Simultaneous symmetry breaking in spontaneous Floquet states: Floquet-Nambu-Goldstone modes, Floquet thermodynamics, and the time operator

We study simultaneous symmetry-breaking in a spontaneous Floquet state, focusing on the specific case of an atomic condensate. We first describe the quantization procedure of the Nambu-Goldstone (NG) modes for a stationary state simultaneously breaking several symmetries by invoking the generalized Gibbs ensemble, allowing for a thermodynamical description of the problem. The quantization procedure involves a Berry-Gibbs connection, which depends on the macroscopic conserved charges associated to each broken symmetry and is not invariant under generalized gauge transformations. We extend the formalism to a spontaneous Floquet state simultaneously breaking several symmetries, finding that each broken symmetry now has associated a Floquet-Nambu-Goldstone (FNG) mode with zero quasi-energy. Among them, there is a genuine temporal FNG mode arising from the continuous time-translation symmetry breaking, whose quantum amplitude provides a rare realization of a time operator in Quantum Mechanics. Furthermore, as they conserve energy, spontaneous Floquet states admit a thermodynamic description since they have a conserved Floquet charge. Both the temporal FNG mode and the Floquet thermodynamics are the distinctive features of a spontaneous Floquet state, absent in conventional, driven Floquet systems. We apply our formalism to a particular realization of spontaneous Floquet state, the CES state, which breaks $U(1)$ and time-translation symmetries, representing a time supersolid. Using the Truncated Wigner method, we numerically compute its quantum fluctuations, which are theoretically predicted to be dominated by the temporal FNG mode, observing a remarkable agreement between simulation and theory. Based on these results, we propose a feasible experimental method to observe the temporal FNG mode.