Higher symmetry breaking and non-reciprocity in a driven-dissipative Dicke model

Abstract

Higher symmetries in interacting many-body systems often give rise to new phases and unexpected dynamical behavior. Here, we theoretically investigate a variant of the Dicke model with higher-order discrete symmetry, resulting from complex-valued coupling coefficients between quantum emitters and a bosonic mode. We propose a driven-dissipative realization of this model focusing on optomechanical response of a driven atom tweezer array comprised of $n$ sub-ensembles and placed within an optical cavity, with the phase of the driving field advancing stepwise between sub-ensembles. Examining stationary points and their dynamical stability, we identify a phase diagram for $n\geq 3$ with three distinctive features: a $\mathbb{Z}_n$ ($\mathbb{Z}_{2n}$) symmetry-breaking superradiant phase for even (odd) $n$, a normal unbroken-symmetry phase that is dynamically unstable due to non-reciprocal forces between emitters, and a first-order phase transition separating these phases. This $n$-phase Dicke model may be equivalently realized in a variety of optomechanical or opto-magnonic settings, where it can serve as a testbed for studying high-order symmetry breaking and non-reciprocal interactions in open systems.

Figures (3)

• Figure 1: (a) Schematic of the setup for the $n$-phase Dicke model. Atoms are held in harmonic traps at the nodes of the cavity field. The atoms are divided into $n$ groups and each group is illuminated by a pump laser with a $\phi=2\pi/n$ phase difference between adjacent groups. (b) Real and imaginary quadratures of the steady-state cavity field (solid circles) calculated using the atomic center-of-mass positions found through Lyapunov function minimization, shown for $n=1$ through $n=6$. The results show $\mathbb{Z}_n$ symmetry for even $n$ and $\mathbb{Z}_{2n}$ symmetry for odd $n$. The parameters used are $\nu=30$, $\omega_z=2\pi\times 70$ kHz, $\mathrm{\Delta_{pa}}=-2\pi\times 100$ MHz, $\mathrm{\Delta_{pc}}=-2\pi\times 4$ MHz, $\kappa=0$, $g_0=2\pi\times 3$ MHz, and $\mathrm{\Omega}=2\pi\times20$ MHz.
• Figure 2: (a-c) Contour plots of the forces $\dot{p}_1=0$ (blue) and $\dot{p}_2=0$ (red) for $n=4$, where we have set $z_1=-z_3$ and $z_2=-z_4$. Steady states (teal markers) occur where the blue and red curves intersect. Jacobian eigenvalue analysis is used to determine whether each steady state is unstable (circles) or stable (stars). (d) Phase diagram for $n=4$ showing the steady-state position of one group ($z_3$) at a particular symmetry-broken solution. Orange squares correspond to the parameters used in (a-c). Black lines are phase boundaries calculated by expanding Eq. (\ref{['EOMs']}) to third order around the symmetry-broken solutions. (e) Line cuts corresponding to the dashed lines in (d) showing the first-order transition as a function of $\mathrm{\Omega}$. Dashed lines show the location of the discontinuity for the two values of $\mathrm{\Delta_{pc}}$. Pink points have a slight vertical offset for visibility. In the $|\mathrm{\Delta_{pc}}|\rightarrow \infty$ limit, the transition becomes second-order (continuous). Calculations in figure are done with $\mathrm{\Delta_{pa}}=-2\pi\times 100$ MHz, $\kappa=2\pi\times 0.5$ MHz, $\nu=30$, $g_0=2\pi\times3$ MHz, and $\omega_z=2\pi\times 70$ kHz.
• Figure 3: Density plots of cavity field trajectories for (a) $n=3$, (b) $n=4$, (c) $n=5$, and (d) $n=6$ in a parameter regime predicted to have stable steady states. Different colors correspond to different perturbations in the initial positions of the atoms, which are the starting conditions for the numerical integration. The high concentration of single colors at the vertices of a (a) hexagon, (b) square, (c) decagon, and (d) hexagon indicate that the trajectories have spontaneously broken either a $\mathbb{Z}_{2n}$ or a $\mathbb{Z}_n$ symmetry. All trajectories are integrated over a time span of 6 ms. (e) Time trajectories of the real (dark color) and imaginary (light color) quadratures of the cavity field for $n=4$. Colors correspond to the data in (b). Parameters used for all data in figure are $\mathrm{\Omega}=2\pi\times 20$ MHz, $\mathrm{\Delta_{pc}}=-2\pi\times 4$ MHz, $\mathrm{\Delta_{pa}}=-2\pi\times 100$ MHz, $\kappa=2\pi\times 0.5$ MHz, $\nu=30$, $g_0=2\pi\times3$ MHz, and $\omega_z=2\pi\times 70$ kHz.