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

We propose and characterize protocols to realize discrete  
time crystals (DTCs) in the periodically driven central spin model. 
While period-doubling DTCs have been observed in this system before, we 
uncover a unifying interaction-induced echo mechanism underlying several 
distinct dynamical regimes. We first demonstrate that this echo can 
enable exact period-doubling revivals when the Ising interaction 
strength, $lambda$, between the central spin and the $N_{rm sat}$ 
satellite spins is tuned to $2 pi$. Notably, these revivals persist for 
arbitrary $N_{rm sat}$ and transverse field $g$. This not only 
stabilizes the DTC response over a wide parameter regime, but also leads 
to a dynamical freezing regime. Furthermore, when $lambda=pi$ and  
$g=pi/2$, this echo induces a Clifford group structure. Consequently, 
higher-period revivals emerge that naturally steer the system through an 
entangled manifold of Bell-cat and spin-cat states. We establish that 
due to parity-dependent phases accumulated during the echo, the 
recurrence period is $12 T,, (24 T)$ for even (odd) $N_{rm sat}$. 
Remarkably, the period-12 oscillations for even $N_{rm sat}$ are robust 
to perturbations, thereby promoting these revivals to a `higher-order' 
(HO)-DTC beyond the conventional crypto-equilibrium paradigm. Finally, 
we demonstrate that the multipartite entanglement generated by the 
Clifford dynamics can be harnessed for multiparameter metrology, with 
odd $N_{rm sat}$ enabling Heisenberg-limited sensitivity.