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International Journal of Modern Physics E

THE COLLISIONAL DECAY OF A FERMI SYSTEM INTERACTING WITH A MANY-MODE ELECTROMAGNETIC FIELD

Eliade N. Stefanescu, A. Sãndulescu, W. Scheid

Published Feb 1, 2000• 8 citations
Publisher

Abstract

We consider a system of Z fermions coupled to a dissipative environment through a two-body potential. We represent the system in a basis of single-particle, two-particle, … Z-particle excitated states. Using a procedure for averaging the rapid oscillations of the reduced density matrix in the interaction picture, the master equation of the system takes the form of a series expansion of powers of the dissipative potential matrix elements. The term of the second-order describes single-particle transitions, while the higher-order terms correspond to correlated transitions of the system particles. For the second- and the third-order terms, we derive microscopic expressions of the dissipative coefficients. For dissipative systems, when the state collectivity is broken into pieces through quantum diffusion, we use the quantum master equation of the second-order approximation. This equation satisfies basic physical conditions: particle conservation, Fermi–Dirac or Bose–Einstein distributions as asymptotic solutions of the populations, and entropy increase. On this basis, the decay of a Fermi system interacting with a many-mode electromagnetic field is described in terms of microscopic quantities: the matrix elements of the dissipative potential, the densities of the environment states, and the occupation probabilities of these states. A near-dipode–dipode interaction of the system with other neighbouring systems is taken into account. In addition to the coupling of the polarization with the population, included in the usual equations for two-level systems as a non-linear detuning, in equations for N-level systems two new couplings of the polarizations appear: a coupling due to the proximity potential, and a coupling due to the local field corrections, as a renormalization of the Rabi frequencies.

DOI
10.1142/s0218301300000039

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We consider a system of Z fermions coupled to a dissipative environment through a two-body potential. We represent the system in a basis of single-particle, two-particle, … Z-particle excitated states. Using a procedure for averaging the rapid oscillations of the reduced density matrix in the interaction picture, the master equation of the system takes the form of a series expansion of powers of the dissipative potential matrix elements. The term of the second-order describes single-particle transitions, while the higher-order terms correspond to correlated transitions of the system particles. For the second- and the third-order terms, we derive microscopic expressions of the dissipative coefficients. For dissipative systems, when the state collectivity is broken into pieces through quantum diffusion, we use the quantum master equation of the second-order approximation. This equation satisfies basic physical conditions: particle conservation, Fermi–Dirac or Bose–Einstein distributions as asymptotic solutions of the populations, and entropy increase. On this basis, the decay of a Fermi system interacting with a many-mode electromagnetic field is described in terms of microscopic quantities: the matrix elements of the dissipative potential, the densities of the environment states, and the occupation probabilities of these states. A near-dipode–dipode interaction of the system with other neighbouring systems is taken into account. In addition to the coupling of the polarization with the population, included in the usual equations for two-level systems as a non-linear detuning, in equations for N-level systems two new couplings of the polarizations appear: a coupling due to the proximity potential, and a coupling due to the local field corrections, as a renormalization of the Rabi frequencies.
In a previous paper, we derived a master equation for fermions, of Lindblad's form, with coefficients depending on microscopic quantities. In this paper, we study the properties of the dissipative coefficients taking into account the explicit expressions of: (a) the matrix elements of the dissipative potential, evaluated from the condition that, essentially, this potential induces transitions among the system eigenstates without significantly modifying these states, (b) the densities of the environment states according to the Thomas–Fermi model, and (c) the occupation probabilities of these states taken as a Fermi–Dirac distribution. The matrix of these coefficients correctly describes the system dynamics: (a) for a normal, Fermi–Dirac distribution of the environment population, the decays dominate the excitation processes; (b) for an inverted (exotic) distribution of this population, specific to a clustering state, the excitation processes are dominant.
We derive a quantum master equation for a system of femions coupled to the blackbody radiation field and to many other excitations of the surrounding particles. In comparison with other master equations for many-level systems, the new master equation depends on microscopic coefficients (as functions of physical constants, matrix elements, densitie of the environment states, and occupation probabilities of these states), and includes two additional terms: (1) a term depending on the selfconsistent field of the collective motion of the system or of other surrounding particles, and (2) a non-Markovian term.
An essential problem of the quantum information systems is the controllability and observability of the quantum states, generally described by Lindblad's master equation with phenomenological coefficients. However, in its general form, this equation describes a decay of the mean-values, but not necessarily the expected decaying transitions. We propose a quantum master equation with microscopic coefficients where these transitions are correctly described, and investigate the properties of the dissipative coefficients. Being of Lindblad's form, this equation preserves the quantum-mechanical properties of the density matrix during the whole evolution of the system and is in accordance with the entropy principle. On this basis, generalized Bloch-Feynman equations with microscopic coefficients for N-level systems are obtained.
We established a master equation for a system of fermions coupled by electric dipol interaction with the free electromagnetic field. This equation has a Lindblad equation form, with a hamiltonian part corresponding to the shell model and a dissipative part with microscopic coefficients depending on physical constants, matrix elements, and temperature. The dependence on the transition energy of the dissipative coefficients satisfies asymptothycal conditions for the energy transfer in full agreement with the principle of the detailed balance.
A dissipating system consisting of two strongly interacting subsystems, each coupled to its respective thermal reservoir is considered. The usual method for obtaining a master equation for the reduced density operator of such a system is to assume the factorization of this density operator in the derivation of the irreversible term, retaining the coupling in the reversible term alone. This approach has the serious limitation that it fails to preserve detailed balance in general and yields an incorrect canonical form for the stationary solution of the density operator. A general approach, based on the introduction of a new interaction picture, is presented. This method is illustrated by applying it to a coupled boson system and to a two-level atom interacting with a single mode of the electromagnetic field.
We investigate the long-time behavior of a model consisting of N two-level atoms in a lossless cavity. The Hamiltonian of our system contains the radiation oscillators in addition to the matter Hamiltonian and the usual ∫j·Adv interaction term. In order to treat the system perturbatively, it would be necessary to remove the tremendous degeneracy of the system. Since this is prohibitively difficult, and since we are interested in the long-time behavior of the system, we solve the quantum mechanical Liouville equation directly for a wide class of physically important initial distribution functions. We show the effective expansion parameter is γ̃Nγ̃ where γ̃ is a dimensionless atomic dipole moment and N is the number of atoms. In the lowest order we find the self-consistent field approximation. In the next order, particle-field correlations appear. We explicitly solve the equations of motion for the particle-field correlations in terms of the average quantities that appear in the self-consistent field approximation. We show the self-consistent field approximation consists of five first-order differential equations. Next we show the equations of motion for the density matrix of the system correct to order (γ̃Nγ̃)2 are equivalent to eight first-order differential equations. The three additional equations are needed to describe the three second moments of the density matrix of the electromagnetic field that appear in second order. Our lowest-order microscopic equations are equivalent to semiphenomenological theories and our higher-order equations contain only the measurable second-order moments of the electromagnetic field in addition to the variables that appear in semi-phenomenological theories.