A Bose-Einstein condensate in an oscillating spatially asymmetric potential
is shown to exhibit a directed current for unbiased initial conditions despite
time symmetry. This phenomenon occurs only if the interaction between atoms,
treated in mean-field approximation, exceeds a critical value. Our findings can
be described with a three-mode model (TMM). These TMM results corroborate well
with a many-body study over a time scale which increases with increasing atom
number. The duration of this time scale probes the validity of the used
mean-field approximation.Comment: 4 pages, 5 figure

Baowen Li

Dario Poletti

Giuliano Benenti

Giulio Casati

L. P. Pitaevskii

Peter Hänggi

English

arXiv

Crossref

Steering Bose-Einstein Condensates despite Time Symmetry

Poletti, Dario

Benenti, Giuliano

Casati, Giulio

Hänggi, Peter

Li, Baowen

OPUS - Augsburg University Publication Server

D. Poletti

P. Hanggi

B. W. Li

BENENTI, GIULIANO

CASATI, GIULIO

Archivio istituzionale della ricerca - Università dell'Insubria

Steering Bose-Einstein Condensates despite Time SymmetryDario Poletti,1 Giuliano Benenti,2,3 Giulio Casati,2,3 Peter Ha¨nggi,4,1 and Baowen Li1,51Department of Physics and Centre for Computational Science and Engineering, National University of Singapore,Singapore 117542, Republic of Singapore2CNISM, CNR-INFM, and Center for Nonlinear and Complex Systems, Universita` degli Studi dell’Insubria,Via Valleggio 11, 22100 Como, Italy3Istituto Nazionale di Fisica Nucleare, Sezione di Milano, Via Celoria 16, 20133 Milano, Italy4Institut fu¨r Physik, Universita¨t Augsburg, Universita¨tsstr. 1, D-86135 Augsburg, Germany5NUS Graduate School for Integrative Sciences and Engineering, 117597, Republic of Singapore(Received 20 November 2008; revised manuscript received 15 February 2009; published 3 April 2009)A Bose-Einstein condensate in an oscillating spatially asymmetric potential is shown to exhibit adirected current for unbiased initial conditions despite time symmetry. This phenomenon occurs only ifthe interaction between atoms, treated in mean-field approximation, exceeds a critical value. Our findingscan be described with a three-mode model. These three-mode model results corroborate well with a many-body study over a time scale which increases with increasing atom number. The duration of this time scaleprobes the validity of the used mean-field approximation.DOI: 10.1103/PhysRevLett.102.130604 PACS numbers: 05.60.k, 03.75.Kk, 05.70.Ln, 67.85.HjThe realization of Bose-Einstein condensates (BECs)[1] of dilute atomic gases has opened new possibilitiesfor the investigation of interesting physical phenomenainduced by atom-atom interactions [2]. For example, fora BEC in a double-well potential, it was found that thetunneling can be suppressed due to interaction-inducedself-trapping [3–5], thus maintaining population imbalancebetween the two wells. A similar phenomenon has beenpredicted also for a periodically driven BEC in a doublewell [6–10] and shows the importance of interaction in thecoherent control of quantum tunneling between wells.In recent years, ample interest has arisen in the field ofdirected transport in classical and quantum systems inabsence of a dc bias (see [11,12] and references therein).In particular it has been shown that by breaking space- andtime-inversion symmetries it is possible to achieve directed(ratchet) transport [13,14]. The study of the role of inter-actions on the quantum ratchet transport, however, trulyremains terra incognita, with only a very few preliminarystudies available [15–17]. For instance, in [15] it has beenshown that atom-atom interactions in a BEC can cause afinite current while the noninteracting system would notexhibit directed transport; however, all these systems pos-sess a Hamiltonian that is both space and time asymmetric.With this Letter, instead, we consider a nondissipative,i.e., Hamiltonian, system in which the time-reversal sym-metry is not broken and the system is driven smoothly. Weshow that, starting out from initial conditions that aresymmetric both in momentum and space, interactions,treated in the limits of a mean-field approximation, inducedirected current in a BEC. In a full many-body analysis ofthe system, symmetry considerations imply that this di-rected current is asymptotically decaying. Despite this factwe show that a finite current persists over a time scalewhich increases with increasing atom number in thecondensate.To start out, we consider a BEC composed of N atomsconfined in a toroidal trap of radius R and cross sectionr2, subjected to the condition r R, so that the motion isessentially one dimensional. The condensate is driven by atime-periodic external potential, readingVExtðx; tÞ ¼ ½V1 cosðxÞ þ V2 cosð2xþÞ cos½!ðt t0Þ;(1)where V1 and V2 denote the potential depths of the twolattice harmonics, their relative phase, and! the angularfrequency of the oscillations of the perturbation. For 6Liand a trap of radius R ¼ 2:5 m our used unit of time ismR2=@ ¼ 6:93 104 s. This potential can be readilyreproduced in experiments [18]. The space symmetry (x!xþ , with  constant [13]), is broken when   0, .The driving is symmetric under t! tþ 2t0 inversionand, in particular, for t0 ¼ 0, under t! t time symmetry.At zero temperature and as long as the number N0 ofcondensate particles is much larger than the number N ofnoncondensate ones, the evolution of a BEC is well de-scribed by the Gross-Pitaevskii (GP) equation [1]. Thus,for our system the evolution of the condensate wave func-tion c ðx; tÞ, normalized to 1, is described by (taking @ ¼m ¼ 1)i@c ðx; tÞ@t¼ 12@2@x2þ gjc ðx; tÞj2 þ VExtðx; tÞc ðx; tÞ;(2)where x 2 ½0; 2Þ is the azimuthal angle of the torus, g ¼8NaR=r2 the scaled strength of the nonlinear interaction(we consider the repulsive case, i.e., g > 0), and a is thes-wave scattering length for elastic atom-atom collisions.PRL 102, 130604 (2009) P HY S I CA L R EV I EW LE T T E R Sweek ending3 APRIL 20090031-9007=09=102(13)=130604(4) 130604-1  2009 The American Physical SocietyAs the initial condition we use a wave function which issymmetric both in space, x, and in momentum p; i.e., it isof the form c ðx; 0Þ ¼ Pnanein cosðnxÞ, where n 2 N andan, n 2 R. For simplicity, and without loss of generality,we choose the initial condition to bec ðx; 0Þ ¼ 1ﬃﬃﬃ2p1ﬃﬃﬃﬃﬃﬃ2p þ eiﬃﬃﬃﬃp cosðxÞ; (3)where  is the relative phase between the homogeneouspart and the last term which corresponds to a cosinusoidalmodulation in the initial density of particles. While in thenoninteracting (g ¼ 0) case the asymptotic current is zero,for any initial condition that is symmetric in x and p, weshow that this is not the case if g  0.As depicted with Fig. 1 (top panel), our model mayexhibit a nonzero asymptotic current, hpiasym ¼limt!1 1tRt0hpiðt0Þdt0, when g  0 and   0 with hpiðt0Þ ¼ iR20 c ðx; t0Þ @@x c ðx; t0Þdx. The asymptotic time-averaged momentum hpiasym vs g [Fig. 1 (bottom panel)]clearly shows that (i) there is a finite threshold value g?such that hpiasym  0 when g > g? and (ii) there exists anoptimal value gopt that maximizes the current.The asymptotic time-averaged current is shown as afunction of the initial phase !t0 of the driving field inFig. 2. We can deduce that there is a strong asymmetryaround !t0 ¼ , in particular, for the coupling strengthg ¼ 0:075. As a result, the directed current survives evenafter averaging over t0 (cf. the inset of Fig. 2).The asymptotic time-averaged momentum is depicted asa function of the relative phase  in Fig. 3 (top panel). Wenote that for certain intervals of the values of  it is pos-sible to obtain a nonzero current, and these regions widenas g increases. There is a symmetry for ! 2 , andhence the average over all  yields zero current.This ! ð2 Þ symmetry can be reasoned as fol-lows: With t0 ¼ 0, let t!  ¼ t and perform complexconjugation in the GP equation (2), to obtaini@ ~c ðx;Þ@¼12@2@x2þgj ~c ðx;Þj2þVExtðx;Þ~c ðx;Þ;(4)where ~c ðx; Þ  c ?ðx;Þ. Therefore, if c ðx; tÞ is a so-lution of the GP equation, so is ~c ðx; Þ. It follows thath~pðÞi of the wave function ~c ðx; Þ is such that h~pðÞi ¼hpðtÞi, where hpðtÞi is the momentum expectation ofc ðx; tÞ. For the initial condition (3), c ðx; 0Þ ! c ?ðx; 0Þ ¼~c ðx; 0Þ when ! ð2 Þ. We thus conclude thathpiasymðÞ ¼ hpiasymð2 Þ. Clearly, if c ð0Þ 2 R[ ¼ 0,  in Eq. (3)], then hpiasym ¼ 0. While in thenoninteracting case the current is asymptotically vanishingfor any initial condition, complex or real [14], in the g  0case a directed current can arise when   0, .In Fig. 3 (bottom panel) we show that a directed currentcan be generated only when the potential is spatiallyasymmetric, that is,   0, . Note that the nonzero cur-rent is obtained and remains stable in two regions aroundthe maximum symmetry violation values  ¼ =2, 3=2.FIG. 1 (color online). Upper panel: Momentum expectationhpi vs time t for  ¼ 0 (black curve) and  ¼ =2 [gray(red) curve]. Used parameter values are g ¼ 0:2, V1 ¼ V2 ¼ 2, ¼ =2, ! ¼ 10, t0 ¼ 0. Lower panel: Asymptotic time-average momentum hpiasym vs the interaction strength g for  ¼=2. Data are obtained from the GP equation (2) (black squares)or from the three-mode model ansatz (5) (red circles). In the firstcase g?  0:065 and gopt  0:15; in the latter g?  0:070 andgopt  0:15.FIG. 2 (color online). Asymptotic time-averaged momentumhpiasym vs !t0 for g ¼ 0:05 (filled black squares), g ¼ 0:075(empty red circles), g ¼ 0:1 (filled blue triangles), and g ¼ 0:2(pink asterisks). Inset: Asymptotic current averaged over t0,hpiasym, as a function of the interaction strength g. Parametervalues: V1 ¼ V2 ¼ 2,  ¼ =2, ! ¼ 10,  ¼ =2.PRL 102, 130604 (2009) P HY S I CA L R EV I EW LE T T E R Sweek ending3 APRIL 2009130604-2From our numerical studies we notice that the only statesthat are significantly excited during the dynamical evolu-tion of the system assume the momenta 0, þ1, or 1. Wehence attempt to reproduce our present results, at least on aqualitative level, with the use of a mean-field three-modemodel (TMM). We use the ansatzc ðx; tÞ ¼ 1ﬃﬃﬃﬃﬃﬃ2p ½AðtÞ þ PðtÞeix þMðtÞeix; (5)where AðtÞ, PðtÞ, and MðtÞ are time-dependent, complexcoefficients such that jAj2 þ jPj2 þ jMj2 ¼ 1. Using (5)we find the effective mean-field Hamiltonian of the TMM,Heff ¼Z 20dxc ?ðx; tÞ 12@2@x2þ g2jc ðx; tÞj2þ VExtðx; tÞc ðx; tÞ; (6)together with the equations of motioni _A ¼ g2½ðjAj2 þ 2jPj2 þ 2jMj2ÞAþ 2APMþ K2ðPþMÞ cosð!tÞ;i _P ¼ P2þ g2½ðjPj2 þ 2jAj2 þ 2jMj2Pþ A2MÞþ K2ðAþ iMÞ cosð!tÞ;i _M ¼ M2þ g2½ðjMj2 þ 2jAj2 þ 2jPj2ÞMþ A2Pþ K2ðA iPÞ cosð!tÞ:(7)This TMM qualitatively reproduces the behavior of theasymptotic time-averaged current [see Fig. 1 (bottompanel)]. In Fig. 4 we depict the evolution of the populationsjAj2, jPj2, and jMj2 of the three modes. For  ¼ 0 ( ¼=2) the time-averaged current is zero (nonzero).The obtained results may appear surprising if one ob-serves that the full second-quantized quantum many-bodyoperator is linear, obeying time-reversal symmetry, andsymmetry arguments necessarily imply a vanishing asymp-totic current [14]. Actually there occurs no contradictionbecause the two limits t! 1 and N ! 1 do not com-mute: a directed current is obtained only in the mean-fieldlimit in which we let first N ! 1 and then t! 1. Theissue for a physical system in which both the number ofparticles and the time of the experimental run are finiteconstitutes the vindication of the regime of validity of theused nonlinear GP equation. For this purpose, we nextstudy the second-quantized many-body problem withinthe three-mode approximation, being also accessible tonumerical investigation.The second-quantized quantummany-body HamiltonianreadsH^ eff ¼Z 20dxc^ y 12@2@x2þ g2Nc^ y c^ þ VExtðx; tÞc^ ;(8)where, within the three-mode approximation,c^ y ¼ 1ﬃﬃﬃﬃﬃﬃ2p ða^y þ p^yeix þ m^yeixÞ; (9)a^y, p^y, and m^y denote the boson creation operators for thestates with momenta 0, þ1, and 1. These operatorssatisfy the commutation relations ½a^; a^y ¼ 1, ½p^; p^y ¼1, ½m^; m^y ¼ 1, and operators corresponding to differentmodes commute. The particle conservation reads ha^ya^þp^yp^þ m^ym^i ¼ N, where N is the total number of par-ticles. As an initial condition, we consider N atoms in theFIG. 3 (color online). Top panel: Asymptotic time-averagedmomentum hpiasym vs  for g ¼ 0:05 (filled black squares), g ¼0:075 (empty red circles), g ¼ 0:15 (filled blue triangles), andg ¼ 0:2 (pink asterisks). The other parameters’ values are V1 ¼V2 ¼ 2,  ¼ =2, ! ¼ 10, and t0 ¼ 0. Bottom panel:Asymptotic time-averaged momentum hpiasym vs  for V1 ¼V2 ¼ 0:2 (filled black squares), V1 ¼ V2 ¼ 0:5 (empty redcircles), V1 ¼ V2 ¼ 1 (filled blue triangles), and V1 ¼ V2 ¼ 2(pink crosses). Other parameters’ values are g ¼ 0:2,  ¼ =2,! ¼ 10, and t0 ¼ 0.FIG. 4. Population of the three levels, jAj2 (black), jPj2(lightgray), and jMj2 (dark gray), which carry momentum 0, þ1, and1, respectively, for  ¼ 0 (top) and =2 (bottom), at g ¼ 0:2,V1 ¼ V2 ¼ 2,  ¼ =2, ! ¼ 10, t0 ¼ 0.PRL 102, 130604 (2009) P HY S I CA L R EV I EW LE T T E R Sweek ending3 APRIL 2009130604-3state1ﬃﬃﬃﬃﬃﬃN!pa^yﬃﬃﬃ2p þ eiðp^y þ m^yÞ2Nj0i: (10)We studied numerically the evolution of the N-bodysystem governed by the second-quantized three-modeHamiltonian (8). The normalized population imbalancebetween levels with momenta þ1 and 1 is depicted inFig. 5. The TMM holds up to increasingly longer times t?as N increases. We estimate t?ðNÞ as the time instant atwhich the N-body population imbalance deviates by morethan 1% from the mean-field treatment value: Our numeri-cal data are well fitted by a logarithmic dependence, t? /lnN (cf. inset of Fig. 5).To check the stability of the condensate, we have diago-nalized the single-particle reduced density matrix, whosematrix elements read hx^yy^i, with x^, y^ being a^, p^ or m^.According to [19], we have a BEC if this matrix possessesone eigenvalue of order N. Defining ~t as the time for whichthe condensed state is highly populated, we have foundnumerically that ~t / N0:45. This means that for large N andtimes t obeying ~t > t > t, the mean-field approach nolonger provides an accurate description of the many-bodysystem despite the presence of a condensate.In conclusion, we have shown that in BEC the interac-tion plays a prominent role in inducing long lasting cur-rents, when otherwise impossible. We note that BECs intoroidal traps have recently been realized in different ex-periments [20–23]. The initial condition (3) can be imple-mented by properly exciting the ground-state of a con-densate in a toroidal trap, and the phase  can be tunedby letting the excited condensate evolve over a given timespan. The interaction strength can be changed by tuning anexternal magnetic field [24]. Extrapolating the dependencegiven in Fig. 5 of t? on the number N of condensate par-ticles, we obtain t?  800 (in an in situ experiment thiscorresponds to a time scale 0.1 s) forN  5 105 particles[22]. Changing the number of condensed particles [25]opens the possibility to explore in situ the range of validityof the GP description via the measurement of our pre-dicted, intriguing directed transport.We acknowledge fruitful discussions with A. Mouritzenand A. Smerzi. This work is supported in part by theAcademic Research Fund, WBS Grant No. 150-000-002-112 (B. L.). Support by the German Excellence Initiativevia the ‘‘Nanosystems Initiative Munich (NIM)’’ (P. H.) isgratefully acknowledged as well.[1] F. Dalfovo et al., Rev. Mod. Phys. 71, 463 (1999); L. P.Pitaevskii and S. Stringari, Bose-Einstein Condensation(Oxford University Press, New York, 2003).[2] O. Morsch and M. Oberthaler, Rev. Mod. Phys. 78, 179(2006).[3] G. J. Milburn et al., Phys. Rev. A 55, 4318 (1997).[4] A. Smerzi et al., Phys. Rev. Lett. 79, 4950 (1997).[5] M. Albiez et al., Phys. Rev. Lett. 95, 010402 (2005).[6] M. Holthaus, Phys. Rev. A 64, 011601(R) (2001).[7] C. Weiss and T. Jinasundera, Phys. Rev. A 72, 053626(2005).[8] X. Luo, Q. Xie, and B. Wu, Phys. Rev. A 76, 051802(R)(2007).[9] Q. Zhang, P. Ha¨nggi, and J. Gong, Phys. Rev. A 77,053607 (2008).[10] L. Morales-Molina and J. Gong, Phys. Rev. A 78, 041403(R) (2008).[11] F. Ju¨licher, A. Ajdari, and J. Prost, Rev. Mod. Phys. 69,1269 (1997); R. D. Astumian and P. Ha¨nggi, Phys. Today55, No. 11, 33 (2002); P. Reimann and P. Ha¨nggi, Appl.Phys. A 75, 169 (2002); P. Ha¨nggi and F. Marchesoni, Rev.Mod. Phys. 81, 1 (2009).[12] P. Reimann, M. Grifoni, and P. Ha¨nggi, Phys. Rev. Lett.79, 10 (1997); J. Lehmann et al., ibid. 88, 228305 (2002);G. G. Carlo et al., ibid. 94, 164101 (2005); I. Dana et al.,Phys. Rev. Lett. 100, 024103 (2008).[13] S. Flach, O. Yevtushenko, and Y. Zolotaryuk, Phys. Rev.Lett. 84, 2358 (2000).[14] S. Denisov et al., Phys. Rev. A 75, 063424 (2007).[15] D. Poletti et al., Phys. Rev. A 76, 023421 (2007).[16] O. Romero-Isart and J. J. Garcı´a-Ripoll, Phys. Rev. A 76,052304 (2007).[17] L. Morales-Molina and S. Flach, New J. Phys. 10, 013 008(2008).[18] T. Salger et al., Phys. Rev. Lett. 99, 190405 (2007).[19] O. Penrose and L. Onsager, Phys. Rev. 104, 576 (1956).[20] S. Gupta et al., Phys. Rev. Lett. 95, 143201 (2005).[21] S. E. Olson et al., Phys. Rev. A 76, 061404(R) (2007).[22] C. Ryu et al., Phys. Rev. Lett. 99, 260401 (2007);K. Helmerson et al., Nucl. Phys. A790, 705c (2007).[23] W.H. Heathcote et al., New J. Phys. 10, 043 012 (2008).[24] E. A. Donley et al., Nature (London) 412, 295 (2001).[25] T. P. Meyrath et al., Phys. Rev. A 71, 041604(R) (2005).FIG. 5 (color online). Population imbalance divided by Nbetween levels with p ¼ þ1 and p ¼ 1 vs time t. Frombottom to top: N ¼ 10, 80, and mean-field TMM, for g ¼ 0:2,V1 ¼ V2 ¼ 2,  ¼ =2, ! ¼ 10, t0 ¼ 0,  ¼ =2. Inset: t vsN (squares) and numerical fit t? ¼ Aþ B lnN, with A  73 andB  54.PRL 102, 130604 (2009) P HY S I CA L R EV I EW LE T T E R Sweek ending3 APRIL 2009130604-4

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Steering Bose-Einstein Condensates despite Time SymmetryDario Poletti,1 Giuliano Benenti,2,3 Giulio Casati,2,3 Peter Hänggi,4,1 and Baowen Li1,51Department of Physics and Centre for Computational Science and Engineering, National University of Singapore,Singapore 117542, Republic of Singapore2CNISM, CNR-INFM, and Center for Nonlinear and Complex Systems, Università degli Studi dell’Insubria,Via Valleggio 11, 22100 Como, Italy3Istituto Nazionale di Fisica Nucleare, Sezione di Milano, Via Celoria 16, 20133 Milano, Italy4Institut für Physik, Universität Augsburg, Universitätsstr. 1, D-86135 Augsburg, Germany5NUS Graduate School for Integrative Sciences and Engineering, 117597, Republic of Singapore(Received 20 November 2008; revised manuscript received 15 February 2009; published 3 April 2009)A Bose-Einstein condensate in an oscillating spatially asymmetric potential is shown to exhibit adirected current for unbiased initial conditions despite time symmetry. This phenomenon occurs only ifthe interaction between atoms, treated in mean-field approximation, exceeds a critical value. Our findingscan be described with a three-mode model. These three-mode model results corroborate well with a many-body study over a time scale which increases with increasing atom number. The duration of this time scaleprobes the validity of the used mean-field approximation.DOI: 10.1103/PhysRevLett.102.130604 PACS numbers: 05.60.k, 03.75.Kk, 05.70.Ln, 67.85.HjThe realization of Bose-Einstein condensates (BECs)[1] of dilute atomic gases has opened new possibilitiesfor the investigation of interesting physical phenomenainduced by atom-atom interactions [2]. For example, fora BEC in a double-well potential, it was found that thetunneling can be suppressed due to interaction-inducedself-trapping [3–5], thus maintaining population imbalancebetween the two wells. A similar phenomenon has beenpredicted also for a periodically driven BEC in a doublewell [6–10] and shows the importance of interaction in thecoherent control of quantum tunneling between wells.In recent years, ample interest has arisen in the field ofdirected transport in classical and quantum systems inabsence of a dc bias (see [11,12] and references therein).In particular it has been shown that by breaking space- andtime-inversion symmetries it is possible to achieve directed(ratchet) transport [13,14]. The study of the role of inter-actions on the quantum ratchet transport, however, trulyremains terra incognita, with only a very few preliminarystudies available [15–17]. For instance, in [15] it has beenshown that atom-atom interactions in a BEC can cause afinite current while the noninteracting system would notexhibit directed transport; however, all these systems pos-sess a Hamiltonian that is both space and time asymmetric.With this Letter, instead, we consider a nondissipative,i.e., Hamiltonian, system in which the time-reversal sym-metry is not broken and the system is driven smoothly. Weshow that, starting out from initial conditions that aresymmetric both in momentum and space, interactions,treated in the limits of a mean-field approximation, inducedirected current in a BEC. In a full many-body analysis ofthe system, symmetry considerations imply that this di-rected current is asymptotically decaying. Despite this factwe show that a finite current persists over a time scalewhich increases with increasing atom number in thecondensate.To start out, we consider a BEC composed of N atomsconfined in a toroidal trap of radius R and cross sectionr2, subjected to the condition r  R, so that the motion isessentially one dimensional. The condensate is driven by atime-periodic external potential, readingVExtðx; tÞ ¼ ½V1 cosðxÞ þ V2 cosð2xþÞ cos½!ðt t0Þ;(1)where V1 and V2 denote the potential depths of the twolattice harmonics, their relative phase, and! the angularfrequency of the oscillations of the perturbation. For 6Liand a trap of radius R ¼ 2:5 m our used unit of time ismR2=@ ¼ 6:93 104 s. This potential can be readilyreproduced in experiments [18]. The space symmetry (x !xþ , with  constant [13]), is broken when   0, .The driving is symmetric under t ! tþ 2t0 inversionand, in particular, for t0 ¼ 0, under t ! t time symmetry.At zero temperature and as long as the number N0 ofcondensate particles is much larger than the number N ofnoncondensate ones, the evolution of a BEC is well de-scribed by the Gross-Pitaevskii (GP) equation [1]. Thus,for our system the evolution of the condensate wave func-tion c ðx; tÞ, normalized to 1, is described by (taking @ ¼m ¼ 1)i@c ðx; tÞ@t¼ 12@2@x2þ gjc ðx; tÞj2 þ VExtðx; tÞc ðx; tÞ;(2)where x 2 ½0; 2Þ is the azimuthal angle of the torus, g ¼8NaR=r2 the scaled strength of the nonlinear interaction(we consider the repulsive case, i.e., g > 0), and a is thes-wave scattering length for elastic atom-atom collisions.PRL 102, 130604 (2009) P HY S I CA L R EV I EW LE T T E R Sweek ending3 APRIL 20090031-9007=09=102(13)=130604(4) 130604-1  2009 The American Physical SocietyAs the initial condition we use a wave function which issymmetric both in space, x, and in momentum p; i.e., it isof the form c ðx; 0Þ ¼ Pnanein cosðnxÞ, where n 2 N andan, n 2 R. For simplicity, and without loss of generality,we choose the initial condition to bec ðx; 0Þ ¼ 1ffiffiffi2p1ffiffiffiffiffiffi2p þ eiffiffiffiffip cosðxÞ; (3)where  is the relative phase between the homogeneouspart and the last term which corresponds to a cosinusoidalmodulation in the initial density of particles. While in thenoninteracting (g ¼ 0) case the asymptotic current is zero,for any initial condition that is symmetric in x and p, weshow that this is not the case if g  0.As depicted with Fig. 1 (top panel), our model mayexhibit a nonzero asymptotic current, hpiasym ¼limt!1 1tRt0hpiðt0Þdt0, when g  0 and   0 with hpiðt0Þ ¼ iR20 c ðx; t0Þ @@x c ðx; t0Þdx. The asymptotic time-averaged momentum hpiasym vs g [Fig. 1 (bottom panel)]clearly shows that (i) there is a finite threshold value g?such that hpiasym  0 when g > g? and (ii) there exists anoptimal value gopt that maximizes the current.The asymptotic time-averaged current is shown as afunction of the initial phase !t0 of the driving field inFig. 2. We can deduce that there is a strong asymmetryaround !t0 ¼ , in particular, for the coupling strengthg ¼ 0:075. As a result, the directed current survives evenafter averaging over t0 (cf. the inset of Fig. 2).The asymptotic time-averaged momentum is depicted asa function of the relative phase  in Fig. 3 (top panel). Wenote that for certain intervals of the values of  it is pos-sible to obtain a nonzero current, and these regions widenas g increases. There is a symmetry for  ! 2 , andhence the average over all  yields zero current.This  ! ð2 Þ symmetry can be reasoned as fol-lows: With t0 ¼ 0, let t !  ¼ t and perform complexconjugation in the GP equation (2), to obtaini@ ~c ðx;Þ@¼12@2@x2þgj ~c ðx;Þj2þVExtðx;Þ~c ðx;Þ;(4)where ~c ðx; Þ  c ?ðx;Þ. Therefore, if c ðx; tÞ is a so-lution of the GP equation, so is ~c ðx; Þ. It follows thath~pðÞi of the wave function ~c ðx; Þ is such that h~pðÞi ¼hpðtÞi, where hpðtÞi is the momentum expectation ofc ðx; tÞ. For the initial condition (3), c ðx; 0Þ ! c ?ðx; 0Þ ¼~c ðx; 0Þ when  ! ð2 Þ. We thus conclude thathpiasymðÞ ¼ hpiasymð2 Þ. Clearly, if c ð0Þ 2 R[ ¼ 0,  in Eq. (3)], then hpiasym ¼ 0. While in thenoninteracting case the current is asymptotically vanishingfor any initial condition, complex or real [14], in the g  0case a directed current can arise when   0, .In Fig. 3 (bottom panel) we show that a directed currentcan be generated only when the potential is spatiallyasymmetric, that is,   0, . Note that the nonzero cur-rent is obtained and remains stable in two regions aroundthe maximum symmetry violation values  ¼ =2, 3=2.FIG. 1 (color online). Upper panel: Momentum expectationhpi vs time t for  ¼ 0 (black curve) and  ¼ =2 [gray(red) curve]. Used parameter values are g ¼ 0:2, V1 ¼ V2 ¼ 2, ¼ =2, ! ¼ 10, t0 ¼ 0. Lower panel: Asymptotic time-average momentum hpiasym vs the interaction strength g for  ¼=2. Data are obtained from the GP equation (2) (black squares)or from the three-mode model ansatz (5) (red circles). In the firstcase g?  0:065 and gopt  0:15; in the latter g?  0:070 andgopt  0:15.FIG. 2 (color online). Asymptotic time-averaged momentumhpiasym vs !t0 for g ¼ 0:05 (filled black squares), g ¼ 0:075(empty red circles), g ¼ 0:1 (filled blue triangles), and g ¼ 0:2(pink asterisks). Inset: Asymptotic current averaged over t0,hpiasym, as a function of the interaction strength g. Parametervalues: V1 ¼ V2 ¼ 2,  ¼ =2, ! ¼ 10,  ¼ =2.PRL 102, 130604 (2009) P HY S I CA L R EV I EW LE T T E R Sweek ending3 APRIL 2009130604-2From our numerical studies we notice that the only statesthat are significantly excited during the dynamical evolu-tion of the system assume the momenta 0, þ1, or 1. Wehence attempt to reproduce our present results, at least on aqualitative level, with the use of a mean-field three-modemodel (TMM). We use the ansatzc ðx; tÞ ¼ 1ffiffiffiffiffiffi2p ½AðtÞ þ PðtÞeix þMðtÞeix; (5)where AðtÞ, PðtÞ, and MðtÞ are time-dependent, complexcoefficients such that jAj2 þ jPj2 þ jMj2 ¼ 1. Using (5)we find the effective mean-field Hamiltonian of the TMM,Heff ¼Z 20dxc ?ðx; tÞ 12@2@x2þ g2jc ðx; tÞj2þ VExtðx; tÞc ðx; tÞ; (6)together with the equations of motioni _A ¼ g2½ðjAj2 þ 2jPj2 þ 2jMj2ÞAþ 2APMþ K2ðPþMÞ cosð!tÞ;i _P ¼ P2þ g2½ðjPj2 þ 2jAj2 þ 2jMj2Pþ A2MÞþ K2ðAþ iMÞ cosð!tÞ;i _M ¼ M2þ g2½ðjMj2 þ 2jAj2 þ 2jPj2ÞMþ A2Pþ K2ðA iPÞ cosð!tÞ:(7)This TMM qualitatively reproduces the behavior of theasymptotic time-averaged current [see Fig. 1 (bottompanel)]. In Fig. 4 we depict the evolution of the populationsjAj2, jPj2, and jMj2 of the three modes. For  ¼ 0 ( ¼=2) the time-averaged current is zero (nonzero).The obtained results may appear surprising if one ob-serves that the full second-quantized quantum many-bodyoperator is linear, obeying time-reversal symmetry, andsymmetry arguments necessarily imply a vanishing asymp-totic current [14]. Actually there occurs no contradictionbecause the two limits t ! 1 and N ! 1 do not com-mute: a directed current is obtained only in the mean-fieldlimit in which we let first N ! 1 and then t ! 1. Theissue for a physical system in which both the number ofparticles and the time of the experimental run are finiteconstitutes the vindication of the regime of validity of theused nonlinear GP equation. For this purpose, we nextstudy the second-quantized many-body problem withinthe three-mode approximation, being also accessible tonumerical investigation.The second-quantized quantummany-body HamiltonianreadsĤ eff ¼Z 20dxĉ y 12@2@x2þ g2Nĉ y ĉ þ VExtðx; tÞĉ ;(8)where, within the three-mode approximation,ĉ y ¼ 1ffiffiffiffiffiffi2p ðây þ p̂yeix þ m̂yeixÞ; (9)ây, p̂y, and m̂y denote the boson creation operators for thestates with momenta 0, þ1, and 1. These operatorssatisfy the commutation relations ½â; ây ¼ 1, ½p̂; p̂y ¼1, ½m̂; m̂y ¼ 1, and operators corresponding to differentmodes commute. The particle conservation reads hâyâþp̂yp̂þ m̂ym̂i ¼ N, where N is the total number of par-ticles. As an initial condition, we consider N atoms in theFIG. 3 (color online). Top panel: Asymptotic time-averagedmomentum hpiasym vs  for g ¼ 0:05 (filled black squares), g ¼0:075 (empty red circles), g ¼ 0:15 (filled blue triangles), andg ¼ 0:2 (pink asterisks). The other parameters’ values are V1 ¼V2 ¼ 2,  ¼ =2, ! ¼ 10, and t0 ¼ 0. Bottom panel:Asymptotic time-averaged momentum hpiasym vs  for V1 ¼V2 ¼ 0:2 (filled black squares), V1 ¼ V2 ¼ 0:5 (empty redcircles), V1 ¼ V2 ¼ 1 (filled blue triangles), and V1 ¼ V2 ¼ 2(pink crosses). Other parameters’ values are g ¼ 0:2,  ¼ =2,! ¼ 10, and t0 ¼ 0.FIG. 4. Population of the three levels, jAj2 (black), jPj2(lightgray), and jMj2 (dark gray), which carry momentum 0, þ1, and1, respectively, for  ¼ 0 (top) and =2 (bottom), at g ¼ 0:2,V1 ¼ V2 ¼ 2,  ¼ =2, ! ¼ 10, t0 ¼ 0.PRL 102, 130604 (2009) P HY S I CA L R EV I EW LE T T E R Sweek ending3 APRIL 2009130604-3state1ffiffiffiffiffiffiN!pâyffiffiffi2p þ eiðp̂y þ m̂yÞ2Nj0i: (10)We studied numerically the evolution of the N-bodysystem governed by the second-quantized three-modeHamiltonian (8). The normalized population imbalancebetween levels with momenta þ1 and 1 is depicted inFig. 5. The TMM holds up to increasingly longer times t?as N increases. We estimate t?ðNÞ as the time instant atwhich the N-body population imbalance deviates by morethan 1% from the mean-field treatment value: Our numeri-cal data are well fitted by a logarithmic dependence, t? /lnN (cf. inset of Fig. 5).To check the stability of the condensate, we have diago-nalized the single-particle reduced density matrix, whosematrix elements read hx̂yŷi, with x̂, ŷ being â, p̂ or m̂.According to [19], we have a BEC if this matrix possessesone eigenvalue of order N. Defining ~t as the time for whichthe condensed state is highly populated, we have foundnumerically that ~t / N0:45. This means that for large N andtimes t obeying ~t > t > t, the mean-field approach nolonger provides an accurate description of the many-bodysystem despite the presence of a condensate.In conclusion, we have shown that in BEC the interac-tion plays a prominent role in inducing long lasting cur-rents, when otherwise impossible. We note that BECs intoroidal traps have recently been realized in different ex-periments [20–23]. The initial condition (3) can be imple-mented by properly exciting the ground-state of a con-densate in a toroidal trap, and the phase  can be tunedby letting the excited condensate evolve over a given timespan. The interaction strength can be changed by tuning anexternal magnetic field [24]. Extrapolating the dependencegiven in Fig. 5 of t? on the number N of condensate par-ticles, we obtain t?  800 (in an in situ experiment thiscorresponds to a time scale 0.1 s) forN  5 105 particles[22]. Changing the number of condensed particles [25]opens the possibility to explore in situ the range of validityof the GP description via the measurement of our pre-dicted, intriguing directed transport.We acknowledge fruitful discussions with A. Mouritzenand A. Smerzi. This work is supported in part by theAcademic Research Fund, WBS Grant No. 150-000-002-112 (B. L.). Support by the German Excellence Initiativevia the ‘‘Nanosystems Initiative Munich (NIM)’’ (P. H.) isgratefully acknowledged as well.[1] F. Dalfovo et al., Rev. Mod. Phys. 71, 463 (1999); L. P.Pitaevskii and S. Stringari, Bose-Einstein Condensation(Oxford University Press, New York, 2003).[2] O. Morsch and M. Oberthaler, Rev. Mod. Phys. 78, 179(2006).[3] G. J. Milburn et al., Phys. Rev. A 55, 4318 (1997).[4] A. Smerzi et al., Phys. Rev. Lett. 79, 4950 (1997).[5] M. Albiez et al., Phys. Rev. Lett. 95, 010402 (2005).[6] M. Holthaus, Phys. Rev. A 64, 011601(R) (2001).[7] C. Weiss and T. Jinasundera, Phys. Rev. A 72, 053626(2005).[8] X. Luo, Q. Xie, and B. Wu, Phys. Rev. A 76, 051802(R)(2007).[9] Q. Zhang, P. Hänggi, and J. Gong, Phys. Rev. A 77,053607 (2008).[10] L. Morales-Molina and J. Gong, Phys. Rev. A 78, 041403(R) (2008).[11] F. Jülicher, A. Ajdari, and J. Prost, Rev. Mod. Phys. 69,1269 (1997); R. D. Astumian and P. Hänggi, Phys. Today55, No. 11, 33 (2002); P. Reimann and P. Hänggi, Appl.Phys. A 75, 169 (2002); P. Hänggi and F. Marchesoni, Rev.Mod. Phys. 81, 1 (2009).[12] P. Reimann, M. Grifoni, and P. Hänggi, Phys. Rev. Lett.79, 10 (1997); J. Lehmann et al., ibid. 88, 228305 (2002);G. G. Carlo et al., ibid. 94, 164101 (2005); I. Dana et al.,Phys. Rev. Lett. 100, 024103 (2008).[13] S. Flach, O. Yevtushenko, and Y. Zolotaryuk, Phys. Rev.Lett. 84, 2358 (2000).[14] S. Denisov et al., Phys. Rev. A 75, 063424 (2007).[15] D. Poletti et al., Phys. Rev. A 76, 023421 (2007).[16] O. Romero-Isart and J. J. Garcı́a-Ripoll, Phys. Rev. A 76,052304 (2007).[17] L. Morales-Molina and S. Flach, New J. Phys. 10, 013 008(2008).[18] T. Salger et al., Phys. Rev. Lett. 99, 190405 (2007).[19] O. Penrose and L. Onsager, Phys. Rev. 104, 576 (1956).[20] S. Gupta et al., Phys. Rev. Lett. 95, 143201 (2005).[21] S. E. Olson et al., Phys. Rev. A 76, 061404(R) (2007).[22] C. Ryu et al., Phys. Rev. Lett. 99, 260401 (2007);K. Helmerson et al., Nucl. Phys. A790, 705c (2007).[23] W.H. Heathcote et al., New J. Phys. 10, 043 012 (2008).[24] E. A. Donley et al., Nature (London) 412, 295 (2001).[25] T. P. Meyrath et al., Phys. Rev. A 71, 041604(R) (2005).FIG. 5 (color online). Population imbalance divided by Nbetween levels with p ¼ þ1 and p ¼ 1 vs time t. Frombottom to top: N ¼ 10, 80, and mean-field TMM, for g ¼ 0:2,V1 ¼ V2 ¼ 2,  ¼ =2, ! ¼ 10, t0 ¼ 0,  ¼ =2. Inset: t vsN (squares) and numerical fit t? ¼ Aþ B lnN, with A  73 andB  54.PRL 102, 130604 (2009) P HY S I CA L R EV I EW LE T T E R Sweek ending3 APRIL 2009130604-4

Steering Bose-Einstein condensates despite time symmetry

Steering Bose-Einstein condensates despite time symmetry

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