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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Study on the fermion systems coupled by electric dipol interaction with the free electromagnetic field

Eliade N. Stefanescu, Andreea Rodica Sterian, Paul E. Sterian

Published Jun 7, 2005• 11 citations
Publisher

Abstract

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.

DOI
10.1117/12.633686

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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 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.
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.
We consider a pair of dipoles such as Rydberg atoms for which direct electrostatic dipole–dipole interactions may be significantly larger than the coupling to transverse radiation. We derive a master equation using the Coulomb gauge, which naturally enables us to include the inter-dipole Coulomb energy within the system Hamiltonian rather than the interaction. In contrast, the standard master equation for a two-dipole system, which depends entirely on well-known gauge-invariant S -matrix elements, is usually derived using the multipolar gauge, wherein there is no explicit inter-dipole Coulomb interaction. We show using a generalised arbitrary-gauge light-matter Hamiltonian that this master equation is obtained in other gauges only if the inter-dipole Coulomb interaction is kept within the interaction Hamiltonian rather than the unperturbed part as in our derivation. Thus, our master equation depends on different S -matrix elements, which give separation-dependent corrections to the standard matrix elements describing resonant energy transfer and collective decay. The two master equations coincide in the large separation limit where static couplings are negligible. We provide an application of our master equation by finding separation-dependent corrections to the natural emission spectrum of the two-dipole system.
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.