Abstract
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.
Highlights
The experimental discovery [1,2] of Floquet time crystals a few years after their theoretical prediction has been a real breakthrough
In conclusion we add quantum fluctuations to a classical and Hamiltonian model of interacting classical angular momenta showing synchronized period doubling in the thermodynamic limit
We study the robustness of synchronized period doubling adding quantum fluctuations in two different ways, realizing two different quantum models
Summary
The experimental discovery [1,2] of Floquet time crystals a few years after their theoretical prediction has been a real breakthrough. We substitute the classical angular momenta with quantum spins of magnitude l and find that, whenever l is finite, the quantum fluctuations destroy the synchronized period-doubling motion. It is recovered only in the limit of infinite spin magnitude l → ∞, when the dynamics becomes classical again. For the chosen initial state, model 2 is equivalent to the first one but the DTWA gives results in quantitative agreement only in the limit l → ∞ For l finite it is only correct in predicting the absence of period doubling in the limit of large N but provides no Rabi oscillations
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