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Journal of Physics B Atomic and Molecular Physics

A quantum-mechanical master equation treatment of the dynamical Stark effect

H. J. Carmichael, Daniel F. Walls

Published Jun 1, 1976• 397 citations
Publisher

Abstract

The description begins with an operator master equation for the atom plus incident field. Reduced atomic matrix elements are derived for arbitrary field strengths. First- and second-order correlation functions in the scattered field are also obtained and discussed in relation to the scattered spectrum and intensity-fluctuation measurements. This formalism has the appealing feature that all information is readily available from the one set of four coupled equations. The deficiencies in both the one-photon approximation and the semiclassical perspective are established in a natural and transparent fashion.

DOI
10.1088/0022-3700/9/8/007

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The description begins with an operator master equation for the atom plus incident field. Reduced atomic matrix elements are derived for arbitrary field strengths. First- and second-order correlation functions in the scattered field are also obtained and discussed in relation to the scattered spectrum and intensity-fluctuation measurements. This formalism has the appealing feature that all information is readily available from the one set of four coupled equations. The deficiencies in both the one-photon approximation and the semiclassical perspective are established in a natural and transparent fashion.
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We have solved a master equation to obtain the diagonal elements of the density matrix for laser light. The master equation represents a generalization of a master equation derived earlier for a single radiation mode interacting with NNN stationary two-level atoms. The generalization takes into account the pumping scheme which characterizes a three-level laser. For positive temperatures and a lossless cavity, the equilibrium solution of the master equation gives the correct statistical-mechanical description of the atoms and radiation. For negative temperatures, the distribution function for the number of photons in the mode is either exponential or peaked, depending on whether the laser is operating below or above threshold. The calculated intensity fluctuations are in good agreement with semiclassical results for lasers operating slightly above threshold.
We have solved a master equation to obtain the diagonal elements of the density matrix for laser light. The master equation represents a generalization of a master equation derived earlier for a single radiation mode interacting with NNN stationary two-level atoms. The generalization takes into account the pumping scheme which characterizes a three-level laser. For positive temperatures and a lossless cavity, the equilibrium solution of the master equation gives the correct statistical-mechanical description of the atoms and radiation. For negative temperatures, the distribution function for the number of photons in the mode is either exponential or peaked, depending on whether the laser is operating below or above threshold. The calculated intensity fluctuations are in good agreement with semiclassical results for lasers operating slightly above threshold.
Introduction In this chapter we will turn to the arena of quantum optics for illustrative examples of the use of the master equation discussed in Chapters 3 and 6. In fact, quantum optics examples have already been utilized in Chapter 2 as introduction to the density matrix. The principal focus here will be quantum damping in these systems, that is, the damping effect on an atom in interaction with the electromagnetic field as a reservoir. Damping is discussed extensively in Chapter 17 in connection with decay-scattering systems. For this system general phase space distribution functions will be reexamined. To some degree, this has already been done in Chapter 2 with the introduction of the Glauber–Sudarshan P (α a * ) function. The micromaser will be discussed as a modern and interesting example of the dynamic interaction of an atom with an electromagnetic cavity not in equilibrium. For use of the student, an appendix to this chapter will briefly review the quantization of the free electromagnetic field and its atomic interaction. There is no possibility of reviewing this extensive and growing field here. Our desire in this chapter is to connect the general topics of this book to this example. The books of Louisell (1973) and Scully and Zubairy (1997) are excellent. We are also indebted to the work of Nussenzweig, Schleich, and Mandel and Wolf (Nussenzweig, 1973; Mandel and Wolf, 1995; Schleich, 2001). We also recall the fine early introduction to this topic by Agarwal (1973). Atomic damping: atomic master equation In this section we shall consider the so-called quantum optics master equation for the reduced atomic density operator, ρ A ( t ), in the Born approximation. The elements of this derivation have already been discussed in Chapter 3 with the derivation of the Pauli equation for 〈α| ρ A |α〉. Here the generalization to off-diagonal contributions only adds complication.
The effect of correlations between an absorber atom and perturbers in the binary-collision approximation are applied to degenerate atomic systems. A generalized absorption profile which specifies the final state of the atom after an absorption event is related to the total intensities of Rayleigh scattering and fluorescence from the atom. It is suggested that additional dynamical information to that obtainable from ordinary absorption experiments is required in order to describe redistributed atomic radiation. The scattering of monochromatic radiation by a degenerate atom is computed in a binary-collision approximation; an equation of motion is derived for the correlation function which is valid outside the quantum-regression regime. Solutions are given for the weak-field conditions in terms of generalized absorption and emission profiles that depend on the indices of the atomic multipoles.