  We study simultaneous symmetry-breaking in a spontaneous Floquet state,
focusing on the specific case of an atomic condensate. We first describe the
quantization of the Nambu-Goldstone (NG) modes for a stationary state
simultaneously breaking several symmetries of the Hamiltonian by invoking the
generalized Gibbs ensemble, which enables 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 whose curvature is not invariant under generalized gauge transformations.
We extend the formalism to Floquet states simultaneously breaking several
symmetries, where Goldstone theorem translates into the emergence of
Floquet-Nambu-Goldstone (FNG) modes with zero quasi-energy. In the case of a
spontaneous Floquet state, 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, since they conserve energy, spontaneous Floquet states can be
shown to possess a conserved Floquet charge. Conventional Floquet systems also
admit a thermodynamic description in terms of the Floquet enthalpy, the
Legendre transform of the energy with respect to the Floquet charge. 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 at long times, observing a remarkable agreement between
simulation and theory. Based on these results, we propose a feasible
experimental scheme to observe the temporal FNG mode of the CES state.
