Tuning towards dynamic freezing using a two-rate protocol

Abstract

We study periodically driven closed quantum systems where two parameters of the system Hamiltonian are driven with frequencies ω1 and ω2 = rω1. 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/ω1 has almost perfect overlap with the initial state. We locate regions in the (ω1, 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 (ω1, 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.

Ramp and periodic dynamics across non-Ising critical points

“…These dipole models have been generalized to higher dimensions 6 . The quantum dynamics of these models has also been studied in details [7][8][9][10][11][12] . It was noted in Ref.…”

Two-rate periodic protocol with dynamics driven through many cycles

“…The freezing region around r = 1 is already known to appear for a driving through a single cycle 22 Moreover the freezing condition remains intact as ω 1 is increased further. However, after a long drive corresponding to n = 100, the freezing scenario wears off as in a long time evolution the evolved state spreads out more within the Hilbert space leaving less chances for the final state to be close to the initial state in the phase space.…”

“…In most of these studies, single-frequency periodic protocols are used to drive the system and examine the resulting dynamics. Recently Kar et al 22 considered a two rate periodic protocol where both the energy bias and detuning in the underlying two level system (TLS) are treated periodically with time. The low frequency regime that they study there, reveals signatures of dynamical freezing as driving for one complete cycle is performed.…”

“…Unlike a linear or any other aperiodic drive, a periodic drive, as mentioned before, unleashes rich set of dynamical phenomena [18][19][20][21][23][24][25] , even more so, if driven by two rate protocols 22 . Repeated passage through the critical points give the system, initially in its ground state, ample chances to move to the excited states and experience the quantum interference of the probability waves at the crossing/ avoided crossing regions.…”

“…The adiabatic-impulse approximation, used initially in Ref. 22 for similar protocol, is extended for stroboscopic long driving and for larger ranges of frequencies. It turns out to capture the final evolved states at small frequencies quite well even after passage through many cycles.…”