Many-body Landau-Zener tunneling in the Bose-Hubbard model

We study a model for ultracold, spinless atoms in quasi-one-dimensional optical lattices and subjected to a tunable tilting force. Statistical tests are employed to quantitatively characterize the spectrum of the Floquet-Bloch operator of the system along the transition from the regular to the quantum chaotic regime. Moreover, we perturbatively include the coupling of the one-band model to the second energy band. This allows us to study the Landau-Zener interband tunneling within a truly many-body description of ultracold atoms. The distributions of the computed tunneling rates provide an independent and experimentally accessible signature of the regular-chaotic transition.

Figure 1

The quasienergy levels $E$ of the FB operator for a system with $N=4$, $L=5$, in the subspace with $\kappa =0$. The WS ladder is seen on the left-hand panel, the perturbative splitting of the first rung is magnified in the lower panel. In the upper panel a single linear function of $1/F$ is added to $2E/F$ to eliminate an overall winding trend, and allow a better visibility of the avoided crossings. The parameters of the Hamiltonian are $J_1=0.038$, $W_1=0.032$, as used also in the subsequent figures.

Figure 2

The distribution $\mathsf{P}\left(s\right)$ and the CDF (insets) for the quasienergy spacing (stairs), the WD (solid), and the Poisson distribution (dashed). The parameters are $N=5$, $L=8$, ${\text{log}}_{10}\left(2\pi /F\right)\simeq 2.0$ (left-hand side, regular) and 2.4 (right-hand side, chaotic).

Figure 3

Statistical tests applied to the system of Fig. 2. (a) ${\chi }^2$-like test of Eq. (5). (b) $T$ test, and (c) variance of number of levels (with mean spacing normalized to 1), for the chaotic (circles) and the regular (pyramids) case. In both panels the solid and dotted-dashed lines are the theoretical predictions for the WD and Poisson statistics of energy levels, respectively.

Figure 4

(Color online) (a,b) The probability distribution $\mathsf{P}$ and (c,d) the CDF for the logarithm of the decay widths $\Gamma$ in the regular regime [(a,c) with ${\text{log}}_{10}\left(2\pi /F\right)\simeq 1.5$] and in the chaotic regime [(b,d) with ${\text{log}}_{10}\left(2\pi /F\right)\simeq 2.1$]. The size of the system is $N=8$, $L=7$. The dashed line is the fit with a log-normal distribution. The inset in panel (d) shows that the log-normal is not appropriate to fit the tails of the distributions in the chaotic regime. Here and in the following figures, $V=1.5$, ${\epsilon }_2-{\epsilon }_1=2.63$, $J_1=0.22$, $J_2=-1.0$, $W_1=0.2$, $W_{\times }=0.1$, $d_F=-0.2$.

Figure 5

(Upper panel) The ${\chi }^2$-like test for the quasienergy spacing $s$ and (lower panel) the spread of the distribution of the decay widths $\Gamma$. The size of the system is $N=8$, $L=7$. The dashed lines are a guide to the eye and suggest that the transition to the chaotic regime can be appreciated by looking at both, the real and the imaginary parts of the FB eigenvalues.

Figure 6

(Color online) (a)–(c) Left-hand side of Eq. (23) in the regular (a) and in the chaotic regime (b,c), for a system with $N=8$, $L=7$. The dashed area in panels (b,c) shows the part of the histogram where the total amount of points is about 40, less then 10% of the full sample. The red solid line is the linear fit to the profile in the scaling region. (d) The exponent of the power law $\mathsf{P}\left(\Gamma \right)\sim {\Gamma }^{-\alpha }$ as obtained from the linear fits shown in (a)–(c), for $N=8$, $L=7$ (black circles), $N=8$, $L=5$ (red squares), $N=7$, $L=8$ (blue diamonds), $N=7$, $L=5$ (green pyramids). To show together different data sets we horizontally translated the first by $+0.1$ and the last by $-0.05$.