The Floquet central spin model: A platform to realize eternal time crystals, entanglement steering, and multiparameter metrology

We propose and characterize protocols to realize eternal discrete time crystals (DTCs) in the periodically driven central spin model. These eternal DTCs exhibit perfect periodic revivals of the initial state at a time $mnT$ (where $n>1$ and $\{m,n\}\in \mathbb{Z}$), when the Ising interaction strength, $\lambda$ between the central spin and the satellite spins is tuned to certain values. The combination of perfect initial-state revival and time-translation-symmetry breaking leads to infinitely long-lived oscillations of the stroboscopic magnetization and the entanglement entropy in these DTCs even for a finite number of satellite spins. We analytically determine the conditions for the existence of these eternal DTCs and prove that the system exhibits eternal period-doubling oscillations ($n=2$) when $\lambda =2\pi$ for an arbitrary number of satellite spins. Furthermore, we propose a protocol to realize eternal higher-order(HO)-DTCs ($n>2$) by tuning $\lambda$ to $\pi$. Intriguingly, this protocol naturally steers the system through an entangled trajectory, thereby leading to the generation of maximally entangled Bell-cat states during the dynamical evolution of the HO-DTC. Finally, we demonstrate that these HO-DTCs can serve as a resource for Heisenberg-limited multiparameter sensing.