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Physical Review A

Positivity of Bloch-Boltzmann equations

Stanisław Kryszewski, Justyna Czechowska

Published Aug 18, 2006• 4 citations
Publisher

Abstract

In a large variety of spectroscopical applications Bloch-Boltzmann equations (BBE) play an essential role. They describe the evolution of the reduced-density operator of an active atom, which is coupled to radiation (Bloch part) and which interacts collisionally with the perturber gas (Boltzmann part). The standard approach to the collisional part is well known from the literature. It preserves hermiticity and normalization, but the question of whether it preserves positivity seems to remain open. The completely positive BBE were recently derived via the general master-equation techniques. These two approaches are applied for a model of $n$-level nondegenerate atom. We show that within this model both approaches to the collisional part of BBE are equivalent (give the same physical predictions). The approach based upon master-equation techniques guarantees the preservation of hermiticity, normalization, and positivity. The proven equivalence ascertains that the standard approach also preserves positivity. Moreover, some aspects of the standard derivation (which atomic states do contribute to the evolution) are clarified by the established equivalence.

DOI
10.1103/physreva.74.022719

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In a large variety of spectroscopical applications Bloch-Boltzmann equations (BBE) play an essential role. They describe the evolution of the reduced-density operator of an active atom, which is coupled to radiation (Bloch part) and which interacts collisionally with the perturber gas (Boltzmann part). The standard approach to the collisional part is well known from the literature. It preserves hermiticity and normalization, but the question of whether it preserves positivity seems to remain open. The completely positive BBE were recently derived via the general master-equation techniques. These two approaches are applied for a model of nnn-level nondegenerate atom. We show that within this model both approaches to the collisional part of BBE are equivalent (give the same physical predictions). The approach based upon master-equation techniques guarantees the preservation of hermiticity, normalization, and positivity. The proven equivalence ascertains that the standard approach also preserves positivity. Moreover, some aspects of the standard derivation (which atomic states do contribute to the evolution) are clarified by the established equivalence.
In a large variety of spectroscopical applications Bloch-Boltzmann equations (BBE) play an essential role. They describe the evolution of the reduced density operator of an active atom which is coupled to radiation (Bloch part) and which interacts collisionally with the perturber gas (Boltzmann part). The standard approach to the collisional part is well-known from the literature. It preserves hermiticity and normalization, but the question whether it preserves positivity seems to remain open. The completely positive BBE were recently derived via the general master equation techniques. These two approaches are applied for a model of n-level nondegenerate atom. We show that within this model both approaches to the collisional part of BBE are equivalent -- give the same physical predictions. The approach based upon master equation techniques guarantees the preservation of hermiticity, normalization and positivity. The proven equivalence ascertains that the standard approach also preserves positivity. Moreover, some aspects of the standard derivation (which atomic states do contribute to the evolution) are clarified by the established equivalence.
In our previous paper [Phys. Rev. A 74, 022719 (2006)] we have analyzed Bloch-Boltzmann equations which govern the evolution of the reduced density operator of an active atom with nondegenerate energy levels. These equations describe the coupling to radiation (Bloch part) and collisional interaction with much denser perturber gas (Boltzmann part). We have shown that the standard approach to the collisional part, which is well known from the literature, is equivalent to the results obtained via master equation techniques. In the present paper we perform a similar comparison, but for an atom with explicitly degenerate energy levels. Master equation guarantees that hermiticity, normalization, and positivity of an atomic density operator are preserved. The equivalence previously found does not hold in this more general case. We show that the standard approach still preserves hermiticity of the atomic density operator. However, preservation of normalization and positivity is strongly doubtful. We argue that in practical application it is safer to use the master equation version of the Bloch-Boltzmann equations.
The density operator of the arbitrary physical system must be positive definite. Employing the general master equation technique which preserves this property, we derive equations of motion for the density operator of an active atom which interacts collisionally with the reservoir of perturber atoms. The obtained general relations applied to the two-level atom yield Bloch-Boltzmann equations (BBE) which, as it is the case with master equation approach, are linear in the matrix elements of the active-atom density operator. The obtained BBE guarantee that positivity is preserved, which needs not to be the case with the results known from literature. We argue that our results are the correct ones and as such should be used in practical applications. Moreover, the structure and the terms which appear in our set of BBE seem to allow simpler and more straightforward physical interpretation.
Master equations play a pivotal role in investigating open quantum systems. In particular, the Bloch-Redfield equation stands out due to its relation to a concrete physical environment. However, without further approximations it does not lead to a Lindblad master equation that guarantees that the density matrix stays completely positive, which has raised some concerns regarding its use. This study builds on previous efforts to transform the Bloch-Redfield framework into a mathematically robust Lindblad equation while fully preserving the effects that are lost within the secular approximation that is commonly used to guarantee positivity. These previous approaches introduced two potential deficiencies: The environment-induced energy shift can be non-Hermitian and some decay rates can be negative, violating the assumptions of Lindblad's theorem. Here we propose and evaluate straightforward solutions to both problems. Our approach offers an effective and general procedure for obtaining a Lindblad equation, derived from a concrete physical environment, while mitigating the unphysical dynamics present in the Bloch-Redfield equation.
Positivity preservation is naturally guaranteed in exact non-Markovian master equations for open quantum system dynamics. However, in many approximated non-Markovian master equations, the positivity of the reduced density matrix is not guaranteed. In this paper we provide a general class of time-local, perturbative, and positivity-preserving non-Markovian master equations generated from stochastic Schr\"odinger equations, particularly quantum-state-diffusion equations. Our method has an expanded range of applicability for accommodating a variety of non-Markovian environments. We show the positivity-preserving master equation for a three-level system coupled to a dissipative bosonic environment as a particular example to exemplify our general approach. We illustrate the numerical simulations with an analysis explaining why the previous approximated non-Markovian master equations cannot guarantee positivity. Our work provides a consistent master equation for studying the non-Markovian dynamics in ultrafast quantum processes and strong-coupling systems.
This tutorial article provides a detailed mathematical overview of the Optical Bloch equation describing the interaction of the two level atom with the Electromagnetic radi- ation field. The equations have been derived in the weak atom field coupling limit. The notion of the optical Bloch equations has been introduced in a systematic process starting from the interaction of the two level atom with the semiclassical radiation field which cannot completely the idea of the Spontaneous emission process later which was included theoretically by,Weigner-Weisskopf theory of spontaneous emission .In the later part of the article the optical Bloch equations has been derived exactly by treating the radiation field quantum mechanically. The radiation field has been quantized using the free field quantization techniques in the Coloumb Gauge with which the Lindblad Master equation has been derived systematically in the weak coupling limit. The optical Bloch equations has been derived for the resonantly driven two level atom from the Lindblad type master equation and later those equations has been solved in the steady state limit. The article also discuss the analytical aspects of describing the theory of atom matter interaction using full-fledged quantum mechanical approach.