Modeling spontaneous breaking of time-translation symmetry

Abstract

We show that an ultra-cold atomic cloud bouncing on an oscillating mirror can reveal spontaneous breaking of a discrete time translation symmetry. In many-body simulations we illustrate the process of the symmetry breaking that can be induced by atomic losses or by a measurement of particle positions. The results pave the way for understanding and realization of the time crystal idea where crystalline structures form in the time domain due to spontaneous breaking of continuous time translation symmetry. 03.75.Lm, Symmetries of a quantum many-body Hamiltonian are reflected by properties of its eigenstates. However, there are systems whose eigenstates are extremely vulnerable to any symmetry breaking perturbation. Bose gas in a symmetric double well potential, with attractive particle interactions, is a simple example [1][2][3][4][5]. The ground state of the system reflects the symmetry of the external potential but it cannot be easily prepared in an experiment because it is a macroscopic superposition of two Bose-Einstein condensates (BEC) located in different potential wells. Loss of a particle is sufficient to break the symmetry and accumulate all remaining particles in one of the wells. Breaking of a symmetry due to an infinitesimally weak perturbation is called spontaneous symmetry breaking phenomenon.Spontaneous breaking of continuous spatial translation symmetry to discrete spatial translation symmetry is responsible for formation of space crystals. Recently it has been proposed that similar phenomenon can also occur in the time domain [6,7]. That is, it is possible to invent systems where the ground state of a time-independent Hamiltonian reveals spatially homogeneous flow of particles which under any symmetry breaking perturbation changes spontaneously to periodic motion of spatially inhomogeneous structures. Such a spontaneous breaking of continuous time translation symmetry to a discrete one is termed time crystal formation, see Fig. 1. Two different systems have been proposed: a bright soliton formed by attractively interacting particles on an Aharonov-Bohm ring [6] and ions on a ring in the presence of an external magnetic field [7] (see also [8][9][10][11][12]). These proposals triggered debate in the literature whether time crystal formation is possible. It seems that different assumptions can lead to contradictory conclusions [13][14][15][16][17][18][19][20][21][22].In the present paper we consider a periodically driven many-body system whose description can be reduced to a Hilbert space spanned by two periodically evolving modes. As in the case of a Bose gas in a double well potential [1-5], a spontaneous symmetry breaking process occurs. In our example, eigenstates of the system pos- sess discrete time translation symmetry which is spontaneously broken to another discrete translation symmetry but with a longer period, see Fig. 1. The spontaneous symmetry breaking that is predicted within the mean field approach, can be analyzed in full many-body simulations. Moreover it can be realized in ultra-...

How to Create a Time Crystal

“…In the translation-invariant setting, Floquet eigenstates are short-range correlated and resemble infinite temperature states which cannot exhibit symmetry breaking [15,18,19]. Under certain conditions, however, prethermal time-crystal-like dynamics can persist for long times [20,21] even in the absence of localization before ultimately being destroyed by thermalization [17,22].In this Letter, we present three main results. First, by exploring the interplay between entanglement, many body localization and TTSB, we produce a phase diagram for a discrete time crystal (DTC) [24].…”

“…This seems to fit the picture of TTSB and raises the question: are decoupled spins undergoing "spin echo" a discrete time crystal? The answer lies in the lack of stability to perturbations [21,26,27]. In this decoupled limit, any imperfection in the spin-echo pulse (e.g., ϵ ≠ 0) immediately destroys the ω=2 subharmonic.…”

Subharmonic Temporal Response and Spontaneous Breaking of Discrete Time Symmetry in Plasmonic Distributed Feedback Laser

“…That is, if n > 1, then the symmetry order of the laser's temporal response is lower than the original system's symmetry order. In recent years, the problem of the spontaneous breaking of discrete time symmetry has been intensively considered . Herein, we demonstrate for the first time the possibility of the spontaneous breaking of discrete time symmetry in plasmonic DFB lasers.…”

“…In recent years, the problem of the spontaneous breaking of discrete time symmetry has been intensively considered. [22][23][24][25][26] Herein, we demonstrate for the first time the possibility of the spontaneous breaking of discrete time symmetry in plasmonic DFB lasers. The spontaneous breaking of discrete time symmetry and subharmonic response to pump modulation opens the way to the creation of optoelectronic devices with a controlled temporal response, e.g., frequency converters of optical signals and sensors based on intracavity laser spectroscopy.…”

“…The n-fold increase in the period of the laser's response is a manifestation of spontaneous breaking of discrete time symmetry. [22][23][24][25][26] The temporal profile of the pumping is invariant with respect to a time shift by T, but the laser's temporal response is invariant only with respect to a time shift by nT. When n > 1, the symmetry order of the laser's temporal response is lower than the original system's symmetry order.…”