Quantum Time Crystal Clock and its Performance

Abstract

Understanding different aspects of time is at the core of many areas in theoretical physics. Minimal models of continuous stochastic and quantum clocks have been proposed to explore fundamental limitations on the performance of timekeeping devices. Owing to the level of complexity in the clock structure and its energy consumption, such devices show trade-offs whose characterization remains an open challenge. Indeed, even conceptual designs for thermodynamically efficient quantum clocks are not yet well understood. In condensed matter theory, time-crystals were found as an exciting new phase of matter, featuring oscillations in (pseudo)-equilibrium with first experimental observations appearing recently. This naturally prompts the question: can time crystals be used as quantum clocks and what is their performance from a thermodynamic perspective? We answer this question and find that quantum time crystals are indeed genuine quantum clocks with a performance enhanced by the spontaneous breaking of time-translation symmetry.

Figures (8)

• Figure 1: Illustration for the clock model composed by many identical spins collectively interacting with a non-equilibrium environment. Continuous monitoring provides a time reference via event-counting which is greatly enhanced in the time-crystal phase. In the top plot we show (left) the number (scaled by the factors indicated) of collective emissions $N_-$ accumulated along two arbitrarily chosen realizations of the dynamics, for $\lambda = 0.7$ (gray), and $\lambda = 2$ (light blue), together with (right) the Fourier transform $\tilde{N}_-[\omega] = \int dt N_{-}$$(t)$$e^{-i\omega t}$. Parameters: $S= 50$, $\gamma_0 = 10^{-3} \omega_C$ and $\beta= 2 \omega_C$.
• Figure 2: (a) Accuracy-Resolution tradeoff curves for $\lambda = 0.7$ below the critical value (red) and $\lambda = 1.5$ above the critical value (orange), in a clock with $S = 50$ ($100$ spins). The black-dashed line represents the Poissonian benchmark relation $\mathcal{A} = \mathcal{R}^{-1}$ with $\mathcal{R}$ in units of $\gamma_0$. The inset shows the Fano factor $\mathcal{F}$ (normalized with $\gamma_0$) for $\lambda = 1.5$ as a function of the threshold $M$. The optimal threshold $M = 523$ is denoted by a circular marker. The minima of $\mathcal{F}$ correspond to maxima in the tradeoff curve as signaled with same markers. (b) WTD histograms $P_{{\rm WTD}}[\mathcal{T}]$ above and below the critical point for $S = 25$ (red) and for $S = 50$ (light blue). (c) Accuracy vs. $\lambda$ (Resolution vs. $\lambda$ in the inset) at optimal threshold for $S= 25$ (red) and $S = 50$ (light blue). The gray arrows point at the $\lambda$ values for which the WTD histograms are shown on (b). The black circles in the inset show the fit to the time-crystal frequency $\mathcal{R} \sim \nu = (\gamma_0 /2\pi) \sqrt{\lambda^2 - 1}$ with coefficient $r^2= 0.999989$. Other parameters: $\gamma_0 = 10^{-3} \omega_C$, $\beta= 2 \omega_C$ and $N(t) = N_-(t)$.
• Figure 3: Accuracy curves as a function of the phase transition parameter at optimal thresholds, for $S = 50$ and different counting observables: any jump detection $N(\mathcal{T}) = \mathcal{K}_{\rm tick}(\mathcal{T})$ (red), only emission events $N(\mathcal{T}) = N_-(\mathcal{T})$ (light blue), and heat current $N(\mathcal{T}) = Q(\mathcal{T})/\omega_C$ (orange). The results are compared to the TUR bound $\langle S_{\rm tick}(\mathcal{T}) \rangle/2$ at FPT (dotted line) and the dynamical activity $\langle \mathcal{K}_{\rm tick}(\mathcal{T})\rangle$ of the KUR bound (dot-dashed line). Inset: Accuracy as a function of the average entropy production per tick when fixing $\lambda = 2$ and varying $S$. Other parameters as in Fig. \ref{['fig:sketch']}.
• Figure A1: Convergence of the stopping-time fluctuation theorem for (a) $S_{\rm mar}(\mathcal{T}_1)$, and (b) $S_{\rm tick}(\mathcal{T})$. Ticking times are defined counting collective emissions $N_-$ (light blue), dynamical activity $\mathcal{K}$ (red) and heat current $Q/\omega_C$ (orange), while setting threshold $M = 5$ counts. Parameters: The system has total spin $S=50$ ($n_s = 100$ spins) and parameter $\lambda = 2$. The environment is defined by $\gamma_0 = 10^{-3} \omega_C$ and $\beta= 0.1\, \omega_C$.
• Figure A2: Waiting time distributions $P_{\rm WTD}[\mathcal{T}]$ obtained counting collective emissions $N_-(t)$ and fixing different threshold values $M = 355$ (orange), $M = 650$ (light blue), $M = 705$ (red), and $M = 1100$ (yellow). The inset shows the accuracy of the clock as a function of $M$. The values of $M$ associated to the shown distributions are signaled by '${\rm x}$' markers. Parameters: The system has total spin $S=50$ ($100$ spins) and parameter $\lambda = 2$. Other parameters as in Fig. 1 of the main text.