Journal of Russian Laser Research, Volume 35, Number 1, January, 2014 TIME CRYSTALS IN ULTRACOLD MATTER J. T. Mendon¸ ca 1 and V. V. Dodonov 2 ∗ 1 IPFN, Instituto Superior T´ ecnico 1049-001 Lisboa, Portugal 2 Instituto de F´ ısica, Universidade de Bras´ ılia 70919-970 Bras´ ılia, DF, Brasil ∗ Corresponding author e-mail: vdodonov @ fis.unb.br e-mail: titomend @ ist.utl.pt 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. Keywords: time crystals, dynamical Casimir effect, ultracold gases, effective charge. 1. Introduction The concept of time crystals has been recently proposed in the quantum [1] and classical [2] regimes. This concept emerges from a long series of different studies on time-varying media [3, 4], which also includes time refraction [5] and the temporal beam splitter [6, 7]. In such temporal configurations, time translation symmetry breaks down, thus leading to energy nonconservation effects as manifested by the particle energy (or frequency) shifts, as observed in optical experiments with time-varying media [8]. Here, we consider time crystals in a gas of ultracold atoms, laser cooled and confined in a magneto- optical trap, and study the phonon behavior in such a periodic structure. It has been recently shown that phonons in a laser-cooled gas have specific properties [6, 7, 9], which result from the existence of collective repulsive forces between the atoms [10,11]. Our interest in time crystals in ultracold matter is due to the extremely low temperatures that result in the practical absence of thermal noise. For this reason, the ultracold gas is an ideal medium to observe quantum vacuum processes in different fields, with particular emphasis on the emission of pairs of quasiparticles (phonons, magnons, etc.) [12–22]. Time crystals are closely related to the dynamical Casimir effect (DCE), which is produced by periodic time-varying boundary conditions [23,24] or, alternatively, by an oscillating background medium [3,25– 30]. This effect is basically an instability of the quantum vacuum, which leads to an exponential growth of emitted boson pairs (photons or phonons) when the oscillating frequency of the medium is equal to Manuscript submitted by the authors in English on October 2, 2013. 1071-2836/14/3501-0093 c  2014 Springer Science+Business Media New York 93