Quantum mechanics of rapidly and periodically driven systems

Luca D’Alessio, Anatoli Polkovnikov

Viktor Novičenko, Egidijus Anisimovas, Gediminas Juzeliūnas

We consider a quantum system periodically driven with a strength which varies slowly on the scale of the driving period. The analysis is based on a general formulation of the Floquet theory relying on the extended Hilbert space. It is shown that the dynamics of the system can be described in terms of a slowly varying effective Floquet Hamiltonian that captures the long-term evolution, as well as rapidly oscillating micromotion operators. We obtain a systematic high-frequency expansion of all these operators. Generalizing the previous studies, the expanded effective Hamiltonian is now time dependent and contains extra terms appearing due to changes in the periodic driving. The same applies to the micromotion operators which exhibit a slow temporal dependence in addition to the rapid oscillations. As an illustration, we consider a quantum-mechanical spin in an oscillating magnetic field with a slowly changing direction. The effective evolution of the spin is then associated with non-Abelian geometric phases reflecting the geometry of the extended Floquet space. The developed formalism is general and also applies to other periodically driven systems, such as shaken optical lattices with a time-dependent shaking strength, a situation relevant to the cold-atom experiments.

Steven Mathey, Sebastian Diehl

We address the nature of phase transitions in periodically driven systems coupled to a bath. The latter enables a synchronized nonequilibrium Floquet steady state at finite entropy, which we analyze for rapid drives within a nonequilibrium renormalization group (RG) approach. While the infinitely rapidly driven limit exhibits a second-order phase transition, here we reveal that fluctuations turn the transition first order when the driving frequency is finite. This can be traced back to a universal mechanism, which crucially hinges on the competition of degenerate, near critical modes associated with higher Floquet Brillouin zones. The critical exponents of the infinitely rapidly driven system-including a new, independent one-can yet be probed experimentally upon smoothly tuning towards that limit.

Pavel Bóna

David J. Luitz, Achilleas Lazarides, Yevgeny Bar Lev

Recently it has been shown that interparticle interactions generically destroy dynamical localization in periodically driven systems, resulting in diffusive transport and heating. In this Rapid Communication we rigorously construct a family of interacting driven systems which are dynamically localized and effectively decoupled from the external driving potential. We show that these systems exhibit tunable periodic or quasiperiodic revivals of the many-body wave function and thus of all physical observables. By numerically examining spinless fermions on a one-dimensional lattice we show that the analytically obtained revivals of such systems remain stable for finite systems with open boundary conditions while having a finite lifetime in the presence of static spatial disorder. We find this lifetime to be inversely proportional to the disorder strength.

Walter E Thirring, Evans M. Harrell