Temporal Floquet-Nambu-Goldstone modes, Floquet thermodynamics, and the time operator in spontaneous Floquet states

We study symmetry-breaking in spontaneous Floquet states, focusing on atomic condensates. First, we quantize the Nambu-Goldstone (NG) modes for stationary states spontaneously breaking several symmetries by invoking the generalized Gibbs ensemble. The quantization involves a Berry-Gibbs connection, which depends on the macroscopic conserved charges and whose curvature is not invariant under generalized gauge transformations. We extend the formalism to Floquet states, where Floquet-NG (FNG) modes emerge with zero quasi-energy. For spontaneous Floquet states, there is a genuine temporal FNG mode associated with the broken time-translation symmetry and whose quantum amplitude provides a unique example of time operator in Quantum Mechanics. Those states conserve energy and a novel charge, the Floquet charge, whose conjugate is the frequency, while driven systems conserve the Floquet enthalpy. This establishes a new thermodynamic paradigm for Floquet physics. We apply our formalism to the CES state, which breaks $U(1)$ and time-translation symmetries, thus embodying a time supersolid. We numerically compute its density-density correlations, predicted to be dominated by the temporal FNG mode at long times, and observe a remarkable agreement between simulation and theory.