Floquet resonances in the double kicked top

We study exact quantum resonances in the double kicked top (DKT), a driven spin model that extends the quantum kicked top (QKT) by introducing an additional time-reversal symmetry-breaking kick. By expressing the dynamics in terms of effective parameters (k_{r},k_{θ}), we analytically show the exact periodicity of the Floquet operator for integer and half-odd integer spin j at k_{r}=jπ/2. The analysis is further extended to k_{r}=jπ/4 for integer spin j and proves the absence of recurrences for half-odd-integer spin j. Remarkably, resonances are observed across the time-reversal symmetry (or its breaking) and generalize the QKT resonances. Spectral statistics and entanglement entropy in the pseudoclassical limit reveal qualitatively distinct behavior at the two resonances. Near k_{r}=jπ/2, the level-spacing ratio distribution evolves from Poisson statistics (PS) to Gaussian orthogonal ensemble (GOE) statistics, indicating a crossover from resonant dynamics to quantum-chaotic behavior through an intermediate regime. No comparable intermediate-statistics regime is observed in the case of k_{r}=jπ/4. In the time-reversal symmetric case k_{θ}=0, our computations of the rate function associated with fidelity show the dynamical quantum phase transition (DQPT) only for half-odd-integer j values. Our work demonstrates the DKT as a controllable platform where resonance, integrability, and quantum chaos can be tuned for any system size through the parameters k_{r} and k_{θ}, making the DKT a useful setting for quantum control and information processing applications.