Slow quench dynamics of periodically driven quantum gases

We study the evolution of bosons in a periodically driven optical lattice during a slow change of the driving amplitude. Both the regime of high-frequency and low-frequency driving are investigated. In the low-frequency regime, resonant absorption of energy is observed. In the high-frequency regime, the dynamics is compared to a system with an effective Hamiltonian in which the atoms are “dressed” by the driving field. This “dressing” can dramatically change the amplitude and sign of the effective tunneling. A particular focus of this study is the investigation of the time scales necessary for the evolving quantum state to follow almost adiabatically to the ground state of the effective many-body system.

Figure 1

Time evolution of the compressibility $\kappa$ for different driving frequencies $\omega$. The value of the strength of the driving force $f\left(t\right)$ is linearly increased in time, such that at time $t_f=12\hslash /J$, ${\mathcal{J}}_0(f_t{t}_{f}a/\hslash \omega )=0$. The “undressed” interaction strength is $U/J=4$, and system length $L=48$ with $N=48$ atoms.

Figure 2

Time evolution of the compressibility $\kappa$ for chosen driving frequencies $\omega$ ($\hslash \omega =6J$, stars; $\hslash \omega =12J$, squares; $\hslash \omega =14J$, triangles; $\hslash \omega =18J$, circles; and $\hslash \omega =30$, diamonds) and without driving following the decrease of the effective tunneling amplitude (continuous line). The inset shows the evolution of the effective tunneling amplitude. Same parameters as in Fig. 1.

Figure 3

Compressibility $\kappa$ vs the final time of the quench, $t_f$, for different driving amplitudes $f_t$ such that (a) $J_{\mathrm{eff}}\left(t_f\right)=0$, (b) $J_{\mathrm{eff}}\left(t_f\right)=0.5J$, and (c) $J_{\mathrm{eff}}\left(t_f\right)=-0.4J$. Different curves are $\hslash \omega /J=18$ (dash-dotted line), $\hslash \omega /J=25$ (continuous line), and undriven (dashed line). The symbols denote actual numerical data. The horizontal lines represent the values of the compressibility of the target states, and arrows indicate the value of the compressibility in the initial state. Parameters are $U/J=4$, $N=L=48$.

Figure 4

(a) Single-particle correlations between different sites vs their distance in the presence of the driving amplitude of frequency $\hslash \omega /J=25$ (continuous line with squares) and for the undriven case (dashed line with circles). The target value is shown for comparison (continuous line). (b) Single-particle correlations ($d=1,3,5$) vs the final time of the evolution. Parameters: $U/J=4$, $N=L=48$, $j=25$, and $f_t$ such that $J_{\mathrm{eff}}\left(t_f\right)=0.5J$. Results are taken at exact periods of the driving.

Figure 5

Real part of the nearest-neighbor correlation [$\text{Re}\left(\langle {\widehat{b}}_j^{†}{\widehat{b}}_{j+1}\rangle \right)$] vs time for the driven case (dotted line) and undriven case (continuous line). Squares mark the values at each quarter of a period. Parameters: $U/J=4$, $\hslash \omega /J=25$, $N=L=48$, and $f_t$ such that $J_{\mathrm{eff}}\left(t_f\right)=-0.4J$.

Figure 6

Momentum distribution $\rho \left(k\right)$ vs momentum for $f_t$ such that (i) $J_{\mathrm{eff}}\left(t_f\right)=0$, (ii) $J_{\mathrm{eff}}\left(t_f\right)=0.5J$, and (iii) $J_{\mathrm{eff}}\left(t_f\right)\approx -0.4J$. The continuous line corresponds to the driven case, the dashed line to the undriven case, and the stars to the value of the target state. Parameters: $U/J=4$, $\hslash \omega /J=25$, $N=L=48$, and $t_f\approx 17\hslash /J$.

Figure 7

Compressibility $\kappa$ vs the strength of the interaction $U$. Squares represent the values of the driven case, circles the undriven case, and the continuous line the value of the target state. Inset: Deviations of the compressibility $\kappa$ from that of the target state ${\kappa }_G$. Parameters: $L=N=64$, $t_f\approx 12\hslash /J$, $\hslash \omega /J=25$, and $f_t$ such that $J_{\mathrm{eff}}\left(t_f\right)=0.5J$.

Figure 8

Real part of the nearest-neighbor correlation [$\text{Re}\left(\langle {\widehat{b}}_j^{†}{\widehat{b}}_{j+1}\rangle \right)$] vs time for the driven case with $\hslash \omega /J=25$ (dotted line, squares mark the value at a quarter of a period) and undriven case (continuous line). Parameters: $U/J=4$, $N=L=48$, $j=25$, and $f_t$ such that (a) $J_{\mathrm{eff}}\left(t\right)=0$, (b) $J_{\mathrm{eff}}\left(t\right)=0.5J$, and (c) $J_{\mathrm{eff}}\left(t\right)\approx -0.4J$ for all $t>t_f\approx 6\hslash /J$. $t_f$ is marked, in each case, with a vertical line.