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Physical Review

Atomic Relaxation in the Presence of a Coherent Optical Field

E. A. Sziklas

Published Dec 10, 1969• 10 citations
Publisher

Abstract

Atomic relaxation in the presence of a strong coherent optical field is examined. Contrary to popular practice, we find that the Bloch equations do not apply when the applied field is strong enough to saturate the transition. The component of the induced polarization in phase with the field lives longer than the out-of-phase component. One important consequence is that pulse durations for adiabatic inversion of atomic levels need not be short relative to the transverse relaxation time of the atom if the pulse amplitude is sufficiently large.

DOI
10.1103/physrev.188.700

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Atomic relaxation in the presence of a strong coherent optical field is examined. Contrary to popular practice, we find that the Bloch equations do not apply when the applied field is strong enough to saturate the transition. The component of the induced polarization in phase with the field lives longer than the out-of-phase component. One important consequence is that pulse durations for adiabatic inversion of atomic levels need not be short relative to the transverse relaxation time of the atom if the pulse amplitude is sufficiently large.
A solution is presented for the damped optical Bloch equations under the excitation of a smooth pulse by first deriving three independent third-order equations of the Bloch vector components. Each equation is reduced to quadratures by assuming that the logarithmic time derivative of the field amplitude is small compared to the Rabi frequency. This results in an approximate summation of the infinite-order time-dependent perturbation in the field amplitude. The relaxation-dependent induced damping of the population inversion is calculated. Also calculated are additional relaxation-dependent contributions to the intensity-dependent refractive index. The time-integrated intensity contribution tends to cause line asymmetry, which becomes, at later times, linear in ${\ensuremath{\gamma}}_{2}$ when ${\ensuremath{\gamma}}_{2}\ensuremath{\gg}{\ensuremath{\gamma}}_{1}$ and zero when $2{\ensuremath{\gamma}}_{2}={\ensuremath{\gamma}}_{1}$, where ${\ensuremath{\gamma}}_{1}$ and ${\ensuremath{\gamma}}_{2}$ are the atomic energy and phase-changing relaxations, respectively. The dependence of the spectral broadening on pulse length, pressure, and length of the sample is discussed.
Equations are derived for the atomic density matrix and relaxation operator for a broadband squeezed field in an arbitrary polarization state and resonance atomic energy levels with an arbitrary degree of degeneracy. It is shown that suppression of the relaxation of the quadrature component of the atomic polarization depends strongly on the type of resonance transition and the polarization state of the squeezed and coherent perturbing fields. When the resonance levels are strongly degenerate, the relaxation of the quadrature component of the atomic polarization under conditions of maximum suppression is nonexponential in character. The mathematical apparatus developed here makes it possible to calculate polarization-related aspects of the multifrequency optical behavior of atomic and molecular systems resonantly excited both by coherent light and by broadband squeezed fields.
In a recent experiment [ DevoeR. G.BrewerR. G., Phys. Rev. Lett.50, 1269 ( 1983)], it was found that the optical Bloch equations could not satisfactorily explain the signal that was observed for free-induction decay in the impurity ion cyrstal Pr3+:LaF3. Several theories have been proposed to explain this failure of the Bloch equations. In this paper, the general validity conditions for the optical Bloch equations are examined within the limits of a Markovian relaxation model. The specific problem to be considered is the interaction of an optical field with two-level atoms. The atoms undergo relaxation as a result of coupling to a perturber bath that, itself, is negligibly affected by the relaxation process. First, a simple decay-parameter model is assumed for the relaxation of atomic density-matrix elements. Such a model is found to lead to a set of generalized Bloch equations of which the conventional Bloch equations form a subset. Subsequently, more-realistic models for relaxation in both vapors and solids are considered within the limits of the impact approximation (i.e., the duration of a fluctuation can be viewed as instantaneous with respect to all relevant time scales in the problem). It is found that, even in the impact (Markovian) approximation, the generalized Bloch equations cannot be expected to provide an adequate description of relaxation, owing to effects in which a fluctuation-induced change in the atomic transition frequency persists between fluctuations. In vapors this frequency shift is produced by velocity-changing collisions that change the atomic resonance frequency (as seen in the laboratory frame), whereas in solids it is produced by local-field fluctuations. The limiting conditions under which one can expect both the generalized and the conventional Bloch equations to retain their validity are explored.
The response of a two-level atom in a strong polychromatic field composed of a large number of equidistant frequency components is investigated. We calculate numerically, as well as analytically, the stationary population inversion and show that the saturation of the atomic transition strongly depends on whether or not there is a central (resonant) frequency component in the driving field. We find that, in the presence of the central component, the atom can remain in the ground state even for a strong Rabi frequency of the driving field. In addition, we find that the inversion is sensitive to the relative phase between the frequency components. When the central component is suppressed, the atomic transition saturates with the Rabi frequency independent of the relative phase.