Controlling emergent dynamical behavior via phase-engineered strong symmetries

TL;DR

This work shows that a tunable relative phase in collective light-matter coupling creates a phase-dependent strong symmetry of the Liouvillian in driven-dissipative cavity QED. This symmetry reshapes symmetry sectors, enabling phase-controlled access to non-stationary, time-crystalline regimes at substantially reduced driving strengths in two cavity QED realizations—a two-species spin-1/2 ensemble and a collective three-level gas. The authors derive microscopic models and their mean-field and effective atom-only descriptions, analyze Liouvillian spectra and decoherence-free subspaces, and demonstrate how phase engineering can switch between dark and bright subspaces for robust state preparation. The findings provide a practical route to programmable dissipative phase transitions and symmetry-protected dynamics with potential applications in quantum memories, sensing, and timekeeping.

Abstract

Symmetry constraints provide a powerful means to control the dynamics of open quantum systems. However, the set of accessible control parameters is often limited. Here, we show that a tunable phase in the collective light-matter coupling of a cavity QED system induces a phase-dependent strong symmetry of the Liouvillian, enabling dynamical control of the open quantum system evolution. We demonstrate that tuning this phase substantially reduces the critical driving strength for dissipative phase transitions between stationary and non-stationary phases. We illustrate this mechanism in two experimentally relevant cavity QED settings: a two-species ensemble of two-level atoms and a single-species ensemble of three-level atoms. Our results establish phase control as a versatile tool for engineering dissipative phase transitions, with implications for quantum state preparation.

Figures (8)

• Figure 1: Schematic of the setups. (a) Two species of two-level atoms confined in a single-mode optical cavity with phase-dependent couplings $g_A=g, g_B=ge^{-i\varphi}$ and (b) description of the effective spin interactions mediated by the cavity. (c) Atomic gas where each atom is modeled as a three-level system with state-dependent couplings $g_A=g, g_B=ge^{-i\varphi}$ and (d) effective atom-only description.
• Figure 2: Phase enabled nonstationary states. (a) Mean-field phase diagram showcasing the effect of tunable phase $\varphi$ on the transition between stationary (SS) and nonstationary states (NSS), and (b) the corresponding weight distribution in the Dicke subspace, $w_{S^2_\text{max}}^\varphi$ and its complement $\overline{w}^\varphi$. Finite-size dynamics for highlighted points in (a) illustrating the usual time-crystalline behavior and the lowering of the threshold value, panels (c) and (d), respectively. We consider system sizes $N\in\left(10,20,\dots,60\right)$ (lines with increasing color intensity). The results are benchmarked against the mean-field values (dashed) with $g/\kappa=0.1$ and $\Delta=0$.
• Figure 3: Three-level system frequency response. (a) Phase-dependent steady-state phase diagram showcasing frequency response for a chosen initial state with $c_\pm=(1\pm\sqrt2)/2\sqrt2, \,c_0=\sqrt{1-c_+^2-c_-^2}$, far from the eigenstates $\ket{v_{\pm1}^\varphi}$. The order parameter $\tilde{\omega}_{N_B}$ is the normalized dominant Fourier component of the excited state $B$. (b) Initial state visualization of threshold contours in $(c_+,c_-)$ parameter space for $\varphi=0$. States are defined as nonstationary if $\tilde{\omega}_{N_B}\geq0.01$ and shaded orange. On the $\eta/g=0$ plane we highlight the eigenstates of $\hat{\tau}_j^\varphi$ mentioned in the main text. (c) Parameter space map taken at the $\eta/g=0.5$ slice from (b). (d) Threshold contours for $\varphi=2\pi/3$, and the equivalent parameter space map (e). The white and green markers correspond to the initial state used in (a), falling inside the nonstationary regime as $\varphi$ is tuned. Other parameters used $g/\kappa=0.1, \Delta=0$, for a simulation time of $g t=2\times10^4$.
• Figure S1: Raman level schemes. (a) Two multilevel atomic species ($A,B$) with distinct ground states and (b) with a shared ground state. Frequencies $\Omega_{r,s}$ drive the optical transitions between the ground and a far detuned ($\delta_{r,s}\gg\Omega_{r,s}$) state $\ket{r,s}$. The cavity couples with strength $\lambda_{r,s}$ to a low lying excited state detuned from the ground state by $\delta_{A/B}$ via a consequent emission/absorption of a photon $\hat{a}$$(\hat{a}^\dagger)$. The dynamics of interest is contained in the low-lying energetic states. We can embed a relative phase in the system by setting $\Omega_{r,s}=|\Omega_{r,s}|e^{i\varphi_{r,s}}$: without loss of generality we choose $|\Omega_{r}|=|\Omega_s|$, for $\varphi_r=0$ and $\varphi_s\equiv\varphi$.
• Figure S2: Effect of finite cavity detuning.(a) Semiclassical dynamics of the total magnetization $s_A^z + s_B^z$ deep into the steady state window for the $\Delta_A=\Delta_B=0$, $\varphi = 4\pi/5$, $\eta=0.8g$ point from Fig. 2 in the main text, with and without cavity detuning, and (b) Fourier-domain comparison for each spin species (blue and red) with (dashed) and without (solid) cavity detuning. (c) Dynamics for the excited state populations of the three-level system ($\hat{N}_k=\hat{\Lambda}_k^\dagger\hat{\Lambda}_k$), for an intial state parametrized with $c_+=0.9/\sqrt{2}$ and $c_-=0.7/\sqrt{2}$ for the eigenstate populations introduced in the main text, and $\varphi=2\pi/3$. Here we use $\Delta_A=\Delta_B=0$, and $\eta/g=0.7$. (d) Fourier domain of population dynamics, in both cases the dashed lines are the data with cavity detuning ($\Delta_c/g=0.25$) and solid ones for $\Delta_c=0$. The dynamical features and the frequency peaks remain largely unchanged with finite $\Delta_c$. In both cases $g/\kappa=0.1$.