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Many-particle tunnelling in a driven Bosonic Josephson junction
Chemical Physics 322 (2006) 118–126 www.elsevier.com/locate/chemphys
Many-particle tunnelling in a driven Bosonic Josephson junction Tharanga Jinasundera, Christoph Weiss, Martin Holthaus
*
Institut fu¨r Physik, Carl von Ossietzky Universita¨t, D-26111 Oldenburg, Germany Received 25 March 2005; accepted 5 June 2005 Available online 25 July 2005
Abstract We study the quantum many-body dynamics of a Bose–Einstein condensate in a double well potential subjected to a time-periodic modulation. For modulation frequencies moderately higher than the single-particle tunnelling frequency, this system displays an interplay of self-trapping and coherent tunnelling destruction, allowing one to effectively shut off the tunnelling contact in extended parameter regimes. For modulation frequencies roughly equalling the tunnelling frequency, one encounters chaotic dynamics, and finds sudden population jumps in the regime of low-frequency driving. A driven Bosonic Josephson junction thus constitutes a promising tool for the coherent control of mesoscopic matter waves. Our predictions can be verified under presently accessible experimental conditions. 2005 Elsevier B.V. All rights reserved. PACS: 03.75.Lm; 05.45.Mt Keywords: Bose–Einstein condensation; Tunnelling effect; Many-body dynamics; Quantum Floquet theory
1. Introduction In a recent experiment, Albiez et al. [1] have observed the tunnelling effect exhibited by weakly coupled Bose–Einstein condensates in a double well potential. After preparing a sample of about 1150 Bose–Einstein-condensed 87Rb atoms in an optical double well with a distance of 4.4 lm between the two minima, these authors were able to measure in situ both the density distribution associated with the tunnelling macroscopic wave function, and the relative phase between the condensates in the two wells. Their system, the first realization of a controllable, single Bosonic Josephson junction, thus allows one to investigate in detail the effect of many-body interactions on the tunnelling pro* Corresponding author. Tel.: +49 441 798 3960; fax: +49 441 798 3080. E-mail address: [email protected] (M. Holthaus).
0301-0104/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2005.06.025
cess, and to compare the actual condensate dynamics to the mean-field approximation. Albiez et al. have loaded the condensate into an asymmetric double well, shaped such that an initial tilt prevented unwanted tunnelling during the preparation phase. This tilt could be controlled by adjusting a focussed laser beam by means of a piezo-actuated mirror; the dynamics was initiated by non-adiabatically shifting to zero tilt, so that tunnelling took place in a symmetric configuration [1]. In this paper, we study the quantum dynamics of a Bosonic Josephson junction which is tilted periodically in time. As we will show, such a periodically driven Bosonic Josephson junction displays a rich behaviour; in particular, in certain parameter regimes there exists an interesting interplay between the well-known self-trapping effect [2–5] and the coherent destruction of tunnelling [6,7]. An experimental realization of such a set-up, which should be possible with currently existing laboratory resources, would add a new tool to quantum optics
T. Jinasundera et al. / Chemical Physics 322 (2006) 118–126
with macroscopic matter waves. This tool might be of interest, e.g., for creating highly entangled many-body states, such as Schro¨dinger cat states [8]. We proceed as follows: in Section 2, we provide the details of our model system, and estimate experimentally accessible numerical values of its parameters. In Section 3, we outline an analytical approximation to the systems quasienergy spectrum, similar in spirit to the high-frequency approximation commonly employed in the context of coherent destruction of single-particle tunnelling [9–12]. Section 4 then is devoted to a discussion of the quantum dynamics of the driven Bosonic Josephson junction: we demonstrate the co-operative effect of self-trapping and coherent tunnelling destruction, and also consider the regime of low-frequency driving. Conclusions are drawn in the final Section 5.
2. The model We consider a symmetric double well potential V dw ð~ rÞ, with minima located on the x-axis at ~ rmin ¼ ðd; 0; 0Þt , separated by a distance 2d. The single-particle Hamiltonian h0 for a Boson of mass M then reads h0 ¼
2 h D þ V dw ð~ rÞ. 2M
ð1Þ
We assume that this double well is filled with N weakly interacting Bose particles at temperature T = 0. Moreover, we envision that the potential be modulated periodically in time with angular frequency x, such that the bottoms of the two wells are shifted up and down in phase opposition to each other with amplitude Fd; experimentally realistic values of d, x, and F will be specified later. Introducing the force vector ~ F , of magnitude F and directed along the x-axis, the many-body Hamiltonian for this system, a driven Bosonic Josephson junction, is given by Z H ðtÞ ¼ d3 r Wy ð~ rÞ h0 þ ~ F ~ r sinðxtÞ 1 4pa h2 y W ð~ rÞWð~ rÞ Wð~ rÞ; þ 2 M
ð2Þ
where Wð~ rÞ denotes the usual field operator in the Schro¨dinger picture, obeying the commutation relation ½Wð~ rÞ; Wy ð~ r0 Þ ¼ dð~ r ~ r0 Þ.
ð3Þ
Here, we tacitly assume that the interaction between the particles is described by the standard pseudopotential, the strength of which is proportional to their s-wave scattering length a. Although this pseudopotential is constructed for particles scattering from each other in the absence of a confining potential [13], it remains a good approximation as long as the characteristic length
119
scale over which the trap varies considerably remains substantially larger than a [14,15]. We now focus on the lowest doublet of single-particle energy eigenstates, rÞ ¼ E u ð~ rÞ; h0 u ð~
ð4Þ
assumed to be real, with the energy E+ of the symmetric ground state differing by the tunnel splitting hX from the energy E of the antisymmetric first excited state, E Eþ hX.
ð5Þ
The odd and even linear combinations 1 rÞ ¼ pffiffiffi ðuþ ð~ rÞ þ u ð~ rÞÞ; u1 ð~ 2 1 u2 ð~ rÞ ¼ pffiffiffi ðuþ ð~ rÞ u ð~ rÞÞ 2
ð6Þ
are localized in the first and second well, respectively; a brief calculation provides the single-particle matrix elements Z Z 1 3 d r u1 ð~ rÞh0 u1 ð~ rÞ ¼ d3 r u2 ð~ rÞh0 u2 ð~ rÞ ¼ ðEþ þ E Þ; 2 Z Z 1 d3 r u1 ð~ rÞh0 u2 ð~ rÞ ¼ d3 r u2 ð~ rÞh0 u1 ð~ rÞ ¼ hX. 2 ð7Þ Thus, the two-mode ansatz Wð~ rÞ ¼ a1 u1 ð~ rÞ þ a2 u2 ð~ rÞ
ð8Þ ðyÞ ai
annihilating (creating) a with the Bose operators particle in the ith well (i = 1, 2), leads to Z d3 r Wy ð~ rÞðh0 þ ~ F ~ r sinðxtÞÞWð~ rÞ hX y 1 ða1 a2 þ ay2 a1 Þ þ ðEþ þ E Þðay1 a1 þ ay2 a2 Þ 2 2 þ Fd sinðxtÞðay1 a1 ay2 a2 Þ;
¼
ð9Þ
where we have considered the functions u1;2 ð~ rÞ as approximate eigenfunctions of the position operator with eigenvalues (±d, 0, 0)t. Since the total number of particles is a constant of motion ay1 a1 þ ay2 a2 ¼ N ;
ð10Þ
the second term on the r.h.s. of Eq. (9) constitutes just an irrelevant overall energy shift, which will be neglected in the following. Moreover, we stipulate that the overlap of the functions u1 ð~ rÞ and u2 ð~ rÞ be only minute, implying that the condensates in the two wells are merely weakly coupled, and therefore employ the on-site approximation Z d3 r Wy ð~ rÞWy ð~ rÞWð~ rÞWð~ rÞ Z 4 ¼ d3 r u1 ð~ rÞ ðay1 ay1 a1 a1 þ ay2 ay2 a2 a2 Þ. ð11Þ
120
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Introducing the parameter Z 4pa h2 d3 r u1 ð~ 2hj ¼ rÞ4 M
ð12Þ
which quantifies the interaction energy for one pair of particles trapped in one of the wells, the dynamics of the driven Josephson junction now are given, within the two-mode approximation (8), by the model Hamiltonian [16] X y h ða a2 þ ay2 a1 Þ þ hjðay1 ay1 a1 a1 þ ay2 ay2 a2 a2 Þ 2 1 þ Fd sinðxtÞðay1 a1 ay2 a2 Þ. ð13Þ
H ðtÞ ¼
For estimating the relevant orders of magnitude, we simply assume that the two wells be about isotropic and harmonic around their bottoms, with oscillation frequency x0, and correspondingly set rÞ ¼ u0 ðx dÞu0 ðyÞu0 ðzÞ; u1 ð~
ð14Þ
where u0 ðxÞ ¼
1=4 Mx0 Mx0 2 x exp p h 2 h
ð15Þ
is the familiar harmonic-oscillator ground-state wave function. Herrings formula, which expresses the tunnelling splitting hX in terms of the current through a surface located between the potential minima [17,18], can then be written in the form Z 2M o hX ¼ 2 dA u1 ð~ rÞ u1 ð~ rÞ; ð16Þ ox h x¼0 where the area integral extends over the surface x = 0, leading immediately to 2 2 d d hX ¼ pffiffiffi hx0 exp 2 ð17Þ L p L with the usual oscillator length sffiffiffiffiffiffiffiffiffiffi h . L¼ Mx0 In addition, one has Z 1 4 d3 r u1 ð~ rÞ ¼ ; 3=2 ð2pÞ L3
ð18Þ
ð19Þ
giving 1 a hj ¼ hx0 pffiffiffiffiffiffi . 2p L
ð20Þ
quires that the distance 2d of the two wells be only slightly larger than the extension 2L of the groundstate wave functions, the exponential exp(+d2/L2)/23/2 will roughly be on the order of unity, so that j/X will be on the order of a/d. Since s-wave scattering lengths of alkali atoms usually are on the order of a few nanometers [19] – but can be widely tuned by means of Feshbach resonances [20,21] – and experimentally accessible double-well traps have a size of a few micrometers [1], we infer that the ratio j/X will typically be on the order of 103. In other words, the all-important dimensionless interaction parameter Nj/ X will typically be on the order of unity when there are N = 1000 trapped Bose particles. More specifically, we consider 87Rb atoms, which possess the scattering length a = 5.3 nm for the jF = 1, mF = 1i spin state, and take the distance between the two minima of the potential as p 2dffiffiffi= 4.4 lm, as in the experiment [1]. Assuming L ¼ d= 2, Eq. (18) gives the harmonic frequency x0 2p Æ 48 s1, while Eq. (17) results in the tunnelling frequency X 0.216x0 2p Æ 10.4 s1, corresponding to a single-particle tunnelling time Tspt 96 ms. Hence, tunnelling times on the order of several hundred milliseconds [1] are well within the reach of possibility. In addition, Eq. (20) quantifies the interaction frequency as j 0.41 s1, resulting in j/ X 6.3 · 103. Taking N 1150 atoms, as in the Heidelberg experiment [1], this amounts to Nj/X 7.2. In view of the crudeness of the underlying assumptions, this estimate is remarkably close to the experimentally determined value (Nj/X)exp 7.5 [22]. It is still mandatory to delineate the regime of validity of the two-mode model Hamiltonian (13). The basic two-mode approximation remains viable as long as the interaction energy per particle remains small compared to the harmonic level spacing hx0, thus enforcing Nhj < hx0. By virtue of the estimate (20), one finds the upper limit pffiffiffiffiffiffi N < 2pL=a ð22Þ on the particle number, requiring N < 736 under the conditions considered above. In addition, the potential drop Fd due to the external drive has to remain small compared to hx0. Thus, recalling Eq. (17), the associated Rabi frequency l = Fd/h has to comply with the condition pffiffiffi p expðþðd=LÞ2 Þ; l
In particular, combining Eqs. (17) and (20) now yields 2 j 1 a d ¼ 3=2 exp þ 2 . ð21Þ X 2 d L
which means l < 4.6X in our numerical example. Moreover, the driving frequency x should remain small compared to the harmonic frequency x0, similarly requiring
This elementary estimate already leads to an important conclusion: since an appreciable tunnelling contact re-
pffiffiffi p 2 expðþðd=LÞ Þ. x
ð24Þ
T. Jinasundera et al. / Chemical Physics 322 (2006) 118–126
Thus, driving frequencies x on the order of or even somewhat larger than the tunnelling frequency X may tentatively be regarded as ‘‘safe’’.
3. Quasienergy spectra for the driven Bosonic Josephson junction In order to obtain analytical insight into the dynamics generated by the Hamiltonian (13), we employ Schwingers method of relating Fock states associated with two independent sets of annihilation and creation operators to angular momentum states [23]: defining the operators 1 J x ¼ ðay1 a2 þ ay2 a1 Þ; 2 i J y ¼ ðay1 a2 ay2 a1 Þ; 2 1 J z ¼ ðay1 a1 ay2 a2 Þ; 2
ð25Þ
an elementary calculation verifies that they obey the angular momentum commutation rules ½J l ; J m ¼ ielmn J n ; while the Casimir invariant is given by N N þ1 . J 2x þ J 2y þ J 2z ¼ 2 2
ð26Þ
ð27Þ
Hence, a system of N particles described by the twomode Hamiltonian (13) corresponds to a conserved angular momentum of magnitude ‘ = N/2. Using these operators (25), the Schro¨dinger-picture Hamiltonian takes the form H ðtÞ ¼ H 0 ðtÞ þ H 1 ;
ð28Þ
cosðxtÞ i m jN ðN 2Þt j‘;mi jwm ðtÞi ¼ exp 2ijtm2 þ 2il x 2 i 1 2 ¼ exp 2hjm þ hjN ðN 2Þ t h 2 2l exp i cosðxtÞm j‘;mi; ð32Þ x
which have the form of Floquet states: in general, a Hamiltonian H(t) = H(t + T) depending periodically on time with period T admits, under mild conditions, a set of solutions of the form [9] i jwm ðtÞi ¼ jum ðtÞi exp em t ; ð33Þ h where the functions jum(t)i inherit the period of the Hamiltonian, jum ðtÞi ¼ jum ðt þ T Þi;
with constant coefficients cm; moreover, the Floquet states respect an adiabatic principle similar to the one for energy eigenstates [29,30]. In the present case (31), we have T = 2p/x, and read off from Eq. (32) the unperturbed Floquet functions 2l cosðxtÞm j‘; mi ð36Þ jum ðtÞi ¼ exp i x together with their quasienergies
1 jN ðN 2Þ ð29Þ H 0 ðtÞ ¼ 2 hjJ 2z þ 2 hl sinðxtÞJ z þ h 2 is diagonal in the standard basis {j‘, mi j ‘ = N/2; m = ‘, ‘ + 1, . . . , +‘} of angular momentum states, while the part describing the tunnelling of a particle from one well to the other,
1 emð0Þ ¼ 2hjm2 þ hjN ðN 2Þ. 2
ð30Þ
couples neighbouring states of this basis. In this angular momentum representation, the driven Josephson junction (28) thus resembles a model considered by van Hemmen et al. [24,25] in an investigation of mesoscopic quantum spin tunnelling; it is also related to the ‘‘kicked top’’ studied in great detail by Haake in the context of quantum chaos [26]. The ‘‘unperturbed’’ Schro¨dinger equation i h
o jwðtÞi ¼ H 0 ðtÞjwðtÞi ot
obviously possesses the solutions
ð31Þ
ð34Þ
and the quantities em are designated as ‘‘quasienergies’’ [27,28]. The principal importance of these solutions (33) rests in the fact that every solution jw(t)i to the time-dependent Schro¨dinger equation possesses an expansion X i jwðtÞi ¼ cm jum ðtÞi exp em t ð35Þ h m
where the operator
H 1 ¼ hXJ x ;
121
ð37Þ
The task now is to calculate the quasienergies em for the full Hamiltonian (28). Since quasienergies are obtained as solutions of the eigenvalue equation ðH ðtÞ ihot Þjua ðtÞi ¼ ea jua ðtÞi;
ð38Þ
this task can be accomplished by diagonalizing the ‘‘perturbation’’ H0 in the basis furnished by the states (32). It is at this point that an important technical aspect of the Floquet picture comes into play: the eigenvalue problem (38) is defined in an extended Hilbert space of explicitly time-dependent, T-periodic functions [31]; if the pair (jum(t)i, em) is a solution, so is (jum(t)ieinxt, em + n hx) for any n = 0, ±1, ±2, ±3 . . . That is, the set of solutions which is complete in the extended Hilbert space is obtained by multiplying each of the eigenfunctions jum(t)i by all Fourier factors einxt; functions jum(t)ieinxt with different values of the ‘‘photon index’’ n label different representatives of the same Floquet state. By the same token,
T. Jinasundera et al. / Chemical Physics 322 (2006) 118–126
The calculation of the exact quasienergies pertaining to the driven Josephson junction (28) thus requires the computation of infinitely many matrix elements hhureisxtjH1jumeinxtii. For sufficiently high driving frequencies, a simple approximation scheme suggests itself: At least if x is large compared to the single-particle tunnelling frequency X, and the interaction parameter Nj/X is sufficiently small, the periodic force will not give rise to major resonances. Under this condition one is entitled to neglect the ‘‘photon replicas’’ (that is, the Floquet satellites associated with different values of the index n) altogether, and to consider only the space spanned by the canonical representatives (36) of the unperturbed Floquet states. We are then left with the finite (N + 1) · (N + 1)-matrix Z 1 T hhur jH 1 jum ii ¼ dthur jH 1 jum i T 0 Z T hX 2l cosðxtÞðm rÞ dt exp i ¼ T 0 x h‘; rjJ x j‘; mi.
X ! XJ 0
k¼1
we see that the time average in Eq. (40) simply serves to ‘‘filter out’’ the zero-mode from this expansion, and find 2l ðm rÞ h‘; rjJ x j‘; mi. hhur jH 1 jum ii ¼ hXJ 0 ð42Þ x Taking into account that the angular momentum component Jx couples only nearest neighbours, we have nonzero matrix elements only for m r = ±1. Recalling that the zero-order Bessel function J0 is even, we thus arrive at 2l hhur jH 1 jum ii ¼ hXJ 0 ð43Þ h‘; rjJ x j‘; mi x and finally obtain, within our high-frequency approximation, the quasienergies em for the driven Josephson junction as eigenvalues of the matrix 2l ð0Þ M rm ¼ em drm hXJ 0 ð44Þ h‘; rjJ x j‘; mi. x Remarkably, this equals the matrix which gives the energy eigenvalues of the undriven junction (i.e., for Fd/ h = l = 0), with the replacement
ð45Þ
0.5
0
–0.5
ð40Þ
Recalling the generating function of the ordinary Bessel functions Jk(z), 1 X ixt k eiz cosðxtÞ ¼ ie J k ðzÞ; ð41Þ
2l . x
Thus, high-frequency driving corresponds to a ‘‘renormalization’’ of the single-particle tunnelling splitting X by the zero-order Bessel function J0(2l/x). Fig. 1 shows exact quasienergies for the model (28), computed numerically for N = 20 particles, interaction parameter Nj/X = 0.7, and high driving frequency x/X = 10, while Fig. 2 displays the result of the approximation (44) for the same situation. As expected, the agreement is excellent. These figures also visualize another important feature: since the undriven junction corresponds, apart from an energy shift, to the Hamiltonian [32,33]
ε/(h ω)
the quasienergy spectrum repeats itself periodically on the energy axis, each ‘‘Brillouin zone’’ of width hx containing one quasienergy representative of each state. The scalar product hhÆ j Æii in the extended Hilbert space combines the usual scalar product with time averaging, Z 1 T hh j ii ¼ dthji. ð39Þ T 0
0
5 2µ/ω
10
Fig. 1. One Brillouin zone of exact quasienergies, computed numerically, for the driven Bosonic Josephson junction (13) with N = 20 particles, interaction parameter Nj/X = 0.7, and high driving frequency x/X = 10, as functions of the scaled driving amplitude 2l/x. Observe the characteristic ‘‘pairing’’ and ‘‘unpairing’’ of states in the vicinity of the values 2.4, 5.5, and 8.7 of that amplitude.
0.5
ε/(h ω)
122
0
–0.5
0
5 2µ/ω
10
Fig. 2. Approximate quasienergies, computed according to the highfrequency scheme (44), for the same situation as considered in Fig. 1.
T. Jinasundera et al. / Chemical Physics 322 (2006) 118–126
H un ¼ hXJ x þ 2hjJ 2z ;
ð46Þ
4. Tunnelling dynamics of a driven condensate We start our discussion of condensate tunnelling in a driven Bosonic Josephson junction by monitoring the 0.5
0.5
ε/(h ω)
its eigenvalues are twofold (almost) degenerate (except for the state with m = 0) in the regime where the part 2 hjJ 2z describing the on-site interaction between the particles dominates, while they remain non-degenerate in the regime of weak interaction, where the Josephson coupling hXJx governs the dynamics. This competition is clearly reflected in the figures: when the scaled amplitude 2l/x is close to the zeros j0,1 2.4048, j0,2 5.5201, or j0,3 8.6537 of J0, the ‘‘renormalized’’ tunnelling frequency approaches zero. Hence, the interaction then dominates, regardless of its numerical strength or weakness, and leads to the appearance of pairwise degenerate Floquet states. In Fig. 3 we plot exact quasienergies for a lower frequency x/X = 3, where the ‘‘regular’’ approximation (44) still gives good results. In contrast, Fig. 4 highlights an example where this approximation breaks down: here the driving frequency equals the single-particle tunnelling frequency, x/X = 1, so that the oscillating force gives rise to a multitude of resonances between ac-Stark-shifted (quasi)energy eigenstates, resulting, even for only N = 20 particles, in an intricate avoided crossing-pattern, commonly viewed as an indication of quantum chaos [26]. With N = 1000 particles, each one having a quasienergy representative in each Brillouin zone, the spectrum would be close to a nightmare. In the following section, we will explore the consequences of this change of the quasienergy spectrum from ‘‘regular’’ to ‘‘chaotic’’ for the quantum N-particle dynamics in the driven junction.
0
–0.5
0
ε/(h ω)
–0.5
0
5 2µ/ω
10
Fig. 3. Exact quasienergies for the driven Bosonic Josephson junction (13) with N = 20 particles, interaction parameter Nj/X = 0.7, and moderate driving frequency x/X = 3. Under these conditions, the high-frequency approximation (44) still works reasonably well.
1
2
3
2µ/ω Fig. 4. Exact quasienergies for the driven Bosonic Josephson junction (13) with N = 20 particles, interaction parameter Nj/X = 0.7, and intermediate driving frequency x/X = 1. The plethora of avoided level crossings is a hallmark of quantum chaos [26]. Observe how the scale of the abscissa differs from that of the previous figures.
time evolution of a system with N = 1000 particles initially all stored in one of the wells, say well No. 1. To this end, we expand the many-body wave function jw(t)i in a basis of Fock states {jm, N mijm = 0, . . . , N} with m particles in the first and N m particles in the second well, specify the initial condition j wðt ¼ 0Þi ¼j N ; 0i;
ð47Þ
and solve the time-dependent Schro¨dinger equation numerically. According to the definition (25), the angular momentum component Jz directly corresponds to the imbalance of the particle number in both wells. We therefore visualize the dynamics by plotting the expectation values hJ z i hwðtÞ j J z j wðtÞi
0
123
ð48Þ
as functions of time, with hJzi/N = +1/2 (or 1/2) signalling ‘‘all particles in the first (or second) well’’. Fig. 5 shows the results of such calculations for three different situations; in all cases, the interaction parameter is Nj/X = 0.7. The dashed line refers to the absence of the driving force; here the condensate undergoes the usual Josephson oscillation [1]. As a consequence of the discreteness of the N-particle energy spectrum, these oscillations appear to be damped. Technically speaking, this damping is a collapse which will be followed by revivals after sufficiently long evolution times. However, we surmise that such revivals will not be observable within experimentally realistic timescales; observe that Xt = 100 corresponds to t 1.6 s for X = 2p Æ 10 s1. The lower, irregularly looking full line in Fig. 5, which more or less averages to zero population imbalance in the course of time, corresponds to an oscillating force
124
T. Jinasundera et al. / Chemical Physics 322 (2006) 118–126
responding to the initial condition (47), that critical value is jNj/Xjc = 1. This fact, together with the highfrequency renormalization (45) of the tunnelling frequency X, immediately leads to a striking prediction: in the presence of an oscillating force with sufficiently high frequency, tunnelling of the condensate will be suppressed not only at those isolated parameters where the scaled amplitude 2l/x equals a zero of J0, but rather in those parameter regimes in which Nj ð49Þ XJ ð2l=xÞ > 1.
/ N
0.5
0
0
0
50 Ωt
100
Fig. 5. Time evolution of the population imbalance hJzi/N in a Bosonic Josephson junction occupied by N = 1000 particles, with interaction parameter Nj/X = 0.7. Initially all particles had been stored in the first well, as corresponding to hJzi/N = 0.5 at Xt = 0. The dashed line refers to the case without driving, the upper full line to driving with amplitude 2l/x = 2.40 and frequency x/X = 3.0, while the lower full line corresponds to driving with the same amplitude 2l/x = 2.40, but low frequency x/X = 1.0.
with amplitude 2l/x = 2.40, close to the first zero of the J0 Bessel function, and low driving frequency x/X = 1.0. Under this condition, the high-frequency renormalization (45) of the tunnelling splitting is not effective, as witnessed by the glaring discrepancy between the true, chaotic quasienergy spectrum displayed in Fig. 4 and the regular high-frequency approximation (44). Thus, even at 2l/x = 2.40 the particles are not kept in the initially occupied well, but tunnel to the other to a significant extent, such that both wells tend to be populated equally after some time. In contrast, the situation is quite different for x/X = 3.0 and 2l/x = 2.40: as indicated by the jagged full line at the top of Fig. 5, now ‘‘coherent destruction of tunnelling’’ takes place, confining the condensate almost fully to the well in which it had been prepared. In the absence of the periodic force, the two-mode model for a condensate in a double well exhibits an effect known as ‘‘self trapping’’: if the absolute value of the interaction parameter Nj/X exceeds a certain critical value, the coherent Josephson oscillation is suppressed and the initial population imbalance is almost preserved in time [3–5]. This effect, which has now been verified in the pioneering experiment by Albiez et al. [1], had previously also been discussed in the context of transport on a dimer [2]. It can be deduced from the discrete nonlinear Schro¨dinger equation which corresponds to the Gross–Pitaevskii mean-field approximation for the two-site model; however, it effectively persists even on the N-particle level [34]. The critical value of jNj/Xj above which self-trapping occurs depends on the initial population imbalance; for maximum imbalance, as cor-
for an averaging interval Dt = 100/X. Fig. 6 summarizes the results. Since values of hJzit/N close to +1/2 correspond to long-lasting trapping in the initially occupied well, while values close to zero indicate strong tunnelling, the expectation is fully confirmed: The full line in Fig. 6 corresponds to a weakly interacting gas, Nj/X = 0.1, so that the trapping condition (49) is satisfied only in narrow intervals around the zeros of J0. The short-dashed line is obtained for Nj/X = 0.3; accordingly, the trapping interval around the first J0-zero is wider, and there is no full equilibration of the population for scaled amplitudes between the second and the third zero, since the Bessel function is not large enough there to enable unhindered tunnelling. This tendency is more pronounced still for Nj/X = 0.5 (long dashes);
0.5
t / N
–0.5
To verify this prediction, we start again from the initial condition (47), and compute the time-averaged population imbalance Z Dt 1 hJ z it dthwðtÞjJ z jwðtÞi ð50Þ Dt 0
0.0
0
5 2µ/ω
10
Fig. 6. Time-averaged population imbalance hJzit/N for a Bosonic Josephson junction occupied by N = 100 particles and driven with frequency x/X = 3.0. Initially all particles had been stored in the first well; the averaging interval is D(Xt) = 100. The interaction parameters are Nj/X = 0.1 (full line); 0.3 (short dashes), 0.5 (long dashes), and 0.7 (dots).
T. Jinasundera et al. / Chemical Physics 322 (2006) 118–126
t / N
0.5
0.0
0
3 ω/Ω
6
Fig. 7. Time-averaged population imbalance hJzit/N for a Bosonic Josephson junction occupied by N = 100 particles, when the scaled amplitude 2l/x is kept fixed at the first (full line), second (short dashes) or third zero (long dashes) of the Bessel function J0. Initially all particles had been stored in the first well; the averaging interval is D(Xt) = 100. The interaction parameter is Nj/X = 0.7.
0.5
/ N
for Nj/X = 0.7 (dots) one even comes close to a clearcut yes/no-alternative: for scaled amplitudes below 1.2 the condensate tunnels vividly, but remains trapped for higher amplitudes. The question then remains just when the driving frequency x/X can be considered as ‘‘sufficiently high’’ to effectuate this interplay between coherent tunnelling destruction and self-trapping. The answer is given in Fig. 7, where we have plotted the time-averaged population imbalance (50) as a function of the scaled frequency x/X, for scaled amplitudes equalling the first (full line), second (short dashes), and third zero (long dashes) of J0. As can be seen, for scaled frequencies x/X of about 1.0 tunnelling takes place, with a somewhat irregular dependence on the frequency, while tunnelling suppression sets in roughly at x/X = 1.5. For scaled frequencies of magnitude 2.0 or higher, the suppression is already quite strong, allowing one to prevent the condensate from spilling to the other well. Since the absolute value of J0(2l/x) is less than unity for nonvanishing amplitude, high-frequency driving can only quench the tunnelling effect. This is different for low frequencies: in Fig. 8, we exemplarily display the time evolution of the population imbalance for x/X = 0.25 and interaction parameters Nj/X = 1.0 (dotted), 3.0 (full line), and 5.0 (dot-dashed); the driving amplitude is l/X = 5.0 in all three cases. There are marked population jumps at about those moments when the two wells are energetically aligned, initially accompanied by rapid oscillations. Since in the absence of the periodic force one has self-trapping for Nj/X > 1, and energetic alignment exists only during short time intervals when such a force is present, one might have
125
0
–0.5
0
50 Ωt
100
Fig. 8. Time evolution of the population imbalance hJzi/N in a Bosonic Josephson junction occupied by N = 1000 particles under lowfrequency driving, x/X = 0.25, with amplitude l/X = 5.0. The interaction parameters are Nj/X = 1.0 (dotted line), 3.0 (full line), and 5.0 (dot-dashed). Initially all particles had been stored in the first well.
expected self-trapping to persist a fortiori under the influence of low-frequency driving. However, as Fig. 8 clearly demonstrates, this naive expectation is incorrect: even for Nj/X = 5.0 the population can equilibrate after a few jumps.
5. Conclusion The recent observation [1] of coherent Josephson oscillations, and of the self-trapping effect, with a mesoscopic sample of Bose–Einstein-condensed atoms in an optical double well potential has paved the road to further developments aiming at the coherent control of quantum many-body dynamics. As we have shown in this paper, a suitable periodic modulation of the double well gives rise to several novel features: When the modulation frequency is at least two times higher than the bare single-particle tunnelling frequency, the interplay of self-trapping and coherent destruction of tunnelling allows one to effectively shut off the tunnelling contact between the wells by adjusting the parameters of the drive; this effect is illustrated most clearly in Fig. 6. For driving frequencies about equal to the tunnelling frequency, the dynamics exhibits signatures of quantum chaos, such as the irregular oscillations displayed in Fig. 5. For still lower frequencies, the modulation can induce a sequence of rapid population jumps, as in Fig. 8, leading to a partial population transfer even when there would be strong self-trapping without the modulation. We therefore expect that future experiments along the lines sketched here will yield considerable new insight into the response of Bose–Einstein condensates to nonperturbatively strong forcing.
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The main restriction of the current work lies in the use of the two-mode approximation (8). It should be clear that a frequency as high as x/X = 10.0, employed in Figs. 1 and 2 to demonstrate the consistency of the approximation (44), is likely to fall outside the regime in which the two-mode approximation holds. However, our estimates at the end of Section 2 also indicate that moderately high frequencies are still viable. Moreover, it should be noted that even with only N = 1000 Bose particles the condition (22) is stretched to its very limit in standard situations. But given the enormous tunability of both the optical potentials and the atomic s-wave scattering lengths [20,21], and thus of both parameters X (or L) and j separately, together with the fact that the two-mode approximation has been shown to capture the dynamics at least qualitatively even under the conditions of the Heidelberg experiment [1], one may be confident that the effects predicted in this paper do not defy experimental verification. One further observation deserves to be mentioned. The ratio of the on-site interaction frequency j, defined in Eq. (12), and the single-particle tunnelling frequency X also governs the transition from a superfluid to a Mott insulator undergone by a Bose–Einstein condensate in an optical lattice when that lattice is made deeper [35,36]. The renormalization (45) of the tunnelling frequency, discussed in the present work for the case of a single Josephson link, thus prompts the question whether a similar quenching of X would also be effective in modulated optical lattices, so that the superfluid–insulator transition could actually be controlled by means of such a modulation. This question will be taken up, and answered affirmatively, in a subsequent publication [37].
Acknowledgements We thank M.K. Oberthaler for a discussion of his experiment [1], and A. Eckardt for insightful remarks. This work was supported by the Deutsche Forschungsgemeinschaft through the Priority Programme SPP 1116, Wechselwirkung in ultrakalten Atom- und Moleku¨lgasen.
References [1] M. Albiez, R. Gati, J. Fo¨lling, S. Hunsmann, M. Cristiani, M.K. Oberthaler, Phys. Rev. Lett. 95 (2005) 010402.
[2] V.M. Kenkre, D.K. Campbell, Phys. Rev. B 34 (1986) 4959. [3] G.J. Milburn, J. Corney, E.M. Wright, D.F. Walls, Phys. Rev. A 55 (1997) 4318. [4] A. Smerzi, S. Fantoni, S. Giovanazzi, S.R. Shenoy, Phys. Rev. Lett. 79 (1997) 4950. [5] S. Raghavan, A. Smerzi, S. Fantoni, S.R. Shenoy, Phys. Rev. A 59 (1999) 620. [6] F. Grossmann, T. Dittrich, P. Jung, P. Ha¨nggi, Phys. Rev. Lett. 67 (1991) 516. [7] M. Grifoni, P. Ha¨nggi, Phys. Rep. 304 (1998) 229. [8] C. Weiss, T. Jinasundera, Coherent control of mesoscopic tunneling (Preprint, University of Oldenburg, 2004). [9] J.H. Shirley, Phys. Rev. 138 (1965) B979. [10] J.M. Gomez Llorente, J. Plata, Phys. Rev. A 45 (1992) R6958, Erratum: Phys. Rev. E 49 (1994) 3547. [11] F. Grossmann, P. Ha¨nggi, Europhys. Lett. 18 (1992) 571. [12] M. Holthaus, Z. Phys. B 89 (1992) 251. [13] K. Huang, C.N. Yang, Phys. Rev. 105 (1957) 767. [14] M. Block, M. Holthaus, Phys. Rev. A 65 (2002) 052102. [15] J. Pade, M. Block, M. Holthaus, Phys. Rev. A 68 (2003) 063402. [16] M. Holthaus, Phys. Rev. A 64 (2001) 011601. [17] L.D. Landau, E.M. Lifshitz, Quantum Mechanics, Butterworth– Heinemann, Oxford, 1997, Section 50. [18] C. Herring, Rev. Mod. Phys. 34 (1962) 631. [19] C.J. Pethik, H. Smith, Bose–Einstein Condensation in Dilute Gases, Cambridge University Press, Cambridge, MA, 2002. [20] S. Inouye, M.R. Andrews, J. Stenger, H.-J. Miesner, D.M. Stamper-Kurn, W. Ketterle, Nature 392 (1998) 151. [21] S.L. Cornish, N.R. Claussen, J.L. Roberts, E.A. Cornell, C.E. Wieman, Phys. Rev. Lett. 85 (2000) 1795. [22] In Ref. [1] the parameter K = 2Nj/X is used; the experimental data correspond to K 15. [23] J. Schwinger, On Angular Momentum (AEC Report NYO-3071, 1952). [24] J.L. van Hemmen, A. Su¨t} o, J. Phys.: Condens. Matter 9 (1997) 3089. [25] J.L. van Hemmen, H. Hey, W.F. Wreszinski, J. Phys. A 30 (1997) 6371. [26] F. Haake, Quantum Signatures of Chaos, Springer Series in Synergetics 54, Berlin, 2004. [27] Ya.B. Zeldovich, Sov. Phys. JETP 24 (1967) 1006 [J. Exp. Theor. Phys. (U.S.S.R.) 51, 1492 (1966)]. [28] V.I. Ritus, Sov. Phys. JETP 24 (1967) 1041 [J. Exp. Theor. Phys. (U.S.S.R.) 51, 1544 (1966)]. [29] H.P. Breuer, M. Holthaus, Z. Phys. D 11 (1989) 1. [30] K. Drese, M. Holthaus, Eur. Phys. J. D 5 (1999) 119. [31] H. Sambe, Phys. Rev. A 7 (1973) 2203. [32] G. Kalosakas, A.R. Bishop, Phys. Rev. A 65 (2002) 043616. [33] G. Kalosakas, A.R. Bishop, V.M. Kenkre, Phys. Rev. A 68 (2003) 023602. [34] M. Holthaus, S. Stenholm, Eur. Phys. J. B 20 (2001) 451. [35] M. Greiner, O. Mandel, T. Esslinger, T.W. Ha¨nsch, I. Bloch, Nature 415 (2002) 39. [36] W. Zwerger, J. Opt. B: Quantum Semiclass. Opt. 5 (2003) S9. [37] A. Eckardt, C. Weiss, M. Holthaus, Superfluid–insulator transition in periodically driven optical lattices (Preprint, University of Oldenburg, 2005).
Many-particle tunnelling in a driven Bosonic Josephson junction Tharanga Jinasundera, Christoph Weiss, Martin Holthaus
*
Institut fu¨r Physik, Carl von Ossietzky Universita¨t, D-26111 Oldenburg, Germany Received 25 March 2005; accepted 5 June 2005 Available online 25 July 2005
Abstract We study the quantum many-body dynamics of a Bose–Einstein condensate in a double well potential subjected to a time-periodic modulation. For modulation frequencies moderately higher than the single-particle tunnelling frequency, this system displays an interplay of self-trapping and coherent tunnelling destruction, allowing one to effectively shut off the tunnelling contact in extended parameter regimes. For modulation frequencies roughly equalling the tunnelling frequency, one encounters chaotic dynamics, and finds sudden population jumps in the regime of low-frequency driving. A driven Bosonic Josephson junction thus constitutes a promising tool for the coherent control of mesoscopic matter waves. Our predictions can be verified under presently accessible experimental conditions. 2005 Elsevier B.V. All rights reserved. PACS: 03.75.Lm; 05.45.Mt Keywords: Bose–Einstein condensation; Tunnelling effect; Many-body dynamics; Quantum Floquet theory
1. Introduction In a recent experiment, Albiez et al. [1] have observed the tunnelling effect exhibited by weakly coupled Bose–Einstein condensates in a double well potential. After preparing a sample of about 1150 Bose–Einstein-condensed 87Rb atoms in an optical double well with a distance of 4.4 lm between the two minima, these authors were able to measure in situ both the density distribution associated with the tunnelling macroscopic wave function, and the relative phase between the condensates in the two wells. Their system, the first realization of a controllable, single Bosonic Josephson junction, thus allows one to investigate in detail the effect of many-body interactions on the tunnelling pro* Corresponding author. Tel.: +49 441 798 3960; fax: +49 441 798 3080. E-mail address: [email protected] (M. Holthaus).
0301-0104/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2005.06.025
cess, and to compare the actual condensate dynamics to the mean-field approximation. Albiez et al. have loaded the condensate into an asymmetric double well, shaped such that an initial tilt prevented unwanted tunnelling during the preparation phase. This tilt could be controlled by adjusting a focussed laser beam by means of a piezo-actuated mirror; the dynamics was initiated by non-adiabatically shifting to zero tilt, so that tunnelling took place in a symmetric configuration [1]. In this paper, we study the quantum dynamics of a Bosonic Josephson junction which is tilted periodically in time. As we will show, such a periodically driven Bosonic Josephson junction displays a rich behaviour; in particular, in certain parameter regimes there exists an interesting interplay between the well-known self-trapping effect [2–5] and the coherent destruction of tunnelling [6,7]. An experimental realization of such a set-up, which should be possible with currently existing laboratory resources, would add a new tool to quantum optics
T. Jinasundera et al. / Chemical Physics 322 (2006) 118–126
with macroscopic matter waves. This tool might be of interest, e.g., for creating highly entangled many-body states, such as Schro¨dinger cat states [8]. We proceed as follows: in Section 2, we provide the details of our model system, and estimate experimentally accessible numerical values of its parameters. In Section 3, we outline an analytical approximation to the systems quasienergy spectrum, similar in spirit to the high-frequency approximation commonly employed in the context of coherent destruction of single-particle tunnelling [9–12]. Section 4 then is devoted to a discussion of the quantum dynamics of the driven Bosonic Josephson junction: we demonstrate the co-operative effect of self-trapping and coherent tunnelling destruction, and also consider the regime of low-frequency driving. Conclusions are drawn in the final Section 5.
2. The model We consider a symmetric double well potential V dw ð~ rÞ, with minima located on the x-axis at ~ rmin ¼ ðd; 0; 0Þt , separated by a distance 2d. The single-particle Hamiltonian h0 for a Boson of mass M then reads h0 ¼
2 h D þ V dw ð~ rÞ. 2M
ð1Þ
We assume that this double well is filled with N weakly interacting Bose particles at temperature T = 0. Moreover, we envision that the potential be modulated periodically in time with angular frequency x, such that the bottoms of the two wells are shifted up and down in phase opposition to each other with amplitude Fd; experimentally realistic values of d, x, and F will be specified later. Introducing the force vector ~ F , of magnitude F and directed along the x-axis, the many-body Hamiltonian for this system, a driven Bosonic Josephson junction, is given by Z H ðtÞ ¼ d3 r Wy ð~ rÞ h0 þ ~ F ~ r sinðxtÞ 1 4pa h2 y W ð~ rÞWð~ rÞ Wð~ rÞ; þ 2 M
ð2Þ
where Wð~ rÞ denotes the usual field operator in the Schro¨dinger picture, obeying the commutation relation ½Wð~ rÞ; Wy ð~ r0 Þ ¼ dð~ r ~ r0 Þ.
ð3Þ
Here, we tacitly assume that the interaction between the particles is described by the standard pseudopotential, the strength of which is proportional to their s-wave scattering length a. Although this pseudopotential is constructed for particles scattering from each other in the absence of a confining potential [13], it remains a good approximation as long as the characteristic length
119
scale over which the trap varies considerably remains substantially larger than a [14,15]. We now focus on the lowest doublet of single-particle energy eigenstates, rÞ ¼ E u ð~ rÞ; h0 u ð~
ð4Þ
assumed to be real, with the energy E+ of the symmetric ground state differing by the tunnel splitting hX from the energy E of the antisymmetric first excited state, E Eþ hX.
ð5Þ
The odd and even linear combinations 1 rÞ ¼ pffiffiffi ðuþ ð~ rÞ þ u ð~ rÞÞ; u1 ð~ 2 1 u2 ð~ rÞ ¼ pffiffiffi ðuþ ð~ rÞ u ð~ rÞÞ 2
ð6Þ
are localized in the first and second well, respectively; a brief calculation provides the single-particle matrix elements Z Z 1 3 d r u1 ð~ rÞh0 u1 ð~ rÞ ¼ d3 r u2 ð~ rÞh0 u2 ð~ rÞ ¼ ðEþ þ E Þ; 2 Z Z 1 d3 r u1 ð~ rÞh0 u2 ð~ rÞ ¼ d3 r u2 ð~ rÞh0 u1 ð~ rÞ ¼ hX. 2 ð7Þ Thus, the two-mode ansatz Wð~ rÞ ¼ a1 u1 ð~ rÞ þ a2 u2 ð~ rÞ
ð8Þ ðyÞ ai
annihilating (creating) a with the Bose operators particle in the ith well (i = 1, 2), leads to Z d3 r Wy ð~ rÞðh0 þ ~ F ~ r sinðxtÞÞWð~ rÞ hX y 1 ða1 a2 þ ay2 a1 Þ þ ðEþ þ E Þðay1 a1 þ ay2 a2 Þ 2 2 þ Fd sinðxtÞðay1 a1 ay2 a2 Þ;
¼
ð9Þ
where we have considered the functions u1;2 ð~ rÞ as approximate eigenfunctions of the position operator with eigenvalues (±d, 0, 0)t. Since the total number of particles is a constant of motion ay1 a1 þ ay2 a2 ¼ N ;
ð10Þ
the second term on the r.h.s. of Eq. (9) constitutes just an irrelevant overall energy shift, which will be neglected in the following. Moreover, we stipulate that the overlap of the functions u1 ð~ rÞ and u2 ð~ rÞ be only minute, implying that the condensates in the two wells are merely weakly coupled, and therefore employ the on-site approximation Z d3 r Wy ð~ rÞWy ð~ rÞWð~ rÞWð~ rÞ Z 4 ¼ d3 r u1 ð~ rÞ ðay1 ay1 a1 a1 þ ay2 ay2 a2 a2 Þ. ð11Þ
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Introducing the parameter Z 4pa h2 d3 r u1 ð~ 2hj ¼ rÞ4 M
ð12Þ
which quantifies the interaction energy for one pair of particles trapped in one of the wells, the dynamics of the driven Josephson junction now are given, within the two-mode approximation (8), by the model Hamiltonian [16] X y h ða a2 þ ay2 a1 Þ þ hjðay1 ay1 a1 a1 þ ay2 ay2 a2 a2 Þ 2 1 þ Fd sinðxtÞðay1 a1 ay2 a2 Þ. ð13Þ
H ðtÞ ¼
For estimating the relevant orders of magnitude, we simply assume that the two wells be about isotropic and harmonic around their bottoms, with oscillation frequency x0, and correspondingly set rÞ ¼ u0 ðx dÞu0 ðyÞu0 ðzÞ; u1 ð~
ð14Þ
where u0 ðxÞ ¼
1=4 Mx0 Mx0 2 x exp p h 2 h
ð15Þ
is the familiar harmonic-oscillator ground-state wave function. Herrings formula, which expresses the tunnelling splitting hX in terms of the current through a surface located between the potential minima [17,18], can then be written in the form Z 2M o hX ¼ 2 dA u1 ð~ rÞ u1 ð~ rÞ; ð16Þ ox h x¼0 where the area integral extends over the surface x = 0, leading immediately to 2 2 d d hX ¼ pffiffiffi hx0 exp 2 ð17Þ L p L with the usual oscillator length sffiffiffiffiffiffiffiffiffiffi h . L¼ Mx0 In addition, one has Z 1 4 d3 r u1 ð~ rÞ ¼ ; 3=2 ð2pÞ L3
ð18Þ
ð19Þ
giving 1 a hj ¼ hx0 pffiffiffiffiffiffi . 2p L
ð20Þ
quires that the distance 2d of the two wells be only slightly larger than the extension 2L of the groundstate wave functions, the exponential exp(+d2/L2)/23/2 will roughly be on the order of unity, so that j/X will be on the order of a/d. Since s-wave scattering lengths of alkali atoms usually are on the order of a few nanometers [19] – but can be widely tuned by means of Feshbach resonances [20,21] – and experimentally accessible double-well traps have a size of a few micrometers [1], we infer that the ratio j/X will typically be on the order of 103. In other words, the all-important dimensionless interaction parameter Nj/ X will typically be on the order of unity when there are N = 1000 trapped Bose particles. More specifically, we consider 87Rb atoms, which possess the scattering length a = 5.3 nm for the jF = 1, mF = 1i spin state, and take the distance between the two minima of the potential as p 2dffiffiffi= 4.4 lm, as in the experiment [1]. Assuming L ¼ d= 2, Eq. (18) gives the harmonic frequency x0 2p Æ 48 s1, while Eq. (17) results in the tunnelling frequency X 0.216x0 2p Æ 10.4 s1, corresponding to a single-particle tunnelling time Tspt 96 ms. Hence, tunnelling times on the order of several hundred milliseconds [1] are well within the reach of possibility. In addition, Eq. (20) quantifies the interaction frequency as j 0.41 s1, resulting in j/ X 6.3 · 103. Taking N 1150 atoms, as in the Heidelberg experiment [1], this amounts to Nj/X 7.2. In view of the crudeness of the underlying assumptions, this estimate is remarkably close to the experimentally determined value (Nj/X)exp 7.5 [22]. It is still mandatory to delineate the regime of validity of the two-mode model Hamiltonian (13). The basic two-mode approximation remains viable as long as the interaction energy per particle remains small compared to the harmonic level spacing hx0, thus enforcing Nhj < hx0. By virtue of the estimate (20), one finds the upper limit pffiffiffiffiffiffi N < 2pL=a ð22Þ on the particle number, requiring N < 736 under the conditions considered above. In addition, the potential drop Fd due to the external drive has to remain small compared to hx0. Thus, recalling Eq. (17), the associated Rabi frequency l = Fd/h has to comply with the condition pffiffiffi p expðþðd=LÞ2 Þ; l
In particular, combining Eqs. (17) and (20) now yields 2 j 1 a d ¼ 3=2 exp þ 2 . ð21Þ X 2 d L
which means l < 4.6X in our numerical example. Moreover, the driving frequency x should remain small compared to the harmonic frequency x0, similarly requiring
This elementary estimate already leads to an important conclusion: since an appreciable tunnelling contact re-
pffiffiffi p 2 expðþðd=LÞ Þ. x
ð24Þ
T. Jinasundera et al. / Chemical Physics 322 (2006) 118–126
Thus, driving frequencies x on the order of or even somewhat larger than the tunnelling frequency X may tentatively be regarded as ‘‘safe’’.
3. Quasienergy spectra for the driven Bosonic Josephson junction In order to obtain analytical insight into the dynamics generated by the Hamiltonian (13), we employ Schwingers method of relating Fock states associated with two independent sets of annihilation and creation operators to angular momentum states [23]: defining the operators 1 J x ¼ ðay1 a2 þ ay2 a1 Þ; 2 i J y ¼ ðay1 a2 ay2 a1 Þ; 2 1 J z ¼ ðay1 a1 ay2 a2 Þ; 2
ð25Þ
an elementary calculation verifies that they obey the angular momentum commutation rules ½J l ; J m ¼ ielmn J n ; while the Casimir invariant is given by N N þ1 . J 2x þ J 2y þ J 2z ¼ 2 2
ð26Þ
ð27Þ
Hence, a system of N particles described by the twomode Hamiltonian (13) corresponds to a conserved angular momentum of magnitude ‘ = N/2. Using these operators (25), the Schro¨dinger-picture Hamiltonian takes the form H ðtÞ ¼ H 0 ðtÞ þ H 1 ;
ð28Þ
cosðxtÞ i m jN ðN 2Þt j‘;mi jwm ðtÞi ¼ exp 2ijtm2 þ 2il x 2 i 1 2 ¼ exp 2hjm þ hjN ðN 2Þ t h 2 2l exp i cosðxtÞm j‘;mi; ð32Þ x
which have the form of Floquet states: in general, a Hamiltonian H(t) = H(t + T) depending periodically on time with period T admits, under mild conditions, a set of solutions of the form [9] i jwm ðtÞi ¼ jum ðtÞi exp em t ; ð33Þ h where the functions jum(t)i inherit the period of the Hamiltonian, jum ðtÞi ¼ jum ðt þ T Þi;
with constant coefficients cm; moreover, the Floquet states respect an adiabatic principle similar to the one for energy eigenstates [29,30]. In the present case (31), we have T = 2p/x, and read off from Eq. (32) the unperturbed Floquet functions 2l cosðxtÞm j‘; mi ð36Þ jum ðtÞi ¼ exp i x together with their quasienergies
1 jN ðN 2Þ ð29Þ H 0 ðtÞ ¼ 2 hjJ 2z þ 2 hl sinðxtÞJ z þ h 2 is diagonal in the standard basis {j‘, mi j ‘ = N/2; m = ‘, ‘ + 1, . . . , +‘} of angular momentum states, while the part describing the tunnelling of a particle from one well to the other,
1 emð0Þ ¼ 2hjm2 þ hjN ðN 2Þ. 2
ð30Þ
couples neighbouring states of this basis. In this angular momentum representation, the driven Josephson junction (28) thus resembles a model considered by van Hemmen et al. [24,25] in an investigation of mesoscopic quantum spin tunnelling; it is also related to the ‘‘kicked top’’ studied in great detail by Haake in the context of quantum chaos [26]. The ‘‘unperturbed’’ Schro¨dinger equation i h
o jwðtÞi ¼ H 0 ðtÞjwðtÞi ot
obviously possesses the solutions
ð31Þ
ð34Þ
and the quantities em are designated as ‘‘quasienergies’’ [27,28]. The principal importance of these solutions (33) rests in the fact that every solution jw(t)i to the time-dependent Schro¨dinger equation possesses an expansion X i jwðtÞi ¼ cm jum ðtÞi exp em t ð35Þ h m
where the operator
H 1 ¼ hXJ x ;
121
ð37Þ
The task now is to calculate the quasienergies em for the full Hamiltonian (28). Since quasienergies are obtained as solutions of the eigenvalue equation ðH ðtÞ ihot Þjua ðtÞi ¼ ea jua ðtÞi;
ð38Þ
this task can be accomplished by diagonalizing the ‘‘perturbation’’ H0 in the basis furnished by the states (32). It is at this point that an important technical aspect of the Floquet picture comes into play: the eigenvalue problem (38) is defined in an extended Hilbert space of explicitly time-dependent, T-periodic functions [31]; if the pair (jum(t)i, em) is a solution, so is (jum(t)ieinxt, em + n hx) for any n = 0, ±1, ±2, ±3 . . . That is, the set of solutions which is complete in the extended Hilbert space is obtained by multiplying each of the eigenfunctions jum(t)i by all Fourier factors einxt; functions jum(t)ieinxt with different values of the ‘‘photon index’’ n label different representatives of the same Floquet state. By the same token,
T. Jinasundera et al. / Chemical Physics 322 (2006) 118–126
The calculation of the exact quasienergies pertaining to the driven Josephson junction (28) thus requires the computation of infinitely many matrix elements hhureisxtjH1jumeinxtii. For sufficiently high driving frequencies, a simple approximation scheme suggests itself: At least if x is large compared to the single-particle tunnelling frequency X, and the interaction parameter Nj/X is sufficiently small, the periodic force will not give rise to major resonances. Under this condition one is entitled to neglect the ‘‘photon replicas’’ (that is, the Floquet satellites associated with different values of the index n) altogether, and to consider only the space spanned by the canonical representatives (36) of the unperturbed Floquet states. We are then left with the finite (N + 1) · (N + 1)-matrix Z 1 T hhur jH 1 jum ii ¼ dthur jH 1 jum i T 0 Z T hX 2l cosðxtÞðm rÞ dt exp i ¼ T 0 x h‘; rjJ x j‘; mi.
X ! XJ 0
k¼1
we see that the time average in Eq. (40) simply serves to ‘‘filter out’’ the zero-mode from this expansion, and find 2l ðm rÞ h‘; rjJ x j‘; mi. hhur jH 1 jum ii ¼ hXJ 0 ð42Þ x Taking into account that the angular momentum component Jx couples only nearest neighbours, we have nonzero matrix elements only for m r = ±1. Recalling that the zero-order Bessel function J0 is even, we thus arrive at 2l hhur jH 1 jum ii ¼ hXJ 0 ð43Þ h‘; rjJ x j‘; mi x and finally obtain, within our high-frequency approximation, the quasienergies em for the driven Josephson junction as eigenvalues of the matrix 2l ð0Þ M rm ¼ em drm hXJ 0 ð44Þ h‘; rjJ x j‘; mi. x Remarkably, this equals the matrix which gives the energy eigenvalues of the undriven junction (i.e., for Fd/ h = l = 0), with the replacement
ð45Þ
0.5
0
–0.5
ð40Þ
Recalling the generating function of the ordinary Bessel functions Jk(z), 1 X ixt k eiz cosðxtÞ ¼ ie J k ðzÞ; ð41Þ
2l . x
Thus, high-frequency driving corresponds to a ‘‘renormalization’’ of the single-particle tunnelling splitting X by the zero-order Bessel function J0(2l/x). Fig. 1 shows exact quasienergies for the model (28), computed numerically for N = 20 particles, interaction parameter Nj/X = 0.7, and high driving frequency x/X = 10, while Fig. 2 displays the result of the approximation (44) for the same situation. As expected, the agreement is excellent. These figures also visualize another important feature: since the undriven junction corresponds, apart from an energy shift, to the Hamiltonian [32,33]
ε/(h ω)
the quasienergy spectrum repeats itself periodically on the energy axis, each ‘‘Brillouin zone’’ of width hx containing one quasienergy representative of each state. The scalar product hhÆ j Æii in the extended Hilbert space combines the usual scalar product with time averaging, Z 1 T hh j ii ¼ dthji. ð39Þ T 0
0
5 2µ/ω
10
Fig. 1. One Brillouin zone of exact quasienergies, computed numerically, for the driven Bosonic Josephson junction (13) with N = 20 particles, interaction parameter Nj/X = 0.7, and high driving frequency x/X = 10, as functions of the scaled driving amplitude 2l/x. Observe the characteristic ‘‘pairing’’ and ‘‘unpairing’’ of states in the vicinity of the values 2.4, 5.5, and 8.7 of that amplitude.
0.5
ε/(h ω)
122
0
–0.5
0
5 2µ/ω
10
Fig. 2. Approximate quasienergies, computed according to the highfrequency scheme (44), for the same situation as considered in Fig. 1.
T. Jinasundera et al. / Chemical Physics 322 (2006) 118–126
H un ¼ hXJ x þ 2hjJ 2z ;
ð46Þ
4. Tunnelling dynamics of a driven condensate We start our discussion of condensate tunnelling in a driven Bosonic Josephson junction by monitoring the 0.5
0.5
ε/(h ω)
its eigenvalues are twofold (almost) degenerate (except for the state with m = 0) in the regime where the part 2 hjJ 2z describing the on-site interaction between the particles dominates, while they remain non-degenerate in the regime of weak interaction, where the Josephson coupling hXJx governs the dynamics. This competition is clearly reflected in the figures: when the scaled amplitude 2l/x is close to the zeros j0,1 2.4048, j0,2 5.5201, or j0,3 8.6537 of J0, the ‘‘renormalized’’ tunnelling frequency approaches zero. Hence, the interaction then dominates, regardless of its numerical strength or weakness, and leads to the appearance of pairwise degenerate Floquet states. In Fig. 3 we plot exact quasienergies for a lower frequency x/X = 3, where the ‘‘regular’’ approximation (44) still gives good results. In contrast, Fig. 4 highlights an example where this approximation breaks down: here the driving frequency equals the single-particle tunnelling frequency, x/X = 1, so that the oscillating force gives rise to a multitude of resonances between ac-Stark-shifted (quasi)energy eigenstates, resulting, even for only N = 20 particles, in an intricate avoided crossing-pattern, commonly viewed as an indication of quantum chaos [26]. With N = 1000 particles, each one having a quasienergy representative in each Brillouin zone, the spectrum would be close to a nightmare. In the following section, we will explore the consequences of this change of the quasienergy spectrum from ‘‘regular’’ to ‘‘chaotic’’ for the quantum N-particle dynamics in the driven junction.
0
–0.5
0
ε/(h ω)
–0.5
0
5 2µ/ω
10
Fig. 3. Exact quasienergies for the driven Bosonic Josephson junction (13) with N = 20 particles, interaction parameter Nj/X = 0.7, and moderate driving frequency x/X = 3. Under these conditions, the high-frequency approximation (44) still works reasonably well.
1
2
3
2µ/ω Fig. 4. Exact quasienergies for the driven Bosonic Josephson junction (13) with N = 20 particles, interaction parameter Nj/X = 0.7, and intermediate driving frequency x/X = 1. The plethora of avoided level crossings is a hallmark of quantum chaos [26]. Observe how the scale of the abscissa differs from that of the previous figures.
time evolution of a system with N = 1000 particles initially all stored in one of the wells, say well No. 1. To this end, we expand the many-body wave function jw(t)i in a basis of Fock states {jm, N mijm = 0, . . . , N} with m particles in the first and N m particles in the second well, specify the initial condition j wðt ¼ 0Þi ¼j N ; 0i;
ð47Þ
and solve the time-dependent Schro¨dinger equation numerically. According to the definition (25), the angular momentum component Jz directly corresponds to the imbalance of the particle number in both wells. We therefore visualize the dynamics by plotting the expectation values hJ z i hwðtÞ j J z j wðtÞi
0
123
ð48Þ
as functions of time, with hJzi/N = +1/2 (or 1/2) signalling ‘‘all particles in the first (or second) well’’. Fig. 5 shows the results of such calculations for three different situations; in all cases, the interaction parameter is Nj/X = 0.7. The dashed line refers to the absence of the driving force; here the condensate undergoes the usual Josephson oscillation [1]. As a consequence of the discreteness of the N-particle energy spectrum, these oscillations appear to be damped. Technically speaking, this damping is a collapse which will be followed by revivals after sufficiently long evolution times. However, we surmise that such revivals will not be observable within experimentally realistic timescales; observe that Xt = 100 corresponds to t 1.6 s for X = 2p Æ 10 s1. The lower, irregularly looking full line in Fig. 5, which more or less averages to zero population imbalance in the course of time, corresponds to an oscillating force
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responding to the initial condition (47), that critical value is jNj/Xjc = 1. This fact, together with the highfrequency renormalization (45) of the tunnelling frequency X, immediately leads to a striking prediction: in the presence of an oscillating force with sufficiently high frequency, tunnelling of the condensate will be suppressed not only at those isolated parameters where the scaled amplitude 2l/x equals a zero of J0, but rather in those parameter regimes in which Nj ð49Þ XJ ð2l=xÞ > 1.
0.5
0
0
0
50 Ωt
100
Fig. 5. Time evolution of the population imbalance hJzi/N in a Bosonic Josephson junction occupied by N = 1000 particles, with interaction parameter Nj/X = 0.7. Initially all particles had been stored in the first well, as corresponding to hJzi/N = 0.5 at Xt = 0. The dashed line refers to the case without driving, the upper full line to driving with amplitude 2l/x = 2.40 and frequency x/X = 3.0, while the lower full line corresponds to driving with the same amplitude 2l/x = 2.40, but low frequency x/X = 1.0.
with amplitude 2l/x = 2.40, close to the first zero of the J0 Bessel function, and low driving frequency x/X = 1.0. Under this condition, the high-frequency renormalization (45) of the tunnelling splitting is not effective, as witnessed by the glaring discrepancy between the true, chaotic quasienergy spectrum displayed in Fig. 4 and the regular high-frequency approximation (44). Thus, even at 2l/x = 2.40 the particles are not kept in the initially occupied well, but tunnel to the other to a significant extent, such that both wells tend to be populated equally after some time. In contrast, the situation is quite different for x/X = 3.0 and 2l/x = 2.40: as indicated by the jagged full line at the top of Fig. 5, now ‘‘coherent destruction of tunnelling’’ takes place, confining the condensate almost fully to the well in which it had been prepared. In the absence of the periodic force, the two-mode model for a condensate in a double well exhibits an effect known as ‘‘self trapping’’: if the absolute value of the interaction parameter Nj/X exceeds a certain critical value, the coherent Josephson oscillation is suppressed and the initial population imbalance is almost preserved in time [3–5]. This effect, which has now been verified in the pioneering experiment by Albiez et al. [1], had previously also been discussed in the context of transport on a dimer [2]. It can be deduced from the discrete nonlinear Schro¨dinger equation which corresponds to the Gross–Pitaevskii mean-field approximation for the two-site model; however, it effectively persists even on the N-particle level [34]. The critical value of jNj/Xj above which self-trapping occurs depends on the initial population imbalance; for maximum imbalance, as cor-
for an averaging interval Dt = 100/X. Fig. 6 summarizes the results. Since values of hJzit/N close to +1/2 correspond to long-lasting trapping in the initially occupied well, while values close to zero indicate strong tunnelling, the expectation is fully confirmed: The full line in Fig. 6 corresponds to a weakly interacting gas, Nj/X = 0.1, so that the trapping condition (49) is satisfied only in narrow intervals around the zeros of J0. The short-dashed line is obtained for Nj/X = 0.3; accordingly, the trapping interval around the first J0-zero is wider, and there is no full equilibration of the population for scaled amplitudes between the second and the third zero, since the Bessel function is not large enough there to enable unhindered tunnelling. This tendency is more pronounced still for Nj/X = 0.5 (long dashes);
0.5
–0.5
To verify this prediction, we start again from the initial condition (47), and compute the time-averaged population imbalance Z Dt 1 hJ z it dthwðtÞjJ z jwðtÞi ð50Þ Dt 0
0.0
0
5 2µ/ω
10
Fig. 6. Time-averaged population imbalance hJzit/N for a Bosonic Josephson junction occupied by N = 100 particles and driven with frequency x/X = 3.0. Initially all particles had been stored in the first well; the averaging interval is D(Xt) = 100. The interaction parameters are Nj/X = 0.1 (full line); 0.3 (short dashes), 0.5 (long dashes), and 0.7 (dots).
T. Jinasundera et al. / Chemical Physics 322 (2006) 118–126
0.5
0.0
0
3 ω/Ω
6
Fig. 7. Time-averaged population imbalance hJzit/N for a Bosonic Josephson junction occupied by N = 100 particles, when the scaled amplitude 2l/x is kept fixed at the first (full line), second (short dashes) or third zero (long dashes) of the Bessel function J0. Initially all particles had been stored in the first well; the averaging interval is D(Xt) = 100. The interaction parameter is Nj/X = 0.7.
0.5
for Nj/X = 0.7 (dots) one even comes close to a clearcut yes/no-alternative: for scaled amplitudes below 1.2 the condensate tunnels vividly, but remains trapped for higher amplitudes. The question then remains just when the driving frequency x/X can be considered as ‘‘sufficiently high’’ to effectuate this interplay between coherent tunnelling destruction and self-trapping. The answer is given in Fig. 7, where we have plotted the time-averaged population imbalance (50) as a function of the scaled frequency x/X, for scaled amplitudes equalling the first (full line), second (short dashes), and third zero (long dashes) of J0. As can be seen, for scaled frequencies x/X of about 1.0 tunnelling takes place, with a somewhat irregular dependence on the frequency, while tunnelling suppression sets in roughly at x/X = 1.5. For scaled frequencies of magnitude 2.0 or higher, the suppression is already quite strong, allowing one to prevent the condensate from spilling to the other well. Since the absolute value of J0(2l/x) is less than unity for nonvanishing amplitude, high-frequency driving can only quench the tunnelling effect. This is different for low frequencies: in Fig. 8, we exemplarily display the time evolution of the population imbalance for x/X = 0.25 and interaction parameters Nj/X = 1.0 (dotted), 3.0 (full line), and 5.0 (dot-dashed); the driving amplitude is l/X = 5.0 in all three cases. There are marked population jumps at about those moments when the two wells are energetically aligned, initially accompanied by rapid oscillations. Since in the absence of the periodic force one has self-trapping for Nj/X > 1, and energetic alignment exists only during short time intervals when such a force is present, one might have
125
0
–0.5
0
50 Ωt
100
Fig. 8. Time evolution of the population imbalance hJzi/N in a Bosonic Josephson junction occupied by N = 1000 particles under lowfrequency driving, x/X = 0.25, with amplitude l/X = 5.0. The interaction parameters are Nj/X = 1.0 (dotted line), 3.0 (full line), and 5.0 (dot-dashed). Initially all particles had been stored in the first well.
expected self-trapping to persist a fortiori under the influence of low-frequency driving. However, as Fig. 8 clearly demonstrates, this naive expectation is incorrect: even for Nj/X = 5.0 the population can equilibrate after a few jumps.
5. Conclusion The recent observation [1] of coherent Josephson oscillations, and of the self-trapping effect, with a mesoscopic sample of Bose–Einstein-condensed atoms in an optical double well potential has paved the road to further developments aiming at the coherent control of quantum many-body dynamics. As we have shown in this paper, a suitable periodic modulation of the double well gives rise to several novel features: When the modulation frequency is at least two times higher than the bare single-particle tunnelling frequency, the interplay of self-trapping and coherent destruction of tunnelling allows one to effectively shut off the tunnelling contact between the wells by adjusting the parameters of the drive; this effect is illustrated most clearly in Fig. 6. For driving frequencies about equal to the tunnelling frequency, the dynamics exhibits signatures of quantum chaos, such as the irregular oscillations displayed in Fig. 5. For still lower frequencies, the modulation can induce a sequence of rapid population jumps, as in Fig. 8, leading to a partial population transfer even when there would be strong self-trapping without the modulation. We therefore expect that future experiments along the lines sketched here will yield considerable new insight into the response of Bose–Einstein condensates to nonperturbatively strong forcing.
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The main restriction of the current work lies in the use of the two-mode approximation (8). It should be clear that a frequency as high as x/X = 10.0, employed in Figs. 1 and 2 to demonstrate the consistency of the approximation (44), is likely to fall outside the regime in which the two-mode approximation holds. However, our estimates at the end of Section 2 also indicate that moderately high frequencies are still viable. Moreover, it should be noted that even with only N = 1000 Bose particles the condition (22) is stretched to its very limit in standard situations. But given the enormous tunability of both the optical potentials and the atomic s-wave scattering lengths [20,21], and thus of both parameters X (or L) and j separately, together with the fact that the two-mode approximation has been shown to capture the dynamics at least qualitatively even under the conditions of the Heidelberg experiment [1], one may be confident that the effects predicted in this paper do not defy experimental verification. One further observation deserves to be mentioned. The ratio of the on-site interaction frequency j, defined in Eq. (12), and the single-particle tunnelling frequency X also governs the transition from a superfluid to a Mott insulator undergone by a Bose–Einstein condensate in an optical lattice when that lattice is made deeper [35,36]. The renormalization (45) of the tunnelling frequency, discussed in the present work for the case of a single Josephson link, thus prompts the question whether a similar quenching of X would also be effective in modulated optical lattices, so that the superfluid–insulator transition could actually be controlled by means of such a modulation. This question will be taken up, and answered affirmatively, in a subsequent publication [37].
Acknowledgements We thank M.K. Oberthaler for a discussion of his experiment [1], and A. Eckardt for insightful remarks. This work was supported by the Deutsche Forschungsgemeinschaft through the Priority Programme SPP 1116, Wechselwirkung in ultrakalten Atom- und Moleku¨lgasen.
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