Dark Soliton Formation as a Dark-State Phase Transition in a Dissipative Superfluid Josephson Junction Chain

Abstract

We identify and characterize a first-order dark-state phase transition between a discrete dark soliton and a uniform superfluid in a Bose-Hubbard chain with a single lossy site. Using classical-field (truncated-Wigner) simulations together with a Bogoliubov stability analysis, we show that the dark-state nature of the soliton suppresses fluctuations and shifts the critical point relative to the comparable phenomenon of optical bistability in driven-dissipative Kerr resonators. We then demonstrate that this mechanism quantitatively captures the bistability phase boundary observed in the experiment of R. Labouvie et al. [Phys. Rev. Lett. 116, 235302 (2016)], resolving substantial discrepancies in prior modeling efforts. Our results reveal how driving, dissipation and quantum coherence can interact to induce nonequilibrium phase transitions in ultra-cold atomic gases.

Figures (4)

• Figure 1: (a) Illustration of the two late-time macrostates in a quasi-1D Josephson junction chain with tunneling $J$ and local particle loss $\gamma$ on the central site. As $J/\gamma$ increases there is a phase transition from a dark soliton (bottom) to a uniform superfluid (top). The former is characterized by a $\pi$ phase jump and vanishing density on the central site, and is a dark state that decouples from the dissipation. (b) Central-site occupation $\abs{\psi_0}^2$ (normalized by maximum occupation $N_f$) vs $J$, showing the bistability of numerical mean-field solutions. Dashed lines (green) show the case of a superfluid to dark soliton, which sweeps a bigger area than a comparable transition connecting the superfluid to a symmetric low-density state without $\pi$ phase jump (orange).
• Figure 2: (a) Linear-stability eigenvalues of the discrete dark soliton \ref{['eq:soliton']} in the BH chain for $J/\gamma = 0.97$ (green) and $J/\gamma=1.06$ (orange). (b),(c) Effective Liouvillian gap $\lambda$ as a function of $J/\gamma$ at fixed dissipation rates (b) $\gamma/\mu=0.1$ and (c) $\gamma/\mu=0.3$. Results with the soliton (green dots) are compared to those obtained with imposed mirror-symmetry (orange diamonds), which excludes the possibility of a soliton. Broken lines are fits to guide the eye. Simulations were performed with a fixed particle number per site and $L$ large enough to avoid boundary effects.
• Figure 3: Emergence and stability of a dark soliton in the experimental system. (a) Evolution of the spatial phase profile $\Phi_j$ vs time for a single trajectory with $L = 100$ (outer sites not shown), $|\Delta \Phi_0 (0)| \approx 0.7$, and all sites initially full, setting $\gamma/\mu=0.056$ and $J/\mu=0.007$. A global phase $\bar{\Phi}$ is subtracted at each time for ease of visualization footnotephaseplot. (b) As for (a) but starting from the soliton with initial $\Delta \Phi_0 (0) \approx \pi$ and low occupation on the lossy site, setting $\gamma = 0$ and $J/\mu = 0.08$. (c) Phase difference between driving sites $|\Delta \Phi_0 (t)|$ (left axis) and normalized atom number $N_0/N_f$ on the lossy site (right axis) vs time for the trajectory in (b). (d) Corresponding time evolution of the current contributions from the respective driving sites, $I_{1,0}$ (dashed) and $I_{-1,0}$ (solid).
• Figure 4: (a) Phase diagram in $J$-$\gamma$ parameter space, presented by the late time (normalized) atomic population difference $\Delta N$ between simulations starting from different initial states: the empty and full central site respectively. A finite $\Delta N$ indicates a bistable regime. The experimental phase boundaries from Ref. Labouvie2016 are displayed with points and are fitted by $J \sim \sqrt{\gamma}$ (upper boundary) and $J \sim \gamma$ (lower boundary). The white dot-dash line indicates $J = \gamma/4$ as expected from theory. (b) As in (a) but for a system with imposed mirror symmetry around the central site, which is incapable of hosting a dark soliton. Displayed data corresponds to $t = 80$ ms and 251 lattice sites, averaged over 64 trajectories for each point.