2 Lukin, et al On the other hand, some of the above diffiulties are related to the mirosopi nature of the system and may be avoided if mesosopi system

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1 Quantum information proessing based on avity QED with mesosopi systems? Mikhail Lukin 1, Mihael Fleishhauer 2, and Ata Imamo glu 3 1 ITAMP, Harvard-Smithsonian Center for Astrophysis, Cambridge, MA Sektion Physik, Universität Münhen, D Münhen, Germany 3 Dept. of Eletr. & Comp. Eng., Univ. of California, Santa Barbara, CA Introdution Reent developments in quantum ommuniation and omputing [1 3] stimulated an intensive searh for physial systems that an be used for oherent proessing of quantum information. It is generally believed that quantum entanglement of distinguishable quantum bits (qubits) is at the heart of quantum information proessing. Signifiant efforts have been direted towards the design of elementary logi gates, whih perform ertain unitary proesses on pairs of qubits. These gates must be apable of generating speifi, in general entangled, superpositions of the two qubits and thus require a strong qubit-qubit interation. Using a sequene of single and two-bit operations, an arbitrary quantum omputation an be performed [2]. Over the past few years many systems have been identified for potential implementations of logi gates and several interesting experiments have been performed. Proposals for strong qubit-qubit interation involve e.g. the vibrational oupling of ooled trapped ions [4], near dipole-dipole or spin-spin interations suh as in nulear magneti resonane [5], ollisional interations of onfined ooled atoms [6] or radiative interations between atoms in avity QED [7]. The possibility of simple preparation and measurement of qubit states as well as their relative insensitivity to a thermal environment makes the latter shemes partiularly interesting for quantum information proessing. Most theoretial proposals on avity-qed systems fous on fundamental systems involving a small number of atoms and few photons. These systems are suffiiently simple to allow for a first-priniple desription. Their experimental implementation is however quite hallenging. For example, extremely high-q miro-avities are needed to preserve oherene during all atom-photon interations. Furthermore, single atoms have to be onfined inside the avities for a suffiiently long time. This requires developments of novel ooling and trapping tehniques, whih is in itself a fasinating diretion of urrent researh. Despite these tehnial obstales, a remarkable progress has been made in this area: quantum proessors onsisting of several oupled qubits now appear to be feasible.? This work is dediated to the memory of Professor Dan Walls

2 2 Lukin, et al On the other hand, some of the above diffiulties are related to the mirosopi nature of the system and may be avoided if mesosopi systems are used. Proposals based on mesosopi systems are also very attrative for possible large-sale implementation in the (presumably distant) future. Here olletive (i.e. many-partile) exitations an be used as qubits, but it is in general diffiult to ontrol the oupling between them. Motivated by this we here onsider an approah that ombines elements of avity QED with mesosopi systems. Speifially, we onsider an N-atom system oupled to a few-photon avity field. We investigate the onditions under whih quantum entanglement an be reated and manipulated in this mesosopi system. Although entanglement manipulation involves olletive rather than single-partile exitations, the system is still suffiiently simple to allow for a first priniple desription. The entral feature of our approah is the ability to manipulate olletive exitations of light and matter by oherent ontrol of the atom-field interation using atomi dark resonanes [8]. The present work demonstrates that the essential elements of QED-based quantum information proessing an be implemented and that some of them an be onsiderably improved in a mesosopi system. We show in partiular that (i) quantum information ontained in polarization states of single photons an be stored in olletive atomi exitations; (ii) simple two-bit operations an be performed; (iii) entanglement an easily be transfered and distributed among olletive exitations of distant atomi ensembles. 2 Colletive exitations as qubits A onvenient way of enoding quantum information in optis is via the analogy between spin-1/2 systems and polarization states of light waves. We therefore begin by assoiating qubits with polarization states of single photons, and show that the states of these qubits an be mapped onto olletive exitations of ensembles of atoms. We are here interested in single-photon exitations of avity modes desribed by a superposition of right (j1 + i)and left (j1 i) irularly polarized omponents jψ i i = ff i j1 i;+ i + fi i j1 i; i; (1) with jff i j 2 +jfi i j 2 = 1. In the following we fous on the ase that involves a pair of suh single-photons states, i.e. i =1; 2. For simpliity let us assume that the two photons oupy different frequeny bands and hene are assoiated with different avity modes. In order to manipulate quantum information stored in suh qubits we onsider optial avities filled with N idential multilevel atoms. The frequeny of a partiular pair of transitions is assumed to be lose to resonane frequenies of the avity. The orresponding oupling strengths of the atoms

3 Cavity QED with mesosopi systems 3 to the two avity modes^a 1+ and ^a 1 are assumed to be equal and are denoted by g (see Fig.1a). In addition some time-dependent lassial fields with Rabi-frequenies Ω 1± (t) ouple the lower (meta-stable) states j ±1 i of these atoms to the exited states ja ±1 i as shown. The exited states deay with (equal) deay rate fl and all atoms are initially prepared in a ertain hyperfine sub-level, i.e. in a pure state. (a) a -1 a g g +1 Ω Ω b (b) Ω Fig. 1. (a) Shemati of the system for storing photon qubits in olletive atomi exitations. (b) Quantum ommuniation system based on photon trapping and release. g Φ g Ω The basi Hamiltonian of the avity + atom system an be written in i=1 ^ffi a j; j as terms of olletive operators ^±aj;b = P N i=1 ^ffi a j;b and ^±aj; j = P N ^V = X j μhg^a j ^±aj;b +μhω j (t) ^±aj; j +h::; (2) where j =1±, and^ff i μν = jμi ii hνj is the flip operator of the ith atom between states jμi and jνi. Here and below we work in a frame rotating with the optial frequenies. Of speial interest are ertain superposition states of light and olletive states of matter that do not interat with the optial fields. These soalled dark-states [8] orrespond to elementary exitations of bosoni quasipartiles, so-alled dark-state polaritons [9]. They are defined by the following anonial transformation ^d j = os j ^a j sin j 1 p N ^ff bj ; tan j (t) =g p N=Ω j (t) (3) fulfill Bose ommu- In the limit of small exitation the operators ^d j and ^d y j tation relations. The ^d y j 's reate a family of dark states whih do not have an exited-state omponent and are deoupled from both optial fields: 1 jd ji:: ;n j ;k i :::i = p ^d y n!k!::: j n ^d y k:::j0ijbi1 i :::jbi N ; (4) ^V jd ji:: ;n j ;k i :::i = 0. These states are omposed of avity field states and symmetri Dike-like atomi states j n j k i ::i ontaining n atoms in level j ji,

4 4 Lukin, et al k atoms in level j i i et, and all others in the ground state jbi: jbi jbi 1 :::jbi N ; jji j 2 j i NX l6=m=1 NX l=1 1 p N jbi 1 :::j j i l :::jbi N ; (5) 1 p 2N(N 1) jbi 1 :::j j i l :::j j i m ::::jbi N ; et: (6) We here assumed that the number of atoms is muh larger than the number of photons in the light field. The essene of the present approah is that a quantum bit stored in photo states an be transfered to olletive atomi exitations (and vie versa) by adiabati passage in dark-polariton states. Speifially single-mode dark states (4) have the following asymptoti behavior in the two limiting ases: jd j ;n j i! jn j ijbi; when Ω fl g p N; (7) jd j ;n j i! j0ij n j i; when Ω fi g p N: (8) It is most important that by varying the strength of the driving field Ω(t), the state of the ombined atom+avity system an be hanged from avitylike (in whih exitation is mostly of photon nature) to atom-like (in whih exitations are shared among the atoms). In the latter ase the lifetime of exitations will not be sensitivetoavity deay; it will be limited solely bythe deay of the meta-stable atomi states. In this proess qubit states enoded in the photon field are mapped onto symmetri olletive exitations of atomi ensembles. Sine all dark states are orthogonal to eah other, opying of all states an proeed in parallel. It is known that adiabati following takes plae in the stimulated Raman proess onsidered here, if the harateristi time sale T exeeds the ratio of the optial deay rate fl to the square of the harateristi Rabi-frequeny. For the present system this ondition translates into g 2 N=flT fl 1: One reognizes that using a mesosopi system with N fl 1 onsiderably improves the adibatiity ondition as ompared to the single-atom ase. 3 Quantum entanglement of olletive exitations A pair of qubits stored in olletive exitations an be entangled using a number of different proesses. Here we onsider the resonantly enhaned Kerr effet [10] in ombination with a avity-qed setup to onstrut an elementary logi gate. The resonantly enhaned Kerr interation in a 4-level onfiguration is the basis for the so-alled photon blokade" in a avity onfiguration [11] and results in extremely strong photon-photon interations of pulses [12]. To implement a two-bit gate we onsider a pair of photons resonant with different transitions of the same multi-state atom. We use a level onfiguration and optial fields as indiated in Fig.2.

5 Cavity QED with mesosopi systems 5 (a) (b) b b 2+ Fig. 2. Shemati of the system for an entanglement operation in atomi Rb. Only the oupling to the relevant transitions is shown. In order to entangle qubit states the following sequene of operations an be used. In the first step [13], the photon state j1 1 i = ff 1 j1 1+ i + fi 1 j1 1 i is transfered to olletive atomi states omposed of j 1± i with the adiabati tehnique desribed above. This operation orresponds to: ff 1 j1 1+ i + fi 1 j1 1 i jbi!j0 1 i ff 1 j 1+ i + fi 1 j 1 i : (9) In the next step, the state of the seond photon j1 2 i is mapped onto the different atomi sub-levels j 2± i: ff 2 j1 2+ i + fi 2 j1 2 i ff 1 j 1+ i + fi 1 j 1 i! j0 2 i (10) ff 1 ff 2 j i + ff 1 fi 2 j 1+ 2 i + fi 1 ff 2 j 1 2+ i + fi 1 fi 2 j 1 2 i : We now want to generate a onditional phase shift on only one of the olletive states, say j 1 2+ i.for this we first apply a weak magneti field in suh away that the transition j 1 i!jbi beomes lose to the frequeny of some avity mode (different from the one used for trapping of the photon j1 1 i). Note that this mode also ouples off-resonantly (with detuning ) the transition j 2+ i! je 2 ;M F = 0i, where jei denotes the exited state. The shift of the atomi energy levels will also result in undesired different phase shifts for the omponents of the olletive atomi states. These phase shifts an be ompensated however (e.g. by reversing the diretion of the field for an appropriate time) and shall not be onsidered here. By applying a lassial field of appropriate frequeny we an transfer one omponent of the olletive statej 1 i bak into the photoni mode: j0ij 1 2+ i! j1 1 ij 2+ i; j0ij 1 2 i! j1 1 ij 2 i: (11) At this point the energies of the states j1 1 ij 2+ i and j1 1 ij 2 i differ in a nontrivial way. Namely the state j1 1 ij 2+ i exhibits an AC-Stark shift ffi = g 2 =, sine it is oupled by the off-resonant avity modeontaining one photon. In order to avoid deoherene assoiated with two-photon absorption,

6 6 Lukin, et al should exeed the optial deayratefl. By simply letting the system evolve for a time fi a onditional phase Φ = ffifi is aumulated. By transferring the photoni omponents j1 1 i bak to the atoms and reversing the magneti field for a time appropriate to eliminate the single-bit phase shifts, the following state is obtained: ff 1 ff 2 j i + ff 1 fi 2 j 1+ 2 i +e iφ fi 1 ff 2 j 1 2+ i + fi 1 fi 2 j 1 2 i: (12) In the language of quantum information, this operation orresponds to a universal logi gate (a so-alled phase gate) [1]. It is lear that by seleting a proper value of the onditional phase Φ and by performing independent single bit rotations, arbitrary entangled states of two qubits an be generated. This an be ahieved however only if the system preserves oherene during the harateristi time required to aumulate a large phase shift. Hene, in the present approah g 2 fi= ο 1 is required to ahieve arbitrary entanglement of olletive states. Thus while transfer operations as disussed in the previous setion do not require a strong-oupling regime, two-bit operations still do. 4 Effets of deoherene In this setion we disuss the effet of deoherene on the manipulation of olletive atomi exitations. In general, deoherene mehanisms depend on the partiular implementation. In order to be speifi we onsider an ensemble of laser-ooled Rb atoms in a magneto-opti trap (MOT). The main soures of deoherene and dissipation are then (i) spontaneous emission from the exited states (with the rate fl), (ii) the finite lifetimes of hyperfine and Zeeman oherenes within the ground state (orresponding deay rateisfl g ) and, (iii) the photon deay of the optial avity with rate fl. For the present problem dephasing of the olletive states is of interest. One finds that the states orresponding to single olletive exitations are dephased at the same rate as the average oherenes orresponding to individual atoms. For instane d dt hbjρja ii = d 1 X N p dt N μ=1hbjρja μ i i = flhbjρja ii: (13) By the same argument, oherenes between hyperfine and Zeeman sub-levels deay at a rate fl g. The states ontaining a single photon in aavity mode will deay with an additional rate fl. In the following we assume that fl g is small on the time sales of interest and an be negleted. Both proesses onsidered in the previous setions are affeted by deoherene, but in a different way. In the ase of quantum state transfer, deoherene due to spontaneous emission an be avoided if the transfer time T is suffiiently long suh that the adiabati following ondition is fulfilled. However, in order to avoid deoherene due to avity deay the transfer time T should

7 Cavity QED with mesosopi systems 7 be short ompared to fl. Hene, ideal quantum state transfer between avity mode and olletive exitations is only possible if g 2 N fl fl fl: (14) In the ase of two-bit operations, spontaneous emission auses two-photon absorption at a rate ο g 2 = 2fl. Here, two-photon absorption an be avoided when the detuning (see Fig.2) is suffiiently large fl fl. At the same time, the entanglement generation should be fast ompared to the avity deay fifl fi 1. Hene, in order to aumulate a large onditional phase without dissipation it is neessary that g 2 fl fl fl fl fl: (15) The main onlusion of this setion is that in priniple inreasing the number of atoms does not make it harder to reate quantum entanglement. Other operations suh as the reliable quantum state transfer between light and matter beome muh easier. The reason for this behavior is that the basi deoherene mehanisms are not enhaned as the number of atoms is inreased. At the same time the oupling of the avity mode to the ground state is enhaned by a fator p N. We note that in pratie deoherene mehanisms exist that do sale with the number of atoms. For instane, off-resonant sattering of the external oherent fields on the transition from the ground jbi to the exited states je i i will result in dephasing of the olletive states whih is learly enhaned: ~fl = NflΩ 2 = ~ 2. Here ~ is the (large) detuning of the oupling field from the jbi!je i i transition frequeny. Therefore, in experiments extra are should be taken to avoid these deoherene mehanisms. 5 Entanglement distribution One of the most intriguing aspets of quantum information is the use of entanglement as information resoure for purposes suh as super-dense information transfer [14], quantum teleportation [15] and seure ommuniation [16]. In this setion we show that the quantum state of olletive atomi exitations inluding possible entanglements an be transferred form a given avity system to other systems under muh improved onditions as ompared to single-atom QED systems. The tehnique is based on the possibility to map quantum orrelations from traveling-wave light fields to olletive atomi states and vie versa with nearly 100% effiieny [17,18]. The basi mehanism is again the adiabati proedure disussed in setion 2 with the additional ingredient of a oupling to a ontinuum of free-spae modes. We will outline the basi features for a single traveling-wave quantum field. In a suitable system, this operation an proeed in parallel for several field omponents and the orresponding generalization is straightforward.

8 8 Lukin, et al We onsider a avity withn idential multi-level atoms as before. In addition we inlude the oupling of the avity mode to a 1-D ontinuum of free-spae modes with reation operators b y k desribed by the effetive Hamiltonian ^V = μh Pk»^ay^b k +h::;» being the oupling onstant. We assume that initially all atoms are in the ground state jbi and that there is no photon in the avity. Thus the ombined avity-atom system is initially in the dark-state jd; 0i (see eq.(4)). The initial state of the free field is taken to be jψ in i = P k ο1 k j1 ki + P k;m ο2 k;m j1 k1 m i + :::.Itisonvenient towork with orrelation amplitudes, i.e. Fourier transforms of ο j k:::l : Φ j (t 1 :::t j )=h0j ^E(t1 )::: ^E(tj )jψi; (16) where ^E(t) = L=(2ß) R d!k exp(i! k t)^b k, and L is the quantization length. E.g. Φ 1 desribes the envelope of a single-photon wave paket, Φ 2 is the oinidene amplitude et. We now onsider a broad lass of pulsed fields that are haraterized by a single ommon envelope funtion h(t) suh that Φ j (t 1 ;t 2 ;:::t j )=ff j p j! h(t1 )h(t 2 ):::h(t j ): (17) Any pure state or mixture of suh pulses an be desribed by a single-mode density matrix ρ nm = ff Λ nff m. The orresponding mode funtion is a superposition of plane waves proportional to h(z=) = R d! k ο k e i! kz=. Due to the interation of the avity mode with the environment, the dark states of the avity + atoms system are oupled to the ontinuum states. When only single-photon pulses are involved the evolution equations of the orresponding state amplitudes are [17]: X _D 1 (t) =i» os (t) ο k (t); k (18) _ο k (t) = i k ο k (t)+i» os (t) D(t): (19) D 1 (t) denotes here the amplitude of the dark-state jd; 1i and (t) 1 (t) is defined in eq.(3). We proeed by formally integrating Eq.(19), substituting the result into Eq.(18) and invoking a Markov approximation. Assuming that no photons arrive to the avity before t 0 we find for the dark state amplitude D 1 (t) = iff 1 D(t) with D(t) = r fl L Z t t 0 dfi os (fi) h(fi) exp ρ fl 2 Z t fi dfi 0 os 2 (fi 0 ) ff : (20) Here we have introdued the empty-avity deay ratefl =» 2 L=. Substituting this result bak into eq.(19) one finds that the outgoing field is desribed by the ommon envelope funtion h out (t) =h(t) p fl L= D(t). In order to trap photons we require that the envelope of the outgoing field and its first derivative vanish identially. I.e. h out (t) = _ h out (t) = 0. Differentiating the above relation for h out (t) yields d dt ln os (t)+ d dt ln h(t) =fl 2 os2 (t): (21)

9 Cavity QED with mesosopi systems 9 If Ω(t) is hosen suh that (t) obeys this equation with the asymptoti ondition os! 0 the output field remains zero and the inoming light pulse is ompletely transferred to the atomi system. The above ondition orresponds to a quantum or dynamial impedane mathing [17]. The term on the r.h.s. of Eq.(21) is the effetive avity deay rate redued due to intraavity eletromagnetially indued transpareny (EIT) [19]. The first term on the l.h.s. desribes internal losses" due to oherent Raman adiabati passage and the seond term is due to the timedependene of the input field. As in the ase of lassial impedane mathing [20], Eq. (21) reflets the ondition for omplete destrutive interferene resulting in a vanishing outgoing wave. Solving Eq.(21) yields os 2 (t) = h 2 (t) fl R t 1 dfih2 (fi) ; (22) whih orresponds to D(t! +1)! 1. Hene, by suitable variation of the lassial driving field any single-photon pulse an be trapped ideally, if its pulse length is longer than the bare-avity deay time. Generalizations of the above onsiderations to multi-photon states an proeed along the same lines, but involve more tedious algebra. In partiular, for the two-photon states one finds D 2 (t) = ff 2 D(t) 2, and in general D k (t) =( i) k ff k d(t) k (23) an be proved. Under onditions of quantum impedane mathing D k (t! 1)! ( i) k ff k for arbitrary k. Hene pulsed fields in a generalized single mode with arbitrary quantum state an be mapped onto the atomi ensemble. Releasing the stored quantum state into a pulse of desired shape an be aomplished in a straightforward way. A simple reversal of the time dependene of the ontrol field at a later time t d leads to a perfet mirror-image of the initial pulse. This an be verified diretly from Eqs. (20). Before onluding we note that the quantum transfer protool desribed here is based solely on the adiabati rotation of the dark state desribed in Setion 2. Hene, this operation an be nearly ideal whenever inequality (14) is fulfilled [17]. Therefore, perfet quantum ommuniation an be ahieved in the present system without invoking the strong oupling regime of avity QED. 6 Conlusions In onlusion, we have shown that quantum information stored in olletive exitations of an N-atom system and an be oherently proessed using avity QED tehniques. We showed that ertain network operations suh as the transfer of exitation between atomi and photoni degrees of freedom and entanglement distribution an be performed without invoking the strong

10 10 Lukin, et al oupling ondition of avity QED. However other operations, suh as two-bit rotations resulting in quantum entanglement still require a strong oupling. Studies of possible ways to alleviate these requirements, and to avoid the strong oupling regime altogether are urrently under way. This inludes, for instane, resonant nonlinearities in a traveling wave geometry, so-alled photon-exhange interations or old ollisions. This work was supported by the National Siene Foundation via the grant to the Institute for Theoretial Atomi and Moleular Physis. We thank Susanne Yelin for stimulating disussions and ollaboration on a related projet. One of us (ML) espeially thanks her and little Theodor for their patiene that allowed to omplete this ontribution. Referenes 1. R. P. Feynman, Int. J. Theor. Phys. 21, 467 (1982); D. Deutsh, Pro. R. So. London A 425, 73 (1989). 2. see e.g.: D. P. DiVinenzo, Siene 270, 255 (1995); C. H. Bennett, Phys. Today 48, No. 10, 24 (1995); A. Ekert and R. Josza, Rev. Mod. Phys. 68, 733 (1996); A. Steane Rep. Prog. Phys. bf 61, 117 (1998). 3. D. Deutsh, Pro. R. So. London A 400, 97 (1985). 4. J. I. Cira and P. Zoller, Phys. Rev. Lett. 74, 4091 (1995). 5. N. A. Gershenfeld and I. L. Chuang, Siene 275, 350 (1997). 6. D. Jaksh et al., Phys. Rev. Lett. 82, 1975 (1999). 7. T. Pellizzari et al., Phys. Rev. Lett. 75, 3788 (1995). 8. see e.g.: E. Arimondo, Progr. in Optis 35, 259 (1996); 9. M. Fleishhauer and M. D. Lukin, (preprint quant-ph/ ) 10. H. Shmidt and A. Imamo glu, Opt. Lett. 21, 1936 (1996). 11. A. Imamo glu et al., Phys. Rev. Lett. 79, 1467 (1997); P. Grangier, D. F. Walls, and K. Gheri, Phys. Rev. Lett. 81, 2833 (1998). 12. M. D. Lukin and A. Imamo glu, Phys. Rev. Lett. 84, 1419 (2000). 13. Note that the order in whih the qubits are trapped is in priniple important. Corresponding orretion an be disregarded however when N fl C. H. Bennet and S. J. Wiesner, Phys. Rev. Lett. 69, 2881 (1992); K. Mattle et al., Phys. Rev. Lett. 76, 4656 (1996). 15. C. H. Bennett et al., Phys. Rev. Lett. 70, 1895 (1990); B. Bouwmeester et al., Nature 390, 575 (1997); D. Boshi et al., Phys. Rev. Lett. 80, 1121 (1998); A. Furusawa et al. Siene 282, 706 (1998). 16. C. H. Bennett and G. Brassard, Pro. of IEEE Int. Conf. on Comp. Systems and Signal Proessing, Banglore India (IEEE, New York 1984); A. K. Ekert, Phys. Rev. Lett. 67, 661 (1991). 17. M. Fleishhauer, S. F. Yelin, and M. D. Lukin, Opt. Comm. (2000) in press 18. M. D. Lukin, S. F. Yelin, and M. Fleishhauer, Phys. Rev. Lett. (2000) (preprint quant-ph/ ). 19. M. D. Lukin et al., Opt. Lett. 23, 295 (1998). 20. A. Siegmann, Lasers, (University Siene Books, Mill Valley CA, 1986).