Exact analysis of AC sensors based on Floquet time crystals

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 resonantly induce transitions between macroscopic paired cat states in the FTC sensor, allowing a 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 different relevant initial preparations of the sensor, including ground states, 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. We present our findings for both linear and nonlinear response regimes, and illustrate them for a specific FTC based on the long-range interacting LMG model.