Higher-order discrete time crystals and enhanced sensing in a quantum kicked top

We characterize various dynamical phases of the simplest version of the quantum kicked top model, a paradigmatic system for studying quantum chaos, which exhibits both regular and chaotic behavior depending on the kick strength. In a previous study, the existence of higher-order discrete time crystals (DTCs) was observed in an infinite-range interacting $p$-spin model, where it was proposed that the order of the DTC satisfies the relation $q\le p$. Within this framework, the $p=2$ model is expected to host only a $2$-DTC phase. However, interestingly, we demonstrate here the existence of a robust $4$-DTC phase in the quantum kicked top, which effectively corresponds to a $p=2$ model with infinite-range interactions. We also show that the system hosts robust $2$-DTC and dynamical freezing (DF) phases around alternating rotationally symmetric points. We explain the emergence of higher-order DTC phases through the classical phase portraits of the system, connected with spin coherent states (SCSs), by identifying special islands that arise within a specific parametric regime. Unlike the $2$-DTC phase, the $4$-DTC phase appears only for certain initial states, as demonstrated through exact calculations. The robustness of the $4$-DTC phase is further investigated through the dynamics of the linear entropy as a function of the angular momentum. We also find an emergent conservation law for both the $2$-DTC and DF phases, while no dynamical conservation arises periodically for the $4$-DTC phase. By investigating the quantum Fisher information, we also demonstrate enhanced metrological sensitivity at the boundaries between different dynamical phases for the estimation of system parameters.