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arXiv:quant-ph/0512198

Bohm-Aharonov Type Effects in Dissipative Atomic Systems

A. I. Solomon, Sonia G. Schirmer

Published Dec 1, 2006• 0 citations
arXivHTMLPDF

Abstract

A state in quantum mechanics is defined as a positive operator of norm 1. For finite systems, this may be thought of as a positive matrix of trace 1. This constraint of positivity imposes severe restrictions on the allowed evolution of such a state. From the mathematical viewpoint, we describe the two forms of standard dynamical equations - global (Kraus) and local (Lindblad) - and show how each of these gives rise to a semi-group description of the evolution. We then look at specific examples from atomic systems, involving 3-level systems for simplicity, and show how these mathematical constraints give rise to non-intuitive physical phenomena, reminiscent of Bohm-Aharonov effects. In particular, we show that for a multi-level atomic system it is generally impossible to isolate the levels, and this leads to observable effects on the population relaxation and decoherence.

DOI
10.1142/9789812772527_0038

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A state in quantum mechanics is defined as a positive operator of norm 1. For finite systems, this may be thought of as a positive matrix of trace 1. This constraint of positivity imposes severe restrictions on the allowed evolution of such a state. From the mathematical viewpoint, we describe the two forms of standard dynamical equations - global (Kraus) and local (Lindblad) - and show how each of these gives rise to a semi-group description of the evolution. We then look at specific examples from atomic systems, involving 3-level systems for simplicity, and show how these mathematical constraints give rise to non-intuitive physical phenomena, reminiscent of Bohm-Aharonov effects. In particular, we show that for a multi-level atomic system it is generally impossible to isolate the levels, and this leads to observable effects on the population relaxation and decoherence.
Dissipation is sometimes regarded as an inevitable and regrettable presence in the real evolution of a quantum system. However, the effects may not always be malign, although often non-intuitive and may even be beneficial. In this note we we display some of these effects for N-level systems, where N = 2,3,4. We start with an elementary introduction to dissipative effects on the Bloch Sphere, and its interior, the Bloch Ball, for a two-level system. We describe explicitly the hamiltonian evolution as well as the purely dissipative dynamics, in the latter case giving the t → ∞ limits of the motion. This discussion enables us to provide an intuitive feeling for the measures of control-reachable states. For the three-level case we discuss the impossibility of isolating a two-level (qubit) subsystem; this is a Bohm-Aharonov type consequence of dissipation. We finally exemplify the four-level case by giving constraints on the decay of two-qubit entanglement.
These lecture notes address an audience of physicists or mathematicians who have been exposed to a first course in quantum mechanics. We start with a brief discussion of the general "system-bath" paradigm of quantum dissipative systems, analyze in some detail the simplest example of "pure dephasing" of a two-level system, and review the basic concept of the density matrix. We then treat the general dissipative time-evolution, introducing completely positive maps, their relation to entanglement theory, and their Kraus decomposition. Restricting ourselves to Markovian evolution, we discuss the Lindblad form of master equations. The notes conclude with an overview of topics of current interest that go beyond Lindblad Markov master equations. (These notes were prepared for lectures delivered by F. Marquardt in October 2007 at the Langeoog workshop of the SFB/TR 12, "Symmetries and Universality in Mesoscopic Systems")
We study the constraints imposed on the population and phase relaxation rates by the physical requirement of completely positive evolution for open NNN-level systems. The Lindblad operators that govern the evolution of the system are expressed in terms of observable relaxation rates, explicit formulas for the decoherence rates due to population relaxation are derived, and it is shown that there are additional, nontrivial constraints on the pure dephasing rates for N>2N>2N>2. Explicit, experimentally testable inequality constraints for the decoherence rates are derived for three- and four-level systems, and the implications of the results are discussed for generic ladder, $\ensuremath{\Lambda}$, and V systems and transitions between degenerate energy levels.
A general density-matrix description of energy-dissipation and decoherence phenomena in open quantum systems is presented. More specifically, contrary to the conventional single-particle correlation expansion, we shall investigate the effect of the adiabatic or Markov limit, before considering/performing any reduction procedure. Our fully operatorial approach allows us to better identify the general properties of the scattering superoperators entering our effective quantum-transport theory at various description levels. In particular, we shall show that—contrary to the semiclassical case—the conventional Markov limit does not preserve the positive-definite character of the corresponding density matrix, thus leading to highly non-physical results. To overcome this serious limitation, originally pointed out and partially solved by Davies and co-workers almost three decades ago, we shall propose an alternative and more general adiabatic procedure, which (i) in the semiclassical limit reduces to the standard Fermi's golden rule, and (ii) describes a genuine Lindblad evolution, thus providing a reliable/robust treatment of energy-dissipation and dephasing processes in semiconductor quantum devices
Open quantum systems with nearly degenerate energy levels have been shown to exhibit long-lived metastable states in the approach to equilibrium, even when modelled with certain Lindblad-form quantum master equations. This is a result of dramatic separation of timescales due to differences between Liouvillian eigenvalues. These metastable states often have nonzero coherences that die off only in the long-time limit once the system reaches thermal equilibrium. We examine two distinct situations that give rise to this effect: one in which dissipative dynamics couple together states only within a nearly degenerate subspace, and one in which they give rise to jumps over finite-energy splittings, between nearly degenerate subspaces. We find, in each case, that a change of basis can often lead to a representation that more naturally captures the impact of the system-bath interaction than does the energy eigenbasis, revealing that separate timescales are associated with separate processes (e.g., decoherence into a nonenergy eigenbasis, decay of population correlations to the initial state). This approach is paired with the inspection of quantum trajectories, which further provide intuition as to how open system evolution is characterized when coherent oscillations, thermal relaxation, and decoherence all occur simultaneously. Published by the American Physical Society 2024
The quantum Mpemba effect concerns with anomalous relaxation of quantum states that evolves either under unitary or non-unitary dynamics. In the context of open quantum systems, while most studies focus on quantum states evolving under completely positive trace-presing dynamics described by the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation, we demonstrate that an analogous effect can arise at the level of operators. In particular, we show that operators that evolves under the adjoint Liouvillian -- despite not being a trace-preserving map -- can still exhibit a genuine Mpemba effect. We derive general conditions under which this phenomenon can occur and validate our predictions for three different open quantum setups. Our results broaden the scope of the Mpemba effect in quantum systems and provide a framework for controlling the relaxation of physically relevant observables.