Quantum correlated steady states under competing collective and individual decay

TL;DR

This work investigates how collective dissipation-induced quantum correlations endure in the presence of competing individual decay. By modeling driven ensembles of many spins with both collective and independent decay, the authors uncover a first-order dissipative phase transition and quantum bistability between a CRSS-like correlated phase and an independent-spin phase, formalized as a mixture of two distinct quantum states in the steady state. They show that the correlated phase ρ_+ closely matches the CRSS of purely collective dissipation, displaying spin squeezing and coherent-like radiation, while the independent phase ρ_- resembles a totally mixed state; the switching dynamics between these phases are governed by the Liouvillian gap, which vanishes with system size, enabling long-lived quasi-stationary CRSS-like behavior even with decoherence. The results connect CRSS physics to realistic platforms and provide experimentally testable predictions for observables in cavity QED and collective radiation setups, offering a route to robust steady-state entanglement in the presence of decoherence.

Abstract

Collective dissipation can generate useful quantum correlations, while ubiquitous individual decay destroys them. We study the interplay between these two competing processes considering a driven system of many spins (``atoms") undergoing both collective and individual dissipation (``radiation"). In steady state and depending on drive, we find that the system exhibits a first-order phase transition and quantum bistability: its quantum state is a mixture of two many-body states associated with the two competing decay processes. Accordingly, one of these states closely resembles a correlated ``coherently radiating spin state" (CRSS) -- the solution of purely collective dissipation -- exhibiting spin-squeezing entanglement. We predict dynamical switching between the two stable states, manifest as many-body quantum jumps in the various observables of spin and radiation. Macroscopically, the switching rate tends to vanish and the system can reside in a correlated CRSS for long times. This reveals how correlated dissipative physics emerges at the presence of decorrelating individual decay, opening a path for unlocking collective dissipation phenomena in realistic quantum platforms and applications. We discuss consequences for experiments in collective radiation.

Figures (14)

• Figure 1: (a) Quantum optical realization of the model, Eq. (\ref{['Eq:ME']}). Atoms trapped inside an optical cavity and coherently driven by a laser amplitude $\Omega$. The $N$ atoms decay collectively through the cavity mirror with rate $\gamma_c$ and individually to off-axis modes at rate $\gamma_s$. The photon detectors PD$_{\parallel}$ and PD$_{\perp}$ give access to the collective atomic observables $\hat{S}_{\pm}$ (and hence $\hat{S}_{x,y}$, $\hat{S}_+\hat{S}_-$) and the "magnetization" (population inversion) $\hat{S}_z$, respectively. (b) Mean-field solution of the population inversion per atom $s_z$ from Eq. \ref{['eq:gammaneq0']} (solid and dashed curve for stable and unstable solutions), with $\Omega_c=\Gamma/4$ and taking $\Gamma/\gamma = \gamma_c(N-1)/(\gamma_c+\gamma_s)=15$. Dicke case ($\gamma_s=0$) is plotted for reference (dash-dotted).
• Figure 2: First-order dissipative phase transition revealed in the numerical solution of the quantum master equation \ref{['Eq:ME']} in steady state, for $N = 18$ (red curves) and $N=64$ (blue curves), and for fixed decay rates $\gamma_{c} = 10\gamma_{s}$. The critical point is estimated at $\Omega_{\mathrm{PT}}\approx 0.61\Omega_c$ for $N\gg 1$ (text and Appendix \ref{['app:PTpoint']}). (a) Average population inversion ("magnetization") $\langle \hat{S}_z\rangle$ as a function of the drive $\Omega$ (solid curves with dots) compared with the mean-field solution of Eq. \ref{['eq:gammaneq0']} (thin-line curves). (b) Probability distribution of the magnetization eigenvalues $m$ at $\Omega_{\rm PT}$ for $N = 64$ exhibits a bimodal form centered at the two stable mean-field solutions. (c) Average total angular momentum $\hat{\bm S}^2$ reveals a transition between collective- and individual-physics phases (large and small total angular momenta, respectively; dotted lines mark results of a totally mixed product state of purely individual physics, Appendix \ref{['app:mixed']}). (d) Spin squeezing parameter shows existence of quantum correlations, $\xi^2<1$, at the collective phase. Thin solid curves display analytical results from Eq. (\ref{['eq:xi']}). (e,f) Average intensity $\langle\hat{S}_+\hat{S}_-\rangle$ and second-order coherence $g^{(2)}(0)$ of collectively radiated light.
• Figure 3: Liouvillian gap $\lambda_1$ as a function of the system size $N$ (semi-log scale). Here $\lambda_1$ is extracted from the long-time relaxation dynamics of $\langle \hat{S}_z(t) \rangle$ calculated exactly from Eq. (\ref{['Eq:ME']}) at the critical point $\Omega = \Omega_{\mathrm{PT}}=0.61 \Omega_c$ (with $\gamma_c = 10 \gamma_s$, see text). The closing of the gap with $N$ is consistent with an exponential law.
• Figure 4: Quantum bistability: steady-state density matrix $\rho_s$ expressed as a mixture of two stable states $\rho_{\pm}$, as in Eq. (\ref{['eq:rho_ss_ppm']}). The states $\rho_{\pm}$ are obtained numerically for $N = 18$ and $\gamma_c = 10\gamma_s$. (a) Average magnetization calculated with $\rho_{+}$ ($\rho_{-}$) agrees with the lower- (upper-) branch mean-field solution. (b) The magnetization distributions for $\rho_{\pm}$ are centered around their respective mean-field values, and their weighted sum (using $a_{\pm}$) approximately reproduces the bimodal distribution of $\rho_s$ (here for $\Omega = 0.73 \Omega_c$). (c) Average total angular momentum $\langle\hat{\bm{S}}^2\rangle$ gives direct evidence that $\rho_+$ is associated with the collective phase (consistent with $j\approx N/2$) whereas $\rho_-$ with the individual-spin phase (consistent with low $j$'s). (d) Similar to (c), here for collective radiation intensity $\langle \hat{S}_+ \hat{S}_- \rangle$. The curve for $\rho_+$ exhibits excellent agreement with the result of pure collective dissipation (Dicke case $\gamma_s=0$, shown in magenta dashed-dotted line).
• Figure 5: Emergence of Dicke-like correlated physics at the presence of decorrelating individual decay. (a) The fidelity of the steady-state component $\rho_+$ with the CRSS --- the entangled solution of purely collective dissipation ($\gamma_s=0$, Dicke)--- remains close to unity until the end of the bistability region (while that with the steady-state mixture $\rho_s$ drops already at the critical point). (b) Second-order correlations of collective radiation for $\rho_+$ are close to $1$, indicating coherent-state radiation as expected for CRSS. (c) Spin squeezing entanglement, $\xi^2<1$, is contained in the CRSS-like state $\rho_+$, in excellent agreement with the analytical result, Eq. \ref{['eq:xi']} (black line). Parameters taken in all plots are the same as those of Fig. \ref{['fig:results']}