Quantum Quenches and Off-Equilibrium Dynamical Transition in the Infinite-Dimensional Bose-Hubbard Model

We study the off-equilibrium dynamics of the infinite-dimensional Bose-Hubbard model after a quantum quench. The dynamics can be analyzed exactly by mapping it to an effective Newtonian evolution. For integer filling, we find a dynamical transition separating regimes of small and large quantum quenches starting from the superfluid state. This transition is very similar to the one found for the fermionic Hubbard model by mean field approximations.

Figure 1

Graphical solution for the value of $p$ at the turning points. The trajectories are full lines, and the position at $t=0$ is indicated by a circle for the three trajectories $A$, $B$, and $C$. In case $A$ it is impossible to have a turning point at $p=\pi /2$. Case $B$ corresponds to the dynamical transition and $C$ to unbounded evolution of $p$. $A$, $B$, and $C$ are plotted in Fig. 2.

Figure 2

Evolution as a function of time of $x$, $|{\Psi }_0{|}^2$ (top panels), and $p$ (bottom panels) increasing the amplitude of the quench. $U_f=3.33$ is kept fixed and several $U_i$ are considered, with $A$, $B$, and $C$ corresponding to $U_i=\{1.62,0.838,0.1\}$ (unlike in the text where $U_f$ is varied). Case $B$ corresponds to the transition point.

Figure 3

(a) Evolution of the time average (continuous line) and microcanonical average (dashed line) of $⟨|{\Psi }_0{|}^2⟩$ as a function of $U_f$ for $U_i=0$. (b) Dynamical phase diagram for the model with a maximum of two bosons per site. Quenches from the Mott phase are not considered. Quenches from the superfluid phase are oscillating and similar to $A$ or $C$. The dynamical transition separating the two is displayed as a dashed line; it meets the Mott phase at $U_f=U_c$. The phase diagram for the case of more than two bosons per site is qualitatively similar.

Figure 4

(a) As in Fig. 2 but for $n_b=3$ and $U_i=1$. Left and right: $U_f=2.5$ and 3.29, respectively, below ($E<0$) and above ($E>0$) the dynamical transition at $U_f^d=3.21$. (b) Variation of $⟨|{\Psi }_0{|}^2⟩$ as a function of $U_f$ for $n=1$, $U_i=3$, with $n_b=2$, 3, 4, and 5. The plot of $n_b=2$ is shifted of 0.025 along the $U_f$ axis for comparison.