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Journal of Modern Optics

Relaxation of a two-level system: non-negativity of density matrix and possibility for anomalously long T 2

Alexei Goun, Boris Ya Zel'dovich

Published Mar 1, 2002• 2 citations
Publisher

Abstract

Density matrix equations for the relaxation of a two-level system are studied. Standard dogma, Γ ik ≥ (γi + γk)/2, about the relaxation constant for the off-diagonal Γ ik ) and diagonal (γi) matrix elements is verified for the case of up- and down-transitions with high rates w 21 and w 12. Non-negativity of the density matrix imposes much weaker limitation, Γ21 ≥ (½)[(w 21)½ - (w 12)½]2. An almost ‘physical’ system is suggested, which describes to a certain approximation such decrease of transverse relaxation constant Γ.

DOI
10.1080/09500340110090792

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Density matrix equations for the relaxation of a two-level system are studied. Standard dogma, Γ ik ≥ (γi + γk)/2, about the relaxation constant for the off-diagonal Γ ik ) and diagonal (γi) matrix elements is verified for the case of up- and down-transitions with high rates w 21 and w 12. Non-negativity of the density matrix imposes much weaker limitation, Γ21 ≥ (½)[(w 21)½ - (w 12)½]2. An almost ‘physical’ system is suggested, which describes to a certain approximation such decrease of transverse relaxation constant Γ.
It is shown that, in general, the interaction between relaxing quantum systems and radiation is described by non-Markov transport equations for the density matrix ρ p and the relaxation matrix depends on the properties of the radiation. The usual Markov-type equations for ρ p are obtained from these equations if γτ c ≪1 and (∊ 2 +4 V 2 ) 1/2 τ c ≪1 (γ is the width of the investigated transition in a quantum system, rc is the correlation time of the perturbations causing relaxation, e is the frequency detuning, and V is the energy of the interaction between the system and the radiation field). When the second inequality is disobeyed, the relaxation process deviates strongly from the Markov type. In the case of small values of V , this results in a non-Lorentzian profile of the absorption lines characterized by a steep descent in the wings. In the range of high values of V , when 2 V τ c 1, the whole relaxation process becomes greatly modified: the dependence of the relaxation coefficients on V becomes strong and a coupling appears between the relaxations of the diagonal and the nondiagonal elements of the density matrix. This is manifested, in particular, in the saturation effect: the term describing saturation in the expression for the polarizability changes considerably and the expression for the line width loses the component due to the adiabatic (elastic) perturbation.
Non-adiabatic transitions in two-level systems are investigated theoretically for a random time dependence of ħω, the energy difference, between the levels. We assumed that ω = ω(x) and the coordinate x = x(t) is a random function of time. Diffusion and Poisson processes (both homogeneous and with a source) for x(t) were assumed. The cases of linear crossing terms (ω = γx) and non-linear terms (ω = ω e exp (- αx) + ω0) were considered. Values of the non-adiabatic transition probability per unit time were obtained by perturbation theory for ω1τ c ≪ 1 where τ c is the correlation time and ω1 is the off-diagonal matrix element.
A method is presented for the coherent control of two-level systems when T2 relaxation is significant. The Bloch equations are rewritten as an equation of motion of the stereographic projection, Gamma, of the spin vector. This allows a Schur-type iteration used for the design of shaped pulses in magnetic resonance and coherent optics to be extended to include the effect of T2. In general, the effect of T2 on Gamma cannot be completely compensated for, although in practice it can be to a high degree. An example is presented of a driving field that produces a coherent superposition (no on-diagonal elements of the density matrix) over a chosen band of frequencies, in the presence of relaxation.
We consider a quantum–mechanical two-level system under the influence of both diagonal and off-diagonal stochastic perturbations, and focus on the decay times T1 and T2, which refer to the relaxation to equilibrium of the populations and relative phase of the two levels, respectively. From both theoretical and experimental viewpoints one traditionally expects that T2≤2T1. On the other hand, from a fourth-order cumulant expansion calculation of the asymptotic time dependence of the density matrix elements, Budimir and Skinner [J. Stat. Phys. 49, 1029 (1987)] showed that, in fact, in some instances T2>2T1. In this paper we solve the stochastic model numerically, which leads to the exact time dependence of the density matrix at all times. We find that the analytic prediction that T2>2T1 is not only correct, but also meaningful, in the sense that the density matrix elements decay exponentially after only a short transient time.
Existing Markov models of the relaxation of quantum-mechanical systems are applicable in the approximations where the difference between eigenfrequencies is either much larger than the relaxation rates (global approach) or much lower than these rates (local approach). In this work, a model has been proposed to describe relaxation in systems where the difference between some eigenfrequencies is close to the relaxation rates. It has been shown that the master equation for the density matrix contains coefficients explicitly depending on the time in the interaction representation. Equations for the occupation numbers of the modes of the system have been derived by example of two interacting oscillators. It has been shown that occupation numbers in the proposed approach in the limits of small and large differences between eigenfrequencies compared to the relaxation rates asymptotically match the occupation numbers in the local and global approaches, respectively.