Non-Equilibrium Steady States and Quantum Chaos in a three-site Driven-Dissipative Bose-Hubbard Chains base on Self-Consistent Mean-Field Approach

TL;DR

This work develops a self-consistent Gutzwiller mean-field (GMF) framework to study non-equilibrium steady states in a driven-dissipative Bose-Hubbard chain with Kerr nonlinearity $U$, drive $F$, hopping $J$, and loss $\gamma$. By solving the nonlinear local Liouvillians through a Picard iteration, it reveals two dynamical regimes: a regular quasilinear phase and a chaotic phase where parametric instabilities drive information scrambling, diagnosed via the out-of-time-ordered correlator (OTOC). The analysis connects the chaos onset to the spectrum of an effective Liouvillian $\mathbf{L}_{\text{eff}}$ and supports the chaotic picture with a Bogoliubov–de Gennes-like stability analysis, TWA simulations, and a detailed Green’s-function/T-matrix treatment. Collectively, these tools provide a scalable path to exploring many-body correlations and chaos in larger driven photonic lattices beyond exact diagonalization. The results illuminate how drive, dissipation, and strong interactions cooperate to produce non-equilibrium chaos and information scrambling in open quantum systems with potential experimental realization in circuit QED and coupled cavities.

Abstract

We investigate the non-equilibrium dynamics and steady-state properties of a driven-dissipative Bose-Hubbard chain using a self-consistent Gutzwiller mean-field (GMF) approach. By employing a robust Picard iteration scheme, we solve the non-linear master equation for the non-equilibrium steady state (NESS) in the presence of strong Kerr nonlinearity. We identify two distinct dynamical regimes governed by the interplay between coherent drive, dissipation, and interaction: a regular quasilinear regime and a chaotic regime. Linear stability analysis reveals that the transition to the chaotic regime is triggered by parametric instabilities arising from the drive-induced coherence. Furthermore, we characterize the onset of quantum chaos by calculating the out-of-time-order correlator (OTOC). Our results show that in the strong coupling regime, the OTOC exhibits rapid exponential growth and saturation, providing a clear signature of information scrambling in this open quantum system. The proposed numerical framework offers an efficient pathway to explore many-body correlations in larger photonic lattices.

Figures (3)

• Figure 1: Numerical solution of the NESS and Quantum Chaos diagnostics. The system is solved using the self-consistent Gutzwiller Mean-Field method with Picard iteration. (a) Dynamic convergence of the local particle population $\langle \hat{n}_l \rangle$ for the drive (Site 1, blue), bulk (Site 2, green), and drain (Site 3, red) sites versus iteration steps. The system settles into a non-uniform density profile driven by the source-drain bias. (b) Convergence of the mean-field order parameter magnitude $|\Psi_l|$, showing the stabilization of the coherent field background. (c) The residual error $\epsilon_k = ||\Psi^{(k)} - \Psi^{(k-1)}||$ on a logarithmic scale. The linear slope indicates the exponential convergence of the Picard iteration algorithm to the unique NESS.
• Figure 2: (Left) Time evolution of the local Phase OTOC $D_{3,3}(t)$ calculated on NESS background using the Picard iteration self-consistent solver. The rapid initial growth and subsequent saturation at a non-zero value indicate significant information scrambling and the onset of local quantum chaos induced by the Kerr nonlinearity. (Right) The OTOC solved by exact solver.
• Figure 3: TWA base on 100 trajectories (left) and 500 trajectories (right).