Mode coupling of interaction quenched ultracold few-boson ensembles in periodically driven lattices

The out-of-equilibrium dynamics of interaction quenched finite ultracold bosonic ensembles in periodically driven one-dimensional optical lattices is investigated. It is shown that periodic driving enforces the bosons in the outer wells of the finite lattice to exhibit out-of-phase dipolelike modes, while in the central well the atomic cloud experiences a local breathing mode. The dynamical behavior is investigated with varying driving frequencies, revealing resonantlike behavior of the intrawell dynamics. An interaction quench in the periodically driven lattice gives rise to admixtures of different excitations in the outer wells, enhanced breathing in the center, and amplification of the tunneling dynamics. We then observe multiple resonances between the inter- and the intrawell dynamics at different quench amplitudes, with the position of the resonances being tunable via the driving frequency. Our results pave the way for future investigations of the use of combined driving protocols in order to excite different inter- and intrawell modes and to subsequently control them.

Figure 1

(a) Time evolution of the fidelity $F_{\left\{{\omega }_D\right\}}\left(t\right)$ as a function of the driving frequency ${\omega }_D$. The driving amplitude is $\delta =0.03$ and the initial state corresponds to the ground state of four weakly interacting bosons with $g=0.05$ confined in a triple well. Inset: Mean response ${\overline{F}}_{\left\{{\omega }_D\right\}}$ at ${\omega }_D=0.75$ and at ${\omega }_D=2.75$ for different interparticle repulsions, $g=0.5$ and $g=3.0$, as a function of the particle number $N$ (see legend). (b) Same as (a), but for a fixed interaction quench, with amplitude $\delta g=2.0$, on top of the driven triple well. (c) Deviation from unity of the first natural occupation number, i.e., $\lambda \left(t\right)=1-n_1\left(t\right)$, during the evolution for different driving frequencies ${\omega }_D$ (see legend). The effect of a stronger interparticle repulsion for $g=1.0$ at ${\omega }_D=3.0$ in the fragmentation process is also shown. (d) The same as (c), but for a fixed interaction quench, $\delta g=2.0$, upon the driving.

Figure 2

Time evolution of the one-body density ${\rho }_1(x,t)$ caused by a periodically driven triple well with (a) ${\omega }_D=0.75$ and (b) a simultaneous interaction quench with amplitude $\delta g=2.0$. White contours, at ${\rho }_1(x,t)=0.25$, are plotted on top in order to facilitate a comparison of the atomic motion between the unquenched (a) and the quenched (b) systems. The driving amplitude is fixed at the value $\delta =0.03$ and the initial state corresponds to the ground state of four weakly interacting bosons with $g=0.05$.

Figure 3

(a) Spectrum of the fidelity $F_{\left\{{\omega }_D\right\}}\left(\omega \right)$ as a function of the driving frequency ${\omega }_D$. Black dots correspond to $F_{\{{\omega }_D,\delta g=0.0\}}\left(\omega \right)$, i.e., to the unquenched system, while open red circles refer to $F_{\{{\omega }_D,\delta g=2.0\}}\left(\omega \right)$, i.e., to the case of a simultaneous interaction quench with amplitude $\delta g=2.0$ on top of the driving. Inset ($a_1$): Tunneling probabilities $A_1\left(t\right), A_2\left(t\right),$ and $A_3\left(t\right)$ (see text and legend) at ${\omega }_D=2.75$. (b) Comparison of the single-particle tunneling probabilities $A_1\left(t\right)$ in a periodically driven triple well without and with a simultaneous interaction quench for various driving frequencies ${\omega }_D$ (see legend). The driving amplitude is fixed at the value $\delta =0.03$ and the initial state corresponds to the ground state of four weakly interacting bosons with $g=0.05$.

Figure 4

(a) Spectrum of the intrawell asymmetry for the left well $\mathrm{\Delta }{\rho }_L\left(\omega \right)$ in a driven triple well, with respect to the driving frequency ${\omega }_D$. The white rectangle indicates the region of resonance. (b) Intrawell asymmetry evolution $\mathrm{\Delta }{\rho }_L\left(t\right)$ at resonance (${\omega }_D=2.875$) employing different interaction quenches (see legend). (c) Excitation probability $|B_{\left\{N_i\right\};\left\{I_i\right\}}{|}^2$ (see text) during the evolution for different driving frequencies ${\omega }_D$. (d) The same as (c), at ${\omega }_D=2.75$, for different quenches on the interparticle repulsion (see legend). The driving amplitude is fixed at the value $\delta =0.03$, while the initial state corresponds to the ground state of four weakly interacting bosons with $g=0.05$.

Figure 5

Time evolution of the fidelity $F_{\{{\omega }_D,\delta g\}}\left(t\right)$ in a periodically driven triple well with (a) ${\omega }_D=0.75$ and (b) ${\omega }_D=2.75$ as a function of the quench amplitude. The driving amplitude is $\delta =0.03$, while the initial state corresponds to the ground state of four weakly interacting bosons with $g=0.05$.

Figure 6

The following are shown as a function of the quench amplitude $\delta g$: the fidelity spectrum $F_{\{{\omega }_D,\delta g\}}\left(\omega \right)$ for (a) ${\omega }_D=0.75$ and (b) ${\omega }_D=2.75$; the spectrum of the local-breathing mode $\left[{\sigma }_M^2\left(\omega \right)\right]$ for the middle well of a periodically driven triple well with (c) ${\omega }_D=0.75$ and (d) ${\omega }_D=2.75$; and the spectrum of the local-dipole mode $\left[\mathrm{\Delta }{\rho }_L\left(\omega \right)\right]$ for the left well of a driven triple well with (e) ${\omega }_D=0.75$ and (f) ${\omega }_D=2.75$. Solid and dashed ellipses indicate the positions of the resonances between the tunneling and the breathing or dipole branches (see text). The driving amplitude is $\delta =0.03$, while the initial state corresponds to the ground state of four weakly interacting bosons with $g=0.05$.

Figure 7

Time evolution of the one-body density ${\rho }_1(x,t)$ in a periodically driven 11-well potential for different driving frequencies: (a) ${\omega }_D=1.25$ and (b) ${\omega }_D=2.875$. The driving amplitude is fixed at the value $\delta =0.03$, while the initial state corresponds to the ground state of five weakly interacting bosons with $g=0.05$. (c) Probability of finding all the bosons in the central well $\left[P_M\left(t\right)\right]$ during the evolution for different driving frequencies ${\omega }_D$ (see legend). (d) The same as (c), but for ${\omega }_D=0.75$ and different quench amplitudes $\delta g$ (see legend).

Figure 8

One-body coherence function at different instants in time ($t_1=1.0, t_2=56.0, t_3=123.0$, and $t_4=193.0$) during the evolution caused by a periodically driven 11-well potential with (a)–(d) ${\omega }_D=0.75$ and (e)–(h) ${\omega }_D=3.0$. (i)–(l) Evolution of the one-body coherence in a periodically driven potential with ${\omega }_D=0.75$ and a simultaneous interaction quench with amplitude $\delta g=1.0$. The driving amplitude is fixed at the value $\delta =0.03$ and the initial state corresponds to the ground state of five weakly interacting bosons with $g=0.05$.

Figure 9

(a) Time evolution of the one-body density ${\rho }_1(x,t)$ caused by a periodically driven harmonic trap with ${\omega }_D=0.25$ and a simultaneous interaction quench with amplitude $\delta g=1.6$. The driving amplitude is fixed toat the value $A=0.6$, while the initial state corresponds to the ground state of six weakly interacting bosons with $g=0.05$. We also illustrate the one-body density profiles at certain instants in time (see legend) during the evolution of the periodically driven oscillator with (b) $\delta g=1.6$ and (c) $\delta g=0$.

Figure 10

(a) Time evolution of the one-body density ${\rho }_1(x,t)$ in a triple well for ${\omega }_D=2.75$. The driving amplitude is fixed at the value $\delta =0.03$, while the initial state corresponds to the ground state of four weakly interacting bosons with $g=0.05$. (b) Mean oscillation amplitude $\mathrm{\Lambda }$ of the left well for $N=4$ bosons as a function of the driving frequency ${\omega }_D$ for different interparticle repulsions (see legend). (c) The same as (b), but for fixed interaction $g=0.2$ and different particle numbers (see legend).

Figure 11

Fidelity evolution $F_{{\omega }_D}\left(t\right)$ of a periodically driven triple well with (a) ${\omega }_D=2.5$ and (b) ${\omega }_D=7.5$ with an increasing number of SPFs (see legend). (c), (d) $F_{{\omega }_D}\left(t\right)$ for various SPFs (see legend) with a simultaneous interaction quench of amplitude (c) $\delta g=0.5$ and (d) $\delta g=2.0$ on top of the periodically driven triple well with ${\omega }_D=0.75$.

Figure 12

Fidelity evolution $F_{{\omega }_D}\left(t\right)$ of a periodically driven triple well with (a) ${\omega }_D=2.5$ and (b) ${\omega }_D=7.5$ with an increasing number of grid sizes (see legend). (c), (d) $F_{{\omega }_D}\left(t\right)$ for various grid sizes (see legend) with a simultaneous interaction quench of amplitude (c) $\delta g=0.5$ and (d) $\delta g=2.0$ on top of the periodically driven triple well with ${\omega }_D=0.75$.