses in opticl lttices. Nture Phys. 1, 23 3 (25). 19. Lewenstein, M. et l. Ultrcold tomic gses in opticl lttices: mimicking condensed mtter physics nd eyond. Adv. Phys. 56, (27). 2. Jksch, D., Briegel, H. J., Circ, J. I., Grdiner, C. W. & Zoller, P. Entnglement of toms vi cold controlled collisions. Phys. Rev. Lett. 82, (1999). 21. Mndel, O. et l. Controlled collisions for multiprticle entnglement of opticlly trpped toms. Nture 425, (23). 22. Sey-Strley, J., Anderlini, M., Jessen, P. S. & Porto, J. V. Lttice of doule wells for mnipulting pirs of cold toms. Phys. Rev. A 73, 3365 (26). 23. Anderlini, M. et l. Controlled exchnge interction etween pirs of neutrl toms in n opticl lttice. Nture 448, (27). 24. Fölling, S. et l. Direct oservtion of second-order tom tunnelling. Nture 448, (27). 25. Loss, D. & DiVincenzo, D. P. Quntum computtion with quntum dots. Phys. Rev. A 57, (1998). 26. Pett, J. R. et l. Coherent mnipultion of coupled electron spins in semiconductor quntum dots. Science 39, (25). 27. Hnson, R., Kouwenhoven, L. P., Pett, J. R., Truch, S. & Vndersypen, L. M. K. Spins in few-electron quntum dots. Rev. Mod. Phys. 79, (27). 28. Beugnon, J. et l. Two-dimensionl trnsport nd trnsfer of single tomic quit in opticl tweezers. Nture Phys. 3, (27). 29. Schrder, D. et l. A neutrl tom quntum register. Phys. Rev. Lett. 93, 1551 (24). 3. Miroshnychenko, Y. et l. Precision preprtion of strings of trpped neutrl toms. New J. Phys. 8, 191 (26). 31. Miroshnychenko, Y. et l. An tom-sorting mchine. Nture 442, 151 (26). 32. Nelson, K. D., Li, X. & Weiss, D. S. Imging single toms in three-dimensionl rry. Nture Phys. 3, (27). 33. Cho, J. Addressing individul toms in opticl lttices with stnding-wve driving fields. Phys. Rev. Lett. 99, 252 (27). 34. Joo, J., Lim, Y. L., Beige, A. & Knight, P. L. Single-quit rottions in two-dimensionl opticl lttices with multiquit ddressing. Phys. Rev. A 74, (26). 35. Gorshkov, A. V., Jing, L., Greiner, M., Zoller, P. & Lukin, M. D. Coherent quntum opticl control with suwvelength resolution. Preprint t (27). 36. Briegel, H. J. & Russendorf, R. Persistent entnglement in rrys of intercting prticles. Phys. Rev. Lett. 86, (21). 37. Griessner, A., Dley, A. J., Clrk, S. R., Jksch, D. & Zoller, P. Drk-stte cooling of toms y superfluid immersion. Phys. Rev. Lett. 97, 2243 (26). 38. Griessner, A., Dley, A. J., Clrk, S. R., Jksch, D. & Zoller, P. Dissiptive dynmics of tomic Hurd models coupled to phonon th: drk stte cooling of toms within Bloch nd of n opticl lttice. New J. Phys. 9, 44 (27). 39. Trotzky, S. et l. Time-resolved oservtion nd control of superexchnge interctions with ultrcold toms in opticl lttices. Science 319, (28). 4. Inouye, S. et l. Oservtion of Feshch resonnces in Bose Einstein condenste. Nture 392, (1998). 41. Courteille, P., Freelnd, R. S., Heinzen, D. J., vn Aeelen, F. A. & Verhr, B. J. Oservtion of Feshch resonnce in cold tom scttering. Phys. Rev. Lett. 81, (1998). 42. Micheli, A., Brennen, G. K. & Zoller, P. A toolox for lttice-spin models with polr molecules. Nture Phys. 2, (26). 43. Jksch, D. et l. Fst quntum gtes for neutrl toms. Phys. Rev. Lett. 85, (2). 44. Lukin, M. D. et l. Dipole lockde nd quntum informtion processing in mesoscopic tomic ensemles. Phys. Rev. Lett. 87, 3791 (21). 45. Tong, D. et l. Locl lockde of Ryderg excittion in n ultrcold gs. Phys. Rev. Lett. 93, 631 (24). 46. Singer, K., Reetz-Lmour, M., Amthor, T., Mrcss, L. G. & Weidemüller, M. Suppression of excittion nd spectrl rodening induced y interctions in cold gs of Ryderg toms. Phys. Rev. Lett. 93, 1631 (24). 47. Lieisch, T. C., Reinhrd, A., Bermn, P. R. & Rithel, G. Atom counting sttistics in ensemles of intercting Ryderg toms. Phys. Rev. Lett. 95, 2532 (25). 48. Heidemnn, R. et l. Evidence for coherent collective Ryderg excittion in the strong lockde regime. Phys. Rev. Lett. 99, (27). 49. Hyes, D., Julienne, P. S. & Deutsch, I. H. Quntum logic vi the exchnge lockde in ultrcold collisions. Phys. Rev. Lett. 98, 751 (27). 5. Auerch, A. Intercting Electrons nd Quntum Mgnetism (Springer, New York, 26). 51. Dun, L.-M., Demler, E. & Lukin, M. D. Controlling spin exchnge interctions of ultrcold toms in n opticl lttice. Phys. Rev. Lett. 91, 942 (23). 52. Kuklov, A. B. & Svistunov, B. V. Counterflow superfluidity of two-species ultrcold toms in commensurte opticl lttice. Phys. Rev. Lett. 9, 141 (23). 53. Wider, A. et l. Coherent collisionl spin dynmics in opticl lttices. Phys. Rev. Lett. 95, 1945 (25). 54. Vucher, B., Nunnenkmp, A. & Jksch, D. Cretion of resilient entngled sttes nd resource for mesurement-sed quntum computtion with opticl superlttices. Preprint t (27). 55. Deutsch, D. Quntum computtionl networks. Proc. R. Soc. Lond. A 425, 73 9 (1989). 56. Yo, A. in Proc. 34th Annu. Symp. Found. Comput. Sci (IEEE Computer Soc., Los Almitos, 1993). 57. Brenco, A. et l. Elementry gtes for quntum computtion. Phys. Rev. A 52, (1995). 58. Frhi, E. et l. A quntum ditic evolution lgorithm pplied to rndom instnces of n NP-complete prolem. Science 292, (21). 59. Deutsch, D. Quntum-theory, the Church Turing principle nd the universl quntum computer. Proc. R. Soc. Lond. A 4, (1985). 6. Bernstein, E. & Vzirni, U. in Proc. 25th Annu. ACM Symp. Theor. Comput (ACM Press, New York, 1993). 61. Gottesmn, D. & Chung, I. L. Demonstrting the viility of universl quntum computtion using teleporttion nd single-quit opertions. Nture 42, (1999). 62. Knill, E., Lflmme, R. & Milurn, G. J. A scheme for efficient quntum computtion with liner optics. Nture 49, (21). 63. Nielsen, M. A. Quntum computtion y mesurement nd quntum memory. Phys. Lett. A 38, 96 1 (23). 64. Russendorf, R. & Briegel, H. J. A One-wy quntum computer. Phys. Rev. Lett. 86, (21). 65. Russendorf, R. & Briegel, H. J. Computtionl model underlying the one-wy quntum computer. Qunt. Info. Comput. 2, (22). 66. Wlther, P. et l. Experimentl one-wy quntum computing. Nture 434, (25). 67. Kiesel, N. et l. Experimentl nlysis of four-quit photon cluster stte. Phys. Rev. Lett. 95, 2152 (25). 68. Gross, D., Eisert, J., Schuch, N. & Perez-Grci, D. Mesurement-sed quntum computtion eyond the one-wy model. Phys. Rev. A 76, (27). 69. Vn den Nest, M., Miyke, A., Dür, W. & Briegel, H. J. Universl resources for mesurement-sed quntum computtion. Phys. Rev. Lett. 97, 1554 (26). 7. Vn den Nest, M., Dür, W., Miyke, A. & Briegel, H. J. Fundmentls of universlity in one-wy quntum computtion. New J. Phys. 9, 24 (27). 71. Gross, D. & Eisert, J. Novel schemes for mesurement-sed quntum computtion. Phys. Rev. Lett. 98, 2253 (27). 72. Browne, D. E. et l. Phse trnsition of computtionl power in the resource sttes for one-wy quntum computtion. Preprint t (27). 73. Verstrete, F. & Circ, J. I. Vlence-ond sttes for quntum computtion. Phys. Rev. A 7, 632 (24). 74. Aliferis, P. & Leung, D. W. Computtion y mesurements: unifying picture. Phys. Rev. A 7, (24). 75. Childs, A. M., Leung, D. W. & Nielsen, M. A. Unified derivtions of mesurement-sed schemes for quntum computtion. Phys. Rev. A 71, (25). 76. Jorrnd, P. & Perdrix, S. Unifying quntum computtion with projective mesurements only nd one-wy quntum computtion. Preprint t qunt-ph/44125 (24). 77. Shor, P. W. in Proc. 37th Annu. Symp. Found. Comput. Sci (IEEE Computer Soc., Los Almitos, 1996). 78. Ahronov, D. & Ben-Or, M. in Proc. 29th Annu. ACM Symp. Theor. Comput (ACM Press, New York, 1997). 79. Gottesmn, D. Stilizer Codes nd Quntum Error Correction. PhD thesis, Cliforni Inst. Technol. (1997). 8. Knill, E., Lflmme, R. & Zurek, W. H. Resilient quntum computtion: error models nd thresholds. Proc. R. Soc. Lond. A 454, (1998). 81. Kitev, A. Y. Fult-tolernt quntum computtion y nyons. Ann. Phys. (NY) 33, 2 3 (23). 82. Russendorf, R. & Hrrington, J. Fult-tolernt quntum computtion with high threshold in two dimensions. Phys. Rev. Lett. 98, 1954 (27). 83. Hofstetter, W., Circ, J. I., Zoller, P., Demler, E. & Lukin, M. D. High-temperture superfluidity of fermionic toms in opticl lttices. Phys. Rev. Lett. 89, 2247 (22). 84. Lee, P. A., Ngos, N. & Wen, X.-G. Doping Mott insultor: physics of high-temperture superconductivity. Rev. Mod. Phys. 78, (26). 85. Anderson, P. W. The resonting vlence ond stte in L 2 CuO 4 nd superconductivity. Science 235, (1987). 86. Werner, F., Prcollet, O., Georges, A. & Hssn, S. R. Interction-induced ditic cooling nd ntiferromgnetism of cold fermions in opticl lttices. Phys. Rev. Lett. 95, 5641 (25). Acknowledgements I thnk H. Briegel for discussions, nd the Germn Reserch Foundtion (DFG), the Europen Union (through the OLAQUI nd SCALA projects) nd the Air Force Office of Scientific Reserch (AFOSR) for support. Author Informtion Reprints nd permissions informtion is ville t The uthor declres no competing finncil interests. Correspondence should e ddressed to the uthor (loch@uni-minz.de). 122

21 NATURE Vol June 28 doi:1.138/nture7127 INSIGHT REVIEW The quntum internet H. J. Kimle 1 Quntum networks provide opportunities nd chllenges cross rnge of intellectul nd technicl frontiers, including quntum computtion, communiction nd metrology. The reliztion of quntum networks composed of mny nodes nd chnnels requires new scientific cpilities for generting nd chrcterizing quntum coherence nd entnglement. Fundmentl to this endevour re quntum interconnects, which convert quntum sttes from one physicl system to those of nother in reversile mnner. Such quntum connectivity in networks cn e chieved y the opticl interctions of single photons nd toms, llowing the distriution of entnglement cross the network nd the teleporttion of quntum sttes etween nodes. In the pst two decdes, rod rnge of fundmentl discoveries hve een mde in the field of quntum informtion science, from quntum lgorithm tht plces pulic-key cryptogrphy t risk to protocol for the teleporttion of quntum sttes 1. This union of quntum mechnics nd informtion science hs llowed gret dvnces in the understnding of the quntum world nd in the ility to control coherently individul quntum systems 2. Unique wys in which quntum systems process nd distriute informtion hve een identified, nd powerful new perspectives for understnding the complexity nd sutleties of quntum dynmicl phenomen hve emerged. In the rod context of quntum informtion science, quntum networks hve n importnt role, oth for the forml nlysis nd the physicl implementtion of quntum computing, communiction nd metrology 2 5. A notionl quntum network sed on proposls in refs 4, 6 is shown in Fig. 1. Quntum informtion is generted, processed nd stored loclly in quntum nodes. These nodes re linked y quntum chnnels, which trnsport quntum sttes from site to site with high fidelity nd distriute entnglement cross the entire network. As n extension of this ide, quntum internet cn e envisged; with only moderte processing cpilities, such n internet could ccomplish tsks tht re impossile in the relm of clssicl physics, including the distriution of quntum softwre 7. Aprt from the dvntges tht might e gined from prticulr lgorithm, there is n importnt dvntge in using quntum connectivity, s opposed to clssicl connectivity, etween nodes. A network of quntum nodes tht is linked y clssicl chnnels nd comprises k nodes ech with n quntum its (quits) hs stte spce of dimension k2 n, wheres fully quntum network hs n exponentilly lrger stte spce, 2 kn. Quntum connectivity lso provides potentilly powerful mens to overcome size-scling nd error-correltion prolems tht would limit the size of mchines for quntum processing 8. At ny stge in the development of quntum technologies, there will e lrgest size ttinle for the stte spce of individul quntum processing units, nd it will e possile to surpss this size y linking such units together into fully quntum network. A different perspective of quntum network is to view the nodes s components of physicl system tht interct y wy of the quntum chnnels. In this cse, the underlying physicl processes used for qun tum network protocols re dpted to simulte the evolution of qun tum mny-ody systems 9. For exmple, toms tht re loclized t seprte nodes cn hve effective spin spin interctions ctlysed y single-photon pulses tht trvel long the chnnels etween the nodes 1. This quntum wiring of the network llows wide rnge for the effective hmiltonin nd for the topology of the resultnt lttice. Moreover, from this perspective, the extension of entnglement cross quntum networks cn e relted to the clssicl prolem of percoltion 11. These exciting opportunities provide the motivtion to exmine reserch relted to the physicl processes for trnslting the strct illustrtion in Fig. 1 into relity. Such considertions re timely ecuse scientific cpilities re now pssing the threshold from lerning phse with individul systems nd dvncing into domin of rudimentry functionlity for quntum nodes connected y quntum chnnels. In this review, I convey some sic principles for the physicl implementtion of quntum networks, with the im of stimulting the involvement of lrger community in this endevour, including in systems-level studies. I focus on current efforts to hrness opticl processes t the level of single photons nd toms for the trnsporttion of quntum sttes relily cross complex quntum networks. Two importnt reserch res re strong coupling of single photons nd toms in the setting of cvity quntum electrodynmics (QED) 12 nd quntum informtion processing with tomic ensemles 13, for which crucil elements re long-lived quntum memories provided y the tomic system nd efficient, quntum interfces etween light nd mtter. Mny other physicl systems re lso eing investigted nd re discussed elsewhere (ref. 2 nd wesites for the Quntum Computtion Rodmp ( the SCALA Integrted Project ( nd Quit Applictions ( A quntum interfce etween light nd mtter The min scientific chllenge in the quest to distriute quntum sttes cross quntum network is to ttin coherent control over the interctions of light nd mtter t the single-photon level. In contrst to toms nd electrons, which hve reltively lrge long-rnge interctions for their spin nd chrge degrees of freedom, individul photons typiclly hve interction cross-sections tht re orders of mgnitude too smll for non-trivil dynmics when coupled to single degrees of freedom for mteril system. The opticl physics community egn to ddress this issue in the 199s, with the development of theoreticl protocols for the coherent trnsfer of quntum sttes etween toms nd photons in the setting of cvity QED 6,14,15. Other importnt dvnces hve een mde in the pst 1 Normn Bridge Lortory of Physics 12 33, Cliforni Institute of Technology, Psden, Cliforni 91125, USA. 123

22 g INSIGHT REVIEW NATURE Vol June 28 decde 2,4, including with tomic ensemles 13,16. The reversile mpping of quntum sttes etween light nd mtter provides the sis for quntumopticl interconnects nd is fundmentl primitive (uilding lock) for quntum networks. Although the originl schemes for such interconnects re sensitive to experimentl imperfections, complete set of theoreticl protocols hs susequently een developed for the roust distriution of quntum informtion over quntum networks, including, importntly, the quntum repeter 4,17 nd sclle quntum networks with tomic ensemles 13. A generic quntum interfce etween light nd mtter is depicted in Fig. 1. This interfce is descried y the interction hmiltonin H int (t), where for typicl sttes H int (t) χ(t), with eing h/2π (where h is Plnck s constnt) nd χ(t) eing the time-dependent coupling strength etween the internl mteril system nd the electromgnetic field. Desirle properties for quntum interfce include tht χ(t) should e user controlled for the clocking of sttes to nd from the quntum memory (for exmple, y using n uxiliry lser), tht the physicl processes used should e roust in the fce of imperfections (for exmple, y using ditic trnsfer) nd tht mistkes should e efficiently detected nd fixed (for exmple, with quntum error correction). In qulittive terms, the rte κ, which chrcterizes the ndwidth of the input output chnnel, should e lrge compred with the rte γ, which chrcterizes prsitic losses, nd oth of these rtes should e smll compred with the rte of coherent coupling χ. Exmples of physicl systems for relizing quntum interfce nd distriuting coherence nd entnglement etween nodes re shown in Fig. 1c, d. In the first exmple (Fig. 1c), single toms re trpped in opticl cvities t nodes A nd B, which re linked y n opticl fire. Externl fields control the trnsfer of the quntum stte Ψ stored in the tom t node A to the tom t node B y wy of photons tht propgte from node A to node B 6,18. In the second exmple (Fig. 1d), singlephoton pulse tht is generted t node A is coherently split into two Quntum node k c H int Quntum chnnel c Node A g Y k Node B in W B (t) W out A (t) Y d W B (t) g Node A Node B c H int k k Node C k W C (t) g Figure 1 Quntum networks., Shown is notionl quntum network composed of quntum nodes for processing nd storing quntum sttes nd quntum chnnels for distriuting quntum informtion. Alterntively, such network cn e viewed s strongly correlted mny-prticle system., The quntum interfce etween mtter (coloured cue) nd light (red curves) is depicted. Coherent interctions in the node re chrcterized y the rte χ; coupling etween the node nd photons in the externl chnnel occurs t the rte κ; nd prsitic losses occur t the rte γ. c, Quntum stte trnsfer nd entnglement distriution from node A to node B is shown in the setting of cvity quntum electrodynmics (QED) 6. At node A, pulse of the control field Ω A out (t) cuses the trnsformtion of tomic stte Ψ into the stte of propgting opticl field (tht is, into flying photon). At node B, the pulse Ω in B(t) is pplied to mp the stte of the flying photon into n tom in the cvity, therey relizing the trnsfer of the stte Ψ from node A to node B (ref. 18). d, The distriution of entnglement y using ensemles of lrge numer of toms is shown 13. A single-photon pulse t node A is coherently split into two entngled components tht propgte to node B nd node C nd then re coherently mpped y the control fields Ω in B, C(t) into stte tht is entngled etween collective excittions in ech ensemle t node B nd node C. At lter times, components of the entngled stte cn e retrieved from the quntum memories y seprte control fields, Ω o u t B, C(t) (ref. 19). H int (t), interction hmiltonin;, h/2π (where h is Plnck s constnt). 124

23 NATURE Vol June 28 INSIGHT REVIEW components nd propgtes to nodes B nd C, where the entngled photon stte is coherently mpped into n entngled stte etween collective excittions t ech of the two nodes 13,19. Susequent red-out of entnglement from the memories t node B nd/or node C s photon pulses is implemented t the push of utton. Cvity QED At the forefront of efforts to chieve strong, coherent interctions etween light nd mtter hs een the study of cvity QED 2. In oth the opticl 12,21 nd the microwve domins, strong coupling of single toms nd photons hs een chieved y using electromgnetic resontors of smll mode volume (or cvity volume) V m with qulity fctors Q Extensions of cvity QED to other systems 26 include quntum dots coupled to micropillrs nd photonic ndgp cvities 27, nd Cooper pirs intercting with superconducting resontors (tht is, circuit QED; see ref. 28 for review). Physicl sis of strong coupling Depicted in Fig. 2 is single tom tht is locted in n opticl resontor nd for which strong coupling to photon requires tht single intrcvity photon cretes lrge electric field. Stted more quntittively, if the coupling frequency of one tom to single mode of n opticl resontor is g (tht is, 2g is the one-photon Ri frequency), then ε μ 2 ω g = C (1) 2 ε V m where μ is the trnsition dipole moment etween the relevnt tomic sttes (with trnsition frequency ω A ), nd ω C ω A is the resonnt frequency of the cvity field, with polriztion vector ε. Experiments in cvity QED explore strong coupling with g >> (γ, κ), where γ is the tomic decy rte to modes other thn the cvity mode nd κ is the decy rte of the cvity mode itself. Expressed in the lnguge of trditionl opticl physics, the numer of photons required to sturte the intrcvity tom is n γ 2 /g 2, nd the numer of toms required to hve n pprecile effect on the intrcvity field is N κγ/g 2. Strong coupling in cvity QED moves eyond trditionl opticl physics, for which (n, N ) >> 1, to explore qulittively new regime with (n, N ) << 1 (ref. 12). In the pst three decdes, vriety of pproches hve een used to chieve strong coupling in cvity QED 12,2 25. In the opticl domin, route to strong coupling is the use of high-finesse opticl resontors (F ) nd tomic trnsitions with lrge μ (tht is, oscilltor strengths ner unity). Progress long this pth is illustrted in Fig. 2c, with reserch now fr into the domin (n, N ) << 1. As the cvity volume V m is reduced to increse g (eqution (1)), the requirement for tomic locliztion ecomes more stringent. Not surprisingly, efforts to trp nd loclize toms in high-finesse opticl cvities in regime of strong coupling hve een centrl to studies of cvity QED in the pst decde, nd the initil demonstrtion ws in 1999 (ref. 29). Susequent dvnces include extending the time for which n tom is trpped to 1 s (refs 3, 31); see ref. 32 for review. Quntum control over oth internl degrees of freedom (tht is, the tomic dipole nd the cvity field) nd externl degrees of freedom (tht is, tomic motion) hs now een chieved for strongly coupled tom cvity system 33. And n exciting prospect is cvity QED with single trpped ions, for which the oundry for strong coupling hs een reched 34. Coherence nd entnglement in cvity QED Applying these dvnces to quntum networks hs llowed single photons to e generted on demnd (Box 1). Through strong coupling of the cvity field to n tomic trnsition, n externl control field Ω(t) trnsfers one photon into the cvity mode nd then to free spce y wy of the cvity output mirror, leding to single-photon pulse ϕ 1 (t) s collimted em. The temporl structure (oth mplitude nd phse) of the resultnt flying photon ϕ 1 (t) cn e tilored y wy of 1 µm g c 1 4 the control field Ω(t) (refs 6, 35), with the sptil structure of the wve pcket eing set y the cvity mode. Severl experiments hve confirmed the essentil spects of this process for the deterministic genertion of single photons 3,34,36. Significntly, in the idel (ditic) limit, the excited stte e of the tom is not populted ecuse of the use of drk stte protocol 37. By deterministiclly generting it strem of single-photon pulses from single trpped toms, these experiments re first step in the development of quntum networks sed on flying photons. Compred with the genertion of single photons y vriety of other systems 38, one of the distinguishing spects of the drk-stte protocol (Box 1) is tht it should e reversile. Tht is, photon tht is emitted from system A should e le to e efficiently trnsferred to nother system B y pplying the time-reversed (nd suitly delyed) field Ω(t) to system B (Fig. 1c). Such n dvnce ws mde 18 y implementing the reversile mpping of coherent opticl field to nd from internl sttes of single trpped cesium tom. Although this experiment ws imperfect, it provides the initil verifiction of the fundmentl primitive on which the protocol for the physicl implementtion of quntum networks in ref. 6 is sed (n importnt theoreticl protocol tht hs een dpted to mny theoreticl nd experimentl settings). g Criticl photon numer n k Time (yers) Figure 2 Elements of cvity QED., Shown is simple schemtic of n tom cvity system depicting the three governing rtes (g, κ, γ) in cvity QED, where g χ in Fig. 1. Coherent exchnge of excittion etween the tom nd the cvity field proceeds t rte g, s indicted y the dshed rrow for the tom nd the green rrows for the cvity field., A photogrph of two mirror sustrtes tht form the Fry Pérot cvity, which is lso shown schemticlly. The cvity length l = 1 μm, wist w = 12 μm trnsverse to the cvity xis, nd finesse F The supporting structure llows ctive servo control of the cvity length to δl 1 14 m (ref. 12). Scle r, 3 mm. c, The reduction in the criticl photon numer n over time is shown for series of experiments in cvity QED tht were crried out y the Cltech Quntum Optics Group. These experiments involved either sphericl-mirror Fry Pérot cvities (circles) or the whispering-gllery modes of monolithic SiO 2 resontors (squres). The dt points shown for 26 nd 28 re for microtoroidl SiO 2 resontor 75,76 ; those for 29 nd 211 (open squres) re projections for this type of resontor

24 INSIGHT REVIEW NATURE Vol June 28 Box 1 Mpping quntum sttes etween toms nd photons A Ω 1 (t) Ω 1 (t) e g 1 B l(t) Ω 1 (t) t Ω 2 (t) Ω 2 (t) e g 1 c t Reversile trnsfer of stte etween light nd single trpped tom cn e chieved through the mppings 1 nd 1 for the coherent sorption nd emission of single photons (in pnel A, nd of the figure, respectively) 18. In this cse, nd represent internl sttes of the tom with long-lived coherence (for exmple, tomic hyperfine sttes in the 6S 1/2, F = 3 nd F = 4 mnifolds of tomic cesium), nd nd 1 re Fock sttes of the photons in the intrcvity field with n = nd n = 1 excittions, respectively. The trnsition etween nd e is strongly coupled to mode of n opticl cvity with interction energy g, where g (in green) is the coherent coupling rte of the tom nd the photon. In this simple setting, the interction hmiltonin for tom nd cvity field hs drk stte D (tht is, there is no excited stte component e ) 37, s given y D = cosθ + sinθ 1, where Ω 2 (t) cosθ = 1+ with Ω(t) s clssicl control field 14. For Ω(t = ) =, then D =. By contrst, for Ω(t ) >> g, D 1. Pnel A, of the figure shows tht y diticlly rmping control field Ω 1 (t) >> g from on to off over time t tht is slow compred g 2 1/2 (1) Ω 2 (t + t) with 1/g, the tomic stte is mpped from to with the ccompnying coherent sorption of one intrcvity photon. Conversely, in pnel A, of the figure, y turning control field Ω 2 (t) from off to on, the tomic stte is mpped from to with the trnsfer of one photon into the cvity mode. These two processes cn e comined to chieve the coherent trnsfer of the stte of propgting opticl field λ(t) = ϕ field (t) into nd out of quntum memory formed y the tomic sttes nd (ref. 18; figure, pnel B). In the idel cse, the mpping is specified y ϕ field (t) (c 1 + c ) (storge) (c 1 + c ) ϕ field (t + τ), where the field stte is tken to e coherent superposition of zero (c ) nd one (c 1 ) photon, ϕ field (t) = E(t)[c field + c 1 1 field ]. E(t) is the envelope of the field externl to the cvity, with E(t) 2 dt = 1; t + τ is user-selected time (discussed elow). Given timing informtion for the incoming field ϕ field (t), the first step in this process (figure, pnel B, ) is ccomplished y diticlly rmping the control field Ω 1 (t) from on to off, s in A,. After this step, the internl sttes of the tom provide long-lived quntum memory (figure, pnel B, ). At user-selected lter time t + τ, the finl step is initited (figure, pnel B, c) y turning Ω 2 (t + τ) from off to on (s in A, ), therey coherently mpping the tomic stte c 1 + c ck to the flying field stte β(t) = ϕ field (t + τ). (t) The ditic trnsfer of quntum sttes (s descried in Box 1, s well s relted possiilities 1,35 ) relies on strong coupling etween n tom nd single polriztion of the intrcvity field. However, y extending the ides in Box 1 to the two polriztion eigenmodes of the cvity for given trnsverse nd longitudinl mode orders, it is possile to generte entnglement etween the internl sttes of the tom nd the polriztion stte of coherently generted photon An initil control field Ω 1 (t 1 ) results in entnglement etween internl sttes of the tom, ±, nd the polriztion stte of flying photon ϕ ± field(t 1 ) tht is coherently generted y the coupled tom cvity system. Applying second control field Ω 2 (t 2 ) returns the tom to its initil (unentngled) stte while generting second flying photon ξ ± field(t 2 ), therey leding to entnglement etween the polriztions of the fields, ϕ ± field nd ξ ± field, emitted t times t 1 nd t 2. Such sequence of opertions hs een pplied to single ruidium toms flling through high-finesse opticl cvity 21. In this study, entngled photons were generted with time seprtion τ = t 2 t 1 limited y the tomic trnsit time. Although the toms rrived rndomly into the cvity mode in this cse, the protocol itself is intrinsiclly deterministic. With trpped toms, it will e possile to generte entngled sttes t user selected times (t 1, t 2 ) t the push of utton. Moreover, the scheme is inherently reversile, so the entnglement etween tom nd field cn e used to distriute entnglement to second tom cvity system in network. In roder context, importnt dvnces hve een mde in the genertion nd trnsfer of quntum sttes in other physicl systems, including quntum dots 42 nd circuits 28 coupled to cvities. With the mturtion of experimentl cpilities in cvity QED tht is now evident, mny previously developed theoreticl protocols will ecome possile. These include the sequentil genertion of entngled multiquit sttes 43, the teleporttion of tomic sttes from one node to nother 15, photonic quntum computtion y wy of photon photon interctions t the nodes 35 nd reversile mpping of quntum sttes of tomic motion to nd from light 44. Clerly, new technicl cpilities eyond conventionl (Fry Pérot) cvities will e required to fcilitte such scientific investigtions; severl cndidte systems re discussed in Box 2. Quntum networks with tomic ensemles An re of considerle reserch ctivity in the quest to distriute coherence nd entnglement cross quntum networks hs een the interction of light with tomic ensemles tht consist of lrge collection of identicl toms. For the regime of continuous vriles, entngle ment hs een chieved etween two tomic ensemles, ech of which consists of ~1 12 toms 45, nd the quntum teleporttion of light to mtter hs een demonstrted y mpping coherent opticl sttes to the collective spin sttes of n tomic memory 46. Further reserch of the continuous vriles regime is reviewed elsewhere 47. Here I focus, insted, on the regime of discrete vriles, with photons nd tomic excittions considered one y one. 126

25 NATURE Vol June 28 INSIGHT REVIEW Box 2 A new prdigm for cvity QED c d Out Photons In Atom Out To uild lrge-scle quntum networks 4,6, mny quntum nodes will need to e interconnected over quntum chnnels. Becuse conventionl (Fry Pérot) configurtions re ill suited for this purpose, there hve een efforts to develop lterntive microcvity systems 26, oth for single toms 75,76,78 nd for tom-like systems (such s nitrogen vcncy centres in dimond 79 ). A quntittive comprison of cndidte systems is provided in ref. 77. A remrkle resontor for this purpose is the microtoroidl cvity tht is formed from fused SiO 2 (refs 8,81) (shown in the figure). Such resontor supports whispering-gllery mode 82 circulting round the outer circumference of the toroid (shown in cross-section in grey, in pnel of the figure), with n evnescent field externl to the resontor. The intensity of the resontor mode is indicted y the coloured contours. Becuse of the smll mode volume V m nd lrge qulity fctor Q, n tom (lue) intercting with the evnescent field of whispering-gllery mode cn e fr into the regime of strong coupling, with projected vlues for the criticl photon n nd tom N numers (n 2 x 1 5 nd N 1 6 ) 77 tht re significntly greter thn current 12 nd projected 77 vlues for cvity QED with Fry Pérot cvities (Fig. 2c). Pioneering friction techniques 8,81 lend themselves to the integrtion of mny microtoroidl resontors to form opticl networks, s illustrted in pnel nd c of the figure. Pnel shows photogrph of silicon chip with liner rry of microtoroidl resontors within n ultrhigh-vcuum pprtus 76. The toroids pper s smll scttering centres on silicon chip tht runs verticlly down the centre of the picture. Blck rrows indicte horizontl SiO 2 fire tper for coupling light to nd from one resontor. Scle r, 2 mm. Pnel c is scnning electron microgrph of n rry of microtoroidl resontors ( mgnifiction of the region ounded y the white ox in pnel ), showing toroids of fused SiO 2 on silicon supports 8. These resontors hve the cpility for input output coupling with smll prsitic loss 81 for the configurtion shown in pnel d (scle r, 1 μm), which is microgrph of n individul toroid nd fire tper from pnel 76. Q = hs een relized t λ = 1,55 nm, nd Q 1 8 t λ = 85 nm, with good prospects for improvement to Q 1 1 (ref. 77). For these prmeters, the efficiency ε for coupling quntum fields into nd out of the resontor could pproch ε while remining firmly in the regime of strong coupling 77. Such high efficiency is crucil for the reliztion of complex quntum networks, including for distriuting nd processing quntum informtion 4,6,35 nd for investigting the ssocition etween quntum mny-ody systems nd quntum networks 9,11. The initil step in this quest to relize quntum network ws the demonstrtion of strong coupling etween individul toms nd the field of microtoroidl resontor 75. More recently, non-clssicl fields hve een generted from the interction of single toms with microtoroidl resontor y wy of photon turnstile, for which single tom dynmiclly regultes the trnsport of photons one y one through the microtoroidl resontor 76 (figure, pnel d). Only single photons cn e trnsmitted in the forwrd direction (from right to left in the figure), with excess photons n > 1 dynmiclly rerouted to the ckwrd direction. Writing nd reding collective spin excittions Reserch on discrete quntum vriles is sed on the remrkle theoreticl protocol descried in ref. 13, in which Luming Dun, Mikhil Lukin, Jun Igncio Circ nd Peter Zoller presented relistic scheme for entnglement distriution y wy of quntum-repeter rchitecture 4,17. Fundmentl to this protocol, which is known s the DLCZ protocol, is the genertion nd retrievl of single spin excittions within n ensemle of lrge numer of toms 48 (Box 3). Together with photoelectric detection of field 1, lser pulse ( write pulse) cretes single excittion 1 tht is stored collectively within the tomic ensemle. At lter time, second lser pulse ( red pulse) deterministiclly converts excittion stored within the tomic memory in the stte 1 into propgting field, denoted field 2. The sic processes illustrted in Box 3 cn e extended to crete n entngled pir of ensemles, L nd R (ref. 13; Fig. 3). The entngled stte is generted in proilistic ut herlded 49 mnner from quntum interference in the mesurement process. Tht is, detection of photon from one tomic ensemle or the other in n indistinguishle mnner results in n entngled stte with one collective spin excittion shred coherently etween the ensemles. In the idel cse, nd to lowest-order proility, photoelectric detection event t either of the two detectors projects the ensemles into the entngled stte Ψ L,R = 1 2 ( L 1 R ± e iη1 1 L R ), with the sign (+ or ) set y whether detector 1 or detector 2 records the event. The phse η 1 is determined y the difference etween the phse shifts long the two chnnels, η 1 = β L - β R (ref. 49), which must e stle. Any given tril with write pulse is unlikely to produce detection event t either detector, nd such filed trils require the system to e reinitilized. However, photo electric detection event t either detector unmiguously herlds the cretion of the entngled stte. Limited y the coherence time etween the metstle lower tomic sttes g i nd s i for ll toms i = 1, 2,..., N within the ensemle (ref. 5; Box 3), this entngled stte is stored in the quntum memory provided y the ensemles nd is ville on demnd for susequent tsks, such s entnglement connection 13,51. Although the ove description is for n idel cse nd neglects higher-order terms, the DLCZ protocol is designed to e resilient to 127

26 INSIGHT REVIEW NATURE Vol June 28 Box 3 Writing nd reding single tomic excittions Write Field 1 Write e Field 1 (designted field 1) with frequency nd/or polriztion distinct from the write field. For smll excittion proility p<<1, in most cses nothing hppens s result of the writing pulse, so the resultnt stte ϕ,1 for the tomic ensemle nd field 1 in the idel cse is given y ϕ,1 1 + e iβ p O(p) (1) Field 2 Red Field 2 The DLCZ protocol 13 is sed on ensemles of N identicl toms (lue) with Λ-level configurtion, s shown in the figure. The metstle lower sttes g nd s cn e, for exmple, tomic hyperfine sttes of the electronic ground level to ensure long lifetime for coherence. All toms re initilly prepred in stte g with no excittion (figure, pnel ), nmely i N g i, nd wek off-resonnt write pulse is then sent through the ensemle. This results in smll proility of mplitude p tht one of the N toms will e trnsferred from g to s nd will emit photon into the forwrd-scttered opticl mode g g e s Red s where n 1 is the stte of the forwrd-propgting field 1 with n 1 photons (n 1 = or 1), the phse β is determined y the propgtion phses of the write pulse nd field 1, nd O(p) denotes of order p. The tomic stte 1 in eqution (1) (ove) is collective (entngled) stte with one excittion shred symmetriclly etween the N toms (tht is, one spin flip from g to s ), where in the idel cse 13 N 1 1 = Σ g i s i g N (2) N i=1 Field 1 is directed to single-photon detector, where detection event is recorded with proility p. Such n event for field 1 herlds tht single excittion (or spin flip from g to s ) hs een creted nd stored in the tomic ensemle in the stte 1 with high proility. Higher-order processes with multiple tomic nd field 1 excittions re lso possile nd idelly occur, to lowest order, with proility p 2. After user-defined dely (suject to the finite lifetime of the quntum memory), the collective tomic excittion 1 cn e efficiently converted to propgting em (designted field 2) y wy of strong red pulse (figure, pnel ), where in the idel cse there is one-to-one trnsformtion of tomic excittion to field excittion, 1 to 1 2. In the cse of resonnce with the trnsition from s to e, the reding process utilizes the phenomenon of electromgneticlly induced trnsprency 16,66. importnt sources of imperfections, including losses in propgtion nd detection, nd detector drk counts. Indeed, the scheme functions with uilt-in entnglement purifiction 13 nd enles entnglement to e extended eyond the seprtion of two ensemles in n efficient nd sclle mnner. Theoreticl extensions 52,53 of the DLCZ protocol hve exmined relted network rchitectures for optimizing sclility in view of lortory cpilities (discussed elow). Coherence nd entnglement with tomic ensemles The initil, enling, steps in the implementtion of the DLCZ protocol were oservtions of quntum correltions oth for single photon pirs 54,55 nd for lrge numer of photons ( ) (ref. 56) generted in the collective emission from tomic ensemles. Single photons were generted y the efficient mpping of stored collective tomic excittion to propgting wve pckets for field 2 (refs 57 61; Box 3). Conditionl red-out efficiencies of 5% in free spce 58 nd 84% in cvity 62 were relized for stte trnsfer from single collective spin excittion stored in the tomic ensemle to single photon for field 2. With these cpilities for coherent control of collective tomic emission, herlded entnglement etween ensemles seprted y 3 m ws chieved in 25 (ref. 49). More recent work hs led to the inference tht the concurrence C (ref. 63) of entnglement stored etween the two ensemles in Fig. 3 is C =.9±.3 (ref. 5), with the ssocited density mtrix shown in Fig. 3. The DLCZ protocol is sed on quntum-repeter rchitecture involving independent opertions on prllel chins of quntum systems 13, with sclility relying crucilly on conditionl control of quntum sttes stored in remote quntum memories 64. The experiment shown in Fig. 3c took n importnt step towrds this gol y chieving the miniml functionlity required for sclle quntum networks 65. Aprt from the DLCZ protocol, which involves mesurementinduced entnglement, it is lso possile to chieve deterministic mpping of quntum sttes of light into nd out of tomic ensemles y using electromgneticlly induced trnsprency 16,66. Pioneering work 67,68 demonstrted the storge nd retrievl of clssicl pulses to nd from n tomic ensemle. This work ws then extended into the quntum regime of single photons 69,7. Entnglement etween two ensemles coupled to cvity mode ws chieved y ditic trnsfer of excittion 71, therey providing mens for on-demnd entnglement. In ddition, the reversile mpping of photonic entnglement into nd out of pirs of quntum memories hs een chieved 19 y n electromgneticlly-induced-trnsprency process, which should ssist the distriution of entnglement over quntum networks (Fig. 1d). Contemporry with this work on herlded nd deterministic entnglement, vriety of experiments sed on entnglement s post diction hve een crried out 72 (tht is, for cses in which physicl stte is not ville for use in sclle network ut which re nonetheless significnt). An importnt dvnce in this regrd is the use of pir of ensemles for entnglement genertion to chieve posteriori teleporttion of light to n tomic memory 73. There hs lso een considerle effort devoted to the detiled chrcteriztion of decoherence for stored tomic excittion nd entnglement 5,65,73. Decoherence of entnglement etween distinct tomic ensemles hs een oserved in the decy of the violtion of Bell s inequlity 65 nd of the fidelity for teleporttion 73. By mesuring concurrence C(t), quntittive chrcteriztions of the reltionship etween the glol evolution of the entngled stte nd the temporl dynmics of vrious locl correltions were lso le to e mde 5. Extending entnglement for quntum networks The entngled sttes tht hve een creted so fr oth in cvity QED nd y using the DLCZ protocol re etween pirs of systems (known s iprtite entnglement) for which there re definitive procedures for opertionl verifiction 72. The cretion of more-generl clsses of entngled stte shred etween more thn two nodes would e of gret interest. However, s reserchers progress towrds more-complex quntum networks, the issue of entnglement verifiction ecomes incresingly prolemtic. At present, the theoreticl tools nd experimentl 128

27 NATURE Vol June 28 INSIGHT REVIEW r exp L,R r idel L,R Write (R) 5/5 em splitter Write (L) R 1 L c I Detector 1 Detector 2 Figure 3 Fundmentls of the DLCZ protocol. A relistic scheme for entnglement distriution y wy of quntum-repeter rchitecture ws proposed y Dun, Lukin, Circ nd Zoller nd is known s the DLCZ protocol 13., Mesurement-induced entnglement etween two tomic ensemles 13,49, L nd R, is shown. Synchronized lser pulses incident on the ensemles (denoted write ems, lue rrows) generte smll mplitudes for opticl fields from spontneous Rmn scttering 48 ; these fields re denoted 1 L nd 1 R (red rrows). These fields interfere t 5/5 em splitter, with outputs directed to two single-photon detectors. A mesurement event t either detector (shown for detector 1) projects the ensemles into the entngled stte Ψ L,R with one quntum of excittion shred remotely etween the ensemles. Entnglement is stored in the quntum memory provided y the ensemles nd cn susequently e converted to propgting light pulses y set of red lser pulses (Box 3)., Experimentlly determined components of the density mtrix ρ e x p L, R for entnglement etween two tomic ensemles re shown 5, corresponding to concurrence C =.9±.3, where C = for n unentngled stte. The first numer in ech ket refers to the excittion numer for the ensemle L, nd idel the second is for the ensemle R. For comprison, the density mtrix ρ L, R for the idel stte Ψ L,R is shown, with concurrence C = 1. c, The lortory set-up is shown for the entnglement of two pirs of tomic ensemles to generte the functionl quntum nodes L nd R, which re seprted y 3 m (ref. 65). Ech of the four elongted ovls shows cylinder of 1 5 cesium toms, which forms n tomic ensemle t ech site. Entngled sttes etween the upper u nd lower l pirs t the L nd R nodes, Ψ u L,R Ψ L,R, l re generted nd stored in n synchronous mnner for ech pir (u nd l) s is the cse in pnel. Atomic excittions for the pirs L u, L l nd R u, R l re susequently converted to flying photons t ech node, with polriztion encoding tht results in violtion of Bell s inequlity 65. The entire experiment functions under the quntum control of single photon detection events. u I cpilities for chrcterizing the generl sttes of quntum networks do not exist. Perhps surprisingly, non-trivil tsk will e to find out whether quntum network works. As modertely complex quntum networks re relized in the lortory, it will ecome incresingly more difficult to ssess the chrcteristics of network quntittively, including whether entnglement extends cross the whole network. One strtegy, motivted y the underlying physicl processes of the network, could e to try to determine the density mtrix ρ(t) for the network. However, this pproch would fil ecuse of the exponentil growth in ρ(t) with the size of network. L u R An lterntive strtegy could e sed on more functionl issues of lgorithmic cpility. An ttempt could e mde to implement quntum lgorithm for computtion or communiction to test whether the purported quntum network hs greter cpilities thn ny clssicl counterprt. This course is, however, prolemtic ecuse the dvntge of quntum network might only e relized ove some threshold in the size of the network. Furthermore, from n experimentl perspective, this strtegy does not offer much in the wy of dignostics for fixing the network when it fils. Another, less ovious, pproch might e to dopt more seriously the perspective of quntum network s quntum mny-ody system nd to serch for more physicl chrcteristics of the network (for exmple, the scling ehviour of pir correltion functions nd multiprtite entnglement). Indeed, n ctive re of reserch is the nture of entnglement for systems tht undergo quntum phse trnsitions, nd there hve een pioneering dvnces in the study of one-dimensionl spin chins 74. Conclusion Progress hs een mde towrds the development of quntum networks, ut the current stte of the rt is primitive reltive to tht required for the roust nd sclle implementtion of sophisticted network protocols, whether over short or long distnces. The reliztion of quntum memories, locl quntum processing, quntum repeters nd error-corrected teleporttion re mitious gols. Nevertheless, there is considerle ctivity directed towrds these gols worldwide. Here cvity-qed-sed networks nd networks implemented using the DLCZ protocol were considered seprtely, ut it is cler tht quntum networks will evolve s heterogeneous entities. For exmple, the sme protocol tht cretes the entnglement etween the two ensemles shown in Fig. 3 cn e used to crete n entngled stte with one excittion shred etween n tom in cvity nd n tomic ensemle. A crucil tsk will e the development of unmiguous procedures for verifying entnglement, non-trivil undertking tht hs not lwys een crried out correctly 72. I hve used quntum networks s unifying theme, ut the reserch descried here hs roder vlue, including dvncing the understnding of quntum dynmicl systems nd, for the cses considered here, creting new physics from controlled nonliner interctions of single photons nd toms. These re exciting times in quntum informtion science s reserchers pss from the regime of individul uilding locks (for exmple, single tom cvity system) to the relm of complex quntum systems tht re ssemled lock y lock from mny such units. 1. Nielsen, M. A. & Chung, I. L. Quntum Computtion nd Quntum Informtion (Cmridge Univ. Press, Cmridge, UK, 2). 2. Zoller, P. et l. Quntum informtion processing nd communiction. Strtegic report on current sttus, visions nd gols for reserch in Europe. Eur. Phys. J. D 36, (25). 3. Bennett, C. H., Brssrd, G. & Ekert, A. K. Quntum cryptogrphy. Sci. Am. 267 (4), 5 57 (1992). 4. Bouwmeester, D., Ekert, A. & Zeilinger, A. (eds) The Physics of Quntum Informtion (Springer, Berlin, 2). 5. Giovnnetti, V., Lloyd, S. & Mccone, L. Quntum-enhnced mesurements: eting the stndrd quntum limit. Science 36, (24). 6. Circ, J. I., Zoller, P., Kimle, H. J. & Muchi, H. Quntum stte trnsfer nd entnglement distriution mong distnt nodes in quntum network. Phys. Rev. Lett. 78, (1997). 7. Preskill, J. P. Plug-in quntum softwre. Nture 42, (1999). 8. Copsey, D. et l. Towrd sclle, silicon-sed quntum computing rchitecture. IEEE J. Quntum Electron. 9, (23). 9. Illuminti, D. Light does mtter. Nture Phys. 2, (26). 1. Dun, L.-M., Wng, B. & Kimle, H. J. Roust quntum gtes on neutrl toms with cvity-ssisted photon scttering. Phys. Rev. A 72, (25). 11. Acín, A., Circ, J. I. & Lewenstein, M., Entnglement percoltion in quntum networks. Nture Phys. 3, (27). 12. Miller, R. et l. Trpped toms in cvity QED: coupling quntized light nd mtter. J. Phys. B. 38, S551 S565 (25). 13. Dun, L.-M., Lukin, M. D., Circ, J. I. & Zoller, P. Long-distnce quntum communiction with tomic ensemles nd liner optics. Nture 414, (21). 14. Prkins, A. S., Mrte, P., Zoller, P. & Kimle, H. J. Synthesis of ritrry quntum sttes vi ditic trnsfer of Zeemn coherence. Phys. Rev. Lett. 71, (1993). 129

28 INSIGHT REVIEW NATURE Vol June vn Enk, S. J., Circ, J. I. & Zoller, P. Photonic chnnels for quntum communiction. Science 279, (1998). 16. Lukin, M. D. Trpping nd mnipulting photon sttes in tomic ensemles. Rev. Mod. Phys. 75, (23). 17. Briegel, H.-J., Dür, W., Circ, J. I. & Zoller, P. Quntum repeters: the role of imperfect locl opertions in quntum communiction. Phys. Rev. Lett. 81, (1998). 18. Boozer, A. D., Boc, A., Miller, R., Northup, T. E. & Kimle, H. J. Reversile stte trnsfer etween light nd single trpped tom. Phys. Rev. Lett. 98, (27). 19. Choi, K. S., Deng, H., Lurt, J. & Kimle, H. J. Mpping photonic entnglement into nd out of quntum memory. Nture 452, (28). 2. Bermn, P. (ed.) Cvity Quntum Electrodynmics (Acdemic, Sn Diego, 1994). 21. Wilk, T., Wester, S. C., Kuhn, A. & Rempe, G. Single-tom single-photon quntum interfce. Science 317, (27). 22. Wlther, H. Quntum optics of single toms. Fortschr. Phys. 52, (24). 23. Meschede, D., Wlther, H. & Mueller, G. One-tom mser. Phys. Rev. Lett. 54, (1985). 24. Rimond, J. M. et l. Proing quntum field in photon ox. J. Phys. B 38, S535 S55 (25). 25. Guerlin, C. et l. Progressive field-stte collpse nd quntum non-demolition photon counting. Nture 448, (27). 26. Vhl, K. J. Opticl microcvities. Nture 424, (24). 27. Khitrov, G., Gis, H. M., Kir, M., Koch, S. W. & Scherer, A. Vcuum Ri splitting in semiconductors. Nture Phys. 2, 81 9 (26). 28. Schoelkopf, R. J. & Girvin, S. M. Wiring up quntum systems. Nture 451, (28). 29. Ye, J., Vernooy, D. W. & Kimle, H. J. Trpping of single toms in cvity QED. Phys. Rev. Lett. 83, (1999). 3. Hijlkem, M. et l. A single-photon server with just one tom. Nture Phys. 3, (27). 31. Fortier, K. M., Kim, S. Y., Gions, M. J. Ahmdi, P. & Chpmn, M. S. Deterministic loding of individul toms to high-finesse opticl cvity. Phys. Rev. Lett. 98, (27). 32. Ye, J., Kimle, H. J. & Ktori, H. Quntum stte engineering nd precision metrology using stte-insensitive light trps. Science (in the press). 33. Boozer, A. D., Boc, A., Miller, R., Northup, T. E. & Kimle, H. J. Cooling to the ground stte of xil motion for one tom strongly coupled to n opticl cvity. Phys. Rev. Lett. 97, 8362 (26). 34. Keller, M., Lnge, B., Hysk, K., Lnge, W. & Wlther, H. Continuous genertion of single photons with controlled wveform in n ion-trp cvity system. Nture 431, (24). 35. Dun, L.-M. & Kimle, H. J. Sclle photonic quntum computtion through cvityssisted interctions. Phys. Rev. Lett. 92, (24). 36. McKeever, J. et l. Deterministic genertion of single photons from one tom trpped in cvity. Science 33, (24). 37. Bergmnn, K., Theuer, H. & Shore, B. W. Coherent popultion trnsfer mong quntum sttes of toms nd molecules. Rev. Mod. Phys. 7, (1998). 38. Lounis, B. & Orrit, M. Single-photon sources. Rep. Prog. Phys. 68, (25). 39. Lnge, W. & Kimle, H. J. Dynmic genertion of mximlly entngled photon multiplets y ditic pssge. Phys. Rev. A 61, (2). 4. Dun, L.-M. & Kimle, H. J. Efficient engineering of multitom entnglement through single-photon detections. Phys. Rev. Lett. 9, (23). 41. Sun, B., Chpmn, M. S. & You, L. Atom photon entnglement genertion nd distriution. Phys. Rev. A 69, (24). 42. Englund, D., Fron, A., Zhng, B., Ymmoto, Y. & Vuckovic, J. Genertion nd trnsfer of single photons on photonic crystl chip. Opt. Express 15, (27). 43. Schön, C., Solno, E., Verstrete, F., Circ, J. I. & Wolf, M. M. Sequentil genertion of entngled multiquit sttes. Phys. Rev. Lett. 95, 1153 (25). 44. Prkins, A. S. & Kimle, H. J. Quntum stte trnsfer etween motion nd light. J. Opt. B 1, (1999). 45. Julsgrd, B., Kozhekin, A. & Polzik, E. S. Experimentl long-lived entnglement of two mcroscopic ojects. Nture 413, 4 43 (21). 46. Sherson, J. F. et l. Quntum teleporttion etween light nd mtter. Nture 443, (26). 47. Cerf, N. J., Leuchs, G. & Polzik, E. S. (eds) Quntum Informtion with Continuous Vriles of Atoms nd Light (World Scientific, New Jersey, 27). 48. Rymer, M. G., Wlmsley, I. A., Mostowski, J. & Soolewsk, B. Quntum theory of sptil nd temporl coherence properties of stimulted Rmn scttering. Phys. Rev. A 32, (1985). 49. Chou, C.-W. et l. Mesurement-induced entnglement for excittion stored in remote tomic ensemles. Nture 438, (25). 5. Lurt, J., Choi, K. S., Deng, H., Chou, C.-W. & Kimle, H. J. Herlded entnglement etween tomic ensemles: preprtion, decoherence, nd scling. Phys. Rev. Lett. 99, 1854 (27). 51. Lurt, J. et l. Towrds experimentl entnglement connection with tomic ensemles in the single excittion regime. New J. Phys. 9, (27). 52. Jing, L., Tylor, J. M. & Lukin, M. D. Fst nd roust pproch to long-distnce quntum communiction with tomic ensemles. Phys. Rev. A 76, 1231 (27). 53. Sngourd, N. et l. Roust nd efficient quntum repeters with tomic ensemles nd liner optics. Preprint t (28). 54. Kuzmich, A. et l. Genertion of nonclssicl photon pirs for sclle quntum communiction with tomic ensemles. Nture 423, (23). 55. Blic, V., Brje, D. A., Kolchin, P., Yin, G. Y. & Hrris, S. E. Genertion of pired photons with controllle wveforms. Phys. Rev. Lett. 94, (25). 56. vn der Wl, C. H. et l. Atomic memory for correlted photon sttes. Science 31, (23). 57. Chou, C. W., Polykov, S. V., Kuzmich, A. & Kimle, H. J. Single-photon genertion from stored excittion in n tomic ensemle. Phys. Rev. Lett. 92, (24). 58. Lurt, J. et l. Efficient retrievl of single excittion stored in n tomic ensemle. Opt. Express 14, (26). 59. Thompson, J. K., Simon, J., Loh, H. & Vuletic, V. A high-rightness source of nrrownd, identicl-photon pirs. Science 313, (26). 6. Mtsukevich, D. N. et l. Deterministic single photons vi conditionl quntum evolution. Phys. Rev. Lett. 97, 1361 (26). 61. Chen, S. et l. Deterministic nd storle single-photon source sed on quntum memory. Phys. Rev. Lett. 97, 1734 (26). 62. Simon, J., Tnji, H., Thompson, J. K. & Vuletic, V. Interfcing collective tomic excittions nd single photons. Phys.Rev. Lett. 98, (27). 63. Wootters, W. K. Entnglement of formtion of n ritrry stte of two quits. Phys. Rev. Lett. 8, (1998). 64. Felinto, D. et l. Conditionl control of the quntum sttes of remote tomic memories for quntum networking. Nture Phys. 2, (26). 65. Chou, C.-W. et l. Functionl quntum nodes for entnglement distriution over sclle quntum networks. Science 316, (27). 66. Hrris, S. E. Electromgneticlly induced trnsprency. Phys. Tody 5, 36 4 (1997). 67. Liu, C., Dutton, Z., Behroozi, C. H. & Hu, L. V. Oservtion of coherent opticl informtion storge in n tomic medium using hlted light pulses. Nture 49, (21). 68. Phillips, D. F., Fleischhuer, A., Mir, A., Wlsworth, R. L. & Lukin, M. D. Storge of light in tomic vpor. Phys. Rev. Lett. 86, (21). 69. Chnelière, T. et l. Storge nd retrievl of single photons trnsmitted etween remote quntum memories. Nture 438, (25). 7. Eismn, M. D. et l. Electromgneticlly induced trnsprency with tunle singlephoton pulses. Nture 438, (25). 71. Simon, J., Tnji, H., Ghosh, S. & Vuletic, V. Single-photon us connecting spin-wve quntum memories. Nture Phys. 3, (27). 72. vn Enk, S. J., Lütkenhus, N. & Kimle, H. J. Experimentl procedures for entnglement verifiction. Phys. Rev. A 75, (27). 73. Chen, Y.-A. et l. Memory-uilt-in quntum teleporttion with photonic nd tomic quits. Nture Phys. 4, (28). 74. Vidl, G., Ltorre, J. I., Rico, E. & Kitev, A. Entnglement in quntum criticl phenomen. Phys. Rev. Lett. 9, (23). 75. Aoki, T. et l. Oservtion of strong coupling etween one tom nd monolithic microresontor. Nture 443, (26). 76. Dyn, B. et l. A photon turnstile dynmiclly regulted y one tom. Science 319, (28). 77. Spillne, S. M. et l. Ultrhigh-Q toroidl microresontors for cvity quntum electrodynmics. Phys. Rev. A 71, (25). 78. Trupke, M. et l. Atom detection nd photon production in sclle, open, opticl microcvity. Phys. Rev. Lett. 99, 6361 (27). 79. Prk, Y.-S., Cook, A. K. & Wng, H. Cvity QED with dimond nnocrystls nd silic microspheres. Nno Lett. 6, (26). 8. Armni, D. K., Kippenerg, T. J., Spillne, S. M. & Vhl, K. J. Ultr-high-Q toroid microcvity on chip. Nture 421, (23). 81. Spillne, S. M., Kippenerg, T. J., Pinter, O. J. & Vhl, K. J. Idelity in fier-tpercoupled microresontor system for ppliction to cvity quntum electrodynmics. Phys. Rev. Lett. 91, 4392 (23). 82. Brginsky, V. B., Gorodetsky, M. L. & Ilchenko, V. S. Qulity-fctor nd nonliner properties of opticl whispering-gllery modes. Phys. Lett. A 137, (1989). Acknowledgements I m grteful for the contriutions of memers of the Cltech Quntum Optics Group, especilly K. S. Choi, B. Dyn nd R. Miller. I m indeted to J. P. Preskill nd S. J. vn Enk for criticl insights. My reserch is supported y the Ntionl Science Foundtion, IARPA nd Northrop Grummn Spce Technology. Author Informtion Reprints nd permissions informtion is ville t The uthor declres no competing finncil interests. Correspondence should e ddressed to the uthor (hjkimle@cltech.edu). 13

29 NATURE Vol June 28 doi:1.138/nture7128 INSIGHT REVIEW Superconducting quntum its John Clrke 1,2 & Frnk K. Wilhelm 3 Superconducting circuits re mcroscopic in size ut hve generic quntum properties such s quntized energy levels, superposition of sttes, nd entnglement, ll of which re more commonly ssocited with toms. Superconducting quntum its (quits) form the key component of these circuits. Their quntum stte is mnipulted y using electromgnetic pulses to control the mgnetic flux, the electric chrge or the phse difference cross Josephson junction ( device with nonliner inductnce nd no energy dissiption). As such, superconducting quits re not only of considerle fundmentl interest ut lso might ultimtely form the primitive uilding locks of quntum computers. The theory of quntum mechnics ws originlly developed to ccount for the oserved ehviour of electrons in toms. More thn 8 yers lter, it is eing used to explin the ehviour of superconducting circuits tht cn e hundreds of nnometres wide nd cn contin trillions of electrons. The quntum nture of these circuits is oservle ecuse they cn e engineered to e isolted from the electricl environment nd re thus represented y single degree of freedom. Significnt coupling to other degrees of freedom cuses rpid decoherence, destroying the quntum stte of the circuit so tht it ehves clssiclly. Unlike toms, these circuits cn e designed nd constructed to tilor their chrcteristic frequencies, s well s other prmeters. These frequencies cn e controlled y djusting n externl prmeter, nd the coupling energy etween two quntum its (quits) cn e turned on nd off t will. Superconducting quntum circuits re the suject of intense reserch t present, in prt ecuse they hve opened up new re of fundmentl science nd in prt ecuse of their long-term potentil for quntum computing. In this review, we egin with rief discussion of superconductivity nd two of the superconducting properties tht underlie how quits operte: flux quntiztion nd Josephson tunnelling. The three fundmentl types of superconducting quit flux, chrge nd phse re then descried. This is followed y review of the rel-time, quntum-coherent dynmics of quits nd the limittions imposed y relxtion nd decoherence, s well s the mechnisms of decoherence. We then discuss schemes for controlling the coupling etween two quits, feture tht gretly simplifies the implementtion of proposed quntum-computing rchitectures. And we finish y discussing quntum optics on chip, new reserch direction in which the electromgnetic fields ssocited with control nd red-out signls re treted quntum mechniclly. Flux quntiztion nd Josephson tunnelling Why do superconductors enle tomic-scle phenomen to e oserved t the mcroscopic level? The reson, s explined elegntly y the theory of Brdeen, Cooper nd Schrieffer 1, is tht in given superconductor ll of the Cooper pirs of electrons (which hve chrge 2e, mss 2m e nd spin zero, nd re responsile for crrying supercurrent) re condensed into single mcroscopic stte descried y wvefunction Ψ(r, t) (where r is the sptil vrile nd t is time.) Like ll quntum-mechnicl wvefunctions, Ψ(r, t) cn e written s Ψ(r, t) exp[iϕ(r, t)] (where i = 1): tht is, s the product of n mplitude nd fctor involving the phse ϕ. Furthermore, in conventionl superconductors such s N, P nd Al, the qusiprticles (electronlike nd hole-like excittions) re seprted in energy from the condenste 2 y n energy gp s (T) = 1.76k B T c (where k B is the Boltzmnn constnt nd T c is the superconducting trnsition temperture). Thus, t tempertures T << T c, the density of qusiprticles ecomes exponentilly smll, s does the intrinsic dissiption for frequencies of less thn 2 s ()/h (where h is Plnck s constnt) roughly 1 11 Hz for Al. The mcroscopic wvefunction leds to two phenomen tht re essentil for quits. The first phenomenon is flux quntiztion. When closed ring is cooled through its superconducting trnsition temperture in mgnetic field nd the field is then switched off, the mgnetic flux Φ in the ring mintined y circulting supercurrent is quntized 2 in integer vlues of the flux quntum Φ h/2e T m 2. This quntiztion rises from the requirement tht Ψ(r, t) e single vlued. The second phenomenon is Josephson tunnelling 2. A Josephson junction consists of two superconductors seprted y n insulting rrier of pproprite thickness, typiclly 2 3 nm, through which Cooper pirs cn tunnel coherently. Brin Josephson showed tht the supercurrent I through the rrier is relted to the guge-invrint phse difference δ(t) etween the phses of the two superconductors y the current phse reltionship I = I sinδ (1) Here I is the mximum supercurrent tht the junction cn sustin (tht is, the criticl current). This phse difference is n electrodynmic vrile tht, in the presence of potentil difference V etween the superconductors, evolves in time s δ. = ω = 2eV (2) where = h/2π nd ω is the ngulr frequency t which the supercurrent oscilltes. The dynmicl ehviour of Josephson junctions is descried in Box 1. The vriles hve, so fr, een regrded s eing clssicl, ut to show quntum-mechnicl ehviour, these vriles must e replced y opertors. The two relevnt opertors re tht for δ, which is ssocited with the Josephson coupling energy E j I Φ /2π, nd tht for the Cooper-pir numer difference N cross the cpcitnce, which is ssocited with the chrging energy E c (2e) 2 /2C, where C is the junction cpcitnce. Furthermore just like the fmilir position nd momentum opertors x nd p x the opertors for δ nd for the chrge on the cpcitor Q 1 Deprtment of Physics,366 LeConte Hll, University of Cliforni, Berkeley, Cliforni 9472, USA. 2 Mterils Sciences Division, Lwrence Berkeley Ntionl Lortory, 1 Cyclotron Rod, Berkeley, Cliforni 9472, USA. 3 Institute for Quntum Computing, University of Wterloo, 2 University Avenue West, Wterloo, Ontrio N2L 3G1, Cnd. 131

30 INSIGHT REVIEW NATURE Vol June 28 Box 1 The Josephson junction s nonliner circuit element U(d) I I C Equtions (1) nd (2) contin the crucil informtion tht the Josephson junction is dissiptionless device with nonliner inductnce. It is these unique fetures tht mke the junction the primitive uilding lock of ll superconducting quits. The nonliner inductnce is esily deduced y noting tht the time derivtive of eqution (1) yields İ = (I cosδ)δ. = (I cosδ)ω = V(2eI / )cosδ from eqution (2). Invoking Frdy s lw V = Lİ (where L is the inductnce) then leds to the Josephson inductnce L j = Φ /(2πI cosδ) = Φ /2π(I 2 I 2 ) 1/2 (where I < I ) (6) The Josephson junction, denoted y n X in pnel of the figure, hs n intrinsic cpcitnce C; this comintion is often denoted y n X in ox. I denotes the criticl current. It is immeditely pprent from eqution (6) tht the junction is lso nonliner oscilltor with resonnt ngulr frequency ω p (I) = (L j C) 1/2 = (2πI /Φ C) 1/2 /(1 I 2 /I 2 ) 1/4. Considerle insight into the dynmics of Josephson junction cn e glened y considering the flow of current J through the junction: J = I sinδ + CV.. Writing V. = ( /2e)δ.. nd rerrnging this yields ( C/2e)δ.. = I I sinδ = (2e/ ) U/ δ. U (Φ /2π) U/ δ is the potentil of tilted wshord for prticle of mss C/2e (s illustrted in pnel of the figure). In the sence of fluctutions, for I < I the prticle remins trpped in one of the potentil wells; clssiclly, it oscilltes in the well t the plsm oscilltion frequency ω p (I)/2π. Thus, δ. =, nd the junction is in the zero-voltge stte; in the quntum picture, the energy in the well is quntized, s shown in the inset (figure, pnel ). By contrst, when I is incresed so tht I > I, the prticle runs down the wshord, δ. >, nd there is voltge cross the junction. When I is susequently reduced so tht I < I, the prticle will continue to propgte until I is close to. Thus, the current voltge chrcteristic is hysteretic. re cnoniclly conjugte, s expressed y the commuttor rcket [δ, Q] = i2e. The fct tht δ nd Q re suject to Heisenerg s uncertinty principle hs fr-reching consequences. On the one hnd, when E j >> E c, δ is well defined, nd Q hs lrge quntum fluctutions; therefore, the Josephson ehviour of the junction domintes. On the other hnd, when E j << E c, N is well defined, nd δ hs lrge quntum fluctutions; therefore, the chrging ehviour of the cpcitor domintes. Using these ides, the prmeters of superconducting quntum circuits cn e designed 3. The first evidence of quntum ehviour in Josephson junction cme from experiments in which mcroscopic quntum tunnelling ws found to occur nd energy levels were shown to e quntized. In mcro scopic quntum tunnelling 4,5, the junction tunnels from the ground stte (Box 1 figure), when I < I, through the potentil rrier tht seprtes it from its neighouring energy well, which is t lower energy. Then, the prticle runs freely down the wshord potentil, d 2 1 I I, C generting voltge 2 s /e tht is redily detected. These results 5 were found to e in strong greement with theory 6. Energy quntiztion 7 ws found in the initil well y irrditing the junction with microwves. The escpe rte from the zero-voltge stte ws incresed when the microwve frequency f m corresponded to the energy difference etween two djcent energy levels. A crucil point is tht the nhrmonic nture of the well, which results from the nonliner inductnce of Josephson junctions (eqution (6), Box 1), cuses the energy spcing to decrese s the quntum numer progressively increses, so ech trnsition hs distinct frequency. If the well were hrmonic, the energy spcings would e identicl, nd the quntum cse would not e distinguishle from the clssicl cse. These experiments showed unequivoclly tht δ is quntum vrile. The next step in the demonstrtion of mcroscopic quntum physics ws to implement devices showing the superposition of two quntum sttes Ψ 1 nd Ψ 2 in the form Ψ = α Ψ 1 + β Ψ 2, s first proposed y Anthony Leggett 8 in the 198s in his discussion of mcroscopic quntum coherence in superconducting devices. In 1997, Ysunou Nkmur et l. 9 crried out the first such experiment on chrge quit, showing spectroscopiclly the superposition of the Cooper-pir sttes n nd n + 1, where the integer n is the quntum numer specifying the numer of Cooper pirs. Susequently, in 2, Jonthn Friedmn et l. 1 nd Cspr vn der Wl et l. 11 showed the superposition of sttes in flux quit. A flux quit consists of superconducting loop interrupted y one 1 or three 11 Josephson junctions. The two quntum sttes re flux pointing up nd flux pointing down or, equivlently, supercurrent flowing in n nticlockwise direction nd supercurrent flowing in clockwise direction. In 22, Denis Vion et l. 12 descried quntronium, quit in which two smll junctions re connected y superconducting islnd, involving the superposition of the Cooper-pir sttes n nd n + 1. Also in 22, John Mrtinis et l. 13 demonstrted phse quit, reinvention of the device used erlier to oserve quntized energy levels 7. The relevnt quntum sttes re the ground stte nd the first excited stte. Some of the experimentl difficulties encountered when operting superconducting quits re descried in Box 2. Flux quits A flux quit, s indicted erlier, consists of superconducting loop interrupted y one 1 or three 11 Josephson junctions (Fig. 1). Although oth designs function similrly, we focus on the three-junction design, which hs een dopted more widely. In this device, one junction is smller in re nd thus hs smller criticl current thn the other two, which function to increse the inductnce of the loop. The smll junction hs lrge vlue for E j /E c, typiclly 5, so the phse difference δ (or, equivlently, the mgnetic flux Φ in the loop) is the relevnt quntum vrile. The two quntum sttes re mgnetic flux pointing up nd mgnetic flux pointing down or, equivlently, nticlockwise quit supercurrent I q circulting in the loop nd clockwise supercurrent. The quit is represented y doule-well potentil, which is generlly symmetricl. The two sttes re coupled y the quntum-mechnicl tunnelling of δ through the rrier seprting the wells, giving rise to the superposition of the two sis sttes Ψ = α ± β (3) When the externlly pplied mgnetic flux Φ e = Φ /2, the doule-well potentil ecomes symmetricl (Fig. 1), nd the two eigenfunctions ecome symmetricl nd ntisymmetricl superpositions of the two sis sttes, with α = β = 1/ 2. At this degenercy point, the splitting of the energy levels of the ground stte nd the first excited stte 1 is ; wy from the degenercy point, the energy difference is ν = ( 2 + ε 2 ) 1/2 (4) where ε = 2I q (Φ e Φ /2) (Fig. 1c). The proilities of oserving the sttes nd in the ground nd first excited sttes s function 132

31 NATURE Vol June 28 INSIGHT REVIEW Box 2 Experimentl issues with superconducting quits Experiments on superconducting quits re chllenging. Most superconducting quits re creted y using electron-em lithogrphy, need millikelvin tempertures nd n ultrlow-noise environment to operte, nd cn e studied only y using very sensitive mesurement techniques. Superconducting quits generlly require Josephson junctions with dimensions of the order of.1.1 µm 2 corresponding to self-cpcitnce of out 1 ff nd re ptterned y using shdow evportion nd electron-em lithogrphy 79 ; n exception is the phse quit, which typiclly hs junction of 1 1 µm 2 nd cn e ptterned photolithogrphiclly. The Josephson junctions re usully Al Al x O y Al (where x 2 nd y 3), nd the oxidtion must e controlled to yield reltively precise vlues of E j nd E c. Becuse quit frequencies re usully 5 1 GHz (which corresponds to.25.5 K), the circuits re operted in dilution refrigertors, typiclly t tempertures of 1 3 mk, to minimize therml popultion of the upper stte. Gret efforts re mde to ttenute externl electricl nd mgnetic noise. The experiment is invrily enclosed in Frdy cge either shielded room or the metl Dewr of the refrigertor with contiguous metl ox on top. The electricl leds tht re connected to the quits nd their red-out devices re hevily filtered or ttenuted. For exmple, lines crrying qusisttic is currents usully hve multiple low-pss filters t the vrious temperture stges of the refrigertor. These include oth inductor cpcitor nd resistor cpcitor filters tht operte up to few hundred meghertz, s well s wires running through copper powder, which results in sustntil loss t higher frequencies 5. The overll ttenution is typiclly 2 db. Finlly, the red-out process for proing quntum system is very delicte. of Φ e re shown in Fig. 1d. At the degenercy point, the proility of oserving either stte is ½. As Φ e is reduced, the proility of oserving increses while tht of oserving decreses. The first oservtion of quntum superposition in flux quit ws mde spectroscopiclly. The stte of the flux quit is mesured with d.c. superconducting quntum interference device (SQUID) 14. This device consists of two Josephson junctions, ech with criticl current I, connected in prllel on superconducting loop of inductnce L. The criticl current of the SQUID I c (Φ s ) is periodic in the externlly pplied mgnetic flux Φ s with period Φ. In the limit β L 2LI /Φ << 1 in which the Josephson inductnce domintes the geometricl inductnce, the criticl current for Φ s = (m + ½)Φ (m is n integer) is reduced to lmost zero, nd the flux dependence of the criticl current tkes the pproximte form 14 I c (Φ s ) 2I cos(πφ s /Φ ). Thus, y ising the SQUID with constnt mgnetic flux ner Φ /2, nd mesuring the criticl current, the chnges in flux produced y nery quit cn e mesured with high sensitivity. In most experiments with quits, pulse of current is pplied to the SQUID, which either remins in the zero-voltge stte or mkes trnsition to the voltge stte, producing voltge 2 s /e. Becuse its current voltge chrcteristic is hysteretic, the SQUID remins t this voltge until the current is hs een removed, llowing reserchers to determine whether the SQUID hs switched. For sufficiently smll current pulses, the proility of the SQUID switching is zero, wheres the proility is one for sufficiently lrge pulses. The switching event is stochstic process nd needs to e repeted mny times for the flux in the SQUID to e mesured ccurtely. The first step in spectroscopic oservtion of quntum superposition is to determine the height of the current pulse t which the SQUID switches with, for exmple, proility of ½ s function of Φ e over nrrow rnge (perhps ± 5mΦ ). Susequently, pulse of microwve flux is pplied t frequency f m, which is of sufficient mplitude nd durtion to equlize the popultions of the ground stte nd first excited stte when the energylevel splitting difference ν = hf m. Assuming tht is mesured, then, on resonnce, there will e pek in the switching proility for Φ e < Φ /2 nd corresponding dip for Φ e > Φ /2. An exmple of these results 11,15 is shown in Fig. 2. The configurtion of the quit nd the SQUID is shown in Fig. 2, nd the peks nd dips in the mplitude of the switching current Proility c d E 1.5 I q Proility mplitudes U D F /2 Figure 1 The theory underlying flux quits., Flux quits consist of superconducting loop interrupted y either one or three (shown) Josephson junctions. The two quntum sttes re mgnetic flux Φ pointing up nd Φ pointing down or, equivlently, supercurrent I q circulting in the loop nticlockwise nd I q circulting clockwise., The doule-well potentil (lck) versus totl flux Φ contined in flux quit is shown. The two wells re symmetricl when the externlly pplied mgnetic flux Φ e is (n + ½)Φ, where n is n integer (n = in this cse). The coloured curves re the eigenfunctions (proility mplitudes) for the ground stte (symmetricl; red) nd first excited stte (ntisymmetricl; lue). c, The energy E of the two superpositions sttes in versus the energy is ε = 2I q (Φ e Φ /2) is shown. The digonl dshed lck lines show the clssicl energies. The energy-level splitting is Δ t the degenercy point, ε =, nd is ν for ε. d, The proilities of the quit flux pointing up (green) or down (yellow) in the ground stte versus pplied flux re shown. F F /2 F e F e n D 133

32 INSIGHT REVIEW NATURE Vol June 28 I sw c f m (GHz) Φe Φ Φ res (1 3 Φ Figure 2 Experimentl properties of flux quits., The configurtion of the originl three-junction flux quit is shown. Arrows indicte the current flow in the two quit sttes (green denotes, nd yellow denotes ). Scle r, 3 μm. (Imge courtesy of C. H. vn der Wl, Rijksuniversiteit Groningen, the Netherlnds)., Rdition of microwve frequency f m induces resonnt peks nd dips in the switching current I sw with respect to the externlly pplied mgnetic flux Φ e normlized to the flux quntum Φ. Frequencies rnge from GHz to.85 GHz. Tick mrks on the y xis show steps of.4 na. (Pnel reproduced, with permission, from ref. 15.) c, Microwve frequency f m is plotted ginst hlf of the seprtion in mgnetic flux, Φ res, etween the pek nd the dip t ech frequency. The line is liner fit through the dt t high frequencies nd represents the clssicl energy. The inset is mgnified view of the lower prt of the grph; the curved line in the inset is fit to eqution (4). The devition of the dt points from the stright line demonstrtes quntum coherence of the nd flux sttes. (Pnel reproduced, with permission, from ref. 15.) versus pplied flux re shown in Fig. 2 for succession of microwve frequencies. As expected, the difference in the pplied flux t which the peks nd dips pper, 2 Φ res, ecomes greter s the microwve frequency increses. The microwve frequency versus Φ res is shown in Fig. 2c. The dt hve een fitted to eqution (4) with I q = (½)dν/dΦ e in the limit ν >>, using s fitting prmeter. The dt revel the existence of n nticrossing (tht is, n voided crossing) t Φ e = Φ /2. Chrge quits A chrge quit (lso known s Cooper-pir ox) is shown in Fig. 3,. The key component is tiny superconducting islnd tht is smll enough tht the electrosttic chrging energy required to plce chrge of 2e on the islnd t zero voltge, (2e) 2 /2C Σ, is much greter thn the therml energy k B T (where C Σ = C g + C j is the totl cpcitnce). For T = 1 K, this requires C Σ to e much less thn 1 ff. The Cooper-pir ox is connected to ground y gte cpcitnce C g in series with potentil V g nd y smll Josephson junction with E j << E c. Given their wek connection to the outside world, the numer of Cooper pirs on the islnd is discrete vrile n. The quit sttes correspond to djcent Cooper-pir numer sttes n nd n + 1. To understnd how to control single Cooper pir, it is useful to first exmine the electrosttic prolem with n infinite junction resistnce (E j = ). The totl electrosttic energy of the circuit is E ch = (2e 2 /C g )(n n g ) 2, where n g = C g V g /2e (representing the gte voltge in terms of the gte chrge, nmely the polriztion chrge tht the voltge induces on the gte cpcitor). Although n is n integer, n g is continuous vrile. E ch versus n g is shown in Fig. 3c for severl vlues of n. It should e noted tht the curves for n nd n + 1 cross t n g = n + ½, the chrge degenercy point. At this point, the gte polriztion corresponds to hlf Cooper pir for oth chrge sis sttes. By restoring the Josephson coupling to smll vlue, the ehviour close to these crossing points is modified. The Josephson junction llows Cooper pirs to tunnel onto the islnd one y one. The resultnt coupling etween neighouring chrge sttes n nd n + 1 mkes the quntum superposition of chrge eigensttes nlogous to the superposition of flux sttes in eqution (3) (identifying = n nd = n + 1 ). The next excited chrge stte is higher in energy y E c nd cn sfely e neglected. At the chrge degenercy point, where the Josephson coupling produces n voided crossing, the symmetricl nd ntisymmetricl superpositions re split y n energy E j. By contrst, fr from this point, E c >> E j, nd the eigensttes re very close to eing chrge sttes. Agin, the energy level structure is nlogous to tht of flux quits, with replced with E j nd ε with E c (n g n ½). Similrly, the proilities of mesuring the ground stte or excited stte depend on the gte voltge rther thn the pplied flux. To mke the quit fully tunle, the Josephson junction is usully replced y d.c. SQUID with low inductnce (β L << 1). E j is then djusted y pplying the pproprite mgnetic flux, which is kept constnt throughout the susequent mesurements. The red-out of chrge quit involves detecting the chrge on the islnd to much greter ccurcy thn 2e. This is ccomplished y using single-electron trnsistor (SET), sensitive electrometer 16. The SET (Fig. 3d), lso sed on tiny islnd, is connected to two superconducting leds y two Josephson junctions. When the voltges cross oth junctions re close to the degenercy point (n g = n + ½), chrges cross the junctions to produce net current flow through the SET. Thus, the current ner the degenercy points depends strongly on the gte voltge (Fig. 3c). Cpcitively coupling the Cooper-pir-ox islnd to the SET islnd mkes contriution to the SET gte voltge so tht the SET current strongly depends on the Cooper-pir-ox stte. This scheme converts the mesurement of chrge into mesurement of chrge trnsport through SET. In fct, for smll Josephson junctions, this chrge trnsport is usully dissiptive, ecuse the phse coherence is destroyed y environmentl fluctutions. Thus, the red-out ctully involves mesuring the resistnce of the SET, which depends on the stte of the Cooper-pir ox. The preferred red-out device is rdio-frequency SET 17, in which SET is emedded in resonnt circuit. Thus, the Q of the resonnt circuit is determined y the resistnce of the SET nd ultimtely y the chrge on the Cooper-pir ox. A pulse of microwves slightly detuned from the resonnt frequency is pplied to the rdiofrequency SET, nd the phse of the reflected signl enles the stte of the quit to e determined. Mny of the initil studies of superconducting quits involved chrge quits. Tht crossing is voided t the degenercy point ws first shown spectroscopiclly y studying chrge quit 9, nd chrge mesurements reveled the continuous, quntum-rounded form of the trnsition etween quntum sttes 18. The coherent oscilltions tht occur with time t this voided energy-level crossing were lso first discovered y studying chrge quit

33 I NATURE Vol June 28 INSIGHT REVIEW Cooper-pir oxes re prticulrly sensitive to low-frequency noise from electrons moving mong defects (see the section Decoherence ) nd cn show sudden lrge jumps in n g. The development of more dvnced chrge quits such s the trnsmon 2 nd quntronium 12 hs gretly meliorted this prolem. The trnsmon is smll Cooperpir ox tht is mde reltively insensitive to chrge y shunting the Josephson junction with lrge externl cpcitor to increse E c nd y incresing the gte cpcitor to the sme size. Consequently, the energy nds of the type shown in Fig. 3c re lmost flt, nd the eigensttes re comintion of mny Cooper-pir-ox chrge sttes. For resons tht will e discussed lter (see the section Decoherence ), the trnsmon is thus insensitive to low-frequency chrge noise t ll operting points. At the sme time, the lrge gte cpcitor provides strong coupling to externl microwves even t the level of single photon, gretly incresing the coupling for circuit quntum electrodynmics (QED) (see the section Quntum optics on chip ). The principle y which quntronium opertes is shown in Fig. 4, nd n ctul circuit is shown in Fig. 4. The Cooper-pir ox involves two Josephson junctions, with cpcitnce C g connected to the islnd seprting them. The two junctions re connected cross third, lrger, junction, with higher criticl current, to form closed superconducting circuit to which mgnetic flux Φ e is pplied. The key to eliminting the effects of low-frequency chrge nd flux noise is to mintin the quit t the doule degenercy point t which the two quit sttes re (to first order) insensitive to these noise sources. To chieve insensitivity to chrge noise, the quit is operted t n g = ½, where the energy levels hve zero slope nd the energy-level splitting is E j (Fig. 3c). Insensitivity to flux noise is chieved y pplying n integer numer of hlf-flux qunt to the loop. The success of this optimum working point hs een elegntly shown experimentlly 21. The insensitivity to oth flux nd chrge implies, however, tht the two sttes of the quit cnnot e distinguished t the doule degenercy point. To mesure the quit stte, current pulse tht moves the quit wy from the flux degenercy point is pplied to the loop, nd this produces clockwise or nticlockwise current in the loop, depending on the stte of the quit. The direction of the current is determined y the third (red-out) junction: the circulting current either dds to or sutrcts from the pplied current pulse, so the red-out junction switches out of the zero-voltge stte t slightly lower or slightly higher vlue of the is current, respectively. Thus, the stte of the quit cn e inferred y mesuring the switching currents. With the dvent of quntronium, much longer relxtion nd decoherence times cn e chieved thn with conventionl Cooper-pir ox. Although this switching red-out scheme is efficient, it hs two mjor drwcks. First, the resultnt high level of dissiption destroys the quntum stte of the quit, mking sequentil mesurements of the stte impossile. Second, the temperture of the red-out junction nd sustrte increse ecuse of the energy tht is deposited while the SQUID is in the voltge stte typiclly for 1 µs nd the equilirium is not restored for ~1 ms. This limits the rte t which mesurements cn e mde to ~1 khz, resulting in long dt-cquisition times. These drwcks hve een overcome y the introduction of the Josephson ifurction mplifier (JBA) 22, prticulrly powerful redout device in which there is no dissiption ecuse the junction remins in the zero-voltge stte (Fig. 4c). The JBA exploits the nonlinerity of the Josephson junction when cpcitor is connected cross it, resulting in the formtion of resonnt (or tnk) circuit. When smll-mplitude microwve pulses re pplied to the resonnt circuit, the mplitude nd phse of the reflected signl re detected, with the signl strength oosted y cryogenic mplifier. From this mesurement, the resonnt frequency of the tnk circuit cn e determined, then the inductnce of the junction which depends on the current flowing through it nd, finlly, the stte of the quntronium. For lrger-mplitude microwves, however, the ehviour of the circuit is strongly nonliner, with the resonnce frequency decresing s the mplitude increses. In prticulr, strong driving t frequencies slightly elow the plsm frequency leds to istility: wek, off-resonnce lower rnch during which the prticle does not explore the nonlinerity, nd high-mplitude response t which frequency mtches the driving frequency (Fig. 4d). The two quit sttes cn e distinguished y choosing driving frequencies nd currents tht cuse the JBA to switch to one response or the other, depending on the quit stte. This technique is extremely fst nd, even though it is sed on switching process, it never drives the junction into the voltge stte. Furthermore, the JBA remins in the sme stte fter the mesurement hs een mde. The JBA hs een shown to pproch the quntum non-demolition (QND) limit 22. This limit is reched when the perturtion of the quntum stte during the mesurement does not go eyond tht required y the mesurement postulte of quntum mechnics, so repeted mesurements of the sme eigenstte led to the sme outcome 23. Reching the QND limit is highly desirle for quntum computing. A similr scheme tht pproches the QND limit hs een implemented for the flux quit, with the single Josephson junction replced y red-out SQUID 24. Dispersive red-out for flux quit hs lso een chieved y inductively coupling flux quit to the inductor of resonnt circuit nd then mesuring the flux stte from the shift in the resonnce frequency 25. c d V tr /2 E j n C j C g SET islnd SCB islnd SCB gte Energy E j C j C g Φ e n C int C g SET V g SET V g V g.5 n g 1.5 V tr /2 Figure 3 Chrge quits., A single Cooper-pir-ox (SCB) circuit is shown. The superconducting islnd is depicted in rown nd the junction in lue. E j nd C j re the Josephson coupling energy nd self-cpcitnce, respectively, nd n is the numer of Cooper pirs on the islnd, which is coupled to voltge source with voltge V g y wy of cpcitor with cpcitnce C g. (Pnel reproduced, with permission, from ref. 28.), A microgrph of Cooper-pir ox coupled to single-electron trnsistor (SET) is shown. Scle r, 1 μm. (Pnel reproduced, with permission, from ref. 78.) c, Blck curves show the energy of the Cooperpir ox s function of the scled gte voltge n g = C g V g /2e for different numers (n) of excess Cooper pirs on the islnd. The prol on the fr left corresponds to n = nd the centrl prol to n = 1. Dshed lines indicte the contriution of the chrging energy E ch (n, n g ) lone. The energy-level splitting t n g = ½ is E j. Red curves show the current I through the SET s function of n g. Trnsport is possile t the chrge degenercy points, where the gte strongly modultes the current. (Pnel reproduced, with permission, from ref. 28.) d, A chrge quit with two junctions (left) coupled to SET ised to trnsport voltge V tr (right) is shown. The criticl current of the junctions coupled to the islnd is djusted y mens of n externlly pplied mgnetic flux Ф e. The gte of the SET is coupled to n externlly controlled chrge induced on the cpcitor with cpcitnce C g SET y the voltge V g SET, s well s to the quit chrge y wy of the interction cpcitnce C int. (Pnel reproduced, with permission, from ref. 28.) 135

34 INSIGHT REVIEW NATURE Vol June 28 Phse quits In essence, phse quit 13 consists of single current-ised Josephson junction (Box 1 figure). For is current I just elow the criticl current I, the nhrmonic potentil is pproximtely cuic, nd the energy-level spcing ecomes progressively smller s the quntum numer n increses. As I pproches I, the (clssicl) plsm oscilltion frequency, ω p (I) = 2 1/4 (2πI /Φ C) 1/2 (1 I/I ) 1/4, decreses slowly, while the potentil rrier height, U(I) = (2 2 I Φ /3π)(1 I/I ) 3/2, decreses rpidly. Thus, the proility of escpe from the stte n y mcroscopic quntum tunnelling increses exponentilly s n increses. The quit involves trnsitions etween the ground stte nd the first excited stte 1. To mesure the quntum stte of the quit, microwve pulse is pplied with frequency (E 2 E 1 )/h. If, on the one hnd, the quit is in the stte 1, then the pulse excites trnsition to the stte 2, from which mcroscopic quntum tunnelling cuses the junction to switch to the voltge stte. If, on the other hnd, the junction is initilly in the stte, then no such trnsition occurs. Opertion of the phse quit depends crucilly on the nhrmonicity of the well potentil, which ensures tht E 2 E 1 < E 1 E. The first phse quit tht ws designed involved 1 1 µm 2 N Al x O y N tunnel junction (where x 2 nd y 3), which ws creted photolithogrphiclly. To mesure the occuption proility p 1 of the stte 1, Mrtinis et l. 13 pplied long microwve pulse of ngulr frequency ω 1 = (E 1 E )/, followed y red-out pulse of frequency ω 21 = (E 2 E 1 )/ (Fig. 5). If the stte 1 is occupied, the second pulse switches the junction to the voltge stte, which is detected y low-noise mplifier. If, conversely, the junction is in the stte, the proility of switching is very smll. As the power P 1 in the first pulse is incresed, the proility of 1 eing occupied increses until it reches plteu t.5. The results of the mesurement re shown in Fig. 5, where p 1 is defined s the rtio of the numer of trils in which switching to the voltge stte occurs to the totl numer of trils. As expected, p 1 pproches.5 s P 1 increses. In erly designs of phse quits, the junction switched to the voltge stte, resulting in energy dissiption. In lter, improved, design 26, the quits remin in the zero-voltge stte (Fig. 5, c). The quit junction is emedded in superconducting loop tht is inductively coupled to SQUID nd to line through which sttic nd pulsed currents cn e pssed. With ppropritely chosen prmeters, the potentil energy of the quit displys the two symmetricl wells shown in Fig. 5c. The sttes nd 1 in the left well re the quit sttes; their energy seprtion nd the depth of the well cn e controlled y vrying the flux in the loop. To red out the stte of the phse quit, short ditic pulse tht reduces the depth U of the quit potentil well is pplied to the flux is line. If the quit is in the stte 1, it tunnels rpidly into the right well; in the stte, no tunnelling occurs. Depending on whether tunnelling occurs, the flux in the quit loop differs y single flux quntum, which cn esily e detected susequently y the red-out SQUID. This scheme enles the stte of the quit to e mesured rpidly, typiclly in 5 ns, which is still ditic (slow) on the timescle of trnsitions etween the quit sttes. Susequent mesurement of the flux in these quit loops cn e mde much more slowly. Time-domin mesurements Spectroscopy is importnt for estlishing tht given quit is functionl device, nd it enles energy-level splitting to e mesured s function of relevnt control prmeters. But mesurements in the Cooper-pir ox V C g I 1 E d j rf d d I Z V I V g Flux is c d L s I rf 5 Ω C L j V n Figure 4 Quntronium., A quntronium circuit is depicted. The Cooperpir ox is connected y wy of two Josephson junctions to the detector Josephson junction, which hs Josephson energy E d j (right), nd y wy of cpcitor (with gte cpcitnce C g ) to the sttic voltge is V g nd the rdio-frequency gte voltge V rf tht prepres the stte of the Cooperpir ox. The dshed lines enclose the quit. I is the is current of the detector junction, nd Z is n engineered environmentl impednce. The flux through the loop formed y the three Josephson junctions is controlled y n externl is circuit. The red-out is the phse δ cross the two ox junctions, mesured y comining the is current I with the circulting loop currents I or I 1. (Pnel reproduced, with permission, from ref. 12.), A microgrph of quntronium is shown. The Cooper-pir ox nd leds re depicted in lue, nd the gte electrode in red. (In gold re norml metl films tht re used to remove qusiprticles from the superconducting films.) (Imge courtesy of D. Esteve, Commissrit à l Énergie Atomique, Scly, Frnce.) c, A Josephson ifurction mplifier (JBA) is depicted. In JBA, Josephson junction, represented y the nonliner inductnce L j, is shunted with cpcitnce C vi stry inductnce L S ; I rf is the rdiofrequency current is. The dshed line seprtes the off-chip circuitry (left) from the on-chip circuitry (right). (Pnel reproduced, with permission, from ref. 22.) d, The response curve (voltge V versus frequency ν) of the JBA driven t high rdio-frequency current mplitude t frequency slightly elow resonnce is shown, nd the hysteresis tht results from dynmicl ifurction is indicted (rrows). The red line shows the low-mplitude response of the JBA, nd the green line shows the high-mplitude response; the dshed line indictes metstle sttes. 136

35 NATURE Vol June 28 INSIGHT REVIEW time domin re lso necessry to determine the dynmicl ehviour of quit. These mesurements involve mnipulting the stte of the quit y using pproprite microwve pulses which re lso required to implement single-quit gtes for quntum computing. In rod terms, quits re chrcterized y two times, nmed T 1 nd T 2 y nlogy with nucler mgnetic resonnce (NMR) spectroscopy 27. The relxtion time T 1 is the time required for quit to relx from the first excited stte to the ground stte; this process involves energy loss. The dephsing time T 2 is the time over which the phse difference etween two eigensttes ecomes rndomized. Theoreticlly, oth relxtion nd dephsing re descried y wek coupling to the quntum noise produced y the environment This pproch predicts tht energy relxtion rises from fluctutions t the energy-level splitting frequency of the two sttes in question. The dephsing rte, y contrst, hs two contriutions: 1/T 2 = 1/(2T 1 ) + 1/τ ϕ (5) The first contriution rises from the relxtion process, nd the second, pure dephsing, rises from low-frequency fluctutions with exchnge of infinitesiml energy. (The pure dephsing time is τ ϕ.) The simplest wy to mesure relxtion is to irrdite the quit with microwves t the frequency corresponding to the energy-level splitting etween the ground nd first excited sttes for time much greter thn T 1. After the pulse hs een turned off, the quit hs n equl proility of eing in either stte; the proility p 1 of its eing in the excited stte 1 susequently decys with time t s exp( t/t 1 ). Mesurements of p 1 s function of t yield the vlue of T 1. It should e emphsized tht ech mesurement of p 1 t given time dely involves lrge numer of mesurements, typiclly 1 4 or 1 5. T 1 cn vry from vlues of the order of 1 ns to mny microseconds. To understnd the vrious pulse mesurements, it is useful to consider the Bloch sphere (Fig. 6), which enles ny ritrry quntum superposition of the quntum sttes nd 1 to e considered s vector. The sttes nd 1 point long the positive nd negtive z xis, respectively. The superpositions ± 1 lie long the ± x xes, nd the superpositions ± i 1 long the ± y xes. Thus, given point on the surfce of the sphere defines specific superposition of these sttes. The Bloch sphere cn e used to descrie Ri oscilltions in flux quit. Microwves re pplied t the energy-level splitting frequency for the quit for time τ with the mgnetic-field component long the y xis. During the pulse, the stte vector rottes in the y z plne out the x xis with the Ri frequency ν R, which is proportionl to the microwve mplitude. After time τ, the stte vector is t n ngle 2πν R τ to the z xis. Susequent mesurements of the proility of the quit eing in the stte or 1 yield Ri oscilltions s function of τ. An exmple is shown in Fig. 6. Ri oscilltions re convenient mens of clirting the mplitude of the mgnetic-field component of the microwve field tht is coupled to the quit. In mesuring the dephsing time, it is crucil to distinguish T 2 (eqution (5)) n intrinsic timescle for the decoherence of single quit from T 2 *, the result of n ensemle mesurement. The ensemle is formed ecuse experiments on single quit need to e crried out repetedly so tht sufficiently precise dt re cquired. Even though the different mesurements re nominlly identicl, slow fluctutions on the timescle of single run result in chnge in the operting conditions etween runs. This reduces the oserved coherence time to T 2 * (which is < T 2 ). T 2 * nd T 2 cn e mesured seprtely: T 2 *, which includes the effects of low-frequency noise, y using Rmsey fringes 3 ; nd T 2, y using spin-echo technique 27, which elimintes certin low-frequency contriutions. To oserve Rmsey fringes, π/2 microwve pulse is first pplied t frequency f m with mplitude clirted from the Ri oscilltions tht tips the quit stte vector into the equtoril (x y) plne. The vector precesses freely on the Bloch sphere round the sttic mgnetic field B, with mgnitude tht decreses with time, owing to dephsing. After vrile time dely τ d, second π/2 microwve pulse rings the stte vector to point on the Bloch sphere tht depends on oth f m nd τ d. The susequent mesurement of the quit stte projects the vector onto either or 1. Thus, plot of the switching proility versus τ d for given microwve frequency mps out the free evolution of the quit. For resonnt pulse (f m = ν 1 ), the free evolution nd the microwve pulses re synchronized, nd the mesurement revels coherence mplitude tht decys exponentilly with chrcteristic time T 2 *. To mp out T 2 * over lrger prmeter spce, the π/2 microwve pulses re detuned from ν 1. Thus, the pulse nd evolution re no longer synchronized, nd oscilltions Rmsey fringes re oserved t frequency ν Rmsey = f m ν 1 (Fig. 6c). To remove the slow fluctutions tht differentite T 2 * from T 2, spinecho technique, nlogous to tht used in NMR, cn e used. In this technique, π pulse is pplied t the midpoint in time etween the two π/2 pulses. The π pulse flips the quit stte vector to the opposite side of the equtoril plne; therefore, fluctution tht initilly cused the phse to dvnce now cuses it to lg, nd vice vers. Thus, t the time of the second π/2 pulse, the effects of fluctutions tht occur on timescles longer thn the overll mesurement time re (idelly) completely cncelled out. An exmple is shown in Fig. 6d. Microwve pulses Sttic nd pulsed flux SQUID red-out 1 c p P ns Quit P 21 U Left well 2 1 U Right well w1 w P 1 (ritrry units) Figure 5 Phse quits., The filled circles represent the proility p 1 tht the phse quit occupies the first excited stte versus microwve power P 1 t ngulr frequency ω 1. The solid line is the theoreticl prediction. The inset shows the pulse sequence; the microwves t ngulr frequency ω 1 equlize the proility tht the ground nd excited sttes re occupied, nd the microwves t ngulr frequency ω 21 cuse the quit to switch to the voltge stte if the first excited stte is occupied. (Pnel reproduced, with permission, from ref. 13.), For zero-voltge opertion, the Josephson junction of phse quit is shunted y superconducting loop, coupled Superconducting loop Phse to red-out SQUID, tht llows sttic nd pulsed fluxes to e pplied. The dshed line indictes the components fricted on the silicon chip, which is mintined t 25 mk. (Pnel reproduced, with permission, from ref. 26.) c, The symmetricl doule-well potentil of phse quit is shown. The quit sttes re nd 1. The stte 2 ecomes occupied on the ppliction of microwves t frequency ω 21 provided tht the stte 1 is occupied. The stte 3, ove 2, hs no role in the red-out process. Dots in the right well indicte the intervening energy levels. (Pnel reproduced, with permission, from ref. 26.) 137

36 INSIGHT REVIEW NATURE Vol June 28 c p i i 1 B rf B Time (ns) d I (ritrry units) 1 1 p sw (%) Pulse length (µs) δt (ps) Figure 6 Quit mnipultion in the time domin., The Bloch sphere is depicted, with n pplied sttic mgnetic field B nd rdio-frequency mgnetic field B rf. Any given superposition of the six sttes shown is represented y unique point on the surfce of the sphere., Ri oscilltions in flux quit re shown. The proility p sw tht the detector (SQUID) switches to the norml stte versus pulse length is shown, nd the inset is mgnifiction of the oxed region, showing tht the dense trces re sinusoidl oscilltions. As expected, the excited-stte popultion oscilltes under resonnt driving. (Pnel reproduced, with permission, from ref. 4.) c, Rmsey fringes in phse quit re shown. Coherent oscilltions of the switching proility p 1 etween two detuned π/2 pulses is shown s function of pulse seprtion. (Pnel reproduced, with permission, from ref. 31.) d, The chrge echo in Cooper-pir ox is shown s function of the time difference δt = t 1 t 2, where t 1 is the time etween the initil π/2 pulse nd the π pulse, nd t 2 is the time etween the π pulse nd the second π/2 pulse. The echo peks t δt =. (Pnel reproduced, with permission, from ref. 39.) Mesuring the times T 1, T 2 nd T 2 * provides n importnt initil chrc teriztion of quit coherence. However, other fctors such s pulse in ccurcy, relxtion during mesurement nd more complex decoherence effects result in mesurement errors. A more complete mesure of quit is fidelity, single numer tht represents the difference etween the idel nd the ctul outcome of the experiment. Determining the fidelity involves quntum-process tomogrphy ( repeted set of stte tomogrphies), which chrcterizes quntummechnicl process for ll possile initil sttes. In Rmsey-fringe tomogrphy experiment, Mtthis Steffen et l. 31 found fidelity of ~8%, where 1% of the loss ws ttriuted to red-out errors nd nother 1% to pulse-timing uncertinty. Decoherence Superconducting quits re mcroscopic, so long the lines of Schrödinger s ct they could e expected to e very sensitive to de coherence. In fct, given the unique properties of the superconducting stte, creful engineering hs led to remrkle increses in decoherence times compred with those of erly devices. Idelly, ech type of quit is descried y single degree of freedom. The centrl chllenge is to eliminte ll other degrees of freedom. In rod terms, there re two clsses of decohering element: extrinsic nd intrinsic. Ovious extrinsic sources include electromgnetic signls from rdio nd television trnsmitters; these cn generlly e eliminted y using creful shielding nd enough rodnd filters. A more chllenging extrinsic source to exclude is the locl electromgnetic environment: for exmple, contriutions from the leds tht re coupled to red-out devices or re used to pply flux or chrge ises. These leds llow gret flexiility in control of the system t the expense of considerle coupling to the environ ment. This issue ws recognized in the first proposls of mcroscopic quntum coherence nd lrgely motivted the Cldeir Leggett theory of quntum dissiption 6. This theory mps ny liner dissiption onto th of hrmonic oscilltors. The effects of these oscilltors cn e clculted from the Johnson Nyquist noise tht is generted y the complex impednce of the environment. In the wek-dmping regime, oth T 1 nd τ ϕ cn e computed directly from the power spectrum of this noise, nd then the impednce cn e engineered to minimize decoherence 28,29. The experimentl difficulty is to ensure tht the complex impednces seen y the quit re high over rod ndwidth, for exmple, 1 GHz. It is prticulrly difficult to void resonnces over such rod rnge of frequencies. Clever engineering hs gretly reduced this source of decoherence, ut it would e optimistic to consider tht this prolem hs een completely solved. The min intrinsic limittion on the coherence of superconducting quits results from low-frequency noise, notly 1/f noise (in which the spectrl density of the noise t low frequency f scles s 1/f α, where α is of the order of unity). In the solid stte, mny 1/f noise sources re well descried y the Dutt Horn model s rising from uniform distriution of two-stte defects 32. Ech defect produces rndom telegrph noise, nd superposition of such uncorrelted processes leds to 1/f power spectrum. There re three recognized sources of 1/f noise. The first is criticl-current fluctutions, which rise from fluctutions in the trnsprency of the junction cused y the trpping nd untrpping of electrons in the tunnel rrier 33. All superconducting quits re suject to dephsing y this mechnism. The slow fluctutions modulte energylevel splitting, even t the degenercy point, so ech mesurement is mde on quit with slightly different frequency. The resultnt phse errors led to decoherence. The second source of 1/f noise is chrge fluctutions, which rise from the hopping of electrons etween trps on the surfce of the superconducting film or the surfce of the sustrte. This motion induces chrges onto the surfce of nery superconductors. This decoherence mechnism is prticulrly prolemtic for chrge quits, except t the degenercy point, where the quits re (to first order) insensitive. If the vlue of E c /E j increses, however, the energy nds (Fig. 3c) ecome fltter, nd the quit is correspondingly less sensitive to chrge noise wy from the degenercy point. This mechnism underlies the sustntilly incresed vlues of T 2 in the trnsmon 2. The third source of 1/f noise is mgnetic-flux fluctutions. Although such fluctutions were first chrcterized more thn 2 yers go 34, the mechnism y which these occur remined oscure until recently. It is now thought tht flux noise rises from the fluctutions of unpired electron spins on the surfce of the superconductor or sustrte 35,36, ut the detils of the mechnism remin controversil. Flux noise cuses decoherence in flux quits, except t the degenercy point, s well s in phse quits, which hve no degenercy point. The incresed vlue of T 2 in quntronium results from its insensitivity to oth flux noise nd chrge noise t the doule degenercy point. 138

37 NATURE Vol June 28 INSIGHT REVIEW In generl, ll three low-frequency processes led to decoherence. They do not contriute to relxtion ecuse this process requires n exchnge of energy with the environment t the energy-level splitting frequency of the quit, which is typiclly in the gighertz rnge. However, there is strong evidence tht chrge fluctutions re ssocited with the high-frequency resontors tht hve een oserved, in prticulr, in phse quits 37. Improvements in the qulity of the oxide lyers tht re used in the junctions nd cpcitors hve resulted in lrge reductions in the concentrtion of these high-frequency resontors 38. The strtegy of operting quit t the optimum point, which ws first crried out with quntronium ut is now pplied to ll types of super conducting quit (except for phse quits), hs een successful t incresing phse-coherence times y lrge fctors. Further sustntil improvements hve resulted from the use of chrge- or flux-echo techniques 39,4. In NMR, the spin-echo technique removes the inhomogeneous rodening tht is ssocited with, for exmple, vritions in mgnetic field, nd hence in the NMR frequency, over the smple. In the cse of quits, the vrition is in the quit energy-level splitting frequency from mesurement to mesurement. For some quits, using comintion of echo techniques nd optimum point opertion hs eliminted pure dephsing, so decoherence is limited y energy relxtion (T 2 * = 2T 1 ). In generl, however, the mechnisms tht limit T 1 re unknown, lthough resontors tht re ssocited with defects my e responsile 36,41. The highest reported vlues of T 1, T 2 * nd T 2 re listed in Tle 1. Coupled quits An exceedingly ttrctive nd unique feture of solid-stte quits in generl nd superconducting quits in prticulr is tht schemes cn e implemented tht oth couple them strongly to ech other nd turn off their interction in situ y purely electronic mens. Becuse the coupling of quits is centrl to the rchitecture of quntum computers, this suject hs ttrcted much ttention, in terms of oth theory nd experiment. In this section, we illustrte the principles of coupled quits in terms of flux quits nd refer to nlogous schemes for other superconducting quits. Becuse the flux quit is mgnetic dipole, two neighouring flux quits re coupled y mgnetic dipole dipole interctions. The coupling Tle 1 Highest reported vlues of T 1, T 2 * nd T 2 Quit T 1 (μs) T 2 * (μs) T 2 (μs) Source Flux Y. Nkmur, personl communiction Chrge ref. 77 Phse J. Mrtinis, personl communiction Flux quit d.c. SQUID I p Energy-level splitting (MHz) I p (µa) Figure 7 Controllly coupled flux quits., Two flux quits re shown surrounded y d.c. SQUID. The quit coupling strength is controlled y the pulsed is current I p tht is pplied to the d.c. SQUID efore mesuring the energy-level splitting etween the sttes 1 nd 2., The filled circles show the mesured energy-level splitting of the two coupled flux quits plotted ginst I p. The solid line is the theoreticl prediction, fitted for I p ; there re no fitted prmeters for the energy-level splitting. Error rs, ±1σ. (Pnels reproduced, with permission, from ref. 5.) strength cn e incresed y hving the two quits use common line. Even stronger coupling cn e chieved y including Josephson junction in this line to increse the line s self-inductnce (eqution (6), Box 1). In the cse of chrge nd phse quits, nerest-neighour interctions re medited y cpcitors rther thn inductors. Fixed interction hs een implemented for flux, chrge nd phse quits These experiments show the energy levels tht re expected for the superposition of two pseudospin sttes: nmely, ground stte nd three excited sttes; the first nd second excited sttes my e degenerte. The entnglement of these sttes for two phse quits hs een shown explicitly y mens of quntum-stte tomogrphy 46. The most generl description (including ll imperfections) of the quit stte sed on the four sis sttes of the coupled quits is four-y-four rry known s density mtrix. Steffen et l. 46 crried out mesurement of the density mtrix; they prepred system in prticulr entngled stte nd showed tht only the correct four mtrix elements were non-zero nd tht their mgnitude ws in good greement with theory. This experiment is proof-of-principle demonstrtion of sic function required for quntum computer. Simple quntum gtes hve lso een demonstrted 47,48. Two flux quits cn e coupled y flux trnsformers in essence closed loop of superconductor surrounding the quits enling their interction to e medited over longer distnces. Becuse the superconducting loop conserves mgnetic flux, chnge in the stte of one quit induces circulting current in the loop nd hence flux in the other quit. Flux trnsformers tht contin Josephson junctions enle the interction of quits to e turned on nd off in situ. One such device consists of d.c. SQUID surrounding two flux quits 49 (Fig. 7). The inductnce etween the two quits hs two components: tht of the direct coupling etween the quits, nd tht of the coupling through the SQUID. For certin vlues of pplied is current (elow the criticl current) nd flux, the self-inductnce of the SQUID ecomes negtive, so the sign of its coupling to the two quits opposes tht of the direct coupling. By choosing prmeters ppropritely, the inductnce of the coupled quits cn e designed to e zero or even hve its sign reversed. This scheme hs een implemented y estlishing the vlues of SQUID flux nd is current nd then using microwve mnipultion nd mesuring the energy-level splitting of the first nd second excited sttes 5 (Fig. 7). A relted design tunle flux flux coupling medited y n off-resonnt quit hs een demonstrted 51, nd tunle cpcitors hve een proposed for chrge quits 52. Another pproch to vrile coupling is to fix the coupling strength geometriclly nd tune it y frequency selection. As n exmple, we consider two mgneticlly coupled flux quits ised t their degenercy points. If ech quit is in superposition of eigensttes, then its mgnetic flux oscilltes nd the coupling verges to zero unless oth quits oscillte t the sme frequency, in which cse the quits re coupled. This phenomenon is nlogous to the cse of two pendulums coupled y wek spring. Even if the coupling is extremely wek, the pendulums will e coupled if they oscillte in ntiphse t exctly the sme frequency. Implementing this scheme is prticulrly strightforwrd for two phse quits ecuse their frequencies cn redily e rought in nd out of resonnce y djusting the is currents 37. For other types of quit, the frequency t the degenercy point is set y the s-fricted prmeters, so it is inevitle tht there will e vriility etween quits. As result, if the frequency difference is lrger thn the coupling strength, the quit quit interction cncels out t the degenercy point. Severl pulse sequences hve een proposed to overcome this limittion 53 55, none of which hs een convincingly demonstrted s yet. The twoquit gte demonstrtions were ll crried out wy from the optimum point, where the frequencies cn redily e mtched. On the sis of these coupling schemes, severl rchitectures hve een proposed for scling up from two quits to quntum computer. The centrl ide of most proposls is to couple ll quits to long centrl coupling element, quntum us 56,57 (Fig. 8), nd to use frequency selection to determine which quits cn e coupled This scheme hs een experimentlly demonstrted. As couplers ecome longer, they ecome trnsmission lines tht hve electromgnetic modes. For exmple, two 139

38 INSIGHT REVIEW NATURE Vol June 28 cpsw (%) 1 µm , 1, 1 +, 1, 1 1 cm 1 µm C C g 1, 1, 2, Shift pulse length (ns) d ,, 1 Figure 8 Circuit QED., The upper prt of the pnel depicts microstrip cvity (lue) tht contins chrge quit (green) plced t n ntinode of the electric field. The microstripline cn e used s quntum us. The lower prt depicts this circuit in lumped circuit representtion. (Pnel reproduced, with permission, from ref. 59.) C is the cpcitnce of the coupling cpcitor to the mesurement electronics, nd C g is the cpcitnce of the coupling cpcitor to the chrge quit., The open circles show the mesured vcuum Ri oscilltions of flux quit coupled to lumped resontor. The solid curve is fit to the dt. (Pnel reproduced, with permission, from ref. 68.) c, An energy ldder of quit ground nd excited sttes comined with photon numer n,, n nd 1, n (dshed lines), is shown. With the cvity in resonnce with the quits, the sttes with zero photons split into liner comintions ±, (solid lines), with n energy-level splitting g, nd the sttes with one photon split into liner comintions ±, 1, with n energy-level splitting 2g. The red rrows indicte tht if the system is initilly in one of the sttes represented y dshed lines, it will perform Ri oscilltions etween the quit nd the cvity. (Pnel modified, with permission, from ref. 68.) d, An energy-nd digrm (solid nd dshed lck lines) is shown s function of pplied flux for the mesurement scheme tht led to the results in. The mesurement pulse (π pulse) forces the system from the ground stte (point 1) into stte with n excited quit (point 2) (depicted in lue), which then puts the quit nd the cvity into resonnce t point 3 (depicted in red). After the vcuum Ri oscilltion occurs, the system returns to point 2 or mkes coherent trnsition to point 4, where the quit excittion is converted to cvity photon. (Pnel modified, with permission, from ref. 68.),, 1, F shift Shift pulse π pulse quits hve een coupled y plcing them t the nti nodes of stnding wve on stripline Coupling etween specific pirs of quits cn result in sclle rchitecture 63. By first coupling quit to the stnding-wve mode using frequency selection, photon is excited nd then stored fter decoupling. Susequently, second quit is coupled to the mode, nd the photon trnsfers the quntum stte to the second quit. Architectures for ditic quntum computers re the suject of intense reserch. Aditic quntum computing encodes the solution to hrd prolem in the ground stte of quit system nd uses quntum physics to prepre tht ground stte efficiently. The ground stte of four-quit system with tunle interctions hs een mpped out 64. It should, however, e noted tht there is no proof tht n ditic quntum computer will e fster thn clssicl computer. Quntum optics on chip An importnt new direction in superconducting quit reserch is sed on nlogy etween superconducting circuits nd the fields of tomic physics nd quntum optics. So fr, we hve descried only quits s quntum ojects, nd the control fields nd red-out signls hve een treted s clssicl vriles. Circuit QED, y contrst, ddresses the quntum ehviour of the electromgnetic field, such s tht of single photons. In previous sections, the discussion refers to quntum field in coherent stte in the limit of lrge numers of photons. The key requirement for reching the quntum limit of the electromgnetic field is tht the zero-point fluctution of single mode mesured y the root men squre of the electric field, E rms = E 2 vcuum e strong enough to hve n pprecile coupling strength g = de rms to the quit electric dipole moment d. This requirement is met y incresing the mplitude of the field y creting stnding wve in resontor nd plcing the quit t one of the ntinodes 59 (Fig. 8). The resontor cn e either microstripline n on-chip wve guide for microwves or lumped circuit. In the first experiment 65, the resontor ws tunle. The physics is closely relted to cvity QED 66, in which toms couple to n opticl field confined etween two mirrors. A key difference is tht in circuit QED, the tom (tht is, the superconducting quit) does not move inside the cvity, so the tom field interction hs time to ct without losing the tom. Together with the fct tht g/ is lrger thn the rte of photon loss from the cvity, this difference llows the strong coupling limit of QED to e chieved in reltively strightforwrd mnner. The underlying resons re tht g is proportionl to d (which, for Cooper-pir ox, is lrge, out 1 4 tomic units) nd tht E rms is lso lrge ecuse of the increse in the electromgnetic field in the one-dimensionl stripline. Circuit QED cn e operted in two distinct strong-coupling limits: the resonnt regime, nd the off-resonnt dispersive regime. In the resonnt regime, the quit energy-level splitting is in resonnce with the cvity 14

39 NATURE Vol June 28 INSIGHT REVIEW frequency. In this regime, the comined sttes of the quit nd cvity cn e written in the form quit stte, photon numer. On resonnce, the quit nd cvity cn exchnge excittions without losing energy: tht is, the energy of 1, n is equl to the energy of, n + 1. The eigensttes of the system re thus superpositions of the form ±, n = 1, n ±, n + 1, with energies split y g n, leding to the energy spectrum shown in Fig. 8c. This hs striking consequence: suppose tht initilly the quit energy is not in resonnce with the cvity (so the two re decoupled) nd tht the quit is put into n excited stte while the cvity is left in its vcuum stte. When the quit nd the cvity in tht stte re suddenly coupled y using the procedure shown in Fig. 8d, the originl stte ceses to e n eigenstte nd, insted, ecomes n equl superposition of +, nd,. After time t, these cquire reltive phse of gt/ nd mnifest themselves s coherent oscilltion etween 1, nd, 1, even though initilly there ws no photon in the cvity. These vcuum Ri oscilltions hve een shown spectroscopiclly 67 nd in the time domin 68 (Fig. 8). The second cse is the off-resonnt dispersive regime. In this cse, the quit nd cvity eigensttes re not entngled, nd the two systems cnnot shre excittions. The mutul energies, however, re still correlted, ecuse the energy-level splitting of the quit depends on the cvity stte, nd vice vers. Consequently, the cvity cn e used to red out the quit nd to couple quits to ech other 59. Circuit QED hs een highly successful. So fr, experimentl progress hs included ttining the strong coupling limit 67, mpping out the discrete nture of the quntized field 69, generting single photons 7 nd coupling quits using us 61,62. These developments re leding to flexile quntum optics on chip nd open the door to new domin of mesoscopic physics. Sclle rchitectures for quntum computers sed on circuit QED hve een proposed 62. These ides hve led to the recent demonstrtion of superconducting quit lser. The tom chrge quit is wekly coupled to second led. In pproprite is conditions, cyclic process tkes plce: Cooper pirs tht enter the ox re roken into two qusiprticles, which exit through the second led. This cycle results in significnt overpopultion of the first excited quit stte compred with the ground stte tht is, popultion inversion nd the genertion of lser ction 71. Studies in tomic physics hve produced super techniques for ctively cooling toms. Becuse superconducting quits operte t millikelvin tempertures, it might e thought tht further cooling is unnecessry. But oth the preprtion of high-fidelity initil stte nd the supply of quits initilized to the ground stte for error correction cn e fcilitted y ctive cooling. Cooling to 3 mk from n initil temperture of 4 mk hs een chieved y exciting the popultion of the excited stte of flux quit to higher excited stte tht is deloclized in doule-well potentil, nd then llowing the quit to relx to the ground stte 72. Sisyphus cooling hs lso een demonstrted 73 : in this cooling protocol, the energy tht is supplied to the quit from the het th is cycliclly removed y the mgnetic component of suitly tilored microwve field. Outlook Quntum computing is huge driving force for technologicl innovtion. Since mcroscopic quntum coherence ws shown, the progress in the design nd opertion of superconducting quits hs een remrkle. There is now rich vriety of devices tht contin the three quit types, either seprtely or in comintion. Decoherence times hve een incresed from ~1 ns to ~1 µs, nd single-shot nd QND red-outs re close to eing chieved. So, wht chllenges nd prospects now lie hed? On fundmentl level, the next enchmrk is to verify violtion of Bell s inequlity 74. This inequlity, which involves the outcomes of comintion of two-quit mesurements, is oeyed for ny locl theory ut is violted for truly non-locl physics such s quntum mechnics. A vrition is the Leggett Grg inequlity 75, which reltes to tem porl correltions rther thn to two-quit correltions. One importnt spect of quntum mechnics entnglement hs een shown for superconducting quits 46, ut the testing of whether Bell s inequlity is violted poses formidle technologicl chllenges, prticulrly with respect to the fidelity of the mesurement nd the elimintion of cross-tlk. To mke Bell test convincing, the interction etween quits needs to e switched off very ccurtely so tht mesurements re truly independent. An even more convincing test would involve true spce-like seprtion: tht is, mesuring the red-out of two quits in such short time tht no signl hs een le to trvel etween them t the speed of light. Given the confines of dilution refrigertor, however, it seems tht it will not e possile to test superconducting quits in this wy. Another importnt experiment involving entnglement will e to investigte whether teleporttion of stte occurs 76 : tht is, the trnsfer of quntum stte inside n entngled pir of sttes. On the pth to quntum computing, superconducting quits re clerly mong the most promising cndidtes. Nevertheless, the pth is long, nd there re quntittive technologicl ostcles to e overcome, notly incresing the decoherence time nd improving the fidelity of the red-out. The key enchmrk will e to demonstrte simple error correction. To chieve these grnd gols will require technologicl progress, not the lest in the elimintion or t lest the reduction of low-frequency noise. Two-quit coherence in prticulr, the question of whether noise processes re correlted etween quits is lrgely unexplored. Will there ever e superconducting quntum computer? This question cnnot e nswered tody. The error thresholds discussed in fulttolernce reserch 1 error in 1, opertions eing typicl, ut y no mens universl, enchmrk re dunting. However, fulttolernce reserch is n evolving field, nd the computtionl protocols tht hve een discussed so fr (which minimize the numer of physicl quits nd interctions required for given lgorithm) might not e est suited for superconducting quits. Promising lterntives might e error-correction models with more generous thresholds, topologicl computing or other lterntive computtionl models. Aditic quntum computing could lso e n lterntive if it is proved to e fster thn clssicl computing. While ddressing these issues, reserchers re lso likely to gin further insight into mny physicl properties nd processes. 1. Brdeen, J., Cooper, L. N. & Schrieffer, J. R. Theory of superconductivity. Phys. Rev. 18, (1957). 2. Tinkhm, M. Introduction to Superconductivity (McGrw-Hill, New York, 1996). 3. Devoret, M. H. in Quntum Fluctutions: Les Houches Session LXIII (eds Reynud, S., Gicoino, E. & Dvid, F.) (Elsevier, Amsterdm, 1997). 4. Voss, R. F. & We, R. A. Mcroscopic quntum tunneling in 1-µm N Josephson junctions. Phys. Rev. Lett. 47, (1981). 5. Devoret, M. H., Mrtinis, J. M. & Clrke, J. Mesurements of mcroscopic quntum tunneling out of the zero-voltge stte of current-ised Josephson junction. Phys. Rev. Lett. 55, (1985). 6. Cldeir, A. O. & Leggett, A. J. Quntum tunneling in dissiptive system. Ann. Phys. (NY) 149, (1983). 7. Mrtinis, J. M., Devoret, M. H. & Clrke, J. Energy-level quntiztion in the zero-voltge stte of current-ised Josephson junction. Phys. Rev. Lett. 55, (1985). 8. Leggett, A. J. in Chnce nd Mtter: Les Houches Session XLVI (eds Souletie, J., Vnnimenus, J. & Stor, R.) (Elsevier, Amsterdm, 1987). 9. Nkmur, Y., Chen, C. D. & Tsi, J. S. Spectroscopy of energy-level splitting etween two mcroscopic quntum sttes of chrge coherently superposed y Josephson coupling. Phys. Rev. Lett. 79, (1997). 1. Friedmn, J. R., Ptel, V., Chen, W.,Tolpygo, S. K. & Lukens, J. E. Quntum superpositions of distinct mcroscopic sttes. Nture 46, (2). 11. vn der Wl, C. H. et l. Quntum superpositions of mcroscopic persistent current. Science 29, (2). 12. Vion, D. et l. Mnipulting the quntum stte of n electricl circuit. Science 296, (22). 13. Mrtinis, J. M., Nm, S., Aumentdo, J. & Urin, C. Ri oscilltions in lrge Josephsonjunction quit. Phys. Rev. Lett. 89, (22). 14. The SQUID Hndook: Fundmentls nd Technology of SQUIDs nd SQUID Systems Vol. I (eds Clrke, J. & Brginski, A. I.) (Wiley, Weinheim, 24). 15. Wilhelm, F. K. et l. Mcroscopic quntum superposition of current sttes in Josephson junction loop. Usp. Fiz. Nuk 44 (suppl. 171), (21). 16. Devoret, M. H. & Schoelkopf, R., Amplifying quntum signls with the single-electron trnsistor. Nture 46, (2). 17. Schoelkopf, R. J., Whlgren, P., Kozhevnikov, A. A., Delsing, P. & Proer, D. E. The rdiofrequency single-electron trnsistor (RF-SET): fst nd ultrsensitive electrometer. Science 28, (1998). 18. Bouchit, V., Vion, D., Joyez, P., Esteve, D. & Devoret, M. H. Quntum coherence with single Cooper pir. Physic Script T76, (1998). 19. Nkmur, Y., Pshkin, Y. A. & Tsi, J. S., Coherent control of mcroscopic quntum sttes in single-cooper-pir ox. Nture 398, (1999). 141

40 INSIGHT REVIEW NATURE Vol June Koch, J. et l. Chrge-insensitive quit design derived from the Cooper pir ox. Phys. Rev. A 76, (27). 21. Ithier, G. et l. Decoherence in superconducting quntum it circuit Phys. Rev. B 72, (25). 22. Siddiqi, I. et l. Direct oservtion of dynmicl ifurction etween two driven oscilltion sttes of Josephson junction. Phys. Rev. Lett. 94, 275 (25). 23. Brginsky, V. B., Khlili, F. Y. & Thorne, K. S. Quntum Mesurement (Cmridge Univ. Press, Cmridge, UK, 1995). 24. Lupşcu, A. et l. Quntum non-demolition mesurement of superconducting two-level system. Nture Phys. 3, (27). 25. Grjcr, M. et l. Low-frequency mesurement of the tunneling mplitude in flux quit. Phys. Rev. B 69, 651 (24). 26. Cooper, K. B. et l. Oservtion of quntum oscilltions etween Josephson phse quit nd microscopic resontor using fst redout. Phys. Rev. Lett. 93, 1841 (24). 27. Slichter C. P. Principles of Nucler Mgnetic Resonnce 3rd edn (Springer, New York, 199). 28. Mkhlin, Y., Schön, G. & Shnirmn, A. Quntum-stte engineering with Josephsonjunction devices. Rev. Mod. Phys. 73, (21). 29. Wilhelm, F. K., Hrtmnn, U., Storcz, M. J. & Geller, M. R. in Mnipulting Quntum Coherence in Solid Stte Systems (eds Fltté, M. E. & Tifre, I.) (Springer, Dordrecht, 27). 3. Rmsey, N. F. A moleculr em resonnce method with seprted oscillting fields. Phys. Rev. 78, (195). 31. Steffen, M. et l. Stte tomogrphy of cpcitively shunted phse quits with high fidelity. Phys. Rev. Lett. 97, 552 (26). 32. Dutt P. & Horn P. M. Low frequency fluctutions in solids: 1/f noise. Rev. Mod. Phys. 53, (1981). 33. vn Hrlingen, D. J. et l. Decoherence in Josephson-junction quits due to criticl-current fluctutions. Phys. Rev. B 7, (24). 34. Wellstood, F. C., Urin, C. & Clrke, J. Low-frequency noise in dc superconducting quntum interference devices elow 1 K. Appl. Phys. Lett. 5, (1987). 35. Koch, R. H., DiVincenzo, D. P. & Clrke, J. Model for 1/f flux noise in SQUIDs nd quits. Phys. Rev. Lett. 98, 2673 (27). 36. Foro, L. & Ioffe, L. B. Microscopic origin of low-frequency flux noise in Josephson circuits. Preprint t < (27). 37. Simmonds, R. W., Lng, K. M., Hite, D. A., Ppps, D. P. & Mrtinis, J. M. Decoherence in Josephson quits from junction resonnces. Phys. Rev. Lett. 93, 7733 (24). 38. Mrtinis, J. M. et l. Decoherence in Josephson quits from dielectric loss. Phys. Rev. Lett. 95, 2153 (25). 39. Nkmur, Y., Pshkin, Y. A., Ymmoto, T. & Tsi, J. S. Chrge echo in Cooper-pir ox. Phys. Rev. Lett. 88, 4791 (22). 4. Bertet, P. et l. Relxtion nd dephsing in flux-quit. Phys. Rev. Lett. 95, 2572 (25). 41. Astfiev, O., Pshkin, Y. A., Nkmur, Y., Ymmoto, T. & Tsi, J. S. Quntum noise in the Josephson chrge quit. Phys. Rev. Lett. 93, 2677 (24). 42. Berkley, A.J. et l. Entngled mcroscopic quntum sttes in two superconducting quits. Science 3, (23). 43. Mjer, J. B., Puw, F. G., ter Hr, A. C. J., Hrmns, C. J. P. M. & Mooij, J. E. Spectroscopy of coupled flux quits. Phys. Rev. Lett. 94, 951 (25). 44. Pshkin, Y. A. et l. Quntum oscilltions in two coupled chrge quits. Nture 421, (23). 45. McDermott, R. et l. Simultneous stte mesurement of coupled Josephson phse quits. Science 37, (25). 46. Steffen, M. et l. Mesurement of the entnglement of two superconducting quits vi stte tomogrphy. Science 313, (26). 47. Ymmoto, Y., Pshkin, Y. A., Astfiev, O., Nkmur, Y. & Tsi, J. S. Demonstrtion of conditionl gte opertion using superconducting chrge quits. Nture 425, (23). 48. Plntenerg, J. H., de Groot, P. C., Hrmns, C. J. & Mooij, J. E. Demonstrtion of controlled- NOT quntum gtes on pir of superconducting quntum its. Nture 447, (27). 49. Plourde, B. L. T. et l. Entngling flux quits with ipolr dynmic inductnce. Phys. Rev. B 7, 1451 (24). 5. Hime, T. et l. Solid-stte quits with current-controlled coupling. Science 314, (26). 51. Nisknen, A. O. et l. Quntum coherent tunle coupling of superconducting quits. Science 316, (27). 52. Averin, D. V. & Bruder, C. Vrile electrosttic trnsformer controllle coupling of two chrge quits. Phys. Rev. Lett. 91, 573 (23). 53. Bertet, P., Hrmns, C. J. P.M. & Mooij, J. E. Prmetic coupling for superconducting quits. Phys. Rev. B 73, (26). 54. Rigetti, C., Blis, A. & Devoret, M. Protocol for universl gtes in optimlly ised superconducting quits. Phys. Rev. Lett. 94, 2452 (25). 55. Liu, Y.-x., Wei, L. F., Tsi, J. S. & Nori, F. Controllle coupling etween flux quits. Phys. Rev. Lett. 96, 673 (26). 56. Mkhlin, Y., Schön, G. & Shnirmn, A. Josephson-junction quits with controlled couplings. Nture 398, (1999). 57. Wei, L. F., Liu, Y.-x. & Nori, F. Quntum computtion with Josephson quits using currentised informtion us. Phys. Rev. B 71, (25). 58. Lntz, J., Wllquist, M., Shumeiko, V. S. & Wendin, G. Josephson junction quit network with current-controlled interction. Phys. Rev. B 7, 1457 (24). 59. Blis, A., Hung, R.-S., Wllrff, A., Girvin, S. M. & Schoelkopf, R. J. Cvity quntum electrodynmics for superconducting electricl circuits: n rchitecture for quntum computtion. Phys. Rev. A 69, 6232 (24). 6. Helmer, F. et l. Two-dimensionl cvity grid for sclle quntum computtion with superconducting circuits. Preprint t < (27). 61. Mjer, J. B. et l. Coupling superconducting quits vi cvity us. Nture 449, (27). 62. Sillnp, M. A., Prk, J. I. & Simmonds, R. W. Coherent quntum stte storge nd trnsfer etween two phse quits vi resonnt cvity. Nture 449, (27). 63. Fowler, A. G. et l. Long-rnge coupling nd sclle rchitecture for superconducting flux quits. Phys. Rev. B 76, (27). 64. Grjcr, M. et l. Four-quit device with mixed couplings. Phys. Rev. Lett. 96, 476 (26). 65. Devoret, M. H. et l. in Quntum Tunneling in Condensed Medi (eds Kgn, Y. & Leggett, A. J.) (Elsevier, Amsterdm, 1992). 66. Hroche, S. & Kleppner, D. Cvity quntum electrodynmics. Phys. Tody 42, (1989). 67. Wllrff, A. et l. Strong coupling of single photon to superconducting quit using circuit quntum electrodynmics. Nture 431, (24). 68. Johnsson, J. et l. Vcuum Ri oscilltions in mcroscopic superconducting quit LC oscilltor system. Phys. Rev. Lett. 96, 1276 (26). 69. Schuster, D. I. et l. Resolving photon numer sttes in superconducting circuit. Nture 445, (27). 7. Houck, A. A. et l. Generting single microwve photons in circuit. Nture 449, (27). 71. Astfiev, O. et l. Single rtificil-tom lsing. Nture 449, (27). 72. Vlenzuel, S. O. et l. Microwve-induced cooling of superconducting quit. Science 314, (26). 73. Grcjr, M. et l. Sisyphus dmping nd mplifiction y superconducting quit. Preprint t < (27). 74. Cluser, J. F., Horne, M. A., Shimony, A. & Holt, R. A. Proposed experiment to test locl hidden-vrile theories. Phys. Rev. Lett. 23, (1969). 75. Leggett, A. J. & Grg, A. Quntum mechnics versus mcroscopic relism: is the flux there when noody looks? Phys. Rev. Lett. 54, (1985). 76. Bennett, C. H. et l. Teleporting n unknown quntum stte vi dul clssicl nd Einstein Podolsky Rosen chnnels. Phys. Rev. Lett (1993). 77. Schreier, J. A. et l. Suppressing chrge noise decoherence in superconducting chrge quits. Phys. Rev. B 77, 1852 (28). 78. Duty, T., Gunnrson, D., Bldh, K. & Delsing, P. Coherent dynmics of Josephson chrge quit. Phys. Rev. B 69, 1453 (24). 79. Doln, G. J. Offset msks for liftoff photoprocessing. Appl. Phys. Lett. 31, (1977). Acknowledgements Our work is supported y the US Deprtment of Energy (Division of Mterils Sciences nd Engineering, in the Office of Bsic Energy Sciences) (J.C.), nd y the Nturl Sciences nd Engineering Reserch Council of Cnd, QuntumWorks nd EuroSQIP (F.K.W.). Author Informtion Reprints nd permissions informtion is ville t npg.nture.com/reprints. The uthors declre no competing finncil interests. Correspondence should e ddressed to the uthors (jclrke@erkeley.edu; fwilhelm@iqc.c). 142

41 NATURE Vol June 28 doi:1.138/nture7129 INSIGHT REVIEW Coherent mnipultion of single spins in semiconductors Ronld Hnson 1 & Dvid D. Awschlom 2 During the pst few yers, reserchers hve gined unprecedented control over spins in the solid stte. Wht ws considered lmost impossile decde go, in oth conceptul nd prcticl terms, is now relity: single spins cn e isolted, initilized, coherently mnipulted nd red out using oth electricl nd opticl techniques. Progress hs een mde towrds full control of the quntum sttes of single nd coupled spins in vriety of semiconductors nd nnostructures, nd towrds understnding the mechnisms through which spins lose coherence in these systems. These ilities will llow pioneering investigtions of fundmentl quntum-mechnicl processes nd provide pthwys towrds pplictions in quntum informtion processing. In the pst few decdes, the ppliction of nucler mgnetic resonnce nd electron spin resonnce to lrge spin ensemles hs yielded sustntil informtion on spin dynmics in semiconductors. Experimentl dvnces since the 199s hve llowed reserchers to increse their control over single chrges, providing pthwy for studies of single spins. Erly experiments on single spins confined in semiconductor qun tum dots highlighted the opportunity for controlling individul quntum sttes in solid. When quntum informtion processing ecme relistic prospect in the lte 199s, Dniel Loss nd Dvid DiVincenzo proposed quntum computing scheme sed on spins in quntum dots 1, nd Bruce Kne developed proposl for silicon-sed quntum computer 2. It ws pprent from these nd other theoreticl concepts tht, in future quntum computer, the spins must e initilized, mnipulted nd red out one y one 3. At out the sme time, other reserchers were independently developing toolkits of sensitive spin-mnipultion techniques to investigte fundmentl quntum-mechnicl processes in nnostructures such s decoherence on the tomic scle. Ultimtely, round the strt of this century, spintronics emerged 4, field tht seeks to encode clssicl informtion in the spin stte of electrons. Both spintronics nd quntum informtion processing hve een mjor driving forces towrds the control of single-spin systems. Here we review experimentl progress towrds full control of the quntum sttes of single nd coupled spins in different semiconductor systems. We lso discuss the mechnisms tht led to the loss of spin coherence in these systems. Single spins in semiconductors Single-spin systems in semiconductors rodly fll into two ctegories: tomic impurities nd quntum dots. Atomic impurities re routinely dded to semiconductors to control the electricl properties (doping). When the concentrtion of impurities is very low, the possiility of ddressing individul impurities rises. Atomic impurities my hve nucler spin, or they cn ct s potentil trp for electrons or holes. Often they do oth, s in the cse of phosphorus in silicon. If two or more impurities re present, or if there is comintion of impurities nd lttice defects such s vcncy, more complicted centres cn e formed tht often hve excellent properties for single-spin studies. One prime exmple is the nitrogen vcncy (N V) colour centre in dimond, which consists of sustitutionl nitrogen tom next to missing cron (the vcncy) (Fig. 1). This N V centre hs prmgnetic electron spin nd strong opticl trnsition t visile wvelength, which llows opticl imging of single spins. Quntum dots, y contrst, ehve like toms in mny wys, ut they re fricted in the lortory. By engineering the electronic nd structure, reducing the size of the semiconductor crystl in one or more dimensions, or pplying electric fields, chrge crriers cn e confined to smll region of the crystl. If the region is roughly the sme size s the wvelength of the chrge crrier, the energy levels will e quntized s in rel toms. Mny tomic properties, such s shell structure nd opticl selection rules, hve nlogues in quntum dots, giving rise to their nicknme rtificil toms 5 7. In contrst to rel toms, however, quntum dots llow flexile control over the confinement potentil nd tend to e esier to excite opticlly. Quntum dots with lrge tunnel coupling (tht is, strong overlp of their electronic wvefunctions) cn form rtificil molecules. Such covlent onding trnsforms the single-dot oritls into moleculr-like oritls tht spn oth quntum dots. As consequence, spins in neighouring coupled quntum dots overlp strongly nd will form two-prticle wvefunctions such s spin singlet nd triplet sttes 8. Quntum dots come in vrious sizes nd in rnge of mterils. Here we minly focus on the two types of quntum dot in which coherent dynmics hve een oserved t the single-spin level. In the first type, confinement is chieved through the ppliction of electric fields, nd mesurements typiclly involve the trnsport of chrge crriers through the device. Quntum dots with tunle numer of electrons re routinely fricted from two-dimensionl electron gs (2DEG) tht confines the chrge crriers to plne. Confinement in the remining two dimensions is chieved y electric fields, either through metllic surfce gtes ove the 2DEG (Fig. 1) or, if smll pillr hs een prepred y etching, from the edges. Gllium rsenide (GAs) hs een the mteril of choice for mny yers for these devices, s the high level of control hs led to high-purity, flexile devices. More recently, motivted y the detrimentl effect of lttice nucler spins on the coherence times of electron spins, quntum dots hve lso een studied in mterils such s silicon nd cron tht cn e isotopiclly purified to otin lttice tht is free of nucler spins. 1 Kvli Institute of Nnoscience Delft, Delft University of Technology, P.O. Box 546, 26 GA Delft, The Netherlnds. 2 Cliforni Nnosystems Institute, University of Cliforni, Snt Brr, Cliforni 9316, USA. 143

42 INSIGHT REVIEW NATURE Vol June 28 The second type of quntum dot is defined in the semiconductor during the growth of the crystl. For instnce, smll islnds of semiconductor mteril such s indium gllium rsenide (InGAs) cn e creted within mtrix of semiconductor with lrger ndgp, such s GAs (Fig. 1). The difference in ndgp confines chrge crriers to the islnd. Once the mteril is grown, the ndgp profile is fixed. However, chnges to the overll potentil, nd potentil grdients on top of the ndgp profile, cn e induced y electric or mgnetic fields. Another exmple of growth-defined dots is nnocrystl quntum dots, whose smll size confines chrge crriers. Doule dots cn e formed in nnocrystl dots y growing shells of different mterils round the core. Opticl trnsitions in this second type of quntum dot typiclly hve lrge oscilltor strength, nd mny studies use only opticl techniques. Recent yers hve lso seen the dvent of hyrid systems, in which oth electricl trnsport nd opticl excittion nd detection re possile 9. Experiments on single spins in quntum dots In the 199s, mesurements of electron trnsport through single quntum dots yielded informtion out spin sttes 1. The pst five yers hve seen tremendous progress towrds the control of single spins 8. Single-spin dynmics ws first studied in series of pioneering experiments 11 t the NTT Bsic Reserch Lortories in Atsugi, Jpn, in 21 tht mde use of fst voltge pulses on gte electrodes. Toshims Fujisw, Seigo Truch nd co-workers found tht if trnsition etween two sttes ws foridden y spin-selection rules, the cor responding decy time (more thn 2 μs) ws more thn four orders of mgnitude greter thn for trnsitions not involving chnge of spin (out 1 ns). In second experiment, they mde single electron oscillte coherently etween oritls in neighouring coupled dots 12. The oritl ( chrge ) coherence of this oscilltion ws found to dispper in just few nno seconds, wheres theory ws predicting coherence times of severl micro seconds for the spin degree of freedom In 24, Leo Kouwenhoven nd co-workers t the Kvli Institute of Nnoscience in Delft, the Netherlnds, comined the pulse schemes of Fujisw s group with fst chrge sensor tht could tell exctly when n electron ws entering or leving the dot. By mking the tunnelling rte of the electron from the dot dependent on its spin stte, they could determine the spin stte y mesuring the chrge on the dot over time (Fig. 2). Two vritions of this spin-to-chrge conversion were demonstrted to work in single-shot mode 16,17. Agin, relxtion times for single electron nd for two-electron spin sttes were found to e of the order of millisecond. A few yers lter, even longer electron spin relxtion times, of up to second, were found t mgnetic fields of few tesl y Mrc Kstner s group t the Msschusetts Institute of Technology in Cmridge 18. Coherent control over two-electron spin sttes Two electrons in neighouring quntum dots with significnt tunnel coupling form two-prticle spin wvefunction, which cn e spin singlet or spin triplet. The energy difference etween these sttes cn e descried s n effective exchnge splitting, J(t). Control over this exchnge splitting llows dynmicl control of the two-electron spin sttes. If two electrons with opposite spin orienttion in neighouring dots re initilly decoupled, turning on the coupling will result in precession of the two spins in the singlet triplet sis. This leds to periodic swpping of the two spin sttes t integer multiples of the time intervl π /J (where is h/2π nd h is Plnck s constnt), wheres the electrons re entngled for intermedite times 1. In fct, the stte swpping occurs for ritrry initil sttes of the two spins. This twospin control, ppropritely clled SWAP opertion, is n essentil ingredient for mny proposls for quntum computing with spins in dots If logicl quntum its (quits) re encoded in more thn one spin, control over the exchnge splitting is sufficient to uild up ny quntum gte 22. The exchnge opertion hs severl enefits: the control is fully electricl, the interction cn e turned on nd off, nd the resultnt gte opertion times cn e very short (less thn nnosecond). The first step towrds the exchnge opertion ws the oservtion y Truch s group 23 of Puli spin lockde in doule quntum dot. The presence of doule-dot singlet nd triplet sttes ecme pprent when the current ws suppressed in one is direction (Fig. 2c). It ws lter found tht this current lockde cn e lifted y fluctuting fields from the nucler spins tht cuse mixing of the singlet nd triplet spin sttes 24,25. In 25, y using the strength of the exchnge interction to control the mixing, Chrles Mrcus s group t Hrvrd University in Cmridge, Msschusetts, demonstrted coherent oscilltions of two spins 26. Although it ws not yet possile to proe ritrry input sttes, this experiment demonstrted the essence of the SWAP gte. Gte Chrge sensor Depleted region in 2DEG Quntum dots c Quntum dot Ohmic contct to 2DEG InGAs Nitrogen Al x G 1 x As vcncy GAs GAs colour centre 2DEG Nitrogen Cron-13 Cron-12 Figure 1 Single-spin systems. Studies of the coherence of single spin require system in which the spin is loclized nd isolted from environmentl disturnces. In semiconductors, such systems re either impurity toms or quntum dots, which ct s rtificil toms. In the three systems on which this rticle minly focuses, the level of experimentl control is so high tht the dynmics of single spin cn e studied nd mnipulted., A quntum dot defined in two-dimensionl electron gs (2DEG). The electrons re confined in the third dimension y electric fields from the surfce gte electrodes. Electron spins cn e mnipulted using mgnetic resonnce or comintion of electric fields nd position-dependent effective mgnetic field. Interctions etween spins in neighouring tunnel-coupled dots re medited y the exchnge interction. These quntum dots re typiclly mesured t tempertures elow 1 K., A quntum dot defined y growth. The semiconductor of the islnd hs smller ndgp thn tht of the surrounding mtrix, therey confining chrge crriers to the islnd. Spins cn e creted nd controlled opticlly. Additionl gtes cn e used to pply n electric field to the structure to chnge the numer of crriers on the quntum dot. Mesurements re typiclly crried out t round 4 K. Scle r, 5 nm. c, A nitrogen vcncy (N V) colour centre in dimond, consisting of sustitutionl nitrogen tom next to missing cron tom. The N V centre (in the negtively chrged stte) comprises six electrons tht form spin triplet in the electronic ground stte. Strong opticl trnsitions to excited sttes, in comintion with spin-selection rules, llow opticl initiliztion nd red-out of the electron spin. Coherent control of the spin hs een demonstrted with high fidelity t room temperture using mgnetic resonnce. The N V centre intercts with nery electron spins y mens of mgnetic dipolr coupling, nd through hyperfine interction with nery nucler spins. Also, non-locl coupling etween N V centres my e estlished y using the opticl trnsition; photons then ct s meditors of the interction. 144