Equilibrium and dynamical quantum phase transitions in dipolar atomic Josephson junctions

Abstract

An atomic Josephson junction realized with dipolar bosons in a double-well potential can be described by an extended Bose-Hubbard model in which dipolar interactions generate an effective on-site interaction and nearest-neighbor pair tunneling. Using mean-field theory and exact diagonalization, we investigate how this correlated process affects zero-temperature equilibrium and dynamical properties of the system. In equilibrium, we show that pair tunneling induces ground-state parity modulations and significantly reshapes the phase diagram, producing qualitative changes in the quantum phase transitions toward NOON and phase-NOON states, as well as quantitative shifts of the critical points. Out of equilibrium, we demonstrate that it modifies the conditions for macroscopic quantum self-trapping, and assess its impact by comparing mean-field and fully quantum evolution, including the emergence of dynamical quantum phase transitions.

Figures (7)

• Figure 1: Ground state. Distributions of the ground-state probabilities $p_i$ [panels (a)-(b)] and the corresponding entanglement and fragmentation entropies [panels (c)-(d)] as functions of $U/J$, for $N=32$, $U_0 = 2.0$, $J=1.0$, and $V$ varying from 0 to 3.0, i.e. $U=U_0-V$ varying from $2.0$ to $-1.0$. The pair tunneling is $P = 0$ in panels (a) and (c), and $P = 0.1V$ in panels (b) and (d). Horizontal dotted lines indicate $\max S_\text{frag}=1.0$ and $\max S_\text{ent}=\log_2 33\simeq 5.04$, while vertical dashed-dotted lines mark local extrema of $S_\text{ent}$.
• Figure 2: Mean-field phase diagram. Solid lines delineate regions characterized by different stationary points of $\mathcal{E}(\phi, z)$. Thick solid lines indicate boundaries across which the minima change, either continuously (red lines) or discontinuously (black lines), while thin solid lines indicate boundaries across which the maxima change, either continuously (red lines) or discontinuously (black lines). Different colors denote the three dynamical regimes described in Sec. \ref{['sec:MFdyn']}: Josephson regime (green), phase-locked MQST (light blue), and running-phase MQST (dark blue). Above the line $\Pi=\frac{1}{2}$ the same dynamical regimes are present with qualitative differences.
• Figure 3: Equilibrium quantum phase transitions. Thick solid lines (circular markers) denote the energy gap $E_1-E_0$, while thin dashed-dotted lines indicate the ground-state fidelity susceptibility $\chi_{\mathcal{F}}$ for several values of $N$. The dashed vertical line at the center of each panel represents the mean-field critical value of the driving parameter. (a) $\Lambda$-driven continuous QPT to NOON-like states in the weak pair tunneling regime ($\Pi = 0.25$) with $\Lambda_c=-0.75$. (b) $\Pi$-driven continuous QPT to phase-NOON-like states with $\Pi_c=0.5$. (c) $\Lambda$-driven first-order QPT to from phase-NOON-like to NOON-like states in the strong pair tunneling regime ($\Pi=1.0$) with $\Lambda_c=-1.0$. Parameter $J=1.0$.
• Figure 4: Josephson oscillations. Time evolution of the probabilities $|\psi_i(t)|^2$ [panels (a)-(b)], and of the population imbalance $z(t)$ and the relative phase $\phi(t)$ [panels (c)-(d)] for the system initialized in an atomic coherent state with $z_0=0.5$ and $\phi_0=0$. Darker blue corresponds to higher probability. Exact quantum results (bold solid lines) are compared with the corresponding mean-field trajectories (thin lines). (a) and (c): $\Lambda=4.0$, $\Pi=0$. (b) and (d): $\Lambda=4.0$, $\Pi=1.2$. Parameters are $N=64$ and $J=1.0$. Time is in units of $\hbar/J$.
• Figure 5: Phase-locked MQST. Time evolution of the probabilities $|\psi_i(t)|^2$ [panels (a)-(b)], and of the population imbalance $z(t)$ and the relative phase $\phi(t)$ [panels (c)-(d)] for a system initialized in the atomic coherent state with $z_0=0.5$ and $\phi_0=0$. Darker blue corresponds to higher probability. Exact quantum results (bold solid lines) are compared with the corresponding mean-field trajectories (thin lines). (a) and (c): $\Lambda=-1.4$, $\Pi=0$. (b) and (d): $\Lambda=-1.4$, $\Pi=0.6$. Parameters are $N=64$ and $J=1.0$. Time is in units of $\hbar/J$.