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Physical Review A

Systematic corrections to the rotating-wave approximation and quantum chaos

Ronald F. Fox, John C. Eidson

Published Nov 1, 1987• 28 citations
Publisher

Abstract

Quantum chaos has been demonstrated in several studies of the interaction between N two-level quantum systems and a single-mode radiation field interacting in a resonant cavity. It has been shown that the mechanism for this chaos is dynamically identical to a periodically perturbed, near-separatrix motion of a pendulum. Identification of this mechanism followed from examination of the rotating-wave approximation and corrections to this approximation. Our earlier study established the nature of the corrections on the basis of numerical simulation experiments and heuristic arguments. In this paper we invoke an averaging procedure which enables us to systematically establish the form of these corrections. We first apply this procedure to the Rabi model in order to illustrate its utility and exhibit its structure, and then we apply it to the more complicated problem of quantum chaos.

DOI
10.1103/physreva.36.4321

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Quantum chaos has been demonstrated in several studies of the interaction between N two-level quantum systems and a single-mode radiation field interacting in a resonant cavity. It has been shown that the mechanism for this chaos is dynamically identical to a periodically perturbed, near-separatrix motion of a pendulum. Identification of this mechanism followed from examination of the rotating-wave approximation and corrections to this approximation. Our earlier study established the nature of the corrections on the basis of numerical simulation experiments and heuristic arguments. In this paper we invoke an averaging procedure which enables us to systematically establish the form of these corrections. We first apply this procedure to the Rabi model in order to illustrate its utility and exhibit its structure, and then we apply it to the more complicated problem of quantum chaos.
An angular momentum J in a resonant-cavity electromagnetic field exhibits chaos in the time dependence of the expectation value of Jz{\mathit{J}}_{\mathit{z}}Jz​. This fact is transparently demonstrated using the SU(2) generalized coherent-state picture. Using Hale's averaging theorem, the mechanism of chaos is elucidated and found to be equivalent to a periodically modulated, near-separatrix, pendulum motion. The rotating-wave approximation is also elucidated, and its nonchaos is explained. The modulations that cause chaos result from virtual transitions, and the variance of Jz{\mathit{J}}_{\mathit{z}}Jz​ initially grows exponentially.
A mechanism of emergence of Hamiltonian chaos is considered for the model describing the interaction between two-level atoms and their own radiation field in an ideal single-mode cavity. The analysis of the semiclassical Maxwell-Bloch equations shows that the Hamiltonian terms that are neglected in the rotating-wave approximation (RWA) give rise to the formation of a stochastic layer near the RWA-system separatrix. The Mel’nikov method is used to prove that the splitting of the separatrix takes place for arbitrarily small vacuum Rabi frequencies Ω N . The computation of Poincare sections shows that the stochastic layer, which is exponentially narrow for small Ω N , expands with increasing Ω N , and at Ω N ≃ 1, the system exhibits global chaos that manifests itself in irregular oscillation of the atomic population inversion and the broadening of the power spectrum. Promising candidates for observing manifestations of dynamic chaos in this basic quantum-optical system are Rydberg atoms placed in a high-Q superconducting microwave cavity.
Abstract We investigate the applicability of the two major approximations which are most commonly employed in the study of the quantum Rabi model, namely the description of a resonant cavity mode as a single-mode quantized field and the use of the rotating wave approximation. Starting from the Hamiltonian of a two-level system interacting with a multi-mode quantized field, we perform the canonical transformation of the field operators. This allows one to partition the Hamiltonian of the system into two parts. The first part is the interaction of the two-level system with a single collective field mode, while the second one describes the interaction with field fluctuations. The first part is usually associated with the resonant cavity mode. This division enables us to determine the applicability condition of the single-mode approximation. In addition we identify simple approximate relations for the description of the eigenstates, eigenfunctions and the time evolution of the quantum Rabi model beyond the rotating wave approximation.
The dynamics of an ensemble of two-level atoms moving through a single-mode lossless cavity is investigated in the semiclassical and rotating-wave approximations. The dynamical system for the expectation values of the atomic and field observables is considered as a perturbation to one of the following integrable versions: (i) a model with atoms moving through a spatially inhomogeneous resonant field, and (ii) a model with atoms interacting with a nonresonant eigenmode which is assumed to be homogeneous on the cavity size. We find the general exact solutions for both the models and show that they contain special solutions describing a coherent effect of population and radiation trapping. Using the Melnikov method, we prove analytically transverse intersections of stable and unstable manifolds of a hyperbolic fixed point under a small modulation of the vacuum Rabi frequency. These intersections are believed to provide the Smale horseshoe mechanism of Hamiltonian chaos. The analytical results are accompanied with direct computation of topographical maps of maximal Lyapunov exponents that give a representative image of regularity and chaos in the atom-field system in different ranges of its control parameters--the frequency detuning, the number, and the velocity of atoms.
A system of atoms interacting with a radiation field in a resonant cavity is studied under conditions when the dynamics in the classical limit is stochastic. This situation is called quantum chaos. Equations of motion are obtained for the quantum-mechanical expectation values which take into account the quantum correlation functions. It is shown that in a situation corresponding to quantum chaos, the quantum corrections grow exponentially, making the evolution of the system essentially quantal after a certain time tau( variant Planck's over 2pi ) has elapsed. Analytical and numerical analysis show that in this regime the time tau( variant Planck's over 2pi ) obeys the logarithmic law tau( variant Planck's over 2pi ) approximately ln N (N is the number of atoms), and not the law tau( variant Planck's over 2pi ) approximately N(alpha) (alpha is a certain constant of order unity), as would be the case in the absence of chaos.