Resonant ratcheting of a Bose-Einstein condensate

We study the rectification process of interacting quantum particles in a periodic potential exposed to the action of an external ac driving. The breaking of spatio-temporal symmetries leads to directed motion already in the absence of interactions. A hallmark of quantum ratcheting is the appearance of resonant enhancement of the current (Europhys. Lett. 79 (2007) 10007 and Phys. Rev. A 75 (2007) 063424). Here we study the fate of these resonances within a Gross-Pitaevskii equation which describes a mean field interaction between many particles. We find, that the resonance is i) not destroyed by interactions, ii) shifting its location with increasing interaction strength. We trace the Floquet states of the linear equations into the nonlinear domain, and show that the resonance gives rise to an instability and thus to the appearance of new nonlinear Floquet states, whose transport properties differ strongly as compared to the case of noninteracting particles.


INTRODUCTION
The breaking of space-time symmetries, and their role in the generation of directed transport in single particle Hamiltonian ratchets, have been extensively studied in the classical [1,2,3] and quantum regimes [4,5,6,9,10]. Experiments with thermal cold atoms loaded on optical lattices [7] demonstrated the fruitfulness and correctness of the theoretical predictions. Importantly, the latest studies show a resonant enhancement of the current in the quantum regime, due to resonances between Floquet states [5,6]. Real experiments involve many atoms, and interaction between them may be tuned, but will always be left at least at some residual nonzero level. Therefore, the impact of interactions on quantum ratchets has to be addressed.
In this paper, we study, using the mean-field approach, the generation of directed transport of interacting quantum particles in a periodic potential under the action of a two harmonic driving. With this approach we mimic the motion of cold atoms in an optical lattice under the presence of an external force [7], but at much lower temperatures, when a Bose-Einstein condensate may form [8]. To this end, we investigate the continuation of Floquet states of the corresponding linear system into the nonlinear domain. We show that a resonant enhancement of the current in the nonlinear regime takes place, which results from the resonant interaction between Nonlinear Floquet states. We derive an analytical expression for the evolution of quasienergies in the nonlinear regime. Finally we show the relation between the transport properties of nonlinear Floquet states and the asymptotic current of an initial state with zero momentum.

II. MODEL
Experimental realizations of ratchets with cold atoms may tune the temperatures from mK down to µK, such that a Bose-Einstein condensate may form due to interactions between particles [8]. The corresponding general equation to be studied is then given by the onedimensional Gross-Pitaevskii equation (see e.g. [11]) where a s is the s-wave scattering length, M is the atomic mass, k L = π/d is the optical lattice wave number with optical step d, V 0 is the periodic potential depth, and e(τ ) is a periodic driving force. The wave function is normalized to the total number of atoms in the condensate and we define n 0 as the average uniform atomic density [11,12].
Introducing the dimensionless variables x = 2k L X, t = τ /t s , ψ = Ψ/ √ n 0 , and defining 1/µ = M/4 k 2 L t s ; we transform the system (1) to the dimensionless equation [13] iµ ∂ψ(t) ∂t where the dimensionless one-particle Hamiltonian is 11]. The dimensionless ac field E(t + T ) = E(t). As in the linear limit [5,6], we consider is the vector potential [5,6]; we transform the original one-particle Hamiltonian to A. Linear regime Consider the Schrödinger equation, the linear limit of Eqs. (2)(3)(4). Here, we use a tilde to denote wavefunctions, quasienergies and other relevant parameters for the linear regime.
It was shown in [5,6] that, for the appearance of a dc-current in the quantum regime, two symmetries need to be broken. These symmetries are defined in the classical limit as follows Likewise if E(t) possesses the symmetry E(t) = E(−t), then (3) is invariant under the The Hamiltonian Eq.(3) is a periodic function of time. Then the solutions, |ψ(t + t 0 ) = U(t, t 0 )|ψ(t 0 ) , can be characterized by the dynamics of the eigenfunctions of U(T, t 0 ) which satisfy the Floquet theorem: The quasienergiesǫ α (−π <ǫ α < π) and the Floquet eigenstates can be obtained as solutions of the eigenvalue problem of the Floquet operator Due to the discrete translational invariance of Eq.(4) and Bloch's theorem all Floquet states are characterized by a quasimomentum κ with |ψ α (x + 2π) = e i κ |ψ α (x) .
We choose κ = 0 which corresponds to initial states where atoms equally populate all (or many) wells of the spatial potential. This allows us to use periodic boundary conditions for Eq.(2), with spatial period L = 2π, so that the wave function can be expanded in the plane wave eigenbasis of the momentum operatorp, |n = 1 √ 2π e inx , viz.
Thus, the Floquet operator is obtained by solving Eqs. (2)(3)(4) in the linear limit. In the computations we neglect the contribution originating from A(t) 2 , since it only yields a global phase factor. Details of the numerical method are given in [6]. A study of a related problem with nonzero κ has been published in [4], which shows that the essential features of avoided To estimate the net transport, it is necessary to compute the asymptotic current. It is obtained using the expression J(t 0 ) = α p α |C α (t 0 )| 2 [6], where p α are the Floquet states momenta and C α (t 0 ) are the expansion coefficients of the initial wave function in the basis of Floquet states. Breaking the symmetries S a (5) and S b (6), we desymmetrize the Floquet states, i.e. the Floquet states momenta acquire a finite value p α = 0, which results in the appearance of a directed transport.
In general, the current is a function of the initial time t 0 and the relative phase θ, namely J(t 0 , θ). After averaging over the initial time it exhibits the property [5,6]: We focus the analysis on previous computations obtained for µ = 0.2 in [5,6].
Fig.1b shows the current dependence on θ. The computation is performed taking the initial state |0 = 1/ √ 2π, which overlaps with states in the chaotic layer. With this initial condition we mimic a dilute gas of atoms which are spread all over the lattice with zero momentum.
The current has two peaks which are linked to avoided crossings displayed in Fig.1a. In these particular avoided crossings, states from the chaotic layer and transporting states mix, which leads to a leakage from the chaotic layer to the transporting state, thereby enhancing the current.
On the other hand, it was shown in [5,6] that by tuning the amplitude of the second harmonic of the driving force the peaks becomes broader which makes it easier resolving it in experiments (see dashed line in Fig.1b).

B. Nonlinear regime
In the nonlinear case the analysis of the generation of directed transport in the presence of a driving force is much more complicated, due to possible nonintegrability, classical chaos, and mixing [15]. However, one can take the Floquet states of the linear problem, and continue them as periodic orbits into the nonlinear regime. Then these nonlinear Floquet states can be analyzed.
While in the linear regime the evolution of the Floquet state is determined by Eq.(8) with a unitary operatorŨ , in the nonlinear regime the unitarity is lost, and is replaced locally by simplectic maps [16]. Nevertheless, a similar transformation over one period of the ac driving can be defined, viz.
where U is a nonlinear map of the phase space onto itself, defined by integrating a given trajectory over one period of the ac driving [17]. The solutions ψ α constitute generalizations of the linear Floquet states (7).
To compute the nonlinear Floquet states, we use a numerical method implemented in computational studies of periodic orbits [17]. The basics steps are as follows. First, we choose a linear Floquet state as an initial seed. Then taking a small value of the nonlinearity strength, we compute the new solution using a Newton-Raphson iterative procedure, by variying initial seed (see Appendix A for a detailed explanation). The procedure involves conservation of the norm and the variation of the quasienergy of the state, which together enforce the convergence to the desired solution. In each iteration step, the new trial solution is integrated over one time period T . Once a solution is found, we increase the nonlinearity strength again by a small amount and repeat the same procedure. Thus, we trace the solution into the nonlinear domain.
Desymmetrization of the Floquet states, due to breaking of symmetries, leads to the appearance of directed transport in the linear regime with enhancement of transport due to resonant Floquet states. It is therefore worthwhile to investigate what happens with nonlinear Floquet states in the absence of those symmetries, and trace the fate of the abovementioned resonances in the nonlinear regime.

Dimer
To gain insight into the effect of breaking symmetries on nonlinear Floquet states, we utilize a basic model of two coupled BEC states in the presence of an external driving. The equations for a driven two sites model can be written as iµ where f (t) = f 1 sin(ωt) + f 2 sin(2ωt + θ), C is the coupling term, and N 1,2 = |ψ 1,2 | 2 are the populations or number of particles in the sites 1,2. The above equations can be also qualitatively viewed as a restriction of the original case (4) to just two basis states with opposite momenta. That leads to the corresponding different signs of the last terms in the rhs of the above equations.
With f (t) = 0 the equations above are used to describe, on the mean field level, the selftrapping transition of two coupled BEC. It was shown in [18,19] that such a phenomenon occurs when the nonlinearity exceeds a critical value, and the new states are characterized by a population imbalance N 1 − N 2 . The existence of critical or threshold values have been predicted as well [18]. The critical value is determined by the bifurcation point of the stationary solutions. The selftrapped states violate the permutational symmetry of (13), (14), In the presence of nonlinearity close to these linear Floquet states there will be again states, which are 'periodic' in the sense that after one period of the ac driving the state returns to itself, up to a corresponding phase. Note that both the shape of the eigenstate, but also the phase (i.e. the analogue of the quasienergy in the linear case) will smoothly change upon continuation into the nonlinear regime. The continuation process of such a nonlinear Floquet state is thus encoded by the linear Floquet state at g = 0, which is chosen to be continued.
In the case of f 1 , f 2 being nonzero, θ becomes a relevant parameter. For θ = 0, π timereversal antisymmetry is broken, and the nonlinear Floquet solutions will also loose that symmetry, together with the permutational symmetry (see above).
Hereafter, we name by symmetric the case when θ = 0, π and nonsymmetric otherwise. We will also coin linear and nonlinear Floquet states periodic orbits, although they are only periodic up to the above mentioned phase shift. Conversely, if the time-reversal symmetry is broken, a saddle-node bifurcation appears. First of all the state continued from the linear limit, already acquires some nonzero population imbalance, since the symmetry is broken. Fig.2 shows that the strict continuation of that state evolves into a state with a strong population imbalance. Two other states -one which is corresponding to a weak imbalance, and another which has a strong imbalance as wellemerge through the saddle-node bifurcation.
To conclude this part, we may expect that nonlinearity induces Floquet states with nonzero population imbalance via bifurcations, or enhances the already present imbalance (originat-ing from a symmetry breaking) again via bifurcations. We remind the reader here, that the simple dimer model can be obtained from (4) by choosing two momentum basis states with different sign, and replacing the original complicated interaction which is mediated via further basis states, by a direct interaction term. A population imbalance thus means a momentum imbalance as well, i.e. a nonzero current. It may be thus as well expected, that for the full problem, to be treated below, nonlinearity may enhance directed currents via bifurcations. In Fig.3, we present the continuations of two Floquet states depicted in Fig.1, into the nonlinear domain. We observe a bifurcation of one of the periodic states as we increase the nonlinearity strength. Here, as in the dimer, a bifurcation of saddle-node type leads to the formation of three periodic solutions out of one.
Since we want to study the transport of a BEC with zero initial momentum, we focus on the continuation of the state lying in the chaotic layer. The continuation process is indicated by a sequence of Husimi functions in Fig.3a, and can be summarized as follows: i) before the bifurcation, the state is located in the chaotic layer; ii) after the bifurcation, the periodic state transforms into a mixed state due to a resonance with a second transporting state; iii) further increase of the nonlinearity strength transforms it into a transporting state. By contrast, the originally transporting state does not experience significant changes as we increase the nonlinearity strength.
A quantitative measure of this transition process is given by the evolution of the average In order to estimate the shift of the bifurcation point, and to finally predict the new possible resonance positions with increasing nonlinearity, we use a perturbation approach to estimate the dependence ǫ = P (ǫ, g), related to the Aharanov-Anandan phase [23] (see also [16]).
Then for k = 0 we find (see Appendix B for details) where ... = Bifurcations of new states correspond to a coalescence of different families of states, leading to φ j (t) = exp[−iλ j (t)](a jφ1 (t) + b jφ2 (t)), whereφ 1 (t) andφ 2 (t) are the corresponding original Floquet states. The evolution of the quasienergies for the nonlinear periodic states is thus given by where a j and b j are the corresponding weightings of the linear states in the nonlinear state expansion (see Appendix B for details).
In Fig.3b, we plot the quasienergy values computed with Eq.(16), using the nonlinear states continued from the original linear states depicted in Fig.1. We find an excellent agreement with the full numerical results.
For weak nonlinearity, the quasienergies of the states depend linearly on g. A simple perturbation expansion allows to take the wavefunctions of the original linear states and its respective quasienergies with Eq. (15). Then from the quasienergy intersection of the two states, ǫ 1 (θ, g) = ǫ 2 (θ, g), we compute the critical value of g: Inserting the wave function of the original linear states depicted in Fig.1 in Eq. (17), we obtain g = 0.003008, which is a good estimate of the nonlinearity strength at the bifurcation point. Let us now investigate the evolution of the state |0 for κ = 0. In the linear regime, we use the Floquet representation to derive the expression for the asymptotic current. In the nonlinear regime it is no longer possible. Instead, we compute the running average dt withp = ψ|p|ψ , over long times, which becomes the asymptotic current in the limit t → ∞, i.e., J(t 0 ) = lim t→∞ P . To validate the convergence to the asymptotic current we use maximum integration times ranging from 100000 till 300000 time periods.
First we compute the current for θ ≈ −0.99. In such a case, for g = 0, there is an avoided crossing (see above discussion), i.e. a resonance. Further increase of the nonlinearity leads to a decay of the current (Fig.4). However, taking θ = −1.1, −1.2, we observe a corresponding shift of the resonance peak to larger values of the nonlinearity parameter, thus showing a robustness of the resonant current enhancement observed in the linear limit. It was shown for the linear regime [5,6] that the current depends upon the initial time t 0 . In the nonlinear case, the dynamics may exhibit similar behavior to classical chaos [15], making the analysis more complicated. An essential point here is that, while in the linear regime Floquet states mix in narrow parameter regions via avoided crossings, in the nonlinear domain mixed states may survive for a fairly large range of nonlinearity values.
This, along with the fact of having multiple bifurcations with the appearance of new periodic states [24], may lead to classical-like chaotic behavior predicted in [15].
A benchmark for the persistence of directed transport is that the sum of currents for different initial times do not cancel. To check that, we compute the momentum evolution for different initial times with the system in and out of resonances. In Fig.4, we present the current dependence upon the initial time t 0 . The curve depicted by filled circles shows a similar behavior to the one obtained in the linear regime with the system in resonance (cf. Fig.4 in [5]). It also displays a large positive current for all initial times, thus confirming the existence of directed motion. On the contrary, the curve with squares shows a nearly flat profile with smaller current values, recalling the off resonance scenario in the linear limit.

III. CONCLUSIONS
We have studied the rectification of interacting quantum particles in a periodic potential exposed to the action of an external ac driving, using a mean-field approach.
We showed that by tuning the nonlinearity in an optical lattice it is possible to enhance the directed transport of cold atoms. A possible experimental way to achieve it is to vary the scattering length by changing the strength of the magnetic field [25]. These resonances are partly continued from the noninteracting system, but become saddle node bifurcations of more complicated nonlinear Floquet states in the nonlinear system under consideration.
We developed analytical estimates of the nonlinearity strength in the resonance, and showed that the evolution of an initial state with zero momentum carries all signatures of a ratchet state, i.e. its average momentum is nonzero. Therefore, ratchets and quantum resonances of (nonlinear) Floquet states are robust with respect to interactions, and should be observable in real experiments.
Finally, bifurcations of the nonlinear Floquet states have been observed, which may affect the measured currents strongly.  [26]. Hereafter, for simplicity, we drop the index α.
The two vectors X, X(0) are identical for the linear case. Assuming weak nonlinearity, we compute the vector X after one integration period, taking as initial seed the linear state X(0). This implies that after a full integration period our final state deviates from the initial state: G[ X(0)] = X − X(0).
To correct such a deviation we use the Newton-Raphson method. Basically, the method updates the initial seed X(0) after every iteration until each component of G is reduced to a value less than 10 −9 . For every integration over the period T , we use the split operator method.
To successfully accomplish the reduction of the vector difference (A2), some constraints should be fulfilled. First, the Floquet states should preserve the norm. We thus add another component G 2N +1 = X X − X(0) X(0), and the iteration process is also zeroing this component.
To quasienergyẼ (see [27] for definition). It relates to ourǫ asẼ =ǫµ/T , where −µ ω/2 < E < µ ω/2. Then we obtain This expression tells us that the energy of our nonlinear Floquet state is the sum of the corresponding linear and nonlinear contributions. The second term is the time average of the state energy due to the nonlinear interaction.
So far, we have considered the energy evolution of single states without perturbation. However, single states may "coalesce" leading to bifurcations. We take the simplest case with two resonant states. We project vectors on a basis that results from the superposition of the continued original eigenstates, whose continuation lead to resonances in the nonlinear domain. That is, φ j (t) = exp[−iλ j (t)](a jφ1 (t) + b jφ2 (t)) with j = 1, 2; whereφ 1 (t) andφ 2 (t) are the original Floquet states.