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Physical Review A

Coherence in multilevel systems. II. Description of a multilevel system as two levels in contact with a population reservoir

W.G. Breiland, M. D. Fayer, Charles Bonner Harris

Published Jan 1, 1976• 20 citations
Publisher

Abstract

The effects of incoherent feeding and decay on a coherently driven two-level system are investigated using density-matrix techniques. A closed-form solution that includes the effects of relaxation and spontaneous emission between the two levels is obtained for cases in which the sample size is both small and large compared to the wavelength of the coherent field. The formalism developed herein is somewhat more general than the original approach of Bloch, Bloembergen, Purcell, Pound, and Redfield, insofar as it is applicable to cases where the trace of density matrix is not constant in time. As such, several new phenomena are predicted. Specifically, the steady-state solutions reveal that under appropriate conditions a significant off-resonance coherent component can be maintained indefinitely by the combination of driving field and incoherent feeding. We term this effect "kinetic coherence" and develop its formalism for both optical and radio-frequency fields.

DOI
10.1103/physreva.13.383

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The effects of incoherent feeding and decay on a coherently driven two-level system are investigated using density-matrix techniques. A closed-form solution that includes the effects of relaxation and spontaneous emission between the two levels is obtained for cases in which the sample size is both small and large compared to the wavelength of the coherent field. The formalism developed herein is somewhat more general than the original approach of Bloch, Bloembergen, Purcell, Pound, and Redfield, insofar as it is applicable to cases where the trace of density matrix is not constant in time. As such, several new phenomena are predicted. Specifically, the steady-state solutions reveal that under appropriate conditions a significant off-resonance coherent component can be maintained indefinitely by the combination of driving field and incoherent feeding. We term this effect "kinetic coherence" and develop its formalism for both optical and radio-frequency fields.
Pulsed radio-frequency transient phenomena of resonant two-level spins are often described by the Bloch equations where the phenomenological dipole (T2{T}_{2}T2​) and population (T1{T}_{1}T1​) decay times appear. A similar description has been carried over recently into the microwave and infrared regions as well. At visible-ultraviolet wavelengths, however, a more general solution is required because the transition levels do not constitute an isolated system and also spontaneous emission can no longer be disregarded. As a consequence, the lower (1) and upper (2) transition levels are characterized by vastly different decay rates (${\ensuremath{\gamma}}_{1}\ensuremath{\ne}{\ensuremath{\gamma}}_{2}$) and the lifetime T1{T}_{1}T1​ is no longer a defined quantity. In view of laser-induced coherent transient phenomena now being observed, we present general solutions of the density-matrix equations of motion for an optically resonant two-level system in contact with a multilevel reservoir where ${\ensuremath{\gamma}}_{1}\ensuremath{\ne}{\ensuremath{\gamma}}_{2}$. Particular and perturbative solutions involving intensity-dependent damping and frequency shifts are included with applications to echo and nutation phenomena.
This is the second in a series of three papers on nonlinear resonance effects in multilevel quantum systems in which quantum-mechanical systems are considered that are under the influence of any number of classically described driving fields (well defined in amplitude and phase) and are in weak contact with a thermal reservoir, such that relaxation may be described by means of phenomenological decay constants. In the first paper, referred to as I, a projection-operator formalism was developed for separating the equation of motion for the density operator into a part which may be treated by conventional time-dependent perturbation techniques and another part which, because it would give rise to secular terms using a time-dependent perturbation theory, must be treated by more exact means. In this paper three applications of the formalism developed in I are considered. The first application is to a system in which a multiple-quantum transition is taking place between two levels of the system which contains many levels between the two levels under consideration. In particular, the transition probability is calculated for these two levels, which are not directly connected by matrix elements of the interaction but are connected indirectly through intermediate states. The next application considered is the Raman effect in a four-level system. As a final application of the formalism developed in I an expression is derived for the magnetization M+(t)=Mx(t)+iMy(t){M}_{+}(t)={M}_{x}(t)+i{M}_{y}(t)M+​(t)=Mx​(t)+iMy​(t) of a system of NNN noninteracting spins under the influence of a static field in the zzz direction and an arbitrary number of rf fields in the xxx direction. As in I, a phenomenological form is assumed for the relaxation in all of these applications. It is shown in this problem that fields which are not involved in directly causing transitions in this system give rise to frequency shifts in the resonance conditions, which are accounted for in this treatment in a very natural and straightforward manner. For these fields, the rotating as well as the counter-rotating fields give shifts in the resonance condition.
A method for monitoring the time development of the coherent components in a multilevel system using an observable which can measure only the relative populations in the levels is presented. Using standard density matrix formulation, expressions are obtained for the conventional and this new ’’probe’’ pulse method. For several of the more common coherence experiments, optically detected magnetic resonance of molecular excited triplet states is used to illustrate the principles. The effect of coherent averaging of relaxation terms in the spin Hamiltonian is discussed and specifically applied to excited triplet states in zero field. Finally, experimental results of coherence measurements are presented along with a detailed description of the apparatus and methods used.
The resonant response of a multilevel atomic system to strong coherent radiation is analyzed using an extension of the methods of time-independent scattering theory, adapted to the density-matrix formalism in order to include thermal bath effects. General steady-state expressions are derived for self-attenuation and resonance scattering rates in strong saturating fields, as functions of the coupling strengths, relaxation parameters, and frequency detunings. Formal expressions are given for radiative-damping and collision-broadening relaxation parameters, relating the latter to scattering amplitudes. Modifications of collision-broadening parameters by the presence of very strong fields are discussed. The Bloch equations are derived by transformation to the time domain, under specified conditions.
Calculations are made to determine the response (power absorbed, magnetization, etc.) of a multilevel quantum system under the influence of any number of classically described driving fields (well defined in amplitude and phase). A nonperturbative theory is developed to account for those fields that are in resonance with the system. The nonresonant effects, such as virtual energy shifts, are included in the theory. We limit ourselves to a discussion of quantum systems that are in weak contact with a thermal reservoir. The procedure is the following A projection-operator formalism is developed which separates the density matrix of a system into two parts. One part is related to those portions of the driving fields that interact quite strongly with the system. The other part is the nonresonant or weakly interacting portion. An exact equation of motion, which includes the effects of the nonresonant part, is derived for the strongly interacting part of the density matrix. It is then shown that when relaxation can be represented by phenomenological decay constants, a unitary transformation can be found which reduces the equation of motion for the strongly interacting part of the density matrix so that an exact solution is possible. This solution is shown to give the major effects of the driving fields and can be used as the zeroth-order solution in a perturbation treatment of the whole density matrix. Frequency shifts such as the Bloch-Siegert shift and shifts due to couplings between the levels of interest (resonance levels) and other levels (nonresonance levels) all arise in the same natural way in our treatment. Our results in many instances are the extensions of the well-known results for a two-level quantum-mechanical systemm to an NNN-level one, and include resonant multiple quantum effects. As an example, a straightforward application of the method is carried out in order to compute the shifts in the resonance frequencies of a four-level system, driven by two oscillating fields connecting three of the levels, due to the counter-rotating components of the driving fields and the presence of the fourth level, which is not coupled to the other levels by the driving fields.
We present an efficient theoretical method for calculating the time evolution of the density matrix of a multilevel quantum system weakly interacting with incoherent light. The method combines the Bloch-Redfield theory with a partial secular approximation for one-photon coherences, resulting in a master equation that explicitly exposes the reliance on transition rates and the angles between transition dipole moments in the energy basis. The partial secular Bloch-Redfield master equation allows an unambiguous distinction between the regimes of quantum coherent vs. incoherent energy transfer under incoherent light illumination. The fully incoherent regime is characterized by orthogonal transition dipole moments in the energy basis, leading to a dynamical evolution governed by a coherence-free Pauli-type master equation. The coherent regime requires non-orthogonal transition dipole moments in the energy basis and leads to the generation of noise-induced quantum coherences and population-to-coherence couplings. As a first application, we consider the dynamics of excited state coherences arising under incoherent light excitation from a single ground state and observe population-to-coherence transfer and the formation of non-equilibrium quasisteady states in the regime of small excited state splitting. Analytical expressions derived earlier for the V-type system [T. V. Tscherbul and P. Brumer, Phys. Rev. Lett. 113, 113601 (2014)] are found to provide a nearly quantitative description of multilevel excited-state populations and coherences in both the small- and large-molecule limits.