Slow quench dynamics of Mott-insulating regions in a trapped Bose gas

We investigate the dynamics of Mott-insulating regions of a trapped bosonic gas as the interaction strength is changed linearly with time. The bosonic gas considered is loaded into an optical lattice and confined to a parabolic trapping potential. Two situations are addressed: the formation of Mott domains in a superfluid gas as the interaction is increased and their melting as the interaction strength is lowered. In the first case, depending on the local filling, Mott-insulating barriers can develop and hinder the density and energy transport throughout the system. In the second case, the density and local energy adjust rapidly, whereas long-range correlations require a longer time to settle. For both cases, we consider the time evolution of various observables: the local density and energy and their respective currents, the local compressibility, the local excess energy, the heat, and single-particle correlators. The evolution of these observables is obtained using the time-dependent density-matrix renormalization-group technique and comparisons with time evolutions done within the Gutzwiller approximation are provided.

Figure 1

Slow quench from $U_i=4J$ to $U_f=6J$. Evolution of local observables in the presence of a trap as a function of the ramp time $\tau$ and compared with that of a homogeneous system (open symbols) having the same initial local density. Observables are the density $n_l$, compressibility ${\kappa }_l$, occupancy probabilities $P_0$ and $P_1$, neighboring correlation $g_{l,l+1}$, and the particle current $j_{l,l+1}=\langle {\widehat{j}}_{l,l+1}\rangle$. Subplots correspond to two different total numbers of particles, $N=24$ and 48, and two different sites: $l=18$ and the central site $l=32$ (cf. Fig. 2 for the location of these sites).

Figure 2

Final local density and compressibility profiles after a slow quench from $U_i=4J$ to $U_f=6J$ for different ramp times, $\tau$, and for the ground state ($\tau =\infty$) at $U=6J$ in the trapped system. (a, c) $N=24$; (b, d) $N=48$.

Figure 3

Time evolution of the local compressibility ${\kappa }_l$ and current $j_{l,l+1}=\langle {\widehat{j}}_{l,l+1}\rangle$ during a slow quench from $U_i=4J$ to $U_f=6J$ in a time $\tau =7\hslash /J$ for $N=48$ in the trapped system. As the “Mott barriers” are formed, the current in their vicinity weakens.

Figure 4

(a–d) Time evolution of the different contributions to the time derivative of the particle current: (a) Eq. (13a), (b) Eq. (13b), (c) Eq. (13c), and (d) Eq. (13d). (e) Time evolution of the time derivative of the particle current. (f) Time evolution of the particle current. For (a) to (e), the value at $t=0$ is subtracted. The evolution parameters are the same as in Fig. 3.

Figure 5

Final local energy (a, e) and local excess energy (b, f) profiles after a slow quench from $U_i=4J$ to $U_f=6J$ for different ramp times, $\tau$, and for the ground state ($\tau =\infty$) at $U=6J$ for a trapped system. (c) The contribution to $q_l$ due to the external operator [see (20c)]; (d) the contribution from the energy currents [see (20b)]. The dashed line in (b) and (f) corresponds to ${\langle {\widehat{h}}_l\rangle }_{0,i}-{\langle {\widehat{h}}_l\rangle }_{0,f}$.

Figure 6

Time evolution of the different contributions to the energy current: (a) Eq. (15a), (b) Eq. (15b), (c) Eq. (15c), and (d) Eq. (15d). (e) Time evolution of the full energy current. The evolution parameters are the same as in Fig. 3.

Figure 7

Total heat $Q\left(\tau \right)$ vs inverse ramp time $\tau$ for both directions of the quenches for $N=24$ and $N=48$.

Figure 8

Time-evolution, within the Gutzwiller method, of a trapped one-dimensional Bose gas loaded into an optical lattice during a quench from $U_i=6J$ to $U_f=15J$ for two ramp times. Plotted quantities are the local density $\langle {\widehat{n}}_l\rangle$, the local compressibility ${\kappa }_l$, the current $j_{l,l+1}$, and the superfluid order parameter $\langle {\widehat{b}}_l\rangle$. $N=54$, $L=84$, $V_t=0.006J$. Upper panels: $\tau =16\hslash /J$. Middle panel: evolution of the four quantities at $\tilde{l}=28$ (position at which the barrier will develop) and $\tau =16\hslash /J$. Lower panels: $\tau =60\hslash /J$.

Figure 9

Value of the correlator $g_{l,l+d}$ with $l=L/2+1$ after a slow quench from $U_i=4J$ to $U_f=6J$ in a trap for various ramp times $\tau$ and the final ground state ($\tau =\infty$). Left: $N=24$. Right: $N=48$.

Figure 10

Final value of interference pattern $N\left(k\right)$ after a slow quench from $U_i=4J$ to $U_f=6J$ in a trap for different ramp times $\tau$ and for the final ground state ($\tau =\infty$). Left: $N=24$. Right: $N=48$.

Figure 11

Slow quench from $U_i=6J$ to $U_f=4J$; $N=48$. Evolution of local observables in the presence of a trap as a function of the ramp time $\tau$, compared with that of a homogeneous system (open symbols) having the same initial local density. Observables are the density $n_l$, compressibility ${\kappa }_l$, occupancy probabilities $P_0$ and $P_1$, neighboring correlation $g_{l,l+1}$, and particle current $j_{l,l+1}=\langle {\widehat{j}}_{l,l+1}\rangle$.

Figure 12

Final profiles for the local density (a), local compressibility (b), local energy (c), and local excess energy (d) after a slow quench from $U_i=6J$ to $U_f=4J$ for different ramp times, $\tau$, and for the ground state ($\tau =\infty$) at $U=4J$ for a trapped system. (e) The contribution to $q_l$ due to the external operator [see (20c)]; (f) the contribution from the energy currents [see (20b)]. The dashed line in (d) corresponds to ${\langle {\widehat{h}}_l\rangle }_{0,i}-{\langle {\widehat{h}}_l\rangle }_{0,f}$.

Figure 13

Ramp from $U_i=6J$ to $U_f=4J$; $N=48$. (a) Final value for correlator $g_{l,l+d}$ with $l=L/2+1$ for different ramp times, $\tau$, and the final ground state ($\tau =\infty$). (b) Final value for the interference pattern $N\left(k\right)$ [see (2)] for different ramp times and the final ground state ($\tau =\infty$).