The interaction of a quantum system with an oscillating field is studied in a formalism which replaces the semiclassical time-dependent Hamiltonian with a time-independent Hamiltonian represented by an infinite matrix. The formalism is developed as a mathematical equivalent to the semiclassical treatment, and interpreted as a classical approximation to the quantum treatment of the field. Combined with a perturbation theory for two nearly degenerate states, the formalism provides a convenient method for determining resonance transition probabilities including frequency shifts and multiple quantum transitions. The theory is illustrated by a detailed study of the simple case of a two-state system excited by a strong oscillating field.
The interaction of simple, two- and three-level atomic systems with monochromatic and nonmonochromatic laser radiation near resonance transitions is discussed based on the Schroedinger equation with radiation decay and no collisional relaxation, outside the framework of the usual perturbation theory. The mathematical schemes employed include the resonance approximation, exact methods, the quasi-energy method, the adiabatic approximation, etc. A large number of physical phenomena, such as oscillating and inverted populations, self-induced resonances, processes of first order in the weak field, Raman scattering, coherent population transfer, etc. are considered. The review is intended for a broad readership.
The resonance fluorescence spectrum of linear and nonlinear oscillators driven by a monochromatic field of frequency ${\ensuremath{\omega}}_{0}$, near the resonant frequency \ensuremath{\omega}, is calculated both classically and quantum mechanically. The result for the linear oscillator is the same in both calculations: a \ensuremath{\delta} function at ${\ensuremath{\omega}}_{0}$. For the nonlinear oscillator, however, the classical and quantum-mechanical spectra near resonance are qualitatively different: Classically, a \ensuremath{\delta} function at ${\ensuremath{\omega}}_{0}$ is obtained; quantum mechanically, a weak-field calculation leads to a \ensuremath{\delta} function at ${\ensuremath{\omega}}_{0}$ together with finite-width peaks nearby. An intuitive explanation for the difference is suggested by the examination of the quantum-mechanical energy-correlation function for the linear oscillator. This function describes energy oscillations with frequencies within a band centered at ${\ensuremath{\omega}}_{0}$-\ensuremath{\omega}. The oscillations can be interpreted as the beating of the driven oscillation with those produced by the vacuum fluctuations (the latter having a frequency spread of the order of the relaxation constant). Because of the nonlinearity, the energy oscillation modulates the effect of the driving field to produce sidebands. Higher-order processes produce additional peaks. A heuristic picture of vacuum fluctuations is developed.
The present paper treats the electromagnetically induced transparency (EIT), i.e., the absorption spectrum of a weak probe field, near-resonant to a material transition, in the presence of a strong amplitude-phase fluctuating field which couples the upper level of the transition to another excited level. The coupling field is assumed to be chaotic (i.e., a complex Gaussian process with a Lorentzian spectrum) and so strong that the material relaxation and the detuning can be neglected. In addition, we consider the Fourier transform of the spectrum
Uaa(t), which determines absorption of a probe field with a time-dependent intensity. The quantity
Uaa(t) equals a diagonal element of the average reduced evolution operator of the two-level system (TLS), coupled by the strong field. As a starting point, we use the stationary, partially averaged stochastic Schr\"odinger equation, which has the form of a set of two second-order differential equations. The theory is developed by combining analytical and numerical approaches. For the most important regime, when the rms Rabi frequency
V0 is much greater than the coupling-field bandwidth $\ensuremath{\nu},$ we obtain scaling relations for the EIT line shape and the TLS evolution. In particular, the absorption at the minimum (i.e., at the resonance frequency) scales as ${\ensuremath{\nu}}^{1/3}{V}_{0}^{\ensuremath{-}4/3},$ whereas the evolution function
Uaa(t) performs damped oscillations with the decay rate on the order of ${(V}_{0}^{2}\ensuremath{\nu}{)}^{1/3}.$ These results differ significantly from those following from the uncorrelated-jump model. The results of this paper are shown to be applicable to more general models of amplitude-phase fluctuations than the Gaussian-Markovian chaotic field.
The resonant scattering of photons from a perturbed two-level system is studied in detail. Adiabatic perturbations are approximated by a randomly fluctuating term in the level splitting. Off-diagonal perturbations are accounted for in the linewidth and the time dependence of the correlation functions. Explicit expressions for the cross section are obtained in the statistical and motional-narrowing limits. The detailed-balancing symmetry of the scattering is established. Minor modifications of the approximate expressions for the cross sections are proposed in order to preserve this symmetry. The effects of off-diagonal perturbations, which couple the atom to a thermal bath, are investigated at length for a model interaction.