Atomic Three-Body Loss as a Dynamical Three-Body Interaction

Transcription

1 Atomc Three-Body Loss as a Dynamcal Three-Body Interacton The MIT Faculty has made ths artcle openly avalable. Please share how ths access benefts you. Your story matters. Ctaton As Publshed Publsher Daley, A. J. et al. Atomc Three-Body Loss as a Dynamcal Three-Body Interacton. Physcal Revew Letters (2009): The Amercan Physcal Socety. Amercan Physcal Socety Verson Fnal publshed verson Accessed Fr May 04 08:09:04 EDT 2018 Ctable Lnk Terms of Use Detaled Terms Artcle s made avalable n accordance wth the publsher's polcy and may be subject to US copyrght law. Please refer to the publsher's ste for terms of use.

2 week endng Atomc Three-Body Loss as a Dynamcal Three-Body Interacton A. J. Daley, 1 J. M. Taylor, 2 S. Dehl, 1 M. Baranov, 1 and P. Zoller 1 1 Insttute for Theoretcal Physcs, Unversty of Innsbruck, A-6020 Innsbruck, Austra and Insttute for Quantum Optcs and Quantum Informaton of the Austran Academy of Scences, A-6020 Innsbruck, Austra 2 Department of Physcs, Massachusetts Insttute of Technology, Buldng 6C-411, Cambrdge, Massachusetts 02139, USA (Receved 28 October 2008; publshed 30 January 2009) We dscuss how large three-body loss of atoms n an optcal lattce can gve rse to effectve hard-core three-body nteractons. For bosons, n addton to the usual atomc superflud, a dmer superflud can then be observed for attractve two-body nteractons. The nonequlbrum dynamcs of preparaton and stablty of these phases are studed n 1D by combnng tme-dependent densty matrx renormalzaton group technques wth a quantum trajectores method. DOI: /PhysRevLett PACS numbers: Lm, p Cold atomc gases n optcal lattces have proven a test bed for understandng novel quantum phases [1] and nonequlbrum many-body dynamcs [2,3]. Recently, Syassen et al. [4] showed that a strong two-body loss process for molecules n an optcal lattce [1] could produce an effectve, elastc hard-core repulson and thus a Tonks gas [4,5]. Ths s related to the quantum Zeno effect: a large loss dynamcally suppresses processes creatng two-body occupaton on a partcular ste. Whlst elastc two-body nteractons occur n many systems, regmes where elastc three-body nteractons domnate are rare n nature. Here we dscuss how the ubqutous, though normally undesrable three-body losses of atomc physcs experments can nduce effectve three-body nteractons. These are assocated wth nterestng quantum phases, ncludng Pfaffan states [6], and could be used to stablze three-component Ferm mxtures [7], assstng n the producton of a color superflud state [8]. We nvestgate Bosons n an optcal lattce, where a three-body hard-core constrant stablzes the system wth attractve two-body nteractons, and a dmer superflud phase emerges. We focus on the dynamcs of ths ntrnscally tme-dependent system, both testng the hard-core constrant for fnte loss rates, and studyng nonequlbrum propertes ncludng decay. In one dmenson, the exact evoluton s computed by combnng tmedependent densty matrx renormalzaton group methods (t-dmrg) [2,3] wth a quantum trajectores approach from quantum optcs [9]. Three-body recombnaton [10] n an optcal lattce corresponds to decay nto the contnuum of unbound states, and thus loss from the lattce. Ths can be descrbed by a master equaton n the Markov approxmaton [9], whch for atoms n the lowest band of an optcal lattce can be projected onto the correspondng bass of Wanner functons [1,5], assocated wth bosonc annhlaton operators b on ste. We can separate the master equaton nto terms whch conserve partcle number, correspondng to an effectve Hamltonan H eff, and terms whch remove three partcles on a ste: _ ðnþ ¼ ðh eff ðnþ ðnþ H y eff Þþ X 3 2b 3 12 ðnþ3þ ðb y Þ3 ; where ðnþ denotes the system densty operator wth n atoms and ^n ¼ b y b. The domnant loss term s on-ste three-body decay [11] and 3 s the correspondng rate. The effectve Hamltonan s H eff ¼ H X 3 ðb y 12 Þ3 b 3 ; (1) H ¼ J X b y b j þ U X ^n 2 ð ^n 1Þþ X " ^n ; (2) h;j wth J the nearest neghbor tunnelng ampltude, U the elastc two-body nteracton, and " the local potental. The Hamltonan s vald n the lmt where J, ", Un,! wth! the band gap and n the mean densty. In an experment these parameters, n partcular U, may be tuned whle 3 remans constant and large [12]. In Fg. 1(c) we show example values of 3, U, and J usng numbers for Cesum as a functon of lattce depth. If we begn n a pure state wth N partcles, then loss processes lead to heatng, n that they produce a mxed state of dfferent partcle numbers. Wthn a fxed partcle number sector, the dynamcs are descrbed by H eff. Threebody nteractons emerge most clearly n the lmt of rapd decay: 3 J, U,. If we defne the projector P onto the subspace of states wth at most two atoms per ste and Q ¼ 1 P, then n second order perturbaton theory we obtan the effectve model Heff P 2 PHP PHQHP ¼ PHP 6J2 P X c y j 3 c jp; 3 j (3) where c j ¼ðb 2 j = p ffffff P 2 Þ k2n j b k, and N j denotes the set of nearest neghbors of ste j. The term PHP descrbes the Hubbard dynamcs, Eq. (2), supplemented by the hard-core =09=102(4)=040402(4) Ó 2009 The Amercan Physcal Socety

3 week endng FIG. 1 (color onlne). (a) Bosons n an optcal lattce n the presence of three-body loss at a rate 3. (b) Example model parameters estmated for Cs at a magnetc feld of 15 Gauss (where the scatterng length s 20a 0 and the recombnaton length 500a 0 [10] where a 0 s the Bohr radus) as a functon of lattce depth V 0, showng 3 (sold lne), U (dashed), and J (dotted). Values of 3 are obtaned by ntegratng the measured three-body recombnaton rates n free space over a state wth three partcles n a sngle Wanner functon. (c) The probablty that at least one loss event has occurred at tme tj ¼ 2, begnnng wth a sngle partcle on each of 10 stes, and U=J ¼ 3 (sold lne), 5 (dashed), and 10 (dotted), computed usng t-dmrg methods (see text for detals). constrant ðb y Þ3 ¼ 0. Furthermore, the effectve loss rates decrease as J 2 = 3 [13]. Thus, we see the clear emergence of a three-body hardcore constrant n the lmt 3 =J 1. We can study the physcs of the projected model PHP to obtan a qualtatve understandng of the quantum phases assocated wth the projecton. However, the resdual loss processes make ths system ntrnscally tme dependent, and can gve rse to heatng. We therefore study the full nonequlbrum dynamcs, by combnng t-dmrg methods [2,3] wth an expresson of the master equaton as an average over quantum trajectores [9]. Each stochastc trajectory begns from an ntal pure state (sampled from the ntal densty matrx), and can be nterpreted as descrbng a sngle expermental run, n whch losses occurred at partcular tmes t n and on stes n. The evoluton s descrbed by the non-hermtan H eff, except for tmes t n, where losses (or quantum jumps) occur, expectaton values from the master equaton. The latter s performed by stochastc average over both ntal states and over jump events, whch converges rapdly as the number of trajectores s ncreased. The need to smulate many trajectores for convergence s offset by the effcency of smulatng states rather than densty matrces [14], and we can also make use of exstng optmzatons for conserved quanttes. Despte the applcaton of local jump operators, we fnd the evoluton qute effcent, especally for small numbers of jumps [15]. As an example of the suppresson of loss, we consder preparng a homogenous ntal state at unt fllng n a deep optcal lattce where U=J!1. At tme t ¼ 0 we suddenly ramp the lattce to a fnte depth, and observe the probablty p that a sngle three-body loss event has occurred as a functon of tme. In Fg. 1(c) we plot ths probablty for dfferent U=J as a functon of 3 =J. We see a clear suppresson of loss rates for large 3 =J, and also a substantal decrease for larger U=J, resultng from the decreased ampltude for doubly occuped stes. In the lmt of large 3, t s nstructve to study the equlbrum phase dagram of the projected Hamltonan PHP. For U=J > 0, we observe the well-known Mott nsulator (MI) and atomc superflud phases of the Bose- Hubbard model. However, the three-body hard-core condton wll also stablze the system for U=J < 0, where we fnd a dmer superflud phase [see Fg. 2(a)]. Ths s characterzed by the vanshng of the order parameter sgnallng superfludty of sngle atoms (ASF) (hb ¼0), whle d dt jc ðtþ ¼ H effjc ðtþ; jc ðt þ n Þ ¼ C n jc ðt n Þ jjc n jc ðt n Þjj ; where the jump operator C ¼ b 3 corresponds to threebody loss on ste. In stochastc smulaton of the master equaton, the tmes t n are ponts where the norm of the state falls below a randomly chosen threshold. At these tmes, a random jump operator s selected accordng to the probabltes p n /hcðt n ÞjC y C jc ðt n Þ and appled. In ths way we can both nvestgate ndvdual trajectores and compute (4) FIG. 2 (color onlne). Equlbrum analyss of the projected Bose-Hubbard model PHP. (a) Mean-feld phase dagram as a functon of U=ðJzÞ and densty, n. (b,c) Magntude of offdagonal elements of (b) the sngle partcle densty matrx jsð; jþj ¼ jhb y b jj and (c) the dmer densty matrx, jdð; jþj ¼ jhb y by b jb j j, as a functon of j jj, for U=J ¼ 10 (thn sold lne), 5 (dotted), 0 (dot-dash), 5 (dashed), and 10 (thck sold lne). These results are computed for 20 partcles on 20 lattce stes n one dmenson, wth box boundary condtons and þ j ¼ 21, usng magnary tme evoluton n t-dmrg, and plotted on a logarthmc scale.

4 week endng dmer superfludty (DSF) order parameter perssts (hb 2 Þ 0). The superflud regmes are connected va a quantum phase transton assocated wth the spontaneous breakng of a dscrete Z 2 symmetry, remnscent of an Isng transton [16]: the DSF order parameter transforms wth the double phase exp2 compared to the ASF order parameter exp. Consequently, the symmetry! þ exhbted by the DSF order parameter s broken when reachng the ASF phase. We can obtan a qualtatve mean-feld pcture usng a homogeneous Gutzwller ansatz wavefuncton, gven for the projected Hlbert space by j ¼ Q j, where j ¼ r 0 e 0 j0 þr 1 e 1 j1 þr 2 e 2 j2. Normalzaton mples P r 2 ¼ 1, whle the fllng s n ¼ r 2 1 þ 2r To examne the phases, we fnd the energy pffffff E=M d ¼ h jhj, Eðr ; Þ¼Ur 2 2 Jzr2 1 ½r2 0 þ2 2 r2 r 0 cosþ 2r 2 2 Š, where ¼ 2 þ and M d s the number of lattce stes. For any r the energy s mnmzed for an nteger multple of 2.Forr 1, r 0 Þ 0 placng us n the ASF phase wth jhb j 2 ¼ r 2 1 ðr p 0 þ ffffff 2 r2 Þ 2 Þ 0, the phaselockng expresson contrbutes a source term lnear n r 2 to the energy, and consequently the mnmum of the energy cannot be located at r 2 ¼ 0. Thus, a fnte atomc condensate always mples a dmer component jhb 2 j2 ¼ 2ðr 0 r 2 Þ 2, though the reverse s not true. At fxed n, the energy s a functon, e.g., of r 1 alone. For the second order transton found wthn our mean-feld theory, an nstablty for atomc superfludty s ndcated by ts mass term crossng 2 Eðr 1 ¼ 0Þ=@r Ths leads to a crtcal nteracton strength for the ASF-DSF p transton, U c =ðjzþ ¼ 2½1þn=2 þ 2 ffffffffffffffffffffffffffffffffffffffffffffff nð1 n=2þš. Wthn the DSF phase the order parameter obeys jhb 2 j2 ¼ nð1 n=2þ ndependent of the nteracton strength. For n! 2, we approach a MI state n a second order transton. At n ¼ 1 we fnd that the ASF-DSF transton takes place at the same couplng strength as the ASF-MI transton, but wth the opposte sgn. The complete mean-feld phase dagram n the plane of densty and nteracton strength s plotted n Fg. 2(a). From the last term n Eq. (3), we can estmate the ntal loss rate from the ground state Gutzwller wave functon. We obtan the rate eff ¼ 3J 2 z= 3 M d ðh ^n 2 nþ ðn þjhb j 2 Þ, whch s zero n the MI, and / n 2 for the DSF, eff ¼ 3J 2 zm d n 2 = 3. In the DSF phase, the crtcal temperature T c / n 2=3 at low denstes, and the energy densty deposted by a sngle loss, E loss ¼ðzþ 1ÞjUjn=ð2M d Þ. The number of ndependent loss events needed to melt the DSF s then proportonal to T c =E loss, and the meltng tme strongly decreases for ncreasng densty, proportonal to 3 =½jUjðz þ 1ÞJ 2 zn 7=3 Š. These qualtatve features are reproduced n one dmenson, as supported by numercal calculaton of the ground state for PHP. In Fgs. 2(b) and 2(c) we show the charactersaton of the crossover between the ASF and DSF regmes n one dmenson va the off-dagonal elements of the sngle partcle densty matrx, Sð; jþ ¼hb y b j, and the dmer densty matrx Dð; jþ ¼hb y by b jb j. In the MI regme, the off-dagonal elements of Sð; jþ and Dð; jþ decay exponentally. As we enter the superflud regme, quas-long-range order s vsble n the polynomal decay (lnear on the logarthmc scale). As U=J s made more negatve, we see a return to exponental decay for the offdagonal elements of Sð; jþ, but the off-dagonal elements of Dð; jþ stll decay polynomally and, ndeed, ncrease n magntude. Ths characterzes the DSF regme n one dmenson. Here, the transton to the DSF and MI regmes occurs at much smaller ju=jj than n hgher dmensons, but these two transtons agan occur at smlar ju=jj for n ¼ 1. A DSF could be prepared va an adabatc ramp begnnng from states wth very small ampltude of three-body occupaton. We study two such scenaros as llustrated n Fg. 3(a): () Begnnng from a MI, and rampng from U=J ¼ 30 to U=J ¼ 8 to produce a DSF (whch s ntutve, but assocated wth large probablty of decay); or () applyng a superlattce and begnnng from a MI wth two partcles per ste n the lowest wells, then swtchng the nteracton rapdly to U=J ¼ 8 on a tme scale much faster than tunnelng between the lowest wells and rampng down the superlattce. In each case, we compute dynamcs for 3 =J ¼ 250, rampng parameters suffcently FIG. 3 (color onlne). Dynamcs of adabatc ramps nto a dmer superflud regme. (a) We begn wth () a Mott-nsulator state (rampng U=J), and () a state wth preprepared dmers n a superlattce (removng the superlattce). (b)-(c) The sum of knetc (E K ) and nteracton (E I ) energy and (nset) partcle number as a functon of tme for two example trajectores, one wth no loss events (dashed lnes) and one wth several loss events (sold lnes). Here, (b) shows a ramp from U=J ¼ 30 to U=J ¼ 8, wth UðtÞ ¼J=ð100 þ 3tJÞþ, wth and ramp parameters, and (c) shows a ramp wth a superlattce potental, " l ¼ V 0 cosð2l=3þ, where V 0 30J expð 0:1tJÞ, adjusted so that V 0 ðtj ¼ 100Þ ¼0, wth fxed U=J ¼ 8. In each case, 3 ¼ 250J. For (b), we use 20 atoms on 20 lattce stes, for (c), 14 atoms on 23 lattce stes. (d) Plot showng the probablty that no loss event has occurred after tme t for the ramps n (b) (dashed lne) and (c) (sold).

5 week endng equatons. Open questons regardng the nature of the ASF- DSF phase transton wll be addressed wthn a quantum feld theoretcal treatment [17]. We thank L.-M. Duan for dscussons. Work n Innsbruck s supported by the Austran Scence Foundaton (FWF) through SFB 15 and project I118_N16 (EuroQUAM_DQS), and by the DARPA OLE network. J. M. T. s supported by the Pappalardo grant program at MIT. FIG. 4 (color onlne). Comparson of (a) lossless and (b) lossy trajectores from Fg. 3(c). We show the mean densty hn as a functon of poston and tme, and magntude of the dmer correlaton functon jdð; jþj ( Þ j) at the end of the ramp. slowly that wthout loss, the ground state wll be reached wth mnmal presence of excted states. In Fgs. 3(b) and 3(c) we show the tme dependence of the sum of knetc and nteracton energes, and of the total partcle number, for example, trajectores. For each ramp type, we choose a lossless trajectory, where the ground state s reached adabatcally, and a lossy trajectory, where three-body loss events lead to heatng of the system (as holes are produced that correspond to excted states). In Fg. 3(d), we compare the probablty for each type of ramp of producng the lossless trajectory. We see that for the ramp from the MI state, where we pass through a regon of small U=J, the probablty of such a trajectory s essentally zero. For the superlattce ramp, on the other hand, t s much more lkely that we wll obtan the ground state from a randomly chosen trajectory. Ths s because the superlattce both allows us to use large ju=jj, and facltates the choce of a lower densty. In Fg. 4 we show the local densty as a functon of tme and the fnal dmer densty matrx Dð; jþ for (a) the lossless and (b) the lossy trajectores of Fg. 3(c). When a loss occurs t affects not just the densty for the ste on whch t occurs, but also on neghborng stes due to the knowledge that we obtan of the poston of the remanng partcles. We also see clearly the destructon of correlatons n the regon of the system where the loss occurs. Note, however, that a sngle loss event does not always destroy the propertes of the fnal state, and whle the probablty of an ndvdual loss event ncreases wth system sze, a sngle loss event wll change the character of the fnal state less. The three-body nteractons dscussed here could have applcatons to producng Pfaffan-lke states and stablzng three-component mxtures. The theoretcal approach we have used, quantum trajectores combned wth t-dmrg, could also be appled to other classes of master [1] I. Bloch, J. Dalbard, and W. Zwerger, Rev. Mod. Phys. 80, 885 (2008). [2] G. Vdal, Phys. Rev. Lett. 93, (2004); F. Verstraete, V. Murg, and J. I. Crac, Adv. Phys. 57, 143 (2008). [3] A. J. Daley et al., J. Stat. Mech. 04 (2004) P04005; S. R. Whte and A. E. Fegun, Phys. Rev. Lett. 93, (2004); F. Verstraete, J. J. Garca-Rpoll, and J. I. Crac, bd. 93, (2004). [4] N. Syassen et al., Scence 320, 1329 (2008). [5] J. J. Garca-Rpoll et al., arxv: v1. [6] B. Paredes, T. Kelmann, and J. I. Crac, Phys. Rev. A 75, (2007). [7] T. B. Ottensten et al., arxv: v2. [8] A. Rapp, G. Zarand, C. Honerkamp, and W. Hofstetter, Phys. Rev. Lett. 98, (2007). [9] H. J. Carmchael, An Open Systems Approach to Quantum Optcs (Sprnger, Berln, 1993); C. W. Gardner and P. Zoller, Quantum Nose (Sprnger, Berln, 2005). [10] See, e.g., T. Kraemer et al., Nature (London) 440, 315 (2006). [11] Off-ste loss terms can arse from nonzero overlap of the correspondng Wanner functons, n analogy to off-ste elastc nteractons, whch are dscussed n L.-M. Duan, Europhys. Lett. 81, (2008). These are small for 3 values used here. [12] Ths s because nonunversal behavor (where 3 6/ a 4, wth a the scatterng length) s observed away from a Feshbach resonance, where the rate of three-body recombnaton s large and weakly dependent on a [10], whle the two-body scatterng length a can be very small (U / a). [13] Note that off-ste processes can place an upper bound on the useful value of 3, as ncreasng 3 suppresses on-ste loss, but ncreases the rate of loss from processes nvolvng neghborng lattce stes. [14] M. Zwolak and G. Vdal, Phys. Rev. Lett. 93, (2004); F. Verstraete, J. J. Garca-Rpoll, and J. I. Crac, Phys. Rev. Lett. 93, (2004). [15] We typcally take the number of states retaned n bpartte splttngs [2], ¼ 200 n the results presented here. [16] M. W. J. Romans, R. A. Dune, Subr Sachdev, and H. T. C. Stoof, Phys. Rev. Lett. 93, (2004); L. Radzhovsky, J. I. Park, and P. B. Wechman, bd. 92, (2004); L. Radzhovsky, P. B. Wechman, and J. I. Park, Ann. Phys. (Lepzg) 323, 2376 (2008). [17] S. Dehl et al., (to be publshed)