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Asymmetries in the tunneling probability of Bose-Einstein condensate in an accelerating optical lattice

We derive a two-band finite-dimensional model for description of the condensate tunneling in an accelerating optical lattice, taking into account the fine Bloch band structure. The model reveals a very strong dependence of the final band populations on the initial populations and phases. Most importantly, additionally to the known asymmetric dependence on the nonlinearity, there is also a notable asymmetry in the sensitivity of the tunneling probability to the nonliearity-induced initial population of the Bloch band to which the tunneling takes place. This fact can explain the experimentally observed unexpected independence of the upper-to-lower tunneling probablity on the nonlinearity. Finally, we compare the predictions of the two-band model with that of the well-known nonlinear Landau-Zener model and find disagreement when the two bands are initially populated. The disagreement can be qualitative and reveals itself even for a negligible nonlinearity. However, the two models agree remarkably well if just one band is populated initially.


I. INTRODUCTION
Since the successful realization of Bose-Einstein condensation (BEC) in an optical lattice [1,2] the experimental efforts are directed to explore and understand the behavior of BEC in a periodic potential. Many of the BEC phenomena in an optical lattice have an analog in the solid state physics. For instance, Bloch oscillations and Landau-Zener tunneling of BEC were experimentally observed [3,4,5,6] in an accelerating lattice. The atomic interactions in BEC, however, give rise to the essentially nonlinear effects such as the asymmetry of the tunneling probability between the two adjacent bands [5,6] and the Landau and dynamical instabilities [6,7,8,9,10]. Accordingly, two essentially different regimes of BEC dynamics in an optical lattice can be identified [6,11]: the "instability regime", which realizes for a small to intermediate values of the lattice acceleration and the "Landau-Zener regime", which realizes for larger accelerations. When the acceleration is small, the condensate slowly crosses the edge of the Brillouin zone allowing the unstable modes to dominate the order parameter [11,12]. For instance, in one of the bands the modulational instability [13] will result in the bright soliton formation even in a repulsive BEC [14].
For the "Landau-Zener regime" a reduced two-level model (the nonlinear Landau-Zener model) was proposed in Refs. [15,16] to account for the effect of the atomic interactions on the tunneling probability of BEC (see also Refs. [12,17] for further development). This model is derived for a weak optical lattice by an analog of the method of free electrons in the solid state theory (see for instance, Ref. [18]) and is essentially the Landau-Zener model [19] modified by the nonlinear terms due to the atomic interactions in BEC.
The predictions of the nonlinear Landau-Zener model were experimentally tested [5,6].
In accordance with the theoretical prediction the atomic interactions in BEC make the tunneling probability asymmetric, but still a significant disagreement was discovered. Hence, a more accurate model is in order for description of the condensate tunneling.
To account for the modulational instability present in one of the Bloch bands a spatiotemporal model was proposed recently [11]. Essentially, it consists of two linearly coupled nonlinear Schrödinger equations for the slowly varying amplitudes of the two Bloch waves bordering the Brillouin zone edge (i.e. at k = k B ). The key role of the modulational instability of Bloch wave amplitudes in the transition from the reversible tunneling to the asymmetry in the band populations is revealed. Therefore, this model is suitable for descrip-tion of the "instability regime", when the Brillouin zone edge crossing time is comparable to the characteristic time of development of the modulational instability. However, its applicability is restricted to the case of a very small lattice acceleration and thus it cannot be applied to the "Landau-Zener regime" (as defined above).
The aim of this paper is to propose an improved finite-dimensional system (the two-band model) for BEC tunneling in an accelerating optical lattice. We take into account the fine structure of the Bloch bands, which is of crucial importance if the initial conditions are arbitrary. Only the "Landau-Zener regime" is considered. Hence, we can safely neglect the spatial inhomogeneity of the Bloch wave amplitudes. We expand the BEC order parameter over the Bloch waves with a time-dependent wave index. Thus our method is similar to Houston's approach [20] to the accelerating electrons in the solid state theory. The resulting finite-dimensional model can also have application in the study of ultracold atoms in a periodic potential submitted to a constant external force [21,22].
There are two main results. First, the two-band model reveals a dramatic dependence of the final Bloch band populations on the initial populations and the phase difference. Most importantly, additionally to the asymmetry in the dependence of the tunneling probability on the nonlinearity strength, we reveal also the asymmetry in the sensitivity of the tunneling probability to the population of the adjacent Bloch band (i.e. to which the tunneling takes place). This fact, combined with the nonlinearity-induced population of the adjacent band, can explain the independence of the upper-to-lower tunneling probability on the nonlinearity suggested by the experimental results of Ref. [5]. Second, there is a significant disagreement between the predictions of the two-band model and the Landau-Zener model if the two bands are initially populated. The disagreement can be qualitative and reveals itself even for a negligible nonlinearity. The two models agree in the predictions if only one Bloch band is populated initially.
The paper is organized as follows. In the next section a derivation of the two-band model is presented. Some details of the derivation are relegated to appendix A. Then, in section III, the case of a weak lattice is considered. The predictions of the two-band model are compared to those of the nonlinear Landau-Zener model in section IV. The two-band model in the diabatic basis is given in appendix B. Section V contains discussion of the asymmetric BEC tunneling. Finally, a general discussion and perspectives are contained in the concluding section VI.

II. DERIVATION OF THE TWO-BAND MODEL
We consider a quasi one-dimensional BEC, i.e. the transverse width is much smaller than the length of the condensate. For not too large number of atoms the transverse dynamics of BEC is that of a quantum particle confined to the lower energy level of the parabolic trap [24].
In terms of the s-wave scattering length a s , the transverse oscillator length ℓ ⊥ = /mω ⊥ and the linear condensate density For the reduced order parameter ψ(x, t) the Gross-Pitaevskii equation describing the condensate in an accelerated optical lattice reads where g = 4π 2 a s /m, k L is the optical lattice index and x 0 (t) describes the acceleration of the lattice. Here the order parameter is normalized as follows d −1 d 0 dx |ψ| 2 = 1 with d = π/k L being the lattice spacing. The following dimensionless variables will be used below: where E R = 2 k 2 L /(2m) is the recoil energy and ℓ = /mω . We have The normalization now reads (2π) −1 2π 0 dx |ψ| 2 = 1. As we will use only the dimensionless variables of equation (4), we drop all the tildes from x, t, and ψ, for simplicity.
We assume that the quasi-momentum spread of the condensate in the optical lattice is negligible compared to the width of the Brillouin zone. One necessary condition is that the optical lattice be long enough, i.e. the parabolic trap contains many lattice periods: λ ≪ 1.
Then, the effect of the parabolic trap is negligible. We set λ = 0.
An additional condition is that the time of the Brillouin zone edge crossing is much less that the characteristic time of the modulational instability development. This sets a lower limit on the acceleration α. The latter time can be estimated as follows. In the slowly varying envelope approximation [11,13] the spatial Bloch amplitude will be governed by a nonlinear Schrödinger equation with an effective interaction coefficient cχ nn . The characteristic time of the instability development is the inverse of the maximal growth rate t M I ∼ |c| −1 .
Formation of gap solitons is also possible for a nonlinearity of the order of the gap between the Bloch bands [26]. In the following, however, we consider only the weak nonlinearity limit c ≪ v 0 , used in the experiment [5].
Under the conditions that the condensate has a narrow initial momentum spread and the time of the Brillouin zone edge crossing is much less that the characteristic time of the modulational instability development the solution to equation (4) can be expanded in the basis of the Bloch waves with a definite (time-dependent) index q(t) and an overall linear phase due to the acceleration: where v(t) ≡ẋ 0 and Ψ n,q (x) ≡ e iqx u n,q (x) satisfies Due to the normalization and orthogonality of the Bloch functions, It is shown below that, in a vicinity of the Brillouin zone edge q = k B , for a small acceleration and under the condition the effect of the upper bands on the tunneling between the first two Bloch bands. The nonlinear term with c ≪ v 0 can be taken into account by using a perturbation theory.
The expansion (5) is in the spirit of Houston's approach in the theory of electrons accelerating in a crystal lattice [20]. It takes into account, for instance, that in the experiments [5,6] the condensate was in a Bloch state before the acceleration was turned on.
A. The linear limit c = 0 For the moment let us drop the nonlinear term and consider the linear limit of equation (4). The contribution from the nonlinear term is accounted later on. Substituting the expression (5) into equation (4) and dropping the overall imaginary exponent gives Setting q(t) = q 0 − v(t) to cancel the non-periodic in x term and projecting the resulting equation onto the basis function u n,q (x) we get an equation for the amplitude where α =ẍ 0 . Here the inner product over the lattice period is used: being an even function, u * n,q (x) = u n,q (−x) and, hence, the diagonal inner product u n,q |∂ q u n,q = 0, while the non-diagonal is given as The coupling coefficient κ nm is real and satisfies κ nm (q) = −κ mn (q).
Note that the coupling coefficient between the Bloch bands is inversely proportional to . Thus, for a small acceleration and an optical lattice satisfying the influence of the upper bands on BEC tunneling between the first two Bloch bands at the Brillouin zone edge is negligible. For instance, the above condition is satisfied by a weak optical lattice (see section III).
The system of equations (10)-(11) is consistent with the two-band approximation, since it conserves the total number of atoms in the two bands: The linear two-band model (10)-(11) is important in its own right, since it can have direct application in the study of ultracold atoms in a periodic potential submitted to a constant external force [21,22].
In the experiments on BEC tunneling in an optical lattice [5,6] the nonlinearity satisfies the condition c ≪ v 0 . In this case, the nonlinear term can be treated as a perturbation of the optical lattice potential. Here it is pertinent to mention that the nonlinear term was also treated by introduction of an effective optical lattice potential [6,27]. This treatment is of limited validity.
Under the condition c ≪ v 0 the nonlinear term preserves the qualitative Bloch band structure, but modifies the Bloch functions and the band energies. The nonlinear term serves as an additional lattice potential (in general, time-dependent), the same for all eigenfunctions.
In fact, if we define the nonlinear modesû n,k (x, t) as follows then the effect of the nonlinear term can be treated by using the usual perturbation theory for the eigenvaluesÊ n (k, t) of a perturbed ((k, t)-dependent) linear operator. Notice that the nonlinear modesû n,q are orthogonal, û m,k |û n,k = δ n,m , and for c = 0 coincide with the respective Bloch functions. Thus, for small c, they constitute a complete orthogonal basis (in this respect we generalize the approach of Refs. [28,29]). In the first order of the perturbation theory we get with ... = (2π) −1 π −π dx(...), while the eigenfunctionsû n,q (x) = u n,q (x) + O(c) which is a sufficient approximation for the following. In the two-band approximation we get the following expressions for the energies of the first two Bloch bands (the variable t is omitted for simplicity): Here The terms with σ nm can be neglected, since at the Brillouin zone edge these coefficients are exactly zero due the properties of the Bloch waves. Numerics shows that they grow linearly The nonlinear generalization of the system (10)-(11) is derived by using the nonlinear Bloch wavesΨ n,k ≡ e ikxû n,k = Ψ n,k + O(c) in the expansion (5). For a small acceleration the term proportional to αc can be neglected, thus the inter-band coupling coefficient κ 12 in the nonlinear system is given by the same formula as in the linear case. Setting now 12 , we obtain the nonlinear two-band model: i Here we have used the modified Bloch energies from equation (13) and defined The system (14)-(15) depends on the fine k-structure of the Bloch bands, since q(t) scans through the edge of the Brillouin zone. Hence, the expressions for the Bloch waves in a vicinity of the Brillouin zone edge have to be available. For a general optical lattice the coefficients ε(q), κ 12 (q), and γ nn (q) can be computed numerically. However, such expressions are available in the analytical form (see below) for a weak optical lattice used in the experiments [5,6].

III. WEAK LATTICE APPROXIMATION
Similar to the free electrons approach in the solid state theory [18], in the vicinity of the Brillouin zone edge the Bloch waves from the two lowest bands of a weak optical lattice can be sought in the form For the cosine lattice v(x) = v 0 cos(x) we get The corresponding Bloch waves read where ν(q) = 2[k B (k B −q) + ε(q)]/v 0 . Then the remaining two coefficients of the two-band model are For the linear case, the analytical formulae in equations (18) and (20)

IV. COMPARISON WITH THE LANDAU-ZENER MODEL
To make a quantitative comparison with the nonlinear Landau-Zener model introduced in Refs. [15,16], one has to relate the amplitudes a and b of the incident and Bragg-scattered waves with the amplitudes of the Bloch waves from the first two bands.
There are two approaches. The first one is based on the fact that the adiabatic energy levels of the quantum-mechanical Hamiltonian associated with the system (21)- (22), mimic the structure of the first two Bloch bands.
Therefore, the Bloch amplitudes are obtained by projecting the vector (a, b) onto the instantaneous adiabatic eigenvectors of H [30]. We obtain the following relation The second approach is to use the analytical results for a weak lattice and equate directly the order parameters corresponding to the two models. The two-band model (15)- (16) corresponds to the order parameter with θ(t) being an unimportant phase and q(t) = q 0 − v(t). The nonlinear Landau-Zener model (21)- (22) corresponds to the following order parameter [15] It is straightforward to relate the band amplitudes using the expression for the Bloch waves (19) and associating k = q 0 . The common x-independent phase can be discarded. We have The transformation given by equation (27) is linear and unitary. Asymptotically it trivializes.
Indeed, ν → ∞ as t → −∞, while ν → 0 as t → ∞ (here α > 0). We get The overall conclusion to be argued below is that, whatever the way one relates the two models, the predictions of the nonlinear Landau-Zener model (21) i.e. both amplitudes (a and b and, respectively, a 1 and a 2 ) are non-zero initially. The disagreement is apparent even for a negligible nonlinearity, which case is treated below in detail. A remarkable agreement between the two models if just one band is populated initially (i.e. either a 1 or a 2 is zero initially) must also be stressed.
When initially the atoms populated just one band the two models give the same results.
For instance, for c = 0 the probability of tunneling, defined as P = |a n (∞)| 2 for a n (−∞) = 0, is given by the Landau-Zener formula P = e −πv 2 0 /(2α) [19] which is illustrated in fig. 3. When both bands are populated initially, the two models do not agree in the prediction of the final populations. This is illustrated in fig. 4 same results if related by equation (27). However, the same equation leads to significant disagreement in panels (b) and (c). Similar conclusion can be drawn using the first relation scheme for the two models, i.e. equation (24). Here v 0 = 0.1375, α = 0.036, and c = 0. We have used the following initial condition (a 1 , a 2 ) = The tunneling probability of BEC at the edge of the Brillouin zone in an accelerated optical lattice was found to have an asymmetric dependence on the nonlinearity coefficient c [5]. The asymmetry in the tunneling probability was first theoretically predicted in Ref.
[16] within the nonlinear Landau-Zener model. The two-band model (15) The experimental results (see fig. 2 in Ref. [5]) also suggest that the upper-to-lower tunneling probability, in contrast to the lower-to-upper one, is unaffected by the nonlinearity (there is, however, a large error due to imperfections in the experiment). Below we argue that, indeed, there is an additional asymmetry in the sensitivity of the tunneling probability to the initial nonlinearity-induced population of the adjacent Bloch band, which, together with the asymmetric dependence on the nonlinearity coefficient, can give a complete explanation of the experimental results.
In section IV (see fig. 4 However, this argument cannot explain the fact that the strength of the sensitivity to the population is quite different in the two cases. In conclusion, a very small initial population of the Bloch band to which the tunneling takes place (in fact for r = 0.2 we get |a j | 2 = 0.12) has a dramatic effect on its final population in the case of the upper-to-lower tunneling as compared to that in the case of the lower-to-upper tunneling (if one consider the relative change in the tunneling probability when increasing r). This fact, together with the known asymmetry of the tunneling probability, provides a complete explanation of the experimental results presented in Ref.

VI. CONCLUSION
In the present work a finite-dimensional model has been derived for the condensate tunneling at the edge of the Brillouin zone in an accelerating optical lattice. The fine Bloch band structure was taken into account. A special attention was paid to the case of a weak lattice, which could have seemed a redundant task due to the existence of the nonlinear Landau-Zener model proposed for this case previously [15,16]. However, it turns out that the two-band model derived in this paper and the Landau-Zener model disagree significantly even in the case of a negligible nonlinearity if both adjacent bands are populated initially.
Only in the special case of just one Bloch band being initially populated the two models agree in the prediction of the final populations. In particular, the Landau-Zener result [19] is reproduced by our model.
The two-band model confirms the asymmetry in the tunneling probability dependence on the nonlinearity, predicted theoretically in Ref. [16] and discovered experimentally in Ref. [5]. However, besides the qualitative agreement between the theory and the experiment, a quantitative disagreement was also revealed. For instance, the experimental upper-to-lower tunneling seems to be unaffected by the nonlinearity [5]. So far we have discussed the so-called "Landau-Zener regime" of BEC tunneling neglecting the spatial modulation of the Bloch wave amplitudes. Recently the modulational instability of a Bloch wave amplitude was found to result in the irreversibility of tunneling and the asymmetry in the final band populations [11]. A unified approach, capable to cover the two identified regimes, the "Landau-Zener regime" and the "instability regime" can also be developed. The solution of this problem is left for a future publication. The inner product u n,q |∂ q u m,q can be easily evaluated by using the properties of the Bloch functions u n,q (x). For generality, consider an arbitrary periodic potential v = v(x).

VII. ACKNOWLEDGEMENTS
Define the Hermitian operator L q ≡ 1 2 (−i∂ x + q) 2 + v(x). We have then L q u n,q = E n (q)u n,q and, hence L q ∂ q u n,q = E n (q)∂ q u n,q + dE n (q) dq u n,q + i∂ x u n,q − qu n,q .
Using this result and orthogonality of the functions u n,q we get E n (q) u n,q |∂ q u m,q = u n,q |L q |∂ q u m,q = E m (q) u n,q |∂ q u m,q + i u n,q |∂ x u m,q .
Therefore, for m = n we obtain u n,q |∂ q u m,q = −i u n,q |∂ x u m,q E m (q) − E n (q) .
Now, let us evaluate the product on the r.h.s. of equation (A2). We have Hence, E n (q) u n,q |∂ x u m,q = u n,q |L q |∂ x u m,q = E m (q) u n,q |∂ x u m,q − u n,q | dv(x) dx |u m,q with the result κ nm ≡ u n,q |∂ q u m,q = −i u n,q |∂ x u m,q E m (q) − E n (q) = −i u n,q | dv(x) dx |u m,q (E m (q) − E n (q)) 2 , n = m.
What is left is to consider the diagonal inner product u n,q |∂ q u n,q . Here we use that the potential is even function: v(−x) = v(x). Then the functions u n,q (x) can be selected to satisfy u * n,q (x) = u n,q (−x). We get u n,q |∂ q u n,q = 1 2π π −π dx u n,q (−x)∂ q u n,q (x) = 1 2π π −π dx (−∂ q u n,q (−x)) u n,q (x) = 1 2π π −π dx (−∂ q u n,q (x)) u n,q (−x) = − u n,q |∂ q u n,q , where to replace the q-derivative we have used the independence of the inner product on q: u n,q |u n,q = 1. Hence u n,q |∂ q u n,q = 0. A similar reasoning for u n,q |(−i)∂ x u m,q can be used also to establish that κ nm (q) is real and that κ nm (q) = −κ mn (q). After simple calculations one arrives at the following system in the diabatic basis: i da dt = E(q)a + K(q)b, (B1) where the coefficients are as follows E(q) = 1 − ν 2 (q) 1 + ν 2 (q) ε(q), K(q) = 2ν(q)ε(q) 1 + ν 2 (q) + iα ε(q) ν(q) 1 + ν 2 (q) .
Here the functions ε(q) and ν(q) are taken from section III. (The system (B1)-(B3) was also verified numerically against the two-band model.) It is easy to see that E(q) is an odd function of k B −q, while K(q) is an even function of this variable. The following asymptotic properties of the coefficients can be easily established.
However, an important fact is observed about the system (B1)-(B2): the coupling coefficient K(q) is a complex valued function. Therefore, to obtain an equivalent Landau-Zener model (with a real coupling coefficient), one has to perform the following time-dependent phase transformation a = a (LZ) e iΘ(q) , b = b (LZ) e −iΘ(q) , Θ(q) ≡ arg{K(q)}/2.
It is easy to see that E (LZ) → −αt/2 as t → ±∞ and E (LZ) (k B ) = 0, i.e. we have all the necessary prerequisites for application of the arguments due to Landau and Zener [19] to the system (B6)-(B7). This explains the agreement between the Landau-Zener model and the two-band model when just one band is populated initially.