Landau-Zener Sweeps and Sudden Quenches in Coupled Bose-Hubbard Chains

We simulate numerically the dynamics of strongly correlated bosons in a two-leg ladder subject to a time-dependent energy bias between the two chains. When all atoms are initially in the leg with higher energy, we find a drastic reduction of the interchain particle transfer for slow linear sweeps, in quantitative agreement with recent experiments. This effect is preceded by a rapid broadening of the quasimomentum distribution of atoms, signaling the presence of a bath of low-energy excitations in the chains. We further investigate the scenario of quantum quenches to fixed values of the energy bias. We find that for a large enough density the momentum distribution relaxes to that of an equilibrium thermal state with the same energy.

Figure 1

(a) Sketch of the two-leg ladder described by the Hamiltonian Eq. (1). (b) Instantaneous right-leg population $n_R(t)$ during ground-state (dashed lines) and inverse sweeps (solid lines) with various rates $\alpha$, calculated with $n=0.75$ and $L_s=8$. The Hamiltonian parameters are $|{\Delta }_0|=18.2$, $J_{\parallel }=0.38$, and $U=1.58$ [21]. (c) Transfer efficiency $n_R(T,2\pi /\alpha )$ for inverse sweeps with the same parameters and $n=0.75$, 1, 1.25 (solid lines). The dashed line is the two-level LZ probability $p_{\mathrm{LZ}}(\alpha )$. The circles represent experimental data taken from Ref. 18 for the corresponding set of control parameters [21]. The inset shows the maximum transfer efficiency as a function of the density for $L_s=8$ (open circles) and 12 (filled circles) and the linear extrapolation to higher densities (dashed line and solid line, respectively). The blue horizontal line marks the maximum reached in the experiment.

Figure 2

Broadening of the left- and right-leg momentum distributions during the inverse sweep: (a) Quasimomentum distributions $n_{k\sigma }$ of the left and the right leg as a function of the reduced sweep time $2\pi /\alpha$ as obtained from DMRG for $n=0.75$ and $L_s=16$. The arrows mark peaks in the momentum distributions at $k=\pm \pi$ which appear for $2\pi /\alpha >1$. (b) The corresponding quasimomentum width $⟨k^2{⟩}_{\sigma }$ of both legs (filled symbols) plotted together with the results for $L_s=8$ (open symbols) where slower sweeps are accessible. The dashed vertical line marks the sweep rate for which $n_R$ reaches its maximum [see Fig. 1c]. The Hamiltonian parameters are the same as for Fig. 1.

Figure 3

Quantum dynamics after a sudden quench: (a) Time traces of the right-leg density $n_R(t)$ and (b) the width $⟨k^2{⟩}_R$ of the momentum distribution in the right leg for $n=1.5$ and different values of ${\Delta }_f$. (c) Longtime value $\overline{n_R}$ of the right-leg density versus ${\Delta }_f$ ($n=1.5$). Points and error bars refer to the time-averaged value and amplitude of the last oscillation, respectively [21]. The solid line denotes the right-leg population as obtained for an interacting thermal Bose gas with the same total energy as the initial state in the quenches. The dashed line is the prediction for an ideal Bose gas. The shaded area indicates ${\Delta }_f\le -4J_{\parallel }$. The inset shows the inverse temperatures used in the finite-temperature calculations. (d) Longtime value of the width of the momentum distribution versus detuning for various densities (circles). The solid and dashed lines are the results of the finite-temperature calculations as in (b). For all plots, the size of the system is $L_s=16$ and the Hamiltonian parameters are as in Fig. 1.