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arXiv:2207.02602 · New Journal of Physics

Basis for time crystal phenomena in ultra-cold atoms bouncing on an oscillating mirror

Weronika Golletz, Andrzej Czarnecki, Krzysztof Sacha, Arkadiusz Kuroś

Published Aug 16, 2022• 4 citations
arXivHTMLPDF

Abstract

Abstract We consider classical dynamics of a one-dimensional system of N particles bouncing on an oscillating mirror in the presence of gravitational field. The particles behave like hard balls and they are resonantly driven by the mirror. We identify the manifolds the particles move on and derive the effective secular Hamiltonian for resonant motion of the particles. Proper choice of time periodic oscillations of the mirror allows for engineering of the effective behaviour of the particles. In particular, the system can behave like an N-dimensional fictitious particle moving in an N-dimensional crystalline structure. Our classical analysis constitutes a basis for quantum research of novel time crystal phenomena in ultra-cold atoms bouncing on an oscillating atom mirror.

DOI
10.1088/1367-2630/ac8a0d

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•Dec 30, 2019•0 citations
Abstract We consider classical dynamics of a one-dimensional system of N particles bouncing on an oscillating mirror in the presence of gravitational field. The particles behave like hard balls and they are resonantly driven by the mirror. We identify the manifolds the particles move on and derive the effective secular Hamiltonian for resonant motion of the particles. Proper choice of time periodic oscillations of the mirror allows for engineering of the effective behaviour of the particles. In particular, the system can behave like an N-dimensional fictitious particle moving in an N-dimensional crystalline structure. Our classical analysis constitutes a basis for quantum research of novel time crystal phenomena in ultra-cold atoms bouncing on an oscillating atom mirror.
Abstract We investigate the size of discrete time crystals s (ratio of response period to driving period) that can be created for a Bose–Einstein condensate (BEC) bouncing resonantly on an oscillating mirror. We find that time crystals can be created with sizes in the range s ≈ 20–100 and that such big time crystals are easier to realize experimentally than a period-doubling (s=2) time crystal because they require either a larger drop height or a smaller number of bounces on the mirror. We also investigate the effects of having a realistic soft Gaussian potential mirror for the bouncing BEC, such as that produced by a repulsive light-sheet, which is found to make the experiment easier to implement than a hard-wall potential mirror. Finally, we discuss the choice of atomic system for creating time crystals based on a bouncing BEC and present an experimental protocol for realizing big time crystals. Such big time crystals provide a flexible platform for investigating a broad range of non-trivial condensed matter phenomena in the time domain.
Abstract We review the application of discrete time crystals created in a Bose-Einstein condensate (BEC) of ultracold atoms bouncing resonantly on an oscillating atom mirror to the investigation of condensed matter phenomena in the time dimension. Such a bouncing BEC system can exhibit dramatic breaking of time-translation symmetry, allowing the creation of discrete time crystals having up to about 100 temporal lattice sites and suitable for hosting a broad range of temporal condensed matter phenomena. We first consider single-particle condensed matter phenomena in the time dimension which include Anderson localization due to temporal disorder, topological time crystals, and quasi-crystal structures in time. We then discuss many-body temporal condensed matter phenomena including Mott insulator phases in time, many-body localization in time, many-body topological time crystals and time crystals having long-range exotic interactions. We also discuss the construction of two (or three) dimensional time lattices, involving the bouncing of a BEC between two (or three) orthogonal oscillating mirrors and between two oscillating mirrors oriented at 45∘. The latter configuration supports a versatile Möbius strip geometry which can host a variety of two-dimensional time lattices including a honeycomb time lattice and a Lieb square time lattice. Finally, we discuss the construction of a six-dimensional time-space lattice based on periodically driven BECs trapped in a three-dimensional optical lattice.
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 quantum systems that are able to reveal condensed matter behavior in the time domain. It is known that crystallization in time can be observed in a periodically driven many-body system when interactions between particles force a system to evolve with a period that is an integer multiple of a driving period. This phenomenon is dubbed discrete time crystal formation. Here, we consider ultracold atoms bouncing on an oscillating atom mirror and show that the system can spontaneously form a discrete time crystal where the ratio of a period of its motion and a driving period is a rational number. This kind of discrete time crystal requires higher-order resonant driving, which is analyzed here with the help of an original approach.
We investigate the effect of time-dependent boundary conditions on the dynamics of a quantum bouncer—a particle falling in a homogeneous gravitational field on a moving mirror. We examine more particularly the way a moving mirror modifies the properties of the entire wavefunction of a falling particle. We find that some effects, such as the fact that a quantum particle hitting a moving mirror may bounce significantly higher than when the mirror is fixed, are in line with classical intuition. Other effects, such as the change in relative phases or in the current density in spatial regions arbitrarily far from the mirror are specifically quantum. We further discuss how the effects produced by a moving mirror could be observed in link with current experiments, in particular with cold neutrons.
We report progress towards realizing discrete time crystals using a BEC of potassium-39 atoms bouncing on an oscillating mirror. In the absence of interactions, each bouncing atom performs classical motion in which there are resonant islands in phase space located around periodic orbits whose periods match an integral multiple s of the driving period T of the mirror. If all the atoms are prepared in a single wave-packet, in the absence of interactions they tunnel to neighbouring wave-packets on a timescale determined by the driving amplitude and period of the oscillating mirror. In the presence of a sufficiently strong attractive interaction, the Floquet eigenstates of the periodic system become Schrödinger cat-like states, so that measurement of the position of just a single atom is sufficient to break the timetranslation symmetry and cause the system to collapse into one of the s wave-packets bouncing with period sT. Such a system allows dramatic breaking of discrete time translation symmetry where the symmetry-broken state evolves with a period typically s=40 times longer than the driving period and does not decay. This system provides also a suitable platform for demonstrating a wide range of condensed matter phenomena in the time domain.