Chaos and quantum regimes in $n$-photon driven, dissipative bosonic chains

TL;DR

The paper investigates steady-state dynamics of $n$-photon driven, boundary-dissipative Bose-Hubbard chains using the truncated Wigner approximation. It identifies a universal chaotic hydrodynamic regime where long-wavelength modes restore a local $U(1)$ symmetry and a prethermal domain with photon saturation, as well as a non-chaotic resonant nonlinear wave (RNW) regime where boundary-imposed $\mathbb{Z}_n$ symmetry is transmitted throughout the chain and quantum fluctuations drive phase decoherence. The RNW regime is notably sensitive to boundary conditions, while the hydrodynamic regime exhibits a universal behavior largely independent of drive details. The results have strong implications for quantum state engineering in driven-dissipative circuit QED devices and outline experimental routes to realize nonclassical, correlated steady states.

Abstract

We investigate the steady-state dynamical regimes of boundary-driven, dissipative bosonic chains subjected to $n$-photon drives. Using the truncated Wigner approximation, we explore how multi-photon drives shape the interplay between quantum fluctuations, nonlinear interactions, and dissipative processes in such quantum systems. We identify two main regimes: a chaotic hydrodynamic regime characterized by the restoration of a local $\mathbb{U}(1)$ symmetry, photon saturation due to Kerr nonlinearity, and spatial prethermalization effects; and a non-chaotic resonant nonlinear wave (RNW) regime exhibiting sub-Poissonian photon statistics, persistent $\mathbb{Z}_n$ symmetry, and quantum-driven phase decoherence. Our findings reveal the universal nature of the hydrodynamic regime and highlight the RNW regime's sensitivity to boundary driving conditions, suggesting novel routes for quantum state engineering in driven-dissipative quantum devices. These results are experimentally relevant for state-of-the-art circuit quantum electrodynamics platforms.

Figures (12)

• Figure 1: Sketch of the system and summary of the main results. (a) Chain of $L$ coupled nonlinear resonators subject to boundary single-photon losses and to a $n$-photon drive at the leftmost site, giving to the system Hamiltonian a global $\mathbb{Z}_n$ symmetry. (b) In the presence of chaos, the NESS reaches thermal equilibrium in the spatial direction, where the $\mathbb{Z}_n$ symmetry is replaced by a local $\mathbb{U}(1)$ symmetry. The local quantum states are described by a Gibbs state with effective local temperature and chemical potential. A transition between a prethermal domain characterized by annular local Wigner function, large, positive chemical potential and anomalous heating to a thermal domain with a bell-shaped local Wigner function, negative chemical potential and decreasing temperature is observed. (c) In the absence of chaos, the system reaches a far-from-equilibrium steady state dubbed resonant nonlinear wave regime (RNW) where the symmetry of the boundary drive shapes the local quantum states throughout the chain. In the RNW regime, quantum fluctuations determines the fundamental features of the local states.
• Figure 2: Phase diagram in the NESS as a function of the chain length $L$ and the 2-photon drive amplitude $\zeta$ for the last resonator $\ell=L$. (a) Photon number $n_L$. (b) Photon fluctuations $\delta n_L$, computed according to Eq. \ref{['eqs:photon_fluctuations']}. (c) Saturation value of the semiclassical OTOC $D_{1,L}(\tau\to\infty)$, computed according to Eq. \ref{['eqs:semiclassical_OTOC']}. The four regimes labeled I, II, III, and IV, identified in panel (b) are discussed in the text. Results for panels (a) and (b) are computed by averaging over $N_{\rm traj} = 5\times 10^2$ independent trajectories, while results in panel (c) over $10^2$ trajectories. Statistics are further improved by averaging over a time window $\Delta \tau = 10^3$ after reaching the steady state. Throughout the manuscript, we fix $\gamma=1$ and $U=0.1$. In this section, we instead fix $\Delta=5.6$ and $J=2.2$.
• Figure 3: Two stage relaxation in space. (a) Photon number $n_\ell$, (b) photon fluctuations $\delta n_\ell$, (c) circular variance $\Delta\varphi_\ell$ for different system sizes $L=100, 200, 300, 400$ (from dark to light blue). Results are computed by averaging over $N_{\rm traj} = 225$ independent trajectories. Statistics are further improved by averaging over a time window $\Delta \tau = 10^4$ after reaching the steady state. (d-g) Local Wigner functions $W_\ell(\alpha, \alpha^*)$ for the sites (d) $\ell=1$, (e) $\ell=200$, (f) $\ell=283$, and (g) $\ell=400$ for a $L=400$ chain. The drive amplitude is set to $\zeta=6$. The other parameters are set as in Fig. \ref{['fig:phase_diagram']}.
• Figure 4: Hydrodynamic description. (a) Effective temperature $T_\ell$, (b) effective chemical potential $\mu_\ell$, (c) local entropy $S_\ell$ for different system sizes $L=100, 200, 300, 400$ (from dark to light blue). The vertical grey dashed line separates the portion of the chain where the quantities are extracted imposing the Gibbs state ansatz \ref{['eqs:gibbs']}, or the single-parameter ansatz \ref{['eqs:one_param']}. In this case, the temperature is estimated according to the equipartition theorem as $T_\ell=|\Delta|\langle p_\ell^2\rangle$, being $p_\ell = \sqrt{2}\textrm{Im}(\alpha_\ell)$ the local momentum (see Appendix \ref{['sec:appendix_B']}). Results are computed by averaging over $N_{\rm traj} = 225$ independent trajectories. Statistics are further improved by averaging over a time window $\Delta \tau = 10^4$ after reaching the steady state. The drive amplitude is set to $\zeta=6$. The other parameters are set as in Fig. \ref{['fig:phase_diagram']}.
• Figure 5: Regular RNW regime for $n=2$. (a) Dynamics of the phase variance $\Delta\varphi_\ell^{(2)}$ as a function of time $t$ and site index $\ell$ in a $L=30$ chain. (b) same as in panel (a) but for the first-order coherence function $|g_{1,\ell}^{(1)}|$. (c) Steady-state profile of $\Delta\varphi_\ell^{(m)}$ for $m=1, 2, 3$ (orange, blue and green lines respectively). The black-dashed line corresponds to $\Delta\varphi_\ell^{(1)}$ computed by solving the classical Gross-Pitaevskii equations of motion. (d) Steady-state profile of $|g_{1,\ell}^{(1)}|$. The black-dashed line indicates the Gross-Piatevskii solution. (e-h) Local Wigner functions $W_\ell(\alpha, \alpha^*)$ for (e) $\ell=2$, (f) $\ell=5$, (g) $\ell=20$, and (h) $\ell=30$. The drive amplitude is set to $\zeta=3.5$. The other parameters are set as in Fig. \ref{['fig:phase_diagram']}.