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Optics and Spectroscopy

Derivation of the Kinetic Equation for the Density Matrix of Two-Level Particles by the Green’S Function Method

Е. А. Титов

Published Jun 29, 2004• 6 citations
Publisher

Abstract

On the basis of the Green’s function method for nonequilibrium systems, the kinetic equation for the density matrix of two-level particles used in laser spectroscopy is derived. The assumptions made in deriving the kinetic equation impose certain restrictions on this equation. In the binary approximation, all diagrams are expressed via the reducible vertex part, which can be expressed in terms of the amplitude of scattering of a two-level atom by a buffer gas.

DOI
10.1134/1.1771421

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On the basis of the Green’s function method for nonequilibrium systems, the kinetic equation for the density matrix of two-level particles used in laser spectroscopy is derived. The assumptions made in deriving the kinetic equation impose certain restrictions on this equation. In the binary approximation, all diagrams are expressed via the reducible vertex part, which can be expressed in terms of the amplitude of scattering of a two-level atom by a buffer gas.
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Green's function method is applied to the description of nuclear levels-- superposition of the 1p1h and 2p2h configurations. Irreducible amplitudes for the transitions 1p1h yields 1p1h, 1p1h yields 2p2h, 2p2h yields 1p1h, 2p2h yields 2p2h are introduced to obtain the exact equations for eigerfunctions and eigenvalues of the problem-. The renormalization of the equations made it possible to consider all the processes near the Fermi level using the phenomenological effective amplitudes of quasipanticle interaction. The comparison is made with the generalized random-phase method. The approximations are discussed leading in the Green's function method to the equations for one- and two-particle density matrices as the equations obtained in the random--phase approximation. (auth)
Introduction In this and the next two chapters we lay down the basis to go beyond the Hartree–Fock approximation. We develop a powerful approximation method with two main distinctive features. The first feature is that the approximations are conserving . A major difficulty in the theory of nonequilibrium processes consists in generating approximate Green's functions yielding a particle density n (x, t ) and a current density J (x, t ) which satisfy the continuity equation, or a total momentum P ( t ) and a Lorentz force F ( t ) which is the time-derivative of P ( t ), or a total energy E ( t ) and a total power which is the time-derivative of E ( t ), etc. All these fundamental relations express the conservation of some physical quantity. In equilibrium the current is zero and hence the density is conserved, i.e., it is the same at all times; the force is zero and hence the momentum is conserved; the power fed into the system is zero and hence the energy is conserved, etc. The second important feature is that every approximation has a clear physical interpretation. In Chapter 7 we showed that the two-particle Green's function in the Hartree and Hartree–Fock approximation can be represented diagrammatically as in Fig. 7.1 and Fig. 7.2 respectively. In these diagrams two particles propagate from (1′, 2′) to (1, 2) as if there was no interaction.