TITLE: Quantum Master equation for a system of charged fermions interacting
             with electromagnetic field

ABSTRACT:
 

Quantum master equations represent an essential tool for describing the interplay between dissipative and field-induced processes in a matter-field system. In the long history of this investigation domain, a crucial event was the deduction by Lindblad of a dynamical semigroup generator, and the connection of this generator with the previous phenomenological descriptions performed by Sandulescu and Scutaru. In this description, the quantum-mechanical properties of the density matrix (hermiticity, trace-class and positivity) are preserved during the whole evolution of a dissipative system. However, in this approach, the results are drastically restricted due to the phenomenological parameters of the equation. Recently, important efforts have been devoted to microscopic dissipative models. Along this line, a master equation for a harmonic oscillator in a blackbody radiation field has been derived by Ford, Lewis and O'Connell, and more general equations have been proposed by other authors as Gao, Wiseman and Munro, and Vacchini. However, as Ford and O'Connell pointed out, only the first equation of Ford, Lewis and O'Connell that is in agreement with the phenomenological equation of Sandulescu and Scutaru simultaneously satisfies the positivity and the detailed balance principle. A more general equation satisfying these conditions has only very recently been obtained by Xu, Yan, and Li.

Here, a master equation with fermionic operators and microscopic coefficients is proposed. In comparison with the master equations mentioned above, this equation satisfies the condition of a dynamic detailed balance, leading to Pauli master equations for the energy level populations. More than that, it includes also other physical effects: the self-consistent field of the environment particles, second-order correlations, and non-Markovian effects. The general properties including theprinciples of entropy and of the detailed balance are discussed in detail. The new equation is particularised for a harmonic oscillator, and is compared with the equation of Ford, Lewis and O'Connell or of Sandulescu and Scutaru that does not satisfy the principle of the dynamic detailed balance. Bloch-Feynman equations are derived for the pseudo-spin mean values and the energy transfer between system and environment is analyzed. We find conditions when the environment is cooled while the coherent electromagnetic field is amplified. As a numerical example, the quantum well of a logic gate is considered. The width and the shift of the excitation peak are calculated.

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