Quench-induced resonant tunneling mechanisms of bosons in an optical lattice with harmonic confinement

The nonequilibrium dynamics of small boson ensembles in a one-dimensional optical lattice is explored upon a sudden quench of an additional harmonic trap from strong to weak confinement. We find that the competition between the initial localization and the repulsive interaction leads to a resonant response of the system for intermediate quench amplitudes, corresponding to avoided crossings in the many-body eigenspectrum with varying final trap frequency. In particular, we show that these avoided crossings can be utilized to prepare the system in a desired state. The dynamical response is shown to depend on both the interaction strength as well as the number of atoms manifesting the many-body nature of the tunneling dynamics.

Figure 1

Dependence of the eigenenergies of the Hamiltonian of Eq. (1) on the trap frequency ${\omega }^2$ for a system consisting of $N=4$ bosons trapped in a triple well with additional harmonic confinement. The lowest (a) 15 eigenenergies for $g=1E_{\mathrm{R}}{k}^{-1}$ and (b) 20 eigenenergies for $g=4E_{\mathrm{R}}{k}^{-1}$ are shown. Solid (dashed) lines represent even (odd) parity eigenstates. Very narrow crossings (width smaller than $4\times {10}^{-3}{E}_{\mathrm{R}}^2{\hslash }^{-2}$) are denoted by dotted boxes, narrow avoided crossings are indicated by dashed boxes and wide avoided crossings are denoted by solid boxes. ${\alpha }_i, i=1,...,6$ indicate the position of each avoided crossing (see legend). The dominant number state contribution of each eigenstate at $\omega =0$ is indicated on the left-hand side of each figure.

Figure 2

(a), (c), (e), (g) Time evolution of ${\sigma }_{x,L}^2\left(t\right)$ and (b), (d), (f), (h) the corresponding spectra ${\sigma }_{x,L}^2\left({\omega }_{\mathrm{Fourier}}\right)$, as a function of the trap frequency ${\omega }^2$ after the quench. The system parameters used in each case correspond to (a), (b) $g=0E_{\mathrm{R}}{k}^{-1}, {\omega }_i^2=0.8E_{\mathrm{R}}^2{\hslash }^{-2}$ (initial trap frequency), (c), (d) $g=1E_{\mathrm{R}}{k}^{-1}, {\omega }_i^2=0.8E_{\mathrm{R}}^2{\hslash }^{-2}$, (e), (f) $g=1E_{\mathrm{R}}{k}^{-1}, {\omega }_i^2=0.58E_{\mathrm{R}}^2{\hslash }^{-2}$, and (g), (h) $g=4E_{\mathrm{R}}{k}^{-1}, {\omega }_i^2=0.8E_{\mathrm{R}}^2{\hslash }^{-2}$. ${\alpha }_i, i=1,...,6$ denote the positions of the corresponding avoided crossings (see also Fig. 1). Note that on top of each of the (a), (c), (e), (h) subfigures regions $I, II, III, T_C$ with $C\in \left\{{\alpha }_i\right\}$, are presented. In (d), (f) the branches are organized by solid boxes denoted in Table 1 as segment ${\alpha }_{i,j}$, where the index $j=1,2,3$ introduces an energetically increasing order.

Figure 3

Evolution of the dominant number state populations [see the legend above (a)] for $N=4$ interacting bosons ($g=1E_{\mathrm{R}}{k}^{-1}$) confined in a triple well with additional harmonic confinement (${\omega }_i^2=0.8E_{\mathrm{R}}^2{\hslash }^{-2}$). The dynamics is induced via a quench of the trap frequency to (a) ${\omega }^2=0.745E_{\mathrm{R}}^2{\hslash }^{-2}$ (region $I$), (b) ${\omega }^2=0.58E_{\mathrm{R}}^2{\hslash }^{-2}$ (region $T_{{\alpha }_2}$), (c) ${\omega }^2=0.48E_{\mathrm{R}}^2{\hslash }^{-2}$ (region $T_{{\alpha }_3}$), and (d) ${\omega }^2=0.385E_{\mathrm{R}}^2{\hslash }^{-2}$ (region $T_{{\alpha }_4}$). Note that the scaling of the time axis in (d) is different from the other cases and appears on top of the figure. The evolution of the number state populations is also shown for a quench from ${\omega }_i^2=0.56E_{\mathrm{R}}^2{\hslash }^{-2}$ to (e) ${\omega }^2=0.48E_{\mathrm{R}}^2{\hslash }^{-2}$ (region $I$) and ($f$) ${\omega }^2=0.265E_{\mathrm{R}}^2{\hslash }^{-2}$ (region $T_{\left\{{\alpha }_6{\alpha }_5\right\}}$). For the identification of the different regions see Fig. 2.

Figure 4

${\mathrm{\Delta }}_T\left\{{\sigma }_{x,L}\right\}$ as a function of the postquench frequency ${\omega }^2$. Comparison between (a) different interaction strengths $g$ (see legend), (b) the mean-field (MF) and MCTDHB results (see legend), and (c) different particle numbers $N=6,8$ (see legend) for $g=0.5E_{\mathrm{R}}{k}^{-1}$. The symbol ${}^{*}$ denotes that the system is initialized in the ground state for ${\omega }_{i,*}^2=0.56E_{\mathrm{R}}^2{\hslash }^{-2}$ instead of ${\omega }_i^2=0.8E_{\mathrm{R}}^2{\hslash }^{-2}$ being used otherwise.

Figure 5

(a) $\sqrt{{\rho }^{\left(1\right)}(x;t)}$ for $N=4$ interacting bosons ($g=1E_{\mathrm{R}}{k}^{-1}$) confined in a seven-well lattice with additional harmonic confinement. The initial trap frequency is ${\omega }_i^2=0.56E_{\mathrm{R}}^2{\hslash }^{-2}$ and we quench to $\omega =0$. (b) ${\mathrm{\Delta }}_T\left\{{\sigma }_{x,3w}^2\right\}$ with respect to the postquench ${\omega }^2$ for the triple well and the seven well setup (see legend) with the same parameter values, i.e., ${\omega }_i^2=0.56E_{\mathrm{R}}^2{\hslash }^{-2}$ and $g=1E_{\mathrm{R}}{k}^{-1}$. (c) ${\sigma }_{x,L}^2\left(t\right)$ for varying final trap frequency (see legend) for the seven-well setup. (d) Evolution of the density fraction within the triple-well region ($N_{3w}/N$) for a fifteen-well with an imposed harmonic trap. The system is initialized in the ground state with $N=5, g=1E_{\mathrm{R}}{k}^{-1}$ for varying initial trap frequency, ${\omega }_i^2$ (see legend), and the dynamics is induced by a quench to ${\omega }^2=0.016E_{\mathrm{R}}^2{\hslash }^{-2}$.

Figure 6

${\sigma }_{x,L}^2\left(t\right)$ for a quench from ${\omega }_i^2=0.80E_{\mathrm{R}}^2{\hslash }^{-2}$ to ${\omega }^2=0.40E_{\mathrm{R}}^2{\hslash }^{-2}$ and $g=1E_{\mathrm{R}}{k}^{-1}$ for (a) different number of grid points $M_p$ and $M=9$ SPFs, (b) different number of SPFs $M$ and $M_p=300$ grid points. (c), (d) The same as (a), (b) but for a quench from ${\omega }_i^2=0.80E_{\mathrm{R}}^2{\hslash }^{-2}$ to ${\omega }^2=0.464E_{\mathrm{R}}^2{\hslash }^{-2}$.