Quantum mechanics of systems periodic in time

Abstract

Some expressions for the time evolution of quantum-mechanical systems with Hamiltonians periodic in time, derivable from the work of Shirley and applied by Young, Deal, and Kestner and Haeberlen and Waugh -all for finite-basis-set systems -are derived for a general system (possibly infinite Hilbert space). These results suggest a new type of approximation to the time-evolution operator, one which is exact at multiples of the period of the Hamiltonian. Comparison is made to an exactly soluble problem, namely, a nonrelativistic hydrogen atom in a circularly polarized monochromatic field.

Effective Hamiltonians for periodically driven systems

“…Such systems can be treated using Floquet theory [29,30,31,32,33]. The symmetry with respect to discrete time translations implies that the solutions of the Schrödinger equation…”

Artificial Gauge Fields with Ultracold Atoms in Optical Lattices

“…One can show that for periodic systems the evolution operator fulfills the following properties [117] …”

“…This is an essential part of Floquet's theorem [113,[117][118][119][120], which tells us that the evolution of the system after multiples of one driving period can be described by an effective time-independent Floquet HamiltonianĤ F…”

“…The validity of Eq. (3.12) can be proven using general properties of the evolution operator 13) together with the well known form of Floquet's theorem [117,123] …”

Weak-coupling-like time evolution of driven four-level systems in the strong-coupling regime

“…These formalisms have been used in studies of atomic spectroscopy [22], laser-assisted electron-atom scattering [23], harmonic generation [24], periodically kicked Rydberg atoms [25] and multiphoton excitation and ionization of atoms and molecules [26][27][28][29][30].…”