Modeling spontaneous breaking of time-translation symmetry

We show that an ultracold 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.

Figure 1

The probability of detecting a system at a fixed position vs time. Top: The effects of spontaneous breaking of continuous time-translation symmetry in the course of time crystal formation. Bottom: The situation considered in the present paper where spontaneous symmetry breaking results in a change from a discrete time-translation symmetry to another discrete one, but with a period that is twice as long.

Figure 2

(a) Modulus squared of Floquet states ${\psi }_1(z,t)$ (black solid line) and ${\psi }_2(z,t)$ (red dashed line) of a single particle bouncing on an oscillating mirror in the presence of the gravitational field at $t=0$ [see the Hamiltonian (1)] and for $\omega =1.1$ and $\lambda =0.06$. The presented Floquet states are associated with the $s=2$ classical resonance and consist of two wave packets moving along a periodic orbit. The wave packets propagate with the period $4\pi /\omega$ and with the $2\pi /\omega$ delay with respect to each other, which makes the resulting Floquet states periodic with the period $2\pi /\omega$. (b) The same as (a), but for $t=\pi /\omega$ when the separation of the wave packets is clearly visible (the interference fringes around $z=0$ are related to the reflection of the wave packets from the mirror). (c),(d) A stable solution of the GPE (2), for $g_{0}N=-0.5$, that breaks the original time-translation symmetry—this solution reveals a single wave packet that propagates along the classical periodic orbit with the period $4\pi /\omega$. (c) $t=0$, (d) $t=\pi /\omega$. Black solid lines in (c) and (d) correspond to the results of the numerical integration of (2), while red dashed lines (hardly distinguishable from the solid lines) are related to the results of the two-mode approximation. The gravitational units are used; see the text.

Figure 3

Left column: Time evolution of the single-particle probability density corresponding to the ground state of the Hamiltonian (4) for $g_{0}N=-0.5,N={10}^4$ and the other parameters as in Fig. 2—from top to bottom: $t=0,\pi /\omega ,2\pi /\omega ,3\pi /\omega$, and $4\pi /\omega$. Middle column: Similar data, but in the case when every $\pi /\omega$ period positions of 100 atoms are measured. Initially, the single-particle density ${\rho }_0(z,0)\approx 0.5N\left[\right|{\varphi }_1{(z,0)|}^2+|{\varphi }_2(z,0){|}^2]$ and, because $|{\varphi }_1{(z,0)|}^2={\left|{\varphi }_2(z,0)\right|}^2$, the measurement of particle positions is not able to break the time-translation symmetry. When $t$ increases, $|{\varphi }_1{(z,t)|}^2\ne {\left|{\varphi }_2(z,t)\right|}^2$ and even a measurement of a single particle can break the symmetry and result in a collapse of the system to one of the two propagating wave packets (see the second panel from the top in the middle column). Right column: The results of the measurements of positions of 100 atoms, i.e., at $t=0$, we measure positions of 100 atoms, let the remaining atoms evolve, and, after $\pi /\omega$, we again measure positions of 100 atoms, and so on. The histograms presented in the right column indicate that time-periodic evolution of the system after the spontaneous time-translation symmetry breaking can be observed in a single experimental realization. The gravitational units are used; see the text.