Exact analysis of AC sensors based on Floquet time crystals

TL;DR

This work develops a general, platform-agnostic theory for Floquet time crystals (FTCs) acting as closed-system AC sensors. By analytically deriving the quantum Fisher information dynamics and identifying a Heisenberg-limit scaling $F_h(t) \sim N^2 t^2$ that persists for exponentially long times, the paper shows how resonant transitions between pi-paired cat states under period-doubling resonance enable robust metrological enhancement. A characteristic step-like QFI evolution arises from dephasing across cat subspaces, with the dynamics and scaling modulated by initial state preparation and proximity to Floquet phase transitions. The theory is illustrated in the Lipkin-Meshkov-Glick (LMG) model, revealing how extensive cat-state overlaps yield large QFI, while approaching the phase boundary reduces the signal, and demonstrates the potential for implementing FTC-based sensors in platforms such as trapped ions or prethermal DTCS.

Abstract

We discuss the behavior of general Floquet Time Crystals (FTCs), including prethermal ones, in closed systems acting as AC sensors. We provide an analytical treatment of their quantum Fisher information (QFI) dynamics, which characterizes the ultimate sensor accuracy. By tuning the direction and frequency of the AC field, we show how to induce transitions resonantly between macroscopic paired cat states in the FTC sensor. This allows for robust Heisenberg scaling precision (QFI $\sim N^2 t^2$) for exponentially long times in the system size. The QFI dynamics exhibit, moreover, a characteristic step-like structure in time due to the eventual dephasing along the cat subspaces. The behavior is discussed for various initial sensor preparations, including ground states and low- and high-correlated states. Furthermore, we examine the performance of the sensor along the FTC phase transition; with the QFI capturing its critical exponents. Our findings are presented for both linear and nonlinear response regimes and illustrated for a specific FTC based on the long-range interacting LMG model.

Figures (4)

• Figure 1: (a) Schematic representation of an FTC quantum sensor. The sensor consists of $N$ spins and is exposed to an external AC field that acts as a probe for its amplitude or frequency. (b) Spectrum $\{\epsilon_i\}$ of the Floquet unitary in the FTC sensor, showing the set of ($M=6$) $\pi$-paired quasi-energies corresponding to cat states of opposite parities - the rest of the unpaired spectrum is omitted here for clarity. (c) Dynamics of the QFI for different relevant initial preparations of the sensor: from low to high correlated initial states, and effective Floquet Hamiltonian eigenstates $\{ |E_{i(\bar{i})}\}$. The QFI exhibits a step-like increasing/decreasing characteristic dynamics for each of these classes of preparations, i.e. a sequence of plateaus interspersed by abrupt variations, which occur on timescales proportional to their $\pi$-paired gaps (see Table \ref{['table.states']} for a more detailed summary). The results are illustrated here for an FTC sensor based on the LMG model ($N=40$, $T=1$, $J=1$, $B=0.4$, $T_{\rm ac} = 2T$, linear response $h \to 0$ and sinusoidal signal $f(t)= \sin(\pi t/T)$). See SM-Sec.\ref{['QFI.dynamics.LMG.model.graphs']} for the extended data.
• Figure 2: QFI derivation outline: (a) The FTC sensor, governed by a Floquet Hamiltonian $\hat{H}_F$ and parity operator $\hat{X}$, in put in contact to an AC field $\hat{V}(t)$. Without loss of generality, we assume that the dynamics are driven by the shifted Floquet Hamiltonian, as Eq.\ref{['eq.HF.shift']}. (b) Tuning the AC field frequency in PDR to the sensor simplifies the HSO, leading to a block-diagonal structure along the $\pi$-paired subspaces (Eq.\ref{['eq.heisenb.signal.blockdiagonal']}). (c) Aligning the AC field along the underlying SSB direction achieves an HSO norm (which upper-bounds the QFI) sustaining the Heisenberg limit scaling for exponentially long timescales.
• Figure S1: Numerical solution of the transcendental equation (Eq.\ref{['eq.solve_numericallyfq']}), derived from the derivative of $f_h(t)$ (Eq. \ref{['eq:f_q_func']}) for the physical state with $\varphi = 0$. The first non-trivial root, $\textcolor{black}{\tau_{max}} \approx 2.3311$ where the QFI reaches its maximum value for symmetry-breaking initial state. For $0 < \varphi < \pi/2$, the root $\textcolor{black}{\tau_{max}}$ shifts slightly to smaller value and to bigger value of $\max{f_h}$, indicating a dependence of the QFI peak on the relative phase of the initial state superposition.
• Figure S2: Dynamics of QFI in the FTC sensor based on the LMG model, for different initial preparations, varying system size $N$ and ratio $B/J$. The simulations include four initial states: (green) a system prepared in the ground state of the Floquet Hamiltonian, $|E_1\rangle$; (red) in a superposition of $\pi$-paired eigenstates, $(|E_1\rangle + |E_{\bar{1}}\rangle)/\sqrt{2}$; (orange) in an initial state with all spins aligned in the up direction, $|\uparrow \dots \uparrow\rangle$; or (blue) a highly correlated initial state with a superposition of up and down spins, $(|\uparrow \dots \uparrow\rangle + |\downarrow \dots \downarrow\rangle)/\sqrt{2}$.