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arXiv:2103.03748 · Physical Review A

Spontaneous breaking of a discrete time-translation symmetry

Jasper Smits, Henk T. C. Stoof, P. van der Straten

Published Aug 20, 2021• 11 citations
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Abstract

Spontaneous symmetry breaking is a ubiquitous concept and is well described in many textbooks of physics. However, direct observation of spontaneous symmetry breaking is lacking. Here we present the observation and analysis of a spontaneously broken discrete time-translation symmetry in our driven system. We experimentally find a 50-50 split between two stable and temporal-distinct solutions, indicative of the breaking of a ${\mathbb{Z}}_{2}$ (Ising-like) time-translation symmetry. The experiment allows for further exploration of the symmetry breaking in our discrete time crystal and for engineering excitations in space and time in the quantum domain.

DOI
10.1103/physreva.104.023318

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Spontaneous symmetry breaking is a ubiquitous concept and is well described in many textbooks of physics. However, direct observation of spontaneous symmetry breaking is lacking. Here we present the observation and analysis of a spontaneously broken discrete time-translation symmetry in our driven system. We experimentally find a 50-50 split between two stable and temporal-distinct solutions, indicative of the breaking of a Z2{\mathbb{Z}}_{2}Z2​ (Ising-like) time-translation symmetry. The experiment allows for further exploration of the symmetry breaking in our discrete time crystal and for engineering excitations in space and time in the quantum domain.
We present the theory of spontaneous symmetry breaking (SSB) of discrete time translations as recently realized in the space-time crystals of an atomic Bose-Einstein condensate. The nonequilibrium physics related to such a driven-dissipative system is discussed in both the Langevin as well as the Fokker-Planck formulations. We consider a semiclassical and a fully quantum approach, depending on the dissipation being either frequency independent or linearly dependent on frequency, respectively. For both cases, the Langevin equation and Fokker-Planck equation are derived, and the resulting equilibrium distribution is studied. We also study the time evolution of the space-time crystal and focus in particular on its formation and the associated dynamics of the spontaneous breaking of a Z2{Z}_{2}Z2​ symmetry out of the symmetry-unbroken phase, i.e., the equilibrium Bose-Einstein condensate before the periodic drive is turned on. Finally, we compare our results with experiments and conclude that our theory provides a solid foundation for the observations.
Spontaneous symmetry breaking is one of the central organizing principles in physics. Time crystals have emerged as an exotic phase of matter, spontaneously breaking the time translational symmetry, and are mainly categorized as discrete or continuous. While these distinct types of time crystals have been extensively explored as standalone systems, intriguing effects can arise from their mutual interaction. Here, we demonstrate that a time-independent coupled system of discrete and continuous time crystals induces a simultaneous twofold temporal symmetry breaking, resulting in a hierarchical time crystal phase. Interestingly, one of the subsystems breaks a discrete temporal symmetry that is absent from the time-independent Liouvillian generating the dynamics, but instead emerges dynamically, giving rise to a convoluted non-equilibrium phase. We demonstrate that hierarchical time crystals are robust, emerging for fundamentally different coupling schemes and persisting across wide ranges of system parameters.
Time crystals are physical systems whose time translation symmetry is spontaneously broken. Although the spontaneous breaking of continuous time-translation symmetry in static systems is proved impossible for the equilibrium state, the discrete time-translation symmetry in periodically driven (Floquet) systems is allowed to be spontaneously broken, resulting in the so-called Floquet or discrete time crystals. While most works so far searching for time crystals focus on the symmetry breaking process and the possible stabilising mechanisms, the many-body physics from the interplay of symmetry-broken states, which we call the condensed matter physics in time crystals, is not fully explored yet. This review aims to summarise the very preliminary results in this new research field with an analogous structure of condensed matter theory in solids. The whole theory is built on a hidden symmetry in time crystals, i.e., the phase space lattice symmetry, which allows us to develop the band theory, topology and strongly correlated models in phase space lattice. In the end, we outline the possible topics and directions for the future research.
Time crystals are quantum many-body systems which are able to self-organize their motion in a periodic way in time. Discrete time crystals have been experimentally demonstrated in spin systems. However, the first idea of spontaneous breaking of discrete time translation symmetry, in ultracold atoms bouncing on an oscillating mirror, still awaits experimental demonstration. Here, we perform a detailed analysis of the experimental conditions needed for the realization of such a discrete time crystal. Importantly, the considered system allows for the realization of dramatic breaking of discrete time translation symmetry where a symmetry broken state evolves with a period tens of times longer than the driving period. Moreover, atoms bouncing on an oscillating mirror constitute a suitable system for the realization of dynamical quantum phase transitions in discrete time crystals and for the demonstration of various nontrivial condensed-matter phenomena in the time domain. We show that Anderson localization effects, which are typically associated with spatial disorder and exponential localization of eigenstates of a particle in configuration space, can be observed in the time domain when ultracold atoms are bouncing on a randomly moving mirror.
Time crystals are a phase of matter, for which the discrete time symmetry of the driving Hamiltonian is spontaneously broken. The breaking of discrete time symmetry has been observed in several experiments in driven spin systems. Here, we show the observation of a space-time crystal using ultracold atoms, where the periodic structure in both space and time is directly visible in the experimental images. The underlying physics in our superfluid can be described ab initio and allows for a clear identification of the mechanism that causes the spontaneous symmetry breaking. Our results pave the way for the usage of space-time crystals for the discovery of novel nonequilibrium phases of matter.
Abstract Time crystals are physical systems whose time translation symmetry is spontaneously broken. Although the spontaneous breaking of continuous time-translation symmetry in static systems is proved impossible for the equilibrium state, the discrete time-translation symmetry in periodically driven (Floquet) systems is allowed to be spontaneously broken, resulting in the so-called Floquet or discrete time crystals. While most works so far searching for time crystals focus on the symmetry breaking process and the possible stabilising mechanisms, the many-body physics from the interplay of symmetry-broken states, which we call the condensed matter physics in time crystals, is not fully explored yet. This review aims to summarise the very preliminary results in this new research field with an analogous structure of condensed matter theory in solids. The whole theory is built on a hidden symmetry in time crystals, i.e., the phase space lattice symmetry, which allows us to develop the band theory, topology and strongly correlated models in phase space lattice. In the end, we outline the possible topics and directions for the future research.