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Physical Review

Transition Operators in Radiative Damping Theory

Robert H. Lehmberg

Published May 5, 1969• 30 citations
Publisher

Abstract

The application of transition operators ${\mathcal{P}}_{m,{m}^{\ensuremath{'}}}(t)=\mathrm{exp}(\mathrm{iHt})|m〉〈{m}^{\ensuremath{'}}|\mathrm{exp}(\ensuremath{-}iHt)$ is studied for the problem of an atomic system $S$ with eigenstates ${m}$ interacting with one or more damping reservoirs $R$. The average value of these operators gives the reduced density matrix ${{\ensuremath{\rho}}_{{m}^{\ensuremath{'}},m}}^{(S)}(t)$ for $S$. If $R$ consists of broad-band distributions of harmonic oscillators, (e.g., radiative damping), then damped equations of motion can be derived for all ${\mathcal{P}}_{m,{m}^{\ensuremath{'}}}(t)$, even if $S$ is a multilevel system. One need not specify the initial states of $R$, nor restrict the treatment to second order in the S-R coupling. The formalism is illustrated for the case where $S$ consists of (i) a four-level atom in a resonant cavity (with broad-band modes also present), and (ii) a collection of atoms that can be treated as a multilevel spin system. Density-matrix equations are obtained for the case where no damping radiation is present initially. In (ii), the formalism is used to derive a two-time corrleation function without the aid of the fluctuation-regression theorem.

DOI
10.1103/physrev.181.32

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The application of transition operators Pm,m\ensuremath′(t)=exp(iHt)∣m〉〈m\ensuremath′∣exp(\ensuremath−iHt){\mathcal{P}}_{m,{m}^{\ensuremath{'}}}(t)=\mathrm{exp}(\mathrm{iHt})|m〉〈{m}^{\ensuremath{'}}|\mathrm{exp}(\ensuremath{-}iHt)Pm,m\ensuremath′​(t)=exp(iHt)∣m〉〈m\ensuremath′∣exp(\ensuremath−iHt) is studied for the problem of an atomic system SSS with eigenstates m{m}m interacting with one or more damping reservoirs RRR. The average value of these operators gives the reduced density matrix \ensuremathρm\ensuremath′,m(S)(t){{\ensuremath{\rho}}_{{m}^{\ensuremath{'}},m}}^{(S)}(t)\ensuremathρm\ensuremath′,m​(S)(t) for SSS. If RRR consists of broad-band distributions of harmonic oscillators, (e.g., radiative damping), then damped equations of motion can be derived for all Pm,m\ensuremath′(t){\mathcal{P}}_{m,{m}^{\ensuremath{'}}}(t)Pm,m\ensuremath′​(t), even if SSS is a multilevel system. One need not specify the initial states of RRR, nor restrict the treatment to second order in the S-R coupling. The formalism is illustrated for the case where SSS consists of (i) a four-level atom in a resonant cavity (with broad-band modes also present), and (ii) a collection of atoms that can be treated as a multilevel spin system. Density-matrix equations are obtained for the case where no damping radiation is present initially. In (ii), the formalism is used to derive a two-time corrleation function without the aid of the fluctuation-regression theorem.
In many problems in quantum optics, damping plays an important role. These include, for example, the decay of an atom in an excited state to a lower state and the decay of the radiation field inside a cavity with partially transparent mirrors. In general, damping of a system is described by its interaction with a reservoir with a large number of degrees of freedom. We are interested, however, in the evolution of the variables associated with the system only. This requires us to obtain the equations of motion for the system of interest only after tracing over the reservoir variables. There are several different approaches to deal with this problem. In this chapter, we present a theory of damping based on the density operator in which the reservoir variables are eliminated by using the reduced density operator for the system in the Schrödinger (or interaction) picture. We also present a ‘quantum jump’ approach to damping. In the next chapter, the damping of the system will be considered using the noise operator method in the Heisenberg picture. An insight into the damping mechanism is obtained by considering the decay of an atom in an excited state inside a cavity. The atom may be considered as a single system coupled to the radiation field inside the cavity. Even in the absence of photons in the cavity, there are quantum fluctuations associated with the vacuum state. As discussed in Chapter 1, the field may be visualized as a large number of harmonic oscillators, one for each mode of the cavity.
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.
In the previous chapter, we developed the equation of motion for a system as it evolved under the influence of an unobserved (reservoir) system. We used the density matrix approach and worked in the interaction picture. In this chapter, we consider the same problem of the system-reservoir interaction using a quantum operator approach. We again eliminate the reservoir variables. The resulting equations for the system operators include, in addition to the damping terms, the noise operators which produce fluctuations. These equations have the form of classical Langevin equations, which describe, for example, the Brownian motion of a particle suspended in a liquid. The Heisenberg–Langevin approach discussed in this chapter is particularly suitable for the calculation of two-time correlation functions of the system operator as is, for example, required for the determination of the natural linewidth of a laser. We first consider the damping of the harmonic oscillator by an interaction with a reservoir consisting of many other simple harmonic oscillators. This system describes, for example, the damping of a single-mode field inside a cavity with lossy mirrors. The reservoir, in this case, consists of a large number of phonon-like modes in the mirrors. We also consider the decay of the field due to its interaction with an atomic reservoir. An interesting application of the theory of the system–reservoir interaction is the evolution of an atom inside a damped cavity. It is shown that the spontaneous transition rate of the atom can be substantially enhanced if it is placed in a resonant cavity.
We study the dynamics of a system consisting of a two-level atom and a quantum oscillator coupled by the vacuum radiation field. The oscillator and the atom are assumed to have the same resonance frequency. In contrast to the two-atom or two-oscillator case, the atom-oscillator system leads to an unclosed hierarchy of equations for the operators that determine the radiation pattern emitted by the atom-oscillator system. Instead of using the operator equations, we formulate the problem in terms of density-matrix equations which can be solved once the initial conditions are specified. As an example, we looked at the radiation pattern, the decay dynamics, and the spectrum for an initial condition in which the oscillator is in its n=1n=1n=1 state and the atom is in its excited state. Dressed states (dressed by the static interaction between the atom and oscillator) prove to be especially useful for interpreting the results and comparing them with the two-atom and two-oscillator systems. All calculations are carried out using the resonance or rotating-wave approximation. The relationship of our work to damped Jaynes-Cummings models is noted.
In describing the nonlinear interaction of radiation with matter it is well known that the perturbation expansion technique is used to calculate the atomic wave functions and the transitions between the states of a system. However the technique, though involving all states of the system, is valid only in the limit of weak interaction fields and non-resonant interactions. For strong fields and resonant interactions, density matrix or dressed atom formulisms are commonly used though these exact treatments are restricted to simple two level schemes or approximations of these; their application to general resonant-near resonant multilevel interactions being prohibitively complex.
Recently Cardimona and Stroud emphasized that, when the energy levels of an atom are degenerate or nearly so, there can exist a strong radiative interaction between different spontaneous transitions of the atom. This interaction is described by the generalized damping terms and generalized frequency-shift terms in the equation of motion for the atomic density matrix. We here point out that the generalized damping terms can play an important role in the absence of degeneracy for systems other than atoms, and cite the harmonic oscillator as an example. Specifically, we show that these terms are needed for a proper classical limit of radiation damping, and that a small anharmonicity can have a very large effect on the radiative decay rate of the oscillator's amplitude.