Quench-induced dynamical phase transitions and $\pi$-synchronization in the Bose-Hubbard model

We investigate the nonequilibrium behavior of a fully connected (or all-to-all coupled) Bose-Hubbard model after a Mott to superfluid quench, in the limit of large boson densities and for an arbitrary number $V$ of lattice sites, with potential relevance in experiments ranging from cold atoms to superconducting qubits. By means of the truncated Wigner approximation, we predict that crossing a critical quench strength the system undergoes a dynamical phase transition between two regimes that are characterized at long times either by an inhomogeneous population of the lattice (i.e., macroscopical self-trapping) or by the tendency of the mean-field bosonic variables to split into two groups with phase difference $\pi$, that we refer to as $\pi$-synchronization. We show the latter process to be intimately connected to the presence, only for $V\ge 4$, of a manifold of infinitely many fixed points of the dynamical equations. Finally, we show that no fine-tuning of the model parameters is needed for the emergence of such $\pi$-synchronization, that is in fact found to vanish smoothly in presence of an increasing site-dependent disorder, in what we call a synchronization crossover.

Figure 1

Graphical representation of the system configuration for $V=100$ lattice sites. In the complex plane, the blue markers represent the mean-field bosonic variables $\{{\psi }_j=\sqrt{{\rho }_j}{e}^{i{\theta }_j}\}$ (one marker per each site), a red dot represents the complex dynamical order parameter $\mathrm{\Psi }=r{e}^{i\varphi }$ and a black circle of radius $\sqrt{{\rho }_0}=1$ is drawn as a reference. (a) Generic configuration, for which the ${\rho }_j$ are spread around ${\rho }_0$ and $r\ne 0$. (b) Example of initial configuration ($t=0$) for a MI to SF quench [Eq. (15)]. A polar histogram with ${10}^{\circ }$ wide bins illustrates the distribution of the phases $\left\{{\theta }_j\right\}$. The proximity of the red dot to the origin of the plane reflects the fact that $r\ll 1$ for a MI and $V\gg 1$.

Figure 2

Schematic representation of the representative FPs of the GPE (13) for an even $V\ge 4$. For $V=20$ sites we show the UC (a), two other FPs with $r=0$ [(b) and (c)], the SPAC (d), one $\pi$-aligned configuration (e), and the SFC (f). The circular arrows indicate that the phases of a FP are in general rotating at some constant rate $\mathrm{\Omega }$. (g) The $r=0$ FPs constitute a manifold in the phase space that ranges from the UC to the SPAC. Notice that also for an odd $V\ge 5$ there is an analog manifold of infinitely many $r=0$ FPs, just lacking of the SPAC.

Figure 3

Graphical representation of the DC, defined for $V=4$ lattice sites in (27). The DC is a parametric FP that, depending on the value of the parameter $\mathrm{\Delta }\in (\pi /2,\pi )$, spans the entire manifold of $r=0$ FPs, ranging from the UC (for $\mathrm{\Delta }=\pi /2$) to the SPAC (for $\mathrm{\Delta }=\pi$).

Figure 4

Dynamics of the two-site model in the phase space with coordinates $\theta ={\theta }_1-{\theta }_2$ and $\delta ={\rho }_1-{\rho }_2$. (a) Semiclassical energy (14) in the surroundings of the SPAC ($\delta =0,\theta =\pi$) together with some relevant trajectories (in blue) for $\eta =1<{\eta }_c^{\mathrm{SPAC}}=2$. (b) Phase portrait for $\eta =1$: we show the FPs (red dots), some relevant nonequilibrium trajectories (blue lines), and the flow (green arrows) associated to the GPE (32). [(c) and (d)] Energy landscape and phase portrait for $\eta =3>{\eta }_c^{\mathrm{SPAC}}$. The SPAC is either a saddle or a maximum of the semiclassical Hamiltonian depending on $\eta <{\eta }_c$ (a) or $\eta >{\eta }_c$ (c) and corresponding to a saddle (b) and a nonlinear center (d) of the dynamics, respectively. Instead, the SFC ($\theta =\delta =0$) and the $\pi$-aligned configurations are nonlinear centers for any $\eta >0$ and for $0<\eta <{\eta }_c$. respectively [(b) and (d)]. Noticeably, the isolation (from the other FPs) of the SPAC is in stark contrast with the $V\ge 4$ case.

Figure 5

Depending on the value of the dimensionless hopping strength $\eta$, for a number of lattice sites $V\ge 4$, the behavior of a system initialized close to a $r=0$ FP changes sharply in what we refer to as a DPT. (a) The DC (defined for $V=4$) is a saddle (linear center) of the dynamics if $\eta <{\eta }_c^{\mathrm{DC}}=2[1+sin\left(\mathrm{\Delta }\right)]$ ($\eta >{\eta }_c^{\mathrm{DC}}$). If and only if $\eta <{\eta }_c^{\mathrm{UC}}$ ($\eta <{\eta }_c^{\mathrm{SPAC}}$), for a system initialized in the proximity of the UC (SPAC), the dynamical order parameter $r$ will grow exponentially at short times as $r\sim e^{t/{\tau }^{\mathrm{UC}}}$ ($r\sim e^{t/{\tau }^{\mathrm{SPAC}}}$). (b) Inverse of the characteristic time ${\tau }^{\mathrm{FP}}$ for both the UC (blue dashed line) and SPAC (red continuous line). At the critical ${\eta }_c^{\mathrm{UC}}=4$ and ${\eta }_c^{\mathrm{SPAC}}=2$, the characteristic time of the UC and SPAC, respectively, diverges.

Figure 6

Exact numerical solution of the nonlinear dynamics (13) up to long times for $V=500$ sites and a MI to SF quench. [(a)–(c)] Graphical representation of the mean-field state for a single simulation with initial condition given by Eq. (15). For graphical clarity, only the blue markers of 300 out of the $V=500$ bosonic variables are represented. At time $t=0$, the phases are randomly distributed (a), whereas at $t=1000$ either the $\left\{{\rho }_j\right\}$ are spread around ${\rho }_0$ (b) or the $\left\{{\theta }_j\right\}$ are $\pi$-synchronized (c). (d) Dynamics of the expected value $\langle r\rangle$ obtained according to Eq. (16) as an average over 3000 simulations [each one for a different realization of the random initial phases (15)]. For $\eta =2<{\eta }_c^{\mathrm{UC}}=4$ (blue continue line), $\langle r\rangle$ grows exponentially at short times ($\langle r\rangle \sim exp[t/{\tau }^{\mathrm{UC}}]$, see inset with logarithmic ordinate axis) and relaxes to a finite value 0.38 at long times, indicating MQST and reflected in the spread of the $\left\{{\rho }_j\right\}$ in (b). For $\eta >{\eta }_c^{\mathrm{UC}}$ (red dashed line), $\langle r\rangle$ remains instead small. (b) Dynamics of the expected value $\langle S\rangle$ of the $\pi$-synchronization parameter $S$ obtained as an average over 3000 simulations. For $\eta =5>{\eta }_c^{\mathrm{UC}}$, $\langle S\rangle$ asymptotically relaxes to a finite value 0.35, indicating a robust shift of the system towards the SPAC.

Figure 7

[(a) and (b)] Schematic representation of the trajectories of the system (in blue) in the phase space for $\eta =2$ and $\eta =5$, respectively, after a MI to SF quench, under the nonlinear GPE, up to long times and for an even $V\gg 1$. The red line and dots represent the FPs, and the most relevant configurations are displayed. The blue dot represents the initial condition. For $\eta =2$ ($\eta =5$), the system drifts away from (orbits close to) the manifold of the $r=0$ FPs, finally relaxing to a state characterized by a finite $r$ ($S$). For instance, in reference to Fig. 6, the state at long time is characterized by $r\approx 0.38$ ($S\approx 0.35$).

Figure 8

The disorder competes against the tendency of the phases $\left\{{\theta }_j\right\}$ to $\pi$-synchronize. We numerically solve the GPE (44) for $V=300$ lattice sites and a MI to SF quench. [(a) and (c)] Graphical representation of the mean-field state of the system for a single simulation at initial time (a), and at time $t=200$ for $\sigma =0$ (b) and $\sigma =45\times {10}^{-3}$ (c). The polar histogram of the phases $\left\{{\theta }_j\right\}$ helps to visualize the reduction of $\pi$-synchronization due to the disorder. (d) Time dynamics of the expected value $\langle S\rangle$ of the $\pi$-synchronization parameter $S$ computed as an average over 1000 simulations [each one with a different realization of the initial random phases (15)] according to Eq. (16). $\langle S\rangle$ is smaller for larger disorder strengths ($\sigma =0,15,30,45,60\times {10}^{-3}$). (e) Asymptotic value $\langle S(t\to \infty )\rangle$ vs the disorder strength $\sigma$. For increasing disorder, the $\pi$-synchronization is progressively broken in a synchronization crossover with onset decreasing with $\eta$.