Long-lived Z₂ and U(1) superradiance of a mesoscopic spin chain in a two-mode cavity

We investigate the steady-state Dicke phase transition of mesoscopic ensembles with N=10N=10N=10–34 atomic spins coupled to a high-finesse optical cavity supporting two near-degenerate polarization modes. The atom array is aligned along the cavity axis under a longitudinal magnetic field and transversely pumped with π\piπ-polarized light. Depending on the cavity mode structure, the system is effectively described by a two-mode Dicke–Hepp–Lieb Hamiltonian with either a continuous U(1)U(1)U(1) symmetry for degenerate modes or a discrete Z2Z_2Z2​ symmetry in the presence of a small linear birefringence, as in our cavity. In this regime, coherent interference between the cavity and pump fields generates a local circularly polarized component at each atom, providing positive feedback to the spin bifurcation through a vector AC Stark shift that reshapes the energy landscape and is cavity-cooled via collective decay, as well as through local optical pumping arising from individual atomic dissipation.

We experimentally observe three phases. (i) A normal (dark) phase, in which all spins remain in their ground state and both cavity modes are in vacuum. (ii) A U(1) superradiant phase, where the system enters a Kerr-rotation limit cycle: the cavity field becomes linearly polarized, with the polarization axis precessing in time perpendicular to a simultaneously precessing collective spin. (iii) A Z_2​ superradiant phase, where cavity linear birefringence dominates over atom-induced circular birefringence, causing the collective spin to lock to one of two opposite orientations and the cavity-field dynamics to slow into a telegraphic pitchfork bifurcation.

We focus on three key aspects: (1) the atom-number dependence of the steady-state phase diagram, revealing mesoscopic behavior of open-system phase transitions; (2) fluctuation properties in the two superradiance​ phases, including distinct noise amplitudes of Goldstone versus Higgs modes and two forms of telegraphic switching; and (3) the role of local atomic dissipation in stabilizing superradiant states, extending their lifetime to the vacuum-limited, second-scale regime.