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We study the long time dynamics in closed quantum systems periodically drivenvia time dependent parameters with two frequencies $\omega_1$ and $\omega_2=r\omega_1$. Tuning of the ratio $r$ there can unleash plenty of dynamicalphenomena to occur. Our study includes integrable models like Ising and XYmodels in $d=1$ and Kitaev model in $d=1$ and $2$ and can also be extended toDirac fermions in graphene. We witness the wave-function overlap or dynamicfreezing to occur within some small/ intermediate frequency regimes in the$(\omega_1 ,r)$ plane (with $r\ne0$) when the ground state is evolved throughsingle cycle of driving. However, evolved states soon become steady with longdriving and the freezing scenario gets rarer. We extend the formalism ofadiabatic-impulse approximation for many cycle driving within our two-rateprotocol and show the near-exact comparisons at small frequencies. An extensionof the rotating wave approximation is also developed to gather an analyticalframework of the dynamics at high frequencies. Finally we compute theentanglement entropy in the stroboscopically evolved states within the gappedphases of the system and observe how it gets tuned with the ratio $r$ in ourprotocol. The minimally entangled states are found to fall within the regime ofdynamical freezing. In general, the results indicate that the entanglemententropy in our driven short-ranged integrable systems follow genuine non-arealaw of scaling and show a convergence (with a $r$ dependent pace) towardsvolume scaling behavior as the driving is continued for long time.

Two rate periodic protocol with dynamics driven through many cycles