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Journal of Physics A Mathematical and General

Liouville space theory of sequential quantum processes. II. Application to a system with an internal reservoir

Bryan J. Dalton

Published Jul 1, 1982• 7 citations
Publisher

Abstract

For pt.I see ibid., vol.15, p.2157 (1982). A system consisting of discrete states and continuum states (which form a so-called internal reservoir) is treated, illustrating the theory of sequential quantum processes in Liouville space developed in the preceding paper. The populations and coherences associated with the discrete states satisfy Markovian master equations when the interaction matrix elements between discrete and continuum states are significant over a broad band of continuum states. The population of a single discrete state decays exponentially with time, whilst the population of two coupled discrete states (one only coupled to the continuum states) may exhibit Rabi oscillations. For the latter case of two coupled discrete levels, the population of particular continuum states approaches a two-peak form for long times (Autler-Townes splitting).

DOI
10.1088/0305-4470/15/7/027

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For pt.I see ibid., vol.15, p.2157 (1982). A system consisting of discrete states and continuum states (which form a so-called internal reservoir) is treated, illustrating the theory of sequential quantum processes in Liouville space developed in the preceding paper. The populations and coherences associated with the discrete states satisfy Markovian master equations when the interaction matrix elements between discrete and continuum states are significant over a broad band of continuum states. The population of a single discrete state decays exponentially with time, whilst the population of two coupled discrete states (one only coupled to the continuum states) may exhibit Rabi oscillations. For the latter case of two coupled discrete levels, the population of particular continuum states approaches a two-peak form for long times (Autler-Townes splitting).
The theory of sequential quantum processes has been extended to Liouville space via the use of non-Hermitian projection operators in order to treat the evolution of the quantum density operator and to enable physically important matrix elements of the density operator to be calculated. The formal relationship of master equation methods to the theory of sequential quantum processes is established, and a new set of coupled master equations is derived. Special choices of projection operators lead to further simplification of the results. The Markoff approximation is also examined.
Non-Markovian behaviour in atomic systems coupled to a structured reservoir of quantum EM field modes, such as in high Q cavities, is treated using a quasimode description, and the pseudo mode theory for single quantum reservoir excitations is obtained via Fano diagonalisation. The atomic transitions are coupled to a discrete set of (cavity) quasimodes, which are also coupled to a continuum set of (external) quasimodes with slowly varying coupling constants. Each pseudomode corresponds to a cavity quasimode, and the original reservoir structure is obtained in expressions for the equivalent atom-true mode coupling constants. Cases of multiple excitation of the reservoir are now treatable via Markovian master equations for the atom-discrete quasimode system.
This paper deals with non-Markovian behavior in atomic systems coupled to a structured reservoir of quantum electromagnetic field modes, with particular relevance to atoms interacting with the field in high-Q cavities or photonic band-gap materials. In cases such as the former, we show that the pseudomode theory for single-quantum reservoir excitations can be obtained by applying the Fano diagonalization method to a system in which the atomic transitions are coupled to a discrete set of (cavity) quasimodes, which in turn are coupled to a continuum set of (external) quasimodes with slowly varying coupling constants and continuum mode density. Each pseudomode can be identified with a discrete quasimode, which gives structure to the actual reservoir of true modes via the expressions for the equivalent atom-true mode coupling constants. The quasimode theory enables cases of multiple excitation of the reservoir to now be treated via Markovian master equations for the atom-discrete quasimode system. Applications of the theory to one, two, and many discrete quasimodes are made. For a simple photonic band-gap model, where the reservoir structure is associated with the true mode density rather than the coupling constants, the single quantum excitation case appears to be equivalent to a case with two discrete quasimodes.
Studies the coupling of N discrete levels to a common continuum. The discrete levels, which can include both genuine bound states and autoionisation states, are assumed to be coupled to the continuum and to each other in such a way as to permit a rotating wave approximation in the Schrodinger equation. A general solution for the model is derived with the aid of Laplace transforms and the Weisskopf-Wigner approximation. The author discusses general properties of the discrete level populations and the electron spectrum like population trapping, zeros, crossings, resonances and confluences. As examples the author treats cases where two or three discrete levels interact with a continuum.
In order to describe the evolution of a quantum system that is coupled to a reservoir, a non-phenomenological Kraus map is constructed. At time zero, system and reservoir are not entangled. In the perturbative series for the density operator of the system all reservoir correlation functions are factorised into products of pair-correlation functions. This allows for a resummation of the perturbative series up to infinite order. The density operator can be expressed in terms of an auxiliary system operator that satisfies an analytically tractable integral equation. Hence, the difficulties caused by integral kernels of Nakajima-Zwanzig type are circumvented. Assuming an interaction between system and reservoir of the Jaynes-Cummings form, one shows that the Kraus map is capable of generating Rabi oscillations of a two-level atom. If the reservoir is a continuum, the Kraus map reproduces the Wigner-Weisskopf theory of spontaneous emission.