Fragility of classical Hamiltonian period doubling to quantum fluctuations

We add quantum fluctuations to a classical period-doubling Hamiltonian time crystal, replacing the $N$ classical interacting angular momenta with quantum spins of size $l$. The full permutation symmetry of the Hamiltonian allows a mapping to a bosonic model and the application of exact diagonalization for a quite large system size. In the thermodynamic limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$ the model is described by a system of Gross-Pitaevskii equations whose classical-chaos properties closely mirror the finite-$N$ quantum chaos. For $N\ensuremath{\rightarrow}\ensuremath{\infty}$, and $l$ finite, Rabi oscillations mark the absence of persistent period doubling, which is recovered for $l\ensuremath{\rightarrow}\ensuremath{\infty}$ with Rabi-oscillation frequency tending exponentially to 0. For the chosen initial conditions, we can represent this model in terms of Pauli matrices and apply the discrete truncated Wigner approximation. For finite $l$ this approximation reproduces no Rabi oscillations but correctly predicts the absence of period doubling. Our results show the instability of time-translation symmetry breaking in this classical system even to the smallest quantum fluctuations, because of tunneling effects.