Abstract
We consider the formation of time crystals in a gas of laser-cooled atoms. We show that the phonon frequency is shifted inside a time crystal, and the phonon backscattering resulting from a temporal Bragg diffraction can be observed. A quantum-field description allows us to demonstrate that these effects are accompanied by emission of the phonon pairs from vacuum. The same description can be used for the dynamical Casimir effect, which, in some sense, is as a special case of the time crystal with similar quantum vacuum properties. Due to its low temperature and low noise level, a gas of ultracold atoms seems promising for experimental studies in quantum vacuum phenomena, such as those associated with time crystals.
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