Controlled preparation of phases in two-dimensional time crystals

Open Access

Controlled preparation of phases in two-dimensional time crystals

Arkadiusz Kuroś, Rick Mukherjee, Florian Mintert, and Krzysztof Sacha

Phys. Rev. Research 3, 043203 – Published 21 December 2021

Abstract

The study of phases is useful for understanding novel states of matter. One such state of matter is time crystals which constitute periodically driven interacting many-body systems that spontaneously break time translation symmetry. Time crystals with arbitrary periods (and dimensions) can be realized using the model of Bose-Einstein condensates bouncing on periodically driven mirror(s). In this work, we identify the different phases that characterize the two-dimensional time crystal. By determining the optimal initial conditions and value of system parameters, we provide a practical route to realize a specific phase of the time crystal. These different phases can be mapped to the many-body states existing on a two-dimensional Hubbard lattice model, thereby opening up interesting opportunities for quantum simulation of many-body physics in time lattices.

Received 30 July 2021
Revised 23 November 2021
Accepted 26 November 2021

DOI:https://doi.org/10.1103/PhysRevResearch.3.043203

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Bose-Einstein condensates Exotic phases of matter

2-dimensional systems Atomic gases

Bloch-Floquet theorem Hubbard model Mean field theory Tight-binding model

Atomic, Molecular & OpticalCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Arkadiusz Kuroś ^1,2,*, Rick Mukherjee^3,†, Florian Mintert³, and Krzysztof Sacha ¹

¹Instytut Fizyki Teoretycznej, Uniwersytet Jagielloński, ulica Profesora Stanisława Łojasiewicza 11, PL-30-348 Kraków, Poland
²Institute of Physics, Jan Kochanowski University, Uniwersytecka 7, 25-406 Kielce, Poland
³Blackett Laboratory, Imperial College London, SW7 2AZ, London, United Kingdom

^*arkadiusz.kuros@ujk.edu.pl
^†r.mukherjee@imperial.ac.uk

Article Text

Click to Expand

References

Click to Expand

Issue

Vol. 3, Iss. 4 — December - December 2021

Subject Areas

Reuse & Permissions

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
(a) Ultracold atoms with attractive interactions are initially trapped in the lowest mode of a 2D harmonic trap. On its release, it falls under gravity (whose direction is indicated with ${\vec{F}}_{g}$ ) resulting in the bouncing of the atoms between the two harmonically oscillating orthogonal mirrors. Both mirrors oscillate with frequency $ω$ but with individual amplitudes $λ_{x, y}$ , respectively. In order to obtain stable dynamics, the initial conditions (position and momentum) need to be optimized which depend on heights $(h_{x}, h_{y})$ . (b) Density of noninteracting atoms bouncing between two oscillating mirrors at $t = 2 π / 3 ω$ and corresponding to a resonant Floquet state. (c) The probability density for detecting a particle at fixed position $r = (16, 37)$ in (b) at different times. (d) The system maps to an effective $s_{x} \times s_{y}$ Bose-Hubbard model. By tuning the strength of the attractive interactions, three different phases are identified.
Reuse & Permissions
Figure 2
Three different regimes of the interaction strength (for the case of $s_{x} = 2$ and $s_{y} = 3$ ) characterized by the parameter $a_{max}^{2} = max {| a_{i} |^{2}}$ , cf. Eq. (6). Different regions correspond to the ground state solutions of the Hamiltonian (3) with time translation symmetry (i) being preserved in both lattice directions, (ii) broken only in one of the lattice direction, and (iii) broken in both the lattice directions. Two different strategies have been used to break the symmetry. The first (depicted with red circles) corresponds to isotropic tunneling $J_{x} = J_{y}$ (where $J_{x} = J_{(i_{x}, i_{y}; i_{x} + 1, i_{y})}$ and $J_{y} = J_{(i_{x}, i_{y}; i_{x}, i_{y} + 1)}$ ) with significant nearest-neighbor interactions in one direction, $U_{x} ≫ U_{y}$ (where $U_{x} = U_{(i_{x}, i_{y}; i_{x} + 1, i_{y})}$ and $U_{y} = U_{(i_{x}, i_{y}; i_{x}, i_{y} + 1)}$ ). The other (depicted with blue squares) corresponds to anisotropic tunneling $J_{x} \neq J_{y}$ . The former method results in $1 / 6 < a_{max}^{2} < 1 / 3$ while the latter gives $1 / 6 < a_{max}^{2} < 1 / 2$ in regime (ii). As a consequence of our choice of the system parameters, different symmetry regimes in both strategies coincide with each other—in general they can be located at different ranges of $g_{0} N$ . We use mirror amplitudes $λ_{x} = 0.094$ and $λ_{y} = 0.03$ , frequency $ω = 1.1$ , and relative phase of $δ = 2 π / 3$ to get isotropic tunneling, $J_{x} = J_{y} = 4.8 \times 10^{- 6}$ . For anisotropic tunneling rates, we used $λ_{x} = 0.12$ , $λ_{y} = 0.09$ , $ω = 1.4$ , and $δ = π / 8$ giving $J_{x} = 7.2 \times 10^{- 4}$ , $J_{y} = 3.7 \times 10^{- 5}$ .
Reuse & Permissions
Figure 3
(a),(b) Modulation of the contact interactions between atoms as function of time. (c),(d) Values of the interaction coefficients along the two lattice directions for the Bose-Hubbard model corresponding to (a) and (b), respectively. Small variation of the contact interactions is sufficient to generate non-negligible anisotropic interactions for nearest neighbors.
Reuse & Permissions
Figure 4
Overlap between the instantaneous state and the initial state, $O (t) = | \int d x d y ϕ^{*} {(x, y, t) ϕ (x, y, 0) |}^{2}$ , for interaction strengths corresponding to the three different regimes: $g_{0} N = 0$ [regime (i) in Fig. 2], $g_{0} N = - 0.04$ [regime (ii)], and $g_{0} N = - 0.2$ [regime (iii)]. The initial state has been chosen as $ϕ (x, 0) ϕ (y, 0)$ where $ϕ$ defined in Eq. (8) is obtained by the optimization procedure.
Reuse & Permissions
Figure 5
Density plot of BEC dynamics obtained with optimized parameters for the initial state. The white curves represent classical trajectories. (a)–(c) correspond to short-time dynamics, which is the same for any $g_{0}$ , for three different times (a) $t = 0$ , (b) $t = T / 3$ , and (c) $t = T / 2$ . (d)–(f) correspond to long-time dynamics for different interactions, (d) $g_{0} N = 0$ , (e) $g_{0} N = - 0.04$ , and (f) $g_{0} N = - 0.2$ at fixed time $t = 700 T$ .
Reuse & Permissions

Physical Review Research