Discrete time crystal for periodic-field sensing with quantum-enhanced precision

TL;DR

This work addresses high-precision sensing of a gradient periodic field by exploiting a disorder-free discrete time crystal (DTC) as a quantum sensor. The authors develop a two-chain spin model that realizes a DTC under a binary quench and couple it to a gradient periodic field, deriving an effective Floquet description and demonstrating a quantum-enhanced sensitivity. They show that, within the DTC phase, the quantum Fisher information scales as $\mathcal{F}_Q \propto n^2 L^4$, with sharp phase transitions and robustness to detuning, crosstalk, initialization, and dephasing; importantly, simple measurements can approach this quantum limit. The results include a feasible experimental pathway in ultracold atoms in optical lattices, suggesting that DTC-based metrology can achieve ultimate precision with practical readout and realistic coherence times. Overall, the work provides a scalable route to quantum-enhanced periodic-field sensing using non-equilibrium many-body dynamics.

Abstract

Sensing periodic-fields using quantum sensors has been an active field of research. In many of these scenarios, the quantum state of the probe is flipped regularly by the application of $π$-pulses to accumulate information about the target periodic-field. The emergence of a discrete time crystalline phase, as a nonequilibrium phase of matter, naturally provides oscillations in a many-body system with an inherent controllable frequency. They benefit from long coherence time and robustness against imperfections, which makes them excellent potential quantum sensors. In this paper, through theoretical and numerical analysis, we show that a disorder-free discrete time crystal probe can reach the ultimate achievable precision for sensing a periodic-field. As the amplitude of the periodic-field increases, the discrete time crystalline order diminishes, and the performance of the probe decreases remarkably. Nevertheless, the obtained quantum enhancement in the discrete time crystal phase, which is experimentally accessible using standard projective measurements, shows robustness against different imperfections and dephasing noise in the protocol. Finally, we propose the implementation of our protocol in ultra-cold atoms in optical lattices.

Figures (8)

• Figure 1: (a) Schematic for the DTC sensor introduced here. (b) Imbalance in the total magnetization of the chains $a$ and $b$ denoted as $\mathcal{I}(nT)$, versus the amplitude of the periodic-field $h_{a}$ at different stroboscopic times in a system of size $L{=}6$. (c) Imbalance at even stroboscopic times in systems with various sizes, when $h_a{=}0.25$ and the system operates in the non-DTC region. In both panels, we set $\varepsilon{=}0.1$.
• Figure 2: (a) The QFI as a function of $h_a$ over different stroboscopic times. (b) The dynamical behavior of the QFI versus stroboscopic time $n$ at different values of $h_{a}$. In both panels (a) and (b), we set $L{=}7$ and $\varepsilon{=}0.1$. (c) QFI as a function of $h_a$ for systems of different sizes over $n{=}10$ period cycles when $\varepsilon{=}0.1$. (d) The values of QFI inside the DTC phase, namely $h_{a}{=}10^{-5}$, and in the transition point, namely $h_a{=}h_{a}^{\max}$, as a function of the system size $L$. The markers are the numerical results, and the solid lines provide the best fitting function as $\mathcal{F}_{Q}{\propto}L^{\beta}$ with $\beta{=}4.0902$ and $\beta{=}2.5887$ for a sensor in the DTC phase and in its transition point. The inset is the transition point $h_{a}^{\max}$ as a function of $L$. The best fitting function is obtained as $h_{a}^{\max}{\propto}L^{-1.0916}$.
• Figure 3: (a) The imbalance as a function of $h_a$ for different values of the detuning $\delta f$ over different stroboscopic times. (b) The QFI versus the detuning $\delta f$ for various values of the $h_a$ and $n$. In both panels $L{=}7$. (c) The QFI in systems with different sizes as a function of $n$ when $h_{a}{=}10^{-2}$ and $\delta f{=}10^{-2}$. (d) The QFI as a function of $L$ when the sensor is in the DTC phase and at the transition point and $\delta f{=}10^{-2}$. The numerical results are well described by the fitting function $\mathcal{F}_{Q}{\propto}L^\beta$ with $\beta{=}4.6658$ and $\beta{=}2.2014$ for a sensor deep inside the DTC phase and at the transition point, respectively. In all the panels, we set $\varepsilon{=}0.1$.
• Figure 4: (a) The QFI as a function of $h_a$ over various stereoscopic times, when $\eta{=}10^{-1}$. (b) The QFI as a function of $\eta$ when $n{=50}$. In both plots we set $L{=}7$ and $\varepsilon{=}0.1$.
• Figure 5: (a) The imbalance as a function of $n$ for different values of $\vartheta$ when $h_{a}{=}10^{-2}$. (b) The QFI versus $h_{a}$ over different stroboscopic times when $\vartheta{=}10^{-2}\pi$. In both panels $L{=}7$ and $\varepsilon{=}0.1$.