Tuning towards dynamic freezing using a two-rate protocol

We study periodically driven closed quantum systems where two parameters of the system Hamiltonian are driven with frequencies $ω_1$ and $ω_2=r ω_1$. We show that such drives may be used to tune towards dynamics induced freezing where the wavefunction of the state of the system after a drive cycle at time $T= 2π/ω_1$ has almost perfect overlap with the initial state. We locate regions in the $(ω_1 ,r)$ plane where the freezing is near exact for a class of integrable and a specific non-integrable model. The integrable models that we study encompass Ising and XY models in $d=1$, Kitaev model in $d=2$, and Dirac fermions in graphene and atop a topological insulator surface whereas the non-integrable model studied involves the experimentally realized one-dimensional (1D) tilted Bose-Hubbard model in an optical lattice. In addition, we compute the relevant correlation functions of such driven systems and describe their characteristics in the region of $(ω_1,r)$ plane where the freezing is near-exact. We supplement our numerical analysis with semi-analytic results for integrable driven systems within adiabatic-impulse approximation and discuss experiments which may test our theory.

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