Effect of the lattice alignment on Bloch oscillations of a Bose-Einstein condensate in a square optical lattice

We consider a Bose-Einstein condensate of ultracold atoms loaded into a square optical lattice and subject to a static force. For vanishing atom-atom interactions the atoms perform periodic Bloch oscillations for arbitrary direction of the force. We study the stability of these oscillations for non-vanishing interactions, which is shown to depend on an alignment of the force vector with respect to the lattice crystallographic axes. If the force is aligned along any of the axes, the mean field approach can be used to identify the stability conditions. On the contrary, for a misaligned force one has to employ the microscopic approach, which predicts periodic modulation of Bloch oscillations in the limit of a large forcing.

We consider a Bose-Einstein condensate of ultracold atoms loaded into a square optical lattice and subject to a static force. For vanishing atom-atom interactions the atoms perform periodic Bloch oscillations for arbitrary direction of the force. We study the stability of these oscillations for non-vanishing interactions, which is shown to depend on an alignment of the force vector with respect to the lattice crystallographic axes. If the force is aligned along any of the axes, the mean field approach can be used to identify the stability conditions. On the contrary, for a misaligned force one has to employ the microscopic approach, which predicts periodic modulation of Bloch oscillations in the limit of a large forcing.
A Bose-Einstein condensate (BEC) in optical lattices has intrigued a rapidly growing interest as it provides an experimentally realizable system with controllable interactions. A variety of phenomena concerning different aspects of physics for quantum many-body systems has been studied, such as the superfluid-Mott insulator quantum phase transition [1,2], BEC-BCS crossover for fermionic gases [3], and quantum transport in accelerated lattices, where atoms exhibit fundamental quantum effects such as the Wannier-Stark ladder [4], Landau-Zener tunnelling [5], and Bloch oscillations [4,6] -phenomena usually associated with electron in solid crystall. This work deals with the last mentioned problem, namely, Bloch oscillations (BO) of condensed atoms in optical lattices. We would like to mention that besides pure academic interest this problem also has an applied aspect because BO provide a tool for precision measurement of gravitational field and inter-atomic interaction constant.
Until quite recently almost all theoretical, numerical and experimental studies of BO concerned 1D or quasi 1D lattices (see Ref. [7,8,9] for the contemporary reviews). Nowadays one observes a growing interest in BO in multidimensional lattices [10,11,12,13,14,15]. In the single-particle approach this problem was considered in Refs. [10,11]. It was shown that an increase of the lattice dimensionality introduces new effects not present in the 1D lattice. Some predictions of these works were later on confirmed in the experiment with the array of optical guides [13], where one uses a formal analogy between the Maxwell and Schrödinger equations. The experiment with a BEC of interacting atoms addresses the further questions [15], in particular, the question about the stability of multidimensional BO. Indeed, it is known that a BEC in optical lattices can be dynamically unstable, which quantum-mechanically means decoherence of the BEC [16]. In the present work we study the conditions under which the dynamical instability is suppressed and, hence, multidimensional BO are stable. Unlike 1D lattices, these conditions are shown to involve an alignment of the static force vector with respect to the crystallographic axes of the lattice. We also argue in the work that by changing the angle between the primary lattice vectors and the force vector one may observe a transition from the mean-field to the microscopic Bloch dynamics.
To simplify the equations we shall consider the twodimensional case throughout the paper, -generalization of the results in three dimensions is straightforward. The Bose-Hubbard Hamiltonian of atoms in the tilted 2D lattice reads, where J x,y are the hopping matrix elements in x and y directions, W microscopic atom-atom interaction constant, d lattice period, and F x,y the projections of the static force vector on the lattice axes. The Hilbert space of (1) is spanned by the Fock states |n ≡ |n m,l , where m,l n m,l = N -the total number of atoms. Since in the coordinate representation the Fock states are given by the symmetrized product of the localized Wannier functions, we shall refer to this basis as the Wannier basis. The translational invariance of the system, broken by the static term, can be actually recovered by using the gauge transformation [7]. Then the Hamiltonian (1) takes the form where ω x,y = dF x,y / are the Bloch frequencies associated with x and y component of the static force. We also note that in stead of the Wannier basis one can use the quasimomentum Fock basis |q ≡ |q p,k for the Hamiltonian (2), which we shall refer to as the Bloch basis.
(Needless to say that in the coordinate representation the quasimomentum Fock states are given by the symmetrized product of the extended Bloch functions.) Formally this corresponds to the canonical transformation which implicitly assumes the periodic boundary conditions.
Misaligned Force-We begin with the case of a strong misaligned force dF x , dF y ≫ J x,y > W . In order to illuminate situation, we model BO in a small 2D lattice by numerically solving the time-dependent Schrödinger equation with the Hamiltonian (2) for the specified initial conditions. As those we consider the superfluid state with q p,k = N δ p,0 δ k,0 , which approximates the ground state of the system for F = 0 and W < J x,y . (Substitution of this state by the exact ground state practically does not affect the final result.) Fig. 1 shows the numerical results for a 3 × 3 lattice with 7 atoms inside. The lower panel in Fig. 1 depicts the mean energy of the system, the upper and middle panels show the order parameters e x (t) and e y (t) defined as [18] [By replacing the operators in the bracket in Eq. (3) with â † m,l+1â m,l one obtains a similar expression for the order parameter e y (t).] It is seen in the figure that BO persist in time but are modulated with some characteristic period. We would also like to mention that the Bloch dynamics displayed in Fig. 1 is converged in the thermodynamic limit, i.e., for givenn = N/L 2 the further increase of the system size affects neither the modulation period nor the shape of modulation.
To prove that BO in the misaligned lattice are stable in the limit of strong forcing and to identify the modulation period we proceed as follows. First we introduce the new wave function | Ψ(t) through the relation |Ψ(t) = U 0 (t)| Ψ(t) , where U 0 (t) is the evolution operator for vanishing atom-atom interactions. The function | Ψ(t) obviously obeys the equation, Note that the operator (5) tends to the identity operator for F x , F y → ∞. Substituting U 0 (t) in Eq. (4) by identity matrix and noting that the interaction energy operator is diagonal in the Wannier basis with integer entries, n| m,ln m,l (n m,l − 1)|n = m,l n 2 m,l − N , we conclude that the time evolution of the wave function | Ψ(t) is periodic with the period T W = 2π /W [17]. Coming back to the original wave function this result means the periodic modulation of BO with the frequency ω W = W/ . It is worth stressing that the above proof assumes both F x and F y to be large and, hence, the case of aligned lattices is excluded.
Aligned Force-Next we consider the situation where the force is aligned along one of the crystallographic axes (to be certain, the y-axis in what follows). Within the single-particle approach the static force F y would localize the atoms in the y-direction. Thus one may expect that if F y is large enough the atoms form separate BECs in the planes perpendicular to the force vector, weakly coupled together as a one-dimensional BEC chain. Introducing new operatorsÂ l = 1 √ L mâ m,l andÂ † l , the effective Hamiltonian reads where W ef f = W/L. Thus we have reduced the 2D problem to a 1D problem with the renormalized interaction constant. (If one considers 3D lattices, the renormalization is W ef f = W/L 2 .) Moreover, since the mean number of atoms N in any site of the effective 1D system is given bynL, the occupation numbers will be macroscopically large in the thermodynamic limit N, L → ∞, n = N/L 2 = const, which justifies the mean field approach.
The mean-field Hamiltonian of the system (6) reads (up to the irrelevant constant terms proportional to where A l and A * l are pairs of the canonically conjugated variables and the macroscopic interaction constant g = W ef f N = Wn. Within the mean-field approach the border between stable and unstable (decaying) BO is know exactly [19,20]. Namely, for J/F d > 0.5 the critical value of nonlinearity is a linear function of the static force magnitude, while for J/F d < 0.5 it additionally depends on the value of the hopping matrix elements: Obviously, for a fixed nonlinearity g the condition (8) can also be formulated as a condition on the critical magnitude F cr of the static force. The microscopic analysis of BO in the aligned lattice confirms our working hypothesis. Choosing the parameters in such a way that the 1D mean-field BO are stable, we simulate BO of N = 7 atoms in the 2D lattice with L = 3. The dashed line in the upper panel of Fig. 2 shows the dynamics of the order parameter e x (t). It is seen that e x (t) ≈ −1, -therefore we indeed have in-plane BECs. We also note that the decay and revival of the order parameter e y (t) in the lower panel is an artifact due to the finite size of our lattice. Indeed, it can be shown that the time evolution of e y (t), calculated on the basis of the effective Hamiltonian (6), obeys the equation [21] e y (t) = − exp −2 N 1 − cos W ef f t . Because W ef f = W/L and N =nL, one has e y (t) = −1 in the thermodynamic limit.
The above analysis of BO in the aligned lattice relies on the reduction of a two-dimensional system to an effective one-dimensional mean-field problem. It should be especially stressed that this reduction is possible only if the dynamics of the reduced system is stable. If we choose the parameters in the unstable regime the situation becomes totally different. Figure 3 shows the numerical results for F = 0.2J/d < F cr , where one-dimensional BO suffer from dynamical instability. Unlike in the stable regime, BO along y direction now excite the transverse degree of freedom and we observe decay of the both order parameters towards zero. Thus no reduction to one dimension is possible.
Slightly Misaligned Force-Finally we briefly analyze an experimentally important situation of a small mismatch between the lattice axis and the static field vector, i.e., F x ≪ F y . The solid lines in Fig. 2 show the order parameters for the same dF = 20J but F x = 0.001F . Compared to the case F x = 0 (dashed lines in Fig. 2), we observe the destruction of BEC after time t * ≈ 12T J . This critical time can be understood in terms of the mean-field approach as well. Indeed, it is known that a stationary BEC is unstable for the quasimomentum κ outside the first quarter of the Brillouine zone. Since the static force causes the linear growth of the quasimomentum, κ x,y (t) = κ x,y + F x,y t/ , the system always enter the instability region of the Brillouine zone. However, if the static force is strong enough, the system passes the instability region so quickly that it 'has no time' to decay. [In fact this is a physical argument behind Eq. (8).] In the considered example the strong static force ensures the fast driving along y direction but simultaneously it slowly brings the system to κ x = π/2d along x direction. As soon as this border of instability is reached (t * /T J = J/4dF x ), we observe an irreversible decay of the order parameters.
Conclusion. In summary, we have studied BO of a BEC of atoms in a square lattice for both aligned and misaligned static forces. It is shown that in the case of aligned force the system may be reduced to a onedimensional chain of mini BECs, which we treat by using the mean-field approach. Then the stability diagram of this effective 1D system defines the critical magnitude of the static force above which BO are stable, with no excitations of the transverse degrees of freedom. On the contrary, in the unstable regime, F < F cr , BO induced by the static force excite the transverse modes and, as a consequence, one observes BEC distruction and decay of BO. Our studies also illuminated importance of the alignment. The strong (F > F cr ) but slightly misaligned force is shown to slowly intrigue the transverse modes, which destabilize BO after some well-defined transient time. However, if misalignment is large, BO appear to be stable again. This case corresponds to the quantum (not mean-field) regime of BO, where they are modulated with the frequency defined by the microscopic interaction constant.
Fruitful discussions with A. Buchleitner and financial support by Deutsche Forshungsgemeinschaft under SPP1116 are gratefully acknowledged.