Thermodynamics of coupled time crystals with an application to energy storage

TL;DR

This work develops a thermodynamically consistent framework for two coupled boundary time crystals using a collision-model environment, connecting average dynamics, quantum fluctuations, and correlations across stationary and time-crystal phases. By analyzing two charging schemes, it shows that the time-crystal phase can store more energy with higher efficiency than the stationary phase, and reveals distinct entanglement and correlation patterns in each regime. The study further explores seeding crystallization as a charging mechanism, identifying regions of quasi-periodicity and limit cycles that optimize energy storage and highlighting finite entanglement during certain dynamical phases. Overall, coupled boundary time crystals emerge as promising candidates for quantum batteries in nonequilibrium settings, with practical implications for energy storage in quantum technologies.

Abstract

Open many-body quantum systems can exhibit intriguing nonequilibrium phases of matter, such as time crystals. In these phases, the state of the system spontaneously breaks the time-translation symmetry of the dynamical generator, which typically manifests through persistent oscillations of an order parameter. A paradigmatic model displaying such a symmetry breaking is the boundary time crystal, which has been extensively analyzed experimentally and theoretically. Despite the broad interest in these nonequilibrium phases, their thermodynamics and their fluctuating behavior remain largely unexplored, in particular for the case of coupled time crystals. In this work, we consider two interacting boundary time crystals and derive a consistent interpretation of their thermodynamic behavior. We fully characterize their average dynamics and the behavior of their quantum fluctuations, which allows us to demonstrate the presence of quantum and classical correlations in both the stationary and the time-crystal phases displayed by the system. We furthermore exploit our theoretical derivation to explore possible applications of time crystals as quantum batteries, demonstrating their ability to efficiently store energy.

Figures (8)

• Figure 1: Collision-model setup for a system of two interacting boundary time crystals. (a) A hot ancillary system, in a thermal state with inverse temperature $\beta_1$, interacts with one boundary time crystal (BTC), while a cold ancillary system, in a thermal state with inverse temperature $\beta_2$, interacts with the other BTC. The two BTCs are also driven by two independent lasers and coupled through an interaction Hamiltonian parametrized by the coupling strengths $J$, in the $x$ and $y$ directions, and $J_z$, in the $z$ direction (see main text). (b) Two interacting BTCs working as batteries driven by the lasers. (c) One BTC charging the other one via seeding crystallization in time.
• Figure 2: Time-averaged work input and correlations. (a) Total time-averaged work input for varying $J_z$ and $J$. Region I denotes the stationary phase, while region II the time-crystal phase. The inset shows the phase transition for $J=0$. The other parameters are $\delta = 0$, $n_1 = n_2 = 0$ and $\Omega_1/2 = \Omega_2/2 = \kappa$. The initial state is $\vec{m}^{(j)} = [0, 0, -1/\sqrt{2}]$ for both systems in every point of the phase-diagram. (b) Time-averaged classical correlations $\mathcal{J}$ and (c) time-averaged quantum discord $\mathcal{D}$ and negativity $\mathcal{N}$ as function of $J$, such that $J+J_z=4\kappa$ [see the diagonal in panel (a)]. The mean-field quantities were evolved from $t=0$ to $t\kappa=10^3$ and the average was computed by integrating over the second half of the evolution time. The correlations (see Sec. \ref{['sec:QuantumFluctuations']}) were computed considering trajectories between $t=0$ and $t\kappa=200$, and the integration was done over the second half of such interval. The remaining parameters are $\Omega_1=\Omega_2=2\kappa$, and $n_1=n_2=0$.
• Figure 3: Stored energy and charging efficiency. (a) Time-averaged stored energy in both atomic ensembles, defined as $\bar{\mathcal{E}}= \lim_{t\rightarrow \infty} t^{-1} \int_0^t \mathcal{E}(t^\prime) {\rm d}t^\prime$, in units of $\omega_{\rm at}$. Here, we consider the initial condition to be with all atoms in the ground state. The gray region indicates a bistable regime, that we can access by following adiabatically the time-crystal solution as the interaction parameter $J$ is increased. When starting from the ground state, the systems goes to the stationary phase before the end of the bistable region. The other parameters are $\Omega=2\kappa$ and $J_z=\kappa$. (b)-(c) Stored energy (in units of $\omega_{\rm at}$) and efficiency as function of time for a time-crystal solution (red solid line) and a stationary one (blue dashed line). In both cases, the initial conditions were $\vec{m}=[0,0,-1/\sqrt{2}]$ and for the time-crystal solution we set $J=3.4\kappa$ while for stationary we use $J=3.41\kappa$. The vertical black lines show that the second peak of efficiency correspond to the maximum stored energy. The remaining parameters are: $\Omega=2J_z=2\kappa$ and $\omega_{\rm at}=\nu$.
• Figure 4: Work input, efficiency, and entanglement. (a) Time-integrated average work input as a function of $J$ and $\Omega$. We compute this phase-diagram by considering all atoms initially in the ground state for all points. We identify four regions: stationary (Ia), limit cycle (Ib), oscillatory regime between limit cycle and stationary (Ic), quasi-periodic oscillations (II) and limit cycle (III). (b) A section of the phase diagram (a) for the time-averaged stored energy in units of $\omega_{\rm at}$ as function of $J/\kappa$ for $\Omega=2.5\kappa$. In region II, we have the stored energy in the quasi-periodic regime. In Ic, the system oscillates between limit-cycles and stationary values for small variations of $J/\kappa$. After this intermediate region, it goes to region Ib until it reaches region Ia. Here we time-average trajectories with total time $t\kappa=10^3$. The stored energy is given in units of $\omega_{\rm at}$. (c) The efficiency $\eta(t)$ as function of time, for four values of $J$, as indicated in the legend. Here $\Omega=2.5\kappa$. The blue dashed line correspond to a limit-cycle solution found in region I with initial state being the ground state of $H_{\rm at}$ (d) Time-averaged entanglement negativity between the charger and the battery as function of the occupation number ${n}_1={n}_2={n}$. We further set $\omega_{\rm at}=\nu$.
• Figure 5: Time-averaged work input and quantum correlations for the stationary phase. (a) Time-averaged work input for varying $J_z$ and $J$, for the stationary phase $\Omega_1=\Omega_2=\kappa/2$, respectively. Here, we start with $J=0$ and $J_z=4\kappa$ and we increase $J$ and decrease $J_z$. (b)-(c) Time-averaged classical and quantum correlations, respectively, as function of $J$, such that $J_z + J = 4\kappa$. The mean-field quantities were evolved from $t=0$ to $t\kappa=10^3$ and only their second half were integrated. The correlations were computed considering time intervals $t \in [0, 200]$, and the integration was done over the second half of the interval. The remaining parameters are $\delta=0$, $n_1 = n_2 = 0$ and $\kappa = 2\Omega_1 = 2\Omega_2$. The initial state is $\vec{m}=[0,0,-1/\sqrt{2}]$.