Resonance expansion versus the rotating-wave approximation

Abstract

We propose a general perturbative approach to quantum-optical models without the rotating-wave approximation. We show that a generic Hamiltonian describing interaction between two subsystems can be represented as a series of operators corresponding to different transitions between bare energy levels of the whole system. Under certain relations between frequencies of interacting subsystems one of these transitions becomes resonant. The rotating-wave approximation leads to separation of the resonant transition and to appearance of the integral of motion, which makes the problem exactly solvable in this approximation. ͑Different resonance conditions lead to different integrals of motion.͒ All of the other terms in these expansion can be considered as a perturbation. They result in dynamic Stark shifts and small corrections to the integrals of motion. All possible resonances are classified, and approximate integrals of motion are found for each resonance. Examples of field-field, field-atom, and atom-atom interactions are considered.

Multiphoton Bloch–Siegert shifts and level-splittings in spin-one systems

“…The BlochSiegert shift for the spin 1 case was studied previously by Hermann and Swain [12,13]. Some progress has also been made in the case of the general spin problem [8].…”

Decoherence of two qubits in non-Markovian reservoirs without the rotating-wave approximation

“…Generally, the RWA is justified for small detunings and small ratio of the atom-field coupling divided by the atomic transition frequency [13,14,15]. In atom-field cavity systems, this ratio is typically of the order 10 −7 ∼ 10 −6 .…”

Quantum phase transitions in an effective Hamiltonian: fast and slow systems

“…The above Hamiltonian does not preserve the total excitation number operator N = X 0 + Y 0 and, in the limit ω 1 , ω 2 ≫ g, leads to the appearance of multiphotontype interactions of the form X n + Y m − which, under certain physical conditions on the frequencies ω 1,2 , describe resonant transitions between energy levels of the whole system (see [3] and references therein).…”

“…As it was shown in Ref. [3], the evolution is governed by an effective Hamiltonian describing a certain resonant interaction and the representation space of the total system can be always divided into (almost) invariant subspaces.…”

“…The advantage of this method consists in the possibility of varying the system's parameters, changing relations between them, which allows us to describe different physical regimes using the same mathematical tool. In particular, such an important example as expansion on the resonances in quantum systems not preserving the number of excitations can be obtained [3]. In this case a generic Hamiltonian governing interaction of two subsystems beyond the Rotating Wave Approximation (RWA) can be represented as a series in operators describing all possible transitions in the system.…”