Chaotic and quantum dynamics in driven-dissipative bosonic chains

TL;DR

This work investigates chaos and thermalization in boundary-driven, nonequilibrium Bose-Hubbard chains realized in circuit QED contexts. By deploying the truncated Wigner approximation and semiclassical OTOCs, the authors map the open quantum dynamics to stochastic phase-space trajectories and quantify spatiotemporal chaos, revealing a two-stage spatial relaxation and an extensive prethermal domain with anomalous heating. They identify a strong-drive resonant nonlinear wave (RNW) regime that remains regular in the quantum regime but is destabilized by fluctuations in long chains, leading to metastability and transitions to chaos. The findings illuminate how quantum fluctuations reshape classical driven-dissipative transport, offering a general framework for predicting prethermal chaotic phases in extended driven-dissipative systems and guiding experimental diagnostics in superconducting-circuit architectures.

Abstract

Thermalization in quantum many-body systems typically unfolds over timescales governed by intrinsic relaxation mechanisms. Yet, its spatial aspect is less understood. We investigate this phenomenon in the nonequilibrium steady state (NESS) of a Bose-Hubbard chain subject to coherent driving and dissipation at its boundaries, a setup inspired by current designs in circuit quantum electrodynamics. The dynamical fingerprints of chaos in this NESS are probed using semiclassical out-of-time-order correlators (OTOCs) within the truncated Wigner approximation (TWA). At intermediate drive strengths, we uncover a two-stage thermalization along the spatial dimension: phase coherence is rapidly lost near the drive, while amplitude relaxation occurs over much longer distances. This separation of scales gives rise to an extended hydrodynamic regime exhibiting anomalous temperature profiles, which we designate as a ``prethermal'' domain. At stronger drives, the system enters a nonthermal, non-chaotic finite-momentum condensate characterized by sub-Poissonian photon statistics and a spatially modulated phase profile, whose stability is undermined by quantum fluctuations. We explore the conditions underlying this protracted thermalization in space and argue that similar mechanisms are likely to emerge in a broad class of extended driven-dissipative systems.

Figures (19)

• Figure 1: Boundary-driven dissipative Bose-Hubbard chain: schematics and key results. (a) Tight-binding array of $L$ nonlinear resonators described by the Hamiltonian in Eq. (\ref{['eqs:hamiltonian']}) subject to a drive of amplitude $F$ coherently injecting photons at the leftmost site and to single-photon losses at both ends of the chain. The interplay of interaction, drive, and dissipation leads to a nonequilibrium steady state (NESS). (b) Lyapunov growth between neighboring Wigner trajectories is used to identify chaotic dynamics. (c) In the chaotic regime, the chain hosts three distinct domains illustrated by their steady-state local Wigner functions $W_\ell(\alpha,\alpha^*)$: a nonsymmetric nonthermal domain near the left boundary, an extensive prethermal phase where the $\mathbb{U}(1)$ symmetry of the phase is restored and which hosts a high density of photons only saturated by the Kerr non-linearity, and a $\mathbb{U}(1)$-symmetric thermal domain near the right boundary which is characterized by fluctuations over the vacuum state.
• Figure 2: Nonequilibrium steady-state phase diagram. Steady-state properties of the last resonator in the chain, $\ell = L$, as functions of the chain length $L$ and drive strength $F$. (a) Photon number $n_L$. (b) Photon-number fluctuation $\delta n_L$, defined in Eq. \ref{['eqs:delta_n']}. (c) Saturation value of the steady-state phase OTOC, $D_{1,L}(\tau \to \infty)$, defined in Eq. \ref{['eqs:semiclassical_OTOC']}. The three distinct regimes labeled I, II, and III are discussed in the text. Panels (d), (e), and (f) show cuts of panels (a), (b), and (c), respectively, at fixed chain length $L = 10$. Results are computed by averaging over $N_{\rm traj} = 10^3$ independent Wigner trajectories. Statistics are further improved by averaging over a time window $\Delta \tau$ after reaching the NESS: $\Delta \tau = 25$ for $D_{1,L}(\tau \to \infty)$, and $\Delta \tau = 2 \times 10^3$ for $n_L$ and $\delta n_L$. Throughout the manuscript, the dissipation rate $\gamma$ sets the unit of energy, and the other parameters are fixed to $\Delta = 2.5$, $J = 2$, and $U = 0.1$.
• Figure 3: Chaotic regime: Two-stage relaxation in space. Spatial profiles of equal-time photon statistics in the chaotic NESS. (a) Photon density $n_\ell = \langle |\alpha_\ell|^2 \rangle - 1/2$, showing a growing relaxation length scale with increasing drive strength $F = 5.5,\, 6,\, 6.5,\, 7,\, 7.5,\, 8$ (from yellow to purple), and a plateau across most of the chain at stronger drives. (b) Circular phase variance $\Delta \varphi_\ell := 1 - |\langle {\rm e}^{{\rm i}\varphi_\ell}\rangle |$, which rapidly saturates to unity, indicating a uniform phase distribution. (c) First-order coherence function $|g_{k, \ell}^{(1)}|$ defined in Eq. (\ref{['eqs:g1']}) showing exponential decay of phase correlations on microscopic length scales away from $\ell = k$. (d-f) Normalized local Wigner functions, $\tilde{W}_\ell(\alpha, \alpha^*) := W_\ell(\alpha, \alpha^*)/\textrm{max}[W_\ell(\alpha, \alpha^*)]$ for representative sites in a chain of length $L = 400$. Results are computed by averaging over $N_{\rm traj} = 10^2$ independent Wigner trajectories and over a time window $\Delta\tau=10^4$ once the steady state is reached. The drive strength is set to $F=7.5$ in panels (b-f), and the other parameters are set as in Fig. \ref{['fig:phase_diagram']}.
• Figure 4: Chaotic regime: hydrodynamic description. Spatial profile of the local (a) effective temperature $T_{\ell}$, (b) effective chemical potential $\mu_\ell$, and (c) entropy density $S_\ell := -\textrm{Tr}[ \hat{\rho}_\ell \log \hat{\rho}_\ell]$ across a chain of length $L=400$. The results are obtained by mapping the local steady-state physics at site $\ell$ to the Gibbs ansatz defined in Eq. (\ref{['eq:Gibbs']}). (d) Spatial profile of the dimensionless ratio $\mu_{\ell}/T_\ell$ for the same system. The results are obtained by mapping the local physics at site $\ell$ to the Gibbs ansatz, to the 2-photon impurity ansatz defined in Eqs. (\ref{['eqs:SC_hamiltonian']}-\ref{['eqs:SC_dissipator']}), and to the generalized Scully-Lamb ansatz. Three distinct spatial domains are discussed in the text: 1) nonsymmetric nonthermal domain where all the ansätze fail; 2) prethermal domain where $\mu_{\ell}/T_\ell > 0$; 3) thermal domain with $\mu_{\ell}/T_\ell \leq 0$. (e) Respective sizes of the three domains versus the total chain length $L$. The drive strength is set to $F=7.5$ and the other parameters are set as in Fig. \ref{['fig:phase_diagram']}.
• Figure 5: Onset of the RNW Regime. Spatiotemporal evolution of (a) the photon density $n_\ell$, and (b) the average phase $\langle \varphi_\ell \rangle$ as functions of time $t$ and site index $\ell$ in a $L=100$ chain in the RNW regime. The hatched region in (b) indicates the transient chaotic regime in which the $\mathbb{U}(1)$ symmetry is restored and the average phase becomes ill-defined. Results are obtained upon averaging over $N_{\rm traj} = 5\times 10^3$ independent Wigner trajectories. The drive amplitude is fixed to $F=12.5$. The other parameters are set as in Fig. \ref{['fig:phase_diagram']}.