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arXiv:quant-ph/0312231 · Physical Review A

Constraints on relaxation rates for N -level quantum systems

S. G. Schirmer, A. I. Solomon

Published Aug 30, 2004• 71 citations
arXivHTMLPDF

Abstract

We study the constraints imposed on the population and phase relaxation rates by the physical requirement of completely positive evolution for open $N$-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 $N>2$. 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.

DOI
10.1103/physreva.70.022107

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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, \ensuremathΛ\ensuremath{\Lambda}\ensuremathΛ, and V systems and transitions between degenerate energy levels.
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.
Abstract Relaxation rates provide important characteristics both for classical and quantum processes. Essentially they control how fast the system thermalizes, equilibrates, decoheres, and/or dissipates. Moreover, very often they are directly accessible to be measured in the laboratory and hence they define key physical properties of the system. Experimentally measured relaxation rates can be used to test validity of a particular theoretical model. Here we analyze a fundamental question: does quantum mechanics provide any nontrivial constraint for relaxation rates? We prove the conjecture formulated a few years ago that any quantum channel implies that a maximal rate is bounded from above by the sum of all the relaxation rates divided by the dimension of the Hilbert space. It should be stressed that this constraint is universal (it is valid for all quantum systems with finite number of energy levels) and it is tight (cannot be improved). In addition, the constraint plays an analogous role to the seminal Bell inequalities and the well known Leggett–Garg inequalities (sometimes called temporal Bell inequalities). Violations of Bell inequalities rule out local hidden variable models, and violations of Leggett–Garg inequalities rule out macrorealism. Similarly, violations of the relaxation bound rule out Markovian (meaning CP-divisible) evolution.
A general property of relaxation rates in open quantum systems is discussed. We find an interesting constraint for relaxation rates that universally holds in fairly large classes of quantum dynamics, e.g., weak coupling regimes, as well as for entropy nondecreasing evolutions. We conjecture that this constraint is universal, i.e., it is valid for all quantum dynamical semigroups. The conjecture is supported by numerical analysis. Moreover, we show that the conjectured constraint is tight by providing a concrete model that saturates the bound. This universality marks an essential step toward the physical characterization of complete positivity as the constraint is directly verifiable in experiments. It provides, therefore, a physical manifestation of complete positivity. Our conjecture also has two important implications: it provides (i) a universal constraint for the spectra of quantum channels and (ii) a necessary condition to decide whether a given channel is consistent with Markovian evolution.
Population and phase relaxation in two molecular models is studied. The analysis is based on recent work by Lindblad, in which the time evolution of an N-level quantum system is described using the coresponding generator of a completely positive semigroup. One of the models studied corresponds to a diatomic molecule in a dense medium. New results are obtained for the influence of two quantum transitions on population and phase relaxation. The other model studied corresponds to the case of exchange dephasing in a polyatomic molecule embedded in a dense medium. The formalism permits inclusion of coupling terms neglected by Harris, Shelby, and Cornelius; it is shown that inclusion of these terms leads to a rich range of line shapes, which information is not available from the approximation used by Harris et al.
It is shown that the density matrix equations for a three-level quantum system interacting with external radiation fields can lead to negative populations if arbitrary dephasing rates and shifts are included in these equations. To guarantee non-negative populations, the equations themselves impose certain restrictions on the dephasing widths and shifts. The constraints on the widths are shown to be identical to those that can be derived from a model of Markovian dephasing events, independent of any atom-field interaction.
Relaxation rates are key characteristics of quantum processes, as they determine how quickly a quantum system thermalizes, equilibrates, decoheres, and dissipates. While they play a crucial role in theoretical analyses, relaxation rates are also often directly accessible through experimental measurements. Recently, it was shown that for quantum processes governed by Markovian semigroups, the relaxation rates satisfy a universal constraint: the maximal rate is upper-bounded by the sum of all rates divided by the dimension of the Hilbert space. This bound, initially conjectured a few years ago, was only recently proven using classical Lyapunov theory. In this work, we present a new, purely algebraic proof of this constraint. Remarkably, our approach is not only more direct but also allows for a natural generalization beyond completely positive semigroups. We show that complete positivity can be relaxed to two-positivity without affecting the validity of the constraint. This reveals that the bound is more subtle than previously understood: two-positivity is necessary, but even when further relaxed to Schwarz maps, a slightly weaker-yet still non-trivial-universal constraint still holds. Finally, we explore the connection between these bounds and the number of steady states in quantum processes, uncovering a deeper structure underlying their behavior.