Two rate periodic protocol with dynamics driven through many cycles

We study the long time dynamics in closed quantum systems periodically driven via time dependent parameters with two frequencies $\omega_1$ and $\omega_2=r \omega_1$. Tuning of the ratio $r$ there can unleash plenty of dynamical phenomena to occur. Our study includes integrable models like Ising and XY models in $d=1$ and Kitaev model in $d=1$ and $2$ and can also be extended to Dirac fermions in graphene. We witness the wave-function overlap or dynamic freezing to occur within some small/ intermediate frequency regimes in the $(\omega_1 ,r)$ plane (with $r\ne0$) when the ground state is evolved through single cycle of driving. However, evolved states soon become steady with long driving and the freezing scenario gets rarer. We extend the formalism of adiabatic-impulse approximation for many cycle driving within our two-rate protocol and show the near-exact comparisons at small frequencies. An extension of the rotating wave approximation is also developed to gather an analytical framework of the dynamics at high frequencies. Finally we compute the entanglement entropy in the stroboscopically evolved states within the gapped phases of the system and observe how it gets tuned with the ratio $r$ in our protocol. The minimally entangled states are found to fall within the regime of dynamical freezing. In general, the results indicate that the entanglement entropy in our driven short-ranged integrable systems follow genuine non-area law of scaling and show a convergence (with a $r$ dependent pace) towards volume scaling behavior as the driving is continued for long time.