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Physical Review A

Variational functions in degenerate open quantum systems

Matthias Jakob, Stig Stenholm

Published Apr 20, 2004• 18 citations
Publisher

Abstract

We have derived a Lyapunov functional for a degenerate open atomic system. This functional develops monotonically towards its stationary state. The open system is described by a Lindblad-type master equation. For the construction of the variational functional it is necessary that the Lindblad operator can be diagonalized. Since the generator of motion is non-Hermitian, diagonalization is, in general, only possible if the eigenvalues are nondegenerate. In this paper, we propose that in a physical system the biorthogonal eigenbasis of the Lindblad operator remains complete even when degeneracy is present. Thus diagonalization of the Lindblad operator, and consequently the construction of the variational functional, is still possible. We discuss the reasons and illustrate the theory of the variational functional for a driven $\ensuremath{\Lambda}$-type three-level atom with degenerate ground state. The degeneracy has interesting effects on the variational functional in the steady state with respect to its interpretation as an entropic quantity. In case of the driven three-level atom, the dark state turns out to be an isentropic state.

DOI
10.1103/physreva.69.042105

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We have derived a Lyapunov functional for a degenerate open atomic system. This functional develops monotonically towards its stationary state. The open system is described by a Lindblad-type master equation. For the construction of the variational functional it is necessary that the Lindblad operator can be diagonalized. Since the generator of motion is non-Hermitian, diagonalization is, in general, only possible if the eigenvalues are nondegenerate. In this paper, we propose that in a physical system the biorthogonal eigenbasis of the Lindblad operator remains complete even when degeneracy is present. Thus diagonalization of the Lindblad operator, and consequently the construction of the variational functional, is still possible. We discuss the reasons and illustrate the theory of the variational functional for a driven \ensuremathΛ\ensuremath{\Lambda}\ensuremathΛ-type three-level atom with degenerate ground state. The degeneracy has interesting effects on the variational functional in the steady state with respect to its interpretation as an entropic quantity. In case of the driven three-level atom, the dark state turns out to be an isentropic state.
We consider the Lindblad-type master equation of an open system. We address the question how to construct a functional of the quantum state which displays a monotonic behavior in time. This thus defines uniquely the direction of time in the system. As the generator of time evolution is not a Hermitian operator, the theory requires the considerations of right and left eigenstates. In this paper we assume them to form two complete bases, which allows us to construct the desired quantity. This can be interpreted as a generalized entropy functional. We show how the construction is carried out in the general case, and we illustrate the theory by solving the case of an externally driven and damped two-level system. The treatment is related to earlier work in the field, and its possible relation to time inversion is discussed.
We connect the theory of a Lyapunov functional for an open system with the thermophysical concept of the relative entropy or equivalently with the entropy production rate. In order to do this we consider the environment of the open system as a thermal reservoir, which introduces the temperature as a thermophysical quantity. The general theory of the Lyapunov functional of an open system constructs a metric operator which contains an element of ambiguity in its scaling. This ambiguity can be removed by assuming a thermostatistic environment which fixes the scaling of the metric operator in a unique manner. The Lyapunov functional thus acquires a thermophysical meaning and can be connected with a relative entropy. In this case the metric operator is related to the inverse of the stationary density operator of the system. We illustrate the theory on a two-level atomic system and a degenerate three-level atom which are exposed to a thermal electromagnetic environment.
A unified quantum-statistical framework is proposed for open three-level atomic systems when two lower-energy states are coupled to a single upper state. We suggest a non-Hermitian generalization of the closed model’s Hamiltonian by the analytical continuation of real-valued parameters related to the energies into a complex domain, as well as by considering some terms previously neglected by the procedure of Hermitian conjugation. We show that such systems can have two types of density operator’s evolution, depending on whether they are expected to be sustainable, or in a process of irreversible decay. This approach not only explains previously heuristically introduced terms in equations of motion for the density matrix but also suggests new terms which predict new dissipative effects. We then consider a special case, namely the evolution of pure states described by a generalized Schrödinger equation, which turns out to be nonlinear with respect to the atomic wavefunctions. As a final step toward unification, in addition to the non-Hermitian Hamiltonian generalization, we also include Markovian quantum jumps by adding the Lindblad dissipator. The resulting equations not only expand the range of applicability of open three-level models, but also suggest that dissipative phenomena in quantum-optical systems in vivo are more diverse and significant than previously believed.
The continuous-time differential Lyapunov equation is widely used in linear optimal control theory, a branch of mathematics and engineering. In quantum physics, it is known to appear in Markovian descriptions of linear (quadratic Hamiltonian, linear equations of motion) open quantum systems, typically from quantum master equations. Despite this, the Lyapunov equation is seldom considered a fundamental formalism for linear open quantum systems. In this work we aim to change that. We establish the Lyapunov equation as a fundamental and efficient formalism for linear open quantum systems that can go beyond the limitations of various standard quantum master-equation descriptions, while remaining of much less complexity than general exact formalisms. This also provides valuable insights for non-Hermitian quantum physics. In particular, we derive the Lyapunov equation for the most general number-conserving linear system in a lattice of arbitrary dimension and geometry, connected to an arbitrary number of baths at different temperatures and chemical potentials. Three slightly different forms of the Lyapunov equation are derived via an equation-of-motion approach, by making increasing levels of controlled approximations, without reference to any quantum master equation. Then we discuss their relation with quantum master equations, positivity, accuracy, and additivity issues, the possibility of describing dark states, general perturbative solutions in terms of single-particle eigenvectors and eigenvalues of the system, and quantum regression formulas. Our derivation gives a clear understanding of the origin of the non-Hermitian Hamiltonian describing the dynamics and separates it from the effects of quantum and thermal fluctuations. Many of these results would have been hard to obtain via standard quantum master-equation approaches.
We compare two approaches to open quantum systems, namely, the non-Hermitian dynamics and the Lindblad master equation. In order to deal with more general dissipative phenomena, we propose the unified master equation that combines the characteristics of both of these approaches. This allows us to assess the differences between them as well as to clarify which observed features come from the Lindblad or the non-Hermitian part, when it comes to experiment. Using a generic two-mode single-atom laser system as a practical example, we analytically solve the dynamics of the normalized density matrix operator. We study the two-level model in a number of cases (depending on parameters and types of dynamics), compute different observables and study their physical properties. It turns out that one can not only able to describe the different types of damping in dissipative quantum optical systems but also mimic the undamped anharmonic oscillatory phenomena which happen in quantum systems with more than two levels (while staying within the framework of the analytically simple two-mode approximation).
In this paper I investigate the usability of the characteristic functions for the description of the dynamics of open quantum systems focussing on non-Lindblad-type master equations. I consider, as an example, a non-Markovian generalized master equation containing a memory kernel which may lead to nonphysical time evolutions characterized by negative values of the density matrix diagonal elements [S. M. Barnett and S. Stenholm, Phys. Rev. A 64, 033808 (2001)]. The main result of the paper is to demonstrate that there exist situations in which the symmetrically ordered characteristic function is perfectly well-defined while the corresponding density matrix loses positivity. Therefore, nonphysical situations may not show up in the characteristic function. As a consequence, the characteristic function cannot be considered an alternative complete description of the non-Lindblad dynamics.