Negative-quench-induced excitation dynamics for ultracold bosons in one-dimensional lattices

The nonequilibrium dynamics following a quench of strongly repulsive bosonic ensembles in one-dimensional finite lattices is investigated by employing interaction quenches and/or a ramp of the lattice potential. Both sudden and time-dependent quenches are analyzed in detail. For the case of interaction quenches we address the transition from the strong repulsive to the weakly interacting regime, suppressing in this manner the heating of the system. The excitation modes such as the cradle process and the local breathing mode are examined via local density observables. In particular, the cradle mode is inherently related to the initial delocalization and, following a negative interaction quench, can be excited only for incommensurate setups with filling larger than unity. Alternatively, a negative quench of the lattice depth which favors the spatial delocalization is used to access the cradle mode for setups with filling smaller than unity. Our results shed light on possible schemes to control the cradle and the breathing modes. Finally, employing the notion of fidelity we study the dynamical response of the system after a diabatic or adiabatic parameter modulation for short and long evolution times. The evolution of the system is obtained numerically using the ab initio multilayer multiconfiguration time-dependent Hartree method for bosons, which permits us to follow nonequilibrium dynamics including the corresponding investigation of higher-band effects.

Figure 1

Fragmentation analysis for a system of four bosons in a triple well with $g_{in}=5.0$. Shown are (a) the time evolution of the first three occupations $n_0\left(t\right)$ (upper panel blue lines), $n_1\left(t\right)$ (lower panel red lines), and $n_2\left(t\right)$ (lower panel green lines), for different quench amplitudes $\delta g=-4.9$ (dashed lines), $\delta g=-4.0$ (thick solid lines), and $\delta g=-2.5$ (thin solid lines). (b) Profiles of the lowest natural orbital ${\varphi }_0(x,t)$ for a quench amplitude $\delta g=-4.95$ and different time instants during the evolution $t_1=0.9$ (blue dashed), $t_2=2.6$ (red solid), and $t_3=7.0$ (black dashed-dotted).

Figure 2

(a) Fidelity evolution following negative interaction quenches for $g_{in}=5.0$. (b) The frequency spectrum of the fidelity for $g_f=0.6$ (blue dashed) and $g_f=1.0$ (red solid), which indicates the tunneling modes. The inset shows the dependence of each tunneling branch with respect to the final interaction strength after the quench. We incorporate 150 quenches in the range of $0<g_f<5.0$. The frequency units are normalized as $\omega /\Delta \omega$, with $\Delta \omega =2\pi /T$ and $T$ being the considered evolution time.

Figure 3

The time evolution of the normalized one-body correlation function $g_{ij}^{\left(1\right)}$ after various negative interaction quenches for $g_{in}=5.0$. Shown are different components of the correlation function $g_{ij}^{\left(1\right)}$ with respect to the left well $g_{LL}^{\left(1\right)}$ (blue dashed line), $g_{LM}^{\left(1\right)}$ (green thin solid line), and $g_{LR}^{\left(1\right)}$ (red thick solid line) for final interactions (a) $g_f=3.8$, (b) $g_f=1.6$, and (c) $g_f=0.05$.

Figure 4

Fourier spectrum of the second moment ${\sigma }_M^2\left(\omega \right)$ for the local breathing mode for different quench amplitudes. The initial state corresponds to $g_{in}=5.0$, and the final interactions are $g_f=0.15$ (blue dashed), $g_f=0.8$ (red solid), and $g_f=1.7$ (black dashed-dotted). In the inset we show the $\delta g$ dependence of each breathing branch, where we incorporate 150 quenches in the range of $0<g_f<5.0$. Note that the frequency units are normalized with respect to $\omega /\Delta \omega$, where $\Delta \omega =2\pi /T$ and $T$ is the respective propagation time.

Figure 5

Space-time evolution of the fluctuations $\delta \rho (x,t)$ after a sudden negative quench of the interparticle repulsion from $g_{in}=5.0$ to $g_f=0.07$, thereby approaching the noninteracting limit. We observe the cradle mode in the left and right wells, the local breathing mode in the middle well and the interwell tunneling during the evolution.

Figure 6

The frequency spectrum of the intrawell asymmetry $\Delta {\rho }_L\left(\omega \right)$. (a) The final state of the system is obtained after a sudden negative interaction quench from $g_{in}=5.0$ to $g_f=0.1$ (blue dashed) and $g_f=0.45$ (red solid). The inset shows the evolution of each peak that refers to the cradle as a function of the quench amplitude (we incorporate 150 quenches in the range $0<g_f<5.0)$. (b) The spectrum $\Delta {\rho }_L\left(\omega \right)$ for the same quench amplitude, $\delta g=-4.95$, and different barrier heights $V_0=5.5$ (red solid) and $V_0=3.5$ (blue dashed). (c) Sudden quench to $g_f=0.4$ and the hard-wall boundaries located at $x_{\sigma }=\pm 3\pi /2$ (blue dashed), $x_{\sigma }=\pm 5\pi /4$ (red solid), and $x_{\sigma }=\pm 11\pi /8$ (black dashed-dotted). (d) It is illustrated the spectrum of $\Delta {\rho }_L\left(\omega \right)$ for an imposed harmonic trap $V_{\mathrm{harm}}=0$ (blue dashed), $V_{\mathrm{harm}}=0.02x^2$ (red solid), and $V_{\mathrm{harm}}=0.05x^2$ (black dashed-dotted) on top of the lattice. Finally, note that in each case we use normalized frequency units $\omega /\Delta \omega$, with $\Delta \omega =2\pi /T$ and $T$ being the respective evolution time.

Figure 7

Fidelity evolution for $g_{in}=5.0$ and $g_f=0.1$ as a function of the ramp rate $\tau$ measured in units of the Heisenberg time ${\tau }_H$ (see text). The black dotted line correspond to the situation with $t=\tau$.

Figure 8

The fluctuations $\delta \rho (x,t)$ of the one-body density caused by a negative time-dependent quench of the interparticle repulsion to $g_f=0.07(g_{in}=5.0)$ with a finite ramp rate $\tau =0.8{\tau }_H$. For a direct comparison the quench parameters, here, have been chosen similar with Fig. 5, which refers to the respective sudden quench scenario. We observe that the cradle mode in the left and right wells, the local breathing mode in the middle well, and the interwell tunneling during the evolution persist.

Figure 9

(a) Frequency spectrum of the variance ${\sigma }_M^2\left(\omega \right)$ for time-dependent quenches of the form of Eq. (11) with final interaction $g_f=0.1$ and rates $\tau =0{\tau }_H$ (blue dashed), $\tau =0.8{\tau }_H$ (red solid), and $\tau =3.0{\tau }_H$ (black dashed-dotted). The inset shows each branch of the local breathing mode as a function of the quench rate $\tau$ (we incorporate 160 different rates in the range $0<\tau <120)$. In (b) we present the spectrum of the intrawell asymmetry $\Delta {\rho }_L\left(\omega \right)$ for a time-dependent scenario with quench amplitude $\delta g=-4.90$ and rates $\tau =0{\tau }_H$ (blue dashed), $\tau =0.8{\tau }_H$ (red solid), and $\tau =3.0{\tau }_H$ (black dashed-dotted). The inset demonstrates the dependence of each branch of the cradle mode as a function of the quench rate $\tau$. Finally, note that we have used normalized frequency units $\omega /\Delta \omega$, with $\Delta \omega =2\pi /T$ and $T$ being the respective evolution time.

Figure 10

(a) Fidelity evolution as a function of different sudden negative quenches of the optical lattice depth. The system consists of five strongly interacting bosons $(g=5.0)$ in an eight-well potential with $V_{0;in}=8.0$. (b) Same but for different quenches of the lattice depth and a simultaneous interaction quench to $g_f=0.02$. (c) Profiles of the fidelity evolution for different quench amplitudes $\delta V_0=-3.8$ (blue thin solid), $\delta V_0=-2.5$ (red thick solid), $\delta V_0=-0.9$ (magenta dashed-dotted), $\delta V_0=-0.2$ (black dashed), and a simultaneous interaction quench to $g_f=0.02$. In (d) we present profiles of the fidelity following a negative time-dependent quench of the potential depth to $V_{0;f}=4.0$ with different ramp rates $\tau =0.4{\tau }_H$ (blue thin solid), $\tau =15.0{\tau }_H$ (green thick solid), $\tau =40.0{\tau }_H$ (red thin dashed), $\tau =100.0{\tau }_H$ (magenta thick dashed), $\tau =400.0{\tau }_H$ (yellow thin dashed-dotted), and $\tau =800.0{\tau }_H$ (black thick dashed-dotted) and a simultaneous interaction quench to $g_f=0.02$.

Figure 11

(a) The fluctuations $\delta \rho (x,t)$ of the one-body density, for an eight-well setup with $N=5$, caused by a sudden negative barrier quench from $V_{0;in}=8.0$ to $V_{0;f}=4.0$ and a simultaneous interaction quench from $g_{in}=5.0$ to $g_f=0.02$. (b) The intrawell asymmetry $\Delta {\rho }_2\left(t\right)$ for the second well of the eight-well setup for a barrier quench (red solid curve) to $V_{0;f}=4.0$ and for the combined quench scenario, i.e., barrier and interaction quench, with final parameters $V_{0;f}=4.0$ and $g_f=0.02$ (blue dashed curve).