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arXiv:physics/0202001 · Physical Review A

Completely positive Bloch-Boltzmann equations

Robert Alicki, Stanisław Kryszewski

Published Jul 11, 2003• 26 citations
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Abstract

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.

DOI
10.1103/physreva.68.013809

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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.
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 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 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 authors consider a system of two-level atoms coupled to a radiation field and immersed in a buffer gas. The corresponding Bloch-like equations are augmented by collision operators stemming from the linear Boltzmann equation. Collision operators describe velocity-changing collisions that give rise to various kinetic effects of light. They study the general properties of collision operators, the main one consisting of the fact that such operators are Hermitian in a suitably defined space of velocity distribution functions They introduce a set of functions that serve as approximations to the eigenfunctions. The eigenvalues are connected with transport coefficients (they give an example of such a relation), but they keep eigenvalues as free parameters of the approach. In this way the authors can describe the whole class of collision operators, the Keilson-Storer model being a particular example. Employing the introduced eigenfunctions they derive and formally solve the equations of motion for the expansion coefficients of the elements of the density matrix of active atoms.
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.
The master equation for a two-level atom driven by a strong classical field and damped into a tailored reservoir with nonflat density of modes is derived under the Born-Markov approximation. To derive the master equation, the dressing transformation on the atomic operators is performed first, and, next, the dressed operators are coupled to the reservoir and the corresponding damping rates are calculated. The effects of a strong field and/or structured reservoir are seen as nonstandard terms in the master equation, some of which are reminiscent of terms known for squeezed vacuum reservoirs. The master equation leads to the generalized optical Bloch equations that can easily be solved for the steady state and, together with the quantum regression theorem, allow for analytical expressions for the fluorescence, as well as absorption spectra.