Interaction-induced quantum ratchet in a Bose-Einstein condensate
Dario Poletti,1Giuliano Benenti,2,3Giulio Casati,2,3,1and Baowen Li1,4,5
1Department of Physics and Centre for Computational Science and Engineering, National University of Singapore, Singapore 117542, Republic of Singapore
2CNISM, CNR-INFM and Center for Nonlinear and Complex Systems, Università degli Studi dell’Insubria, Via Valleggio 11, 22100 Como, Italy
3Istituto Nazionale di Fisica Nucleare, Sezione di Milano, via Celoria 16, 20133 Milano. Italy
4Laboratory of Modern Acoustics and Institute of Acoustics, Nanjing University, 210093, People’s Republic of China
5NUS Graduate School for Integrative Sciences and Engineering, 117597, Republic of Singapore 共Received 21 September 2006; revised manuscript received 6 April 2007; published 24 August 2007兲 We study the dynamics of a dilute Bose-Einstein condensate confined in a toroidal trap and exposed to a pair of periodically flashed optical lattices. We first prove that in the noninteracting case this system can present a quantum symmetry which forbids the ratchet effect classically expected. We then show how many-body atom-atom interactions, treated within the mean-field approximation, can break this quantum symmetry, thus generating directed transport.
DOI:10.1103/PhysRevA.76.023421 PACS number共s兲: 42.50.Vk, 05.60.⫺k, 03.75.Kk, 05.45.⫺a
The ratchet effect, that is, the possibility to drive directed transport with the help of zero-mean perturbations, has re- cently gained renewed attention due to its possible relevance for biological transport, molecular motors, and the prospects of nanotechnology 关1,2兴. At the classical level, the ratchet effect can be found in periodic systems due to a broken space-time symmetry关3兴. The ratchet phenomenon has also been discussed in quantum systems关4兴, including the Hamil- tonian limit without dissipation关5兴. Experimental implemen- tations of directed transport range from semiconductor het- erostructures to quantum dots, Josephson junctions, and cold atoms in optical lattices关6兴.
Quantum Hamiltonian ratchets are relevant in systems such as cold atoms in which the high degree of quantum control may allow experimental implementations near the dissipationless limit. Moreover, the realization of Bose- Einstein condensates共BECs兲 of dilute gases has opened new opportunities for the study of dynamical systems in the pres- ence of many-body interactions. Indeed, it is possible to pre- pare initial states with high precision and to tune over a wide range the many-body atom-atom interaction. From the view- point of directed transport, the study of many-body quantum system is, to our knowledge, at the very beginning.
In this paper, we investigate the quantum dynamics of a BEC in a pair of periodically flashed optical lattices. We show how the interaction between atoms in the condensate, studied in the mean-field approximation, can break the quan- tum symmetry present in our model in the noninteracting limit, thus giving rise to the ratchet effect. The role of noise, the validity of the mean-field description, and the possibility to observe experimentally our ratchet model are discussed as well.
We consider N condensed atoms confined in a toroidal trap of radius R and cross sectionr2, with the condition r ⰆR, so that the motion is essentially one dimensional. The dynamics of a dilute condensate in a pair of periodically kicked optical lattices at zero temperature is described by the Gross-Pitaevskii nonlinear equation,
i
t共,t兲 =
冋
−1222+ g兩共,t兲兩2+ V共,,t兲册
共,t兲,共1兲 where is the azimuthal angle, g = 8NaR / r2 is the scaled strength of the nonlinear interaction共we consider the repul- sive case, i.e., g⬎0兲, a is the s-wave scattering length for elastic atom-atom collisions. The kicked potential V共,, t兲 is defined as
V共,,t兲 =兺n 关V1共兲␦共t − nT兲 + V2共,兲␦共t − nT −兲兴,
V1共兲 = k cos, V2共,兲 = k cos共−兲, 共2兲 where k is the kicking strength and T is the period of the kicks. The parameters僆关0,2兴 and僆关0,T兴 are used to break the space and time symmetries, respectively. Note that we setប=1 and that the length and the energy are measured in units of R andប2/ mR2, with m the atomic mass. The wave function normalization reads 兰02d兩共, t兲兩2= 1 and bound- ary conditions are periodic,共+ 2, t兲=共, t兲.
We first consider the noninteracting case g = 0. Here, when
⫽0,and⫽0, T/2, space-time symmetries are broken and there is directed transport, both in the classical limit and, in general, in quantum mechanics 关7兴. However, if we take T = 6 and= 4, then the quantum motion, independently of the kicking strength k, is periodic of period 2T.
In order to prove this periodicity, it is useful to write the initial wave function as 共, 0兲=兺nAnexp共in兲, where An
=21兰02共, 0兲exp共−in兲. After free evolution up to time t, the wave function becomes共, t兲=兺nAnexp共−in22t + in兲. If t = 4 we have 共, t兲=共, 0兲 while, if t=2, we obtain
共, t兲=共+, 0兲. Using these relations we can easily see that the system is periodic with period 12. Indeed,
共,4+兲 = exp关− iV1共兲兴共,0兲,
共,6+兲 = exp关− iV2共,兲兴共+,4+兲
= exp兵− i关V2共,兲 − V1共兲兴其共+,0兲,
共,10+兲 = exp关− iV1共兲兴共,6+兲
= exp关− iV2共,兲兴共+,0兲,
共,12+兲 = exp关− iV2共,兲兴共+,10+兲 =共,0兲, 共3兲 where 共, t+兲 denotes the value of the wave function at time t just after the kick. The momentum 具p共t兲典
= −i兰02d쐓共, t兲共, t兲 also changes periodically with pe- riod 12 共four kicks兲. Therefore the average momentum pav= limt→⬁¯p共t兲共¯p共t兲=1t兰0tdt⬘具p共t⬘兲典兲is given by
pav=4具p共0兲典 + 2具p共4+兲典 + 4具p共6+兲典 + 2具p共10+兲典 12
=具p共0兲典 +k 2
冕
02
关sin共兲 − sin共−兲兴兩共,0兲兩2d. 共4兲
In particular, for the constant initial condition 共, 0兲
= 1 /冑2, which is the ground state of a particle in the trap, the momentum remains zero at any later time. This initial condition has an important physical meaning, as it corre- sponds to the initial condition for a Bose-Einstein conden- sate.
It is therefore interesting to study the case of a BEC be- cause atom-atom interactions may break the above periodic- ity, and this may cause generation of momentum. The nu- merical integration of Eq.共1兲 confirms this expectation: as shown in Fig.1, at g⫽0 the momentum oscillates around a mean value clearly different from zero. Notice that without interactions 共g=0兲 the momentum is exactly zero, so that directed transport is induced by the many-body atom-atom interactions.
In Fig.2, we compare the asymptotic value pav, obtained from long numerical integrations of the Gross-Pitaevskii
equation 共dotted line with triangles兲, with the average of 具p共t兲典 over the first 30 kicks 关p¯共90兲, continuous line with boxes兴. It can be seen that this short-time average is suffi- cient to obtain a good estimate of the average momentum pav, provided that gⲏ0.5. It is interesting to remark that the average momentum after the first kicks grows monotonically with g. Therefore the ratchet current provides a method to measure the interaction strength in an experiment. In the in- set of Fig.2we show the cumulative average p¯共t兲. For strong enough interactions共gⲏ0.5兲 the convergence to the limiting value pavis rather fast as we can already see from the main part of Fig.2.
Since the above complete periodicity of the single particle system共g=0兲 is a very fragile quantum phenomenon, it is important to check the visibility of the ratchet effect when the unavoidable noise leads to a departure from the ideal periodic behavior. For this purpose we consider fluctuations in the kicking period, modeled as random and memoryless variations of the period between consecutive kicks, with the fluctuation amplitude at each kick randomly drawn from a uniform distribution in the interval 关−⑀,⑀兴. We have seen that, when the size of the fluctuations is⑀= T / 200, then the ratchet current generated in the noninteracting case is
¯p共90兲=−0.007. This value of p¯ is much smaller than the genuine many-body ratchet current already shown in Fig.2 for values of gⲏ0.2. A similar conclusion is obtained when the kicks are substituted by more realistic Gaussian pulses of width T / 10. In this case p¯ after 30 kicks is equal to
¯p共90兲=−0.01, again too small to hide or to be confused with the many-body ratchet effect.
The interaction-induced generation of a nonzero current can be understood as follows. We approximate, for small values of g, the free evolution of the BEC by a split-operator method as in关8兴:
共,兲 ⬇ e−i共1/2兲共2/2兲共/2兲e−ig兩˜共,/2兲兩2e−i共1/2兲共2/2兲共/2兲共,0兲, 共5兲 where˜ 共, t +⌬t兲=e−i共1/2兲共/2兲⌬t共, t兲.
FIG. 1. Momentum vs time for different values of interaction strength g, at k⬇0.74 and =−/4: g=0 共dashed line兲, g=0.5 共continuous curve兲, g=1 共dotted curve兲.
FIG. 2. Momentum averaged over the first 30 kicks共solid line with boxes兲 and asymptotic momentum 共dotted line with triangles兲.
Inset: Cumulative average p¯共t兲 as a function of time for different values of g. From bottom to top g = 0.1, 0.2, 0.4, 1.0, 1.5. Parameter values: k⬇0.74,=−/4.
In particular, we obtain
共,4兲 ⬇ exp关− i4g兩共,0兲兩2兴共,0兲 共6兲 and
共,2兲 ⬇ exp兵− ig关兩共,0兲兩2+兩共+,0兲兩2兴其
⫻ 兵cos关F共,0兲兴共+,0兲 − sin关F共,0兲兴共,0兲其, 共7兲 where we have defined F共, 0兲=ig关ⴱ共, 0兲共+, 0兲
−共, 0兲ⴱ共+, 0兲兴. Note that, in the limit g→0, Eqs. 共6兲 and共7兲 become共, 4兲=共, 0兲 and共, 2兲=共+, 0兲, as expected for the noninteracting free evolution. Using this approximation, we compute the evolution of the condensate for the first two kicks, starting from the initial condition
共, 0兲=1/冑2. We obtain
共,4+兲 ⬇ 1
冑2exp关− iV1共兲兴exp共− i2g兲,
共,6+兲 ⬇ 1
冑2exp兵− i关V2共,兲 − V1共兲兴其exp共− i3g兲
⫻ 兵cos关⍀1共兲兴 + sin关⍀1共兲兴exp关− i2V1共兲兴其, 共8兲 where⍀1共兲=g sin共2V1共兲兲. The mechanism of the ratchet effect is now clear: due to atom-atom interactions, the modu- lus square of the wave function at time 6 共before the sec- ond kick兲 is no longer constant in. Instead we have, to first order in g,兩共, 6兲兩2⬇21兵1+g sin关4V1共兲兴其, so that the ini- tial constant probability distribution is modified by a term symmetric under the transformation→−. The current af- ter the kick at time t = 6 is then given by
具p共6+兲典 = −
冕
0 2dV2⬘共,兲兩共,6兲兩2
⬇ gk
冕
0 2dsin共−兲sin共4k cos兲
= − gk sin共兲J1共4k兲, 共9兲 where J1 is the Bessel function of the first kind of index 1.
This current is in general different from zero, provided that V2共,兲 is not itself symmetric under→−, that is, when
⫽0,.
In Fig.3, we show that it is possible to control the direc- tion of transport by varying the phase: the current can be reversed simply by changing→−. This current inversion can be explained by means of the following symmetry con- siderations. The evolution of the wave-function 共, t兲 is given by Eq.共1兲. After substituting in this equation →−, and taking into account that V共−,, t兲=V共, −, t兲, we obtain
i
t˜ 共,t兲 =
冋
−1222+ g兩˜ 共,t兲兩2+ V共,−,t兲册
˜ 共,t兲,共10兲 where˜ 共, t兲⬅共−, t兲. Therefore if 共, t兲 is a solution of the Gross-Pitaevskii equation, then also˜ 共, t兲 is a solution, provided that we substitute→− in the potential V. The momentum 具p˜共t兲典 of the wave function˜ 共, t兲 is obviously given by 具p˜共t兲典=−具p共t兲典, where 具p共t兲典 is the momentum of
共, t兲. This means that, for every共, t兲 whose evolution is ruled by the Gross-Pitaevskii equation with potential V共,, t兲, the wave function˜ 共, t兲 evolves with exactly op- posite momentum if →− in V. Since we start with an even wave function,˜ 共, 0兲=共−, 0兲=共, 0兲, then chang- ing→−changes the sign of the momentum of the wave function at any later time.
When studying the dynamics of a kicked BEC, it is im- portant to take into account the proliferation of noncon- densed atoms. Actually, strong kicks may lead to thermal excitations out of equilibrium and destroy the condensate, rendering the description by the Gross-Pitaevskii equation meaningless关9,10兴. In the following, we show that, for the parameter values considered in this paper, the number of noncondensed particles is negligible compared to the number of condensed ones, thus demonstrating that our theoretical and numerical results based on the Gross-Pitaevskii equation are reliable.
Following the approach developed in关11兴 共see also 关10兴兲, we compute the mean number of noncondensed particles at zero temperature as ␦N共t兲=兺⬁j=1兰02d兩vj共, t兲兩2, where the evolution ofvj共, t兲 is determined by
i
t
冋
uvjj共共,t,t兲兲册
=冋
− HH1共2ⴱ共,t兲,t兲 − HH2共1쐓共,t兲,t兲册 冋
uvjj共共,t,t兲兲册
.共11兲 Here H1共, t兲=H共, t兲−共t兲+gQ共t兲兩共, t兲兩2Q共t兲, H共, t兲
= −1222+ g兩共, t兲兩2+ V共,, t兲 is the mean-field Hamiltonian FIG. 3. Momentum vs time for different values of the parameter
, at k⬇0.74 and g=0.5: =−/4 共continuous curve兲, =0 共dashed line兲,=/4 共dotted curve兲.
that governs the Gross-Pitaevskii Eq. 共1兲, 共t兲 is the chemical potential 关H共, t兲共, t兲=共t兲共, t兲兴, Q共t兲=1
−兩共t兲典具共t兲兩 projects orthogonally to 兩共t兲典, and H2共, t兲
= gQ共t兲2共, t兲Qⴱ共t兲.
We integrate in parallel Eqs.共1兲 and 共11兲. The initial con- dition of the noncondensed part is obtained by diagonalizing the linear operator in Eq.共11兲 关10,11兴. We obtain
冉
uvjj共共,0兲,0兲冊
=12冉
+ 1/− 1/冊
冑e2ij , 共12兲 where=共j2/2+2gj2/2兩共,0兲兩兲1/4, with共, 0兲=1/冑2initial condi- tion of the BEC. The numerical evolution is performed using the split-operator method as in Eq.共5兲, with small integration stepsⰆT.The number␦N of noncondensed particles, depending on the stability or instability of the condensate, grows polyno- mially or exponentially. As shown in Fig.4,␦N grows poly- nomially at small g and exponentially for large g. The tran-
sition from stability to instability takes place at g = gc⬇1.7.
At g⬎gc, thermal particles proliferate exponentially fast,
␦N⬃exp共rt兲, leading to a significant depletion of the con- densate after a time td⬃ln共N兲/r. In this regime, the validity of the mean field description is undermined. On the other hand, for g⬍gc the exponential growth rate r = 0 and the number of noncondensed particles is negligible for up to long times. For instance, as shown in Fig. 4, ␦N⬇0.2 共10兲 after t = 90共30 kicks兲 at g=0.5 共1.5兲, which is much smaller than the total number of particles N⬇103− 105关10,12兴.
Finally, we would like to comment on the experimental feasibility of our proposal. The toruslike potential confining the BEC may be realized by using two two-dimensional cir- cular optical billiards with the lateral dimension being con- fined by two plane optical billiards关13兴, where the circular barrier is obtained by deflecting a laser beam with acousto- optic deflectors fast enough so that this potential can be ap- proximated by a static potential barrier. The toruslike poten- tial may also be realized by using optical-dipole traps produced by red-detuned Laguerre-Gaussian laser beams of varying azimuthal mode index 关14兴. The kicks may be ap- plied using a periodically pulsed strongly detuned laser beam transverse to the ring with the intensity linearly growing in one direction in the plane of the ring while constant in the other two directions, as proposed in关15兴. The feasibility is also supported by the latest progress in the realization of BECs in optical traps such as the87Rb BEC in a quasi-one- dimensional optical box trap, with condensate length
⬃80 m, transverse confinement ⬃5 m, and number of particles N⬃103 关12兴. Sequences of up to 25 kicks have been applied to a BEC of 87Rb atoms confined in a static harmonic magnetic trap, with kicking strength k⬃1 and in the quantum antiresonance case for the kicked oscillator model, T = 2关16兴. Finally, the interaction strength g can be tuned over a very large range using a Feshbach resonance or varying the number of atoms关17兴.
This work was supported in part by the MIUR COFIN-2005.
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FIG. 4. Mean number␦N of noncondensed particles vs time for different values of the interaction strength g: from bottom to top, g = 0.5, 1.5, and 2.0. Inset:␦N vs g after 30 kicks. Parameter values:
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