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Bulletin de la Classe des sciences

Physical energies and lifetimes for a discrete n-level system interacting with radiation

Françoise Hénin

Published Jan 1, 1982• 0 citations
Publisher

Abstract

We derive dispersion equations for the energies and lifetimes in physical representations for a discrete «-level system interacting with radiation. Although the transformation leading from bare to physical representation is not unique, we show that these quantities are intrinsic quantities, the same in all physical representations. Moreover, for a simple case (3 level, RWA, states involving no photons), we show that our results are identical with those obtained by the Green's function method.

DOI
10.3406/barb.1982.57211

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We derive dispersion equations for the energies and lifetimes in physical representations for a discrete «-level system interacting with radiation. Although the transformation leading from bare to physical representation is not unique, we show that these quantities are intrinsic quantities, the same in all physical representations. Moreover, for a simple case (3 level, RWA, states involving no photons), we show that our results are identical with those obtained by the Green's function method.
The interaction is considered between NNN stationary two-level atoms and a radiation field described by a single cavity mode. The state vector for the complete system of atoms plus radiation is expressed as a linear superposition of states constructed from a product of photon states in the nnn representation and products of Pauli spin eigenstates describing all combinations of atoms in the lower and upper energy levels. Equations for the corresponding probability amplitudes are derived by substituting this superposition into the Schr\"odinger equation. The resulting equations are combined into bilinear form and phenomenological damping contributions are added. After the neglect of certain of the bilinear quantities, a master equation is derived which governs the probability Pmn{{P}_{m}}^{n}Pm​n of having mmm atoms in the lower level and nnn photons in the mode. This master equation takes account of multiple single-quantum absorption and emission processes but not of simultaneous multiple processes involving two or more atoms at a time. The equation which governs the expected number of photons 〈n〉〈n〉〈n〉 derived from the master equation bears a close resemblance to a rate equation. The effect of radiation loss from the cavity is incorporated into the master equation. Numerical calculations for a QQQ-spoiled laser show that the statistics of the number of photons in the mode bear a qualitative resemblance to Poisson statistics.
The meaning of perturbed states in the kinetic theory is first discussed. It is shown that the canonical form of the density matrix for a particle in a perturbed level does not correspond to a pure case, in the bare particle representation. The problem of spontaneous emission by an atom in its unperturbed excited state is then discussed. We first show that the results of the kinetic theory can also be obtained by the level shift operator formalism if one goes beyond the usual approximation. It is also shown that these results give a modified Lorentzian shape. Finally, we consider the resonant scattering of a photon. The solution of the kinetic equation gives us the transition probability per unit time. The frequency dependence is given by a modified Lorentz function, analogous to that obtained for the spontaneous emission line.
A model for interacting single-mode electromagnetic radiation and matter consisting of NNN two-level particles is solved exactly using a representation in which the creation and destruction operators for the field act by multiplication and differentiation, respectively. The model includes off-resonant interactions. Closed-form expressions can be given for some special cases. The solutions are transformed to the more familiar Fock and Schr\"odinger representations. Numerical results are given to indicate the behavior of the solutions. An equation for a generating function is derived which would give solutions of the (algebraic) matrix equation to which the problem has been reduced.
In Chap. 1 we introduced the Schrodinger equation and gave solutions for a number of simple cases. The general solution was then obtained by a linear superposition of these solutions. In our work, we are particularly concerned with the way in which atomic systems interact with electromagnetic radiation. These systems are still described by wave functions given by superpositions of simple solutions, but the expansion coefficients Cn (probability amplitudes) become functions of time. In this chapter we derive equations of motion for the expansion coefficients using Schrddinger’s equation and introduce the electric-dipole interaction energy. Then, specializing to a two-level system and a monochromatic electromagnetic field, we solve the equations of motion both by time-dependent perturbation theory and by the more exact Rabi method.