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Physical Review A

Description of n -level systems in cooperative behavior. III. Quantum beats

I. R. Senitzky

Published Jan 1, 1977• 10 citations
Publisher

Abstract

Quantum beats in the radiation field of "three-level" atoms are analyzed by means of a previously developed boson-second-quantization (BSQ) formalism. Two types of atoms are considered: type I, in which beats are due to transitions from each of two closely spaced upper levels to a single lower level, and type II, in which beats are due to transitions between a single upper level and each of two closely spaced lower levels. This classification is motivated by the claim found in the literature that semiclassical theory and quantum theory give qualitatively different results concerning the existence of beats in the case of type-II atoms. The terminology associated with beats is made precise by distinguishing between amplitude modulation at the beat frequency of the transition frequency field $E$, and the existence of a spectral component at the beat frequency in ${E}^{2}$. With respect to the former, no qualitative difference is found in the predictions from semiclassical theory and quantum theory; with respect to the latter, results that appear paradoxical from an intuitive viewpoint are discussed. It is shown that for the case in which the atoms are all in the same coherent superposition of energy states, with all levels occupied, semiclassical and quantum-mechanical results become equivalent as the number of atoms increases. Reference is made to the disturbance caused by the measurement processes in the case of a single atom. The essential difference in spontaneous emission beats between type-I and type-II atoms [with the ground state(s) unoccupied] is explained by the fact that the phase of the beat amplitude may be determined by preparation of the initial state in type-I atoms but may not be determined in type-II atoms.

DOI
10.1103/physreva.15.292

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Quantum beats in the radiation field of "three-level" atoms are analyzed by means of a previously developed boson-second-quantization (BSQ) formalism. Two types of atoms are considered: type I, in which beats are due to transitions from each of two closely spaced upper levels to a single lower level, and type II, in which beats are due to transitions between a single upper level and each of two closely spaced lower levels. This classification is motivated by the claim found in the literature that semiclassical theory and quantum theory give qualitatively different results concerning the existence of beats in the case of type-II atoms. The terminology associated with beats is made precise by distinguishing between amplitude modulation at the beat frequency of the transition frequency field EEE, and the existence of a spectral component at the beat frequency in E2{E}^{2}E2. With respect to the former, no qualitative difference is found in the predictions from semiclassical theory and quantum theory; with respect to the latter, results that appear paradoxical from an intuitive viewpoint are discussed. It is shown that for the case in which the atoms are all in the same coherent superposition of energy states, with all levels occupied, semiclassical and quantum-mechanical results become equivalent as the number of atoms increases. Reference is made to the disturbance caused by the measurement processes in the case of a single atom. The essential difference in spontaneous emission beats between type-I and type-II atoms [with the ground state(s) unoccupied] is explained by the fact that the phase of the beat amplitude may be determined by preparation of the initial state in type-I atoms but may not be determined in type-II atoms.
This paper discusses the suitability of a boson-second-quantization (BSQ) formalism for the description of cooperative atomic behavior---such as that encountered in quantum optics or laser-type phenomena. The physical significance of BSQ energy states and atomic coherent states, as well as the corresponding first quantization descriptions, are discussed. Specialization to the case in which the individual atomic systems are characterized by two levels allows comparison of the BSQ theory with other theories of cooperative behavior, not all of which can be generalized to the case of more than two levels. Passage to the classical limit, an analytical process that must be possible for macroscopic phenomena, is shown to be accomplished very simply in the BSQ formalism by conversion of the small number of collective (quantum mechanical) variables to classical random variables which are prescribed according to the quantum state of the total system. It is found that conventional semiclassical theory is a special case of the fully classical limit of the BSQ formalism, the specialization being the requirement that the system be in an atomic coherent state. The justification for the identification of coherent states with a classical description and energy states with a quantum-mechanical description, found in some of the literature on two-level systems, is questioned.
Quantum beats are analyzed by means of a boson-type second-quantization method which exhibits the collective behavior of nnn-level systems and displays both quantum mechanical and classical (or semiclassical) results. Theoretical criteria for predicting the existence of beats are found which differ from those in the recent literature and indicate a reevaluation of suggested experiments.
Specialization of the previously developed boson-second-quantization (BSQ) formalism for the description of cooperative atomic behavior to the case of a two-level laser allows comparison with and criticism of other theories of cooperative behavior. Application of the BSQ formalism to a model of a three-level laser leads to equations of motion in terms of macroscopic variables without the use of phenomenological modifications. These equations, which can easily be generalized to more than three levels, may be interpreted both classically and quantum-mechanically, and offer the possibility of investigating a number of problems which are of both conceptual and practical interest.
A number of nnn-level systems behaving cooperatively is described by a boson-type second-quantization formalism which has both computational and conceptual utility, and leads naturally to a classical analog. The theory is compared with that for two-level systems and illustrated by the consideration of coupling to general-type fields. Spontaneous emission at all transition frequencies is examined, with both quantum-mechanical and classical results explicitly exhibited.
The term quantum beats refers to a superposed oscillatory behaviour in the light intensity emitted by some suddenly excited atomic systems in their subsequent decay. In this paper we reinvestigate quantum beats for systems where two upper levels may decay to a common ground state by using the recently developed approach of quantum jumps. The treatment turns out to be very simple and intuitive. It gives explicit expressions for the time-dependent intensity of the radiating system and allows us to exhibit several phenomena which so far seem to have been overlooked. We show that quantum beats may also occur if only one of the upper levels is initially excited, a superposition is not needed. A possible experimental set-up for the verification of this effect is proposed. It is also shown that in some cases the beat frequency may differ considerably from the separation delta omega of the upper levels. For equal Einstein coefficients of the upper levels the beats are shown to be unexpectedly absent for small delta omega .
Spontaneous emission from a multilevel atom is studied from the viewpoint of quantum beats. The master equation describing the atomic dynamics is shown to have some parameters that oscillate at the beat frequencies, and because of this a Pauli type of master equation for the diagonal elements of the density matrix is not obtained. As a result, the transition rates in the atomic system show the characteristic beats. The relation of the rate of change of the energy of the atomic system to the total power radiated in the far zone is also established. The normally and antinormally ordered correlation functions of the spontaneously emitted radiation are calculated under the approximation that the level widths are much smaller than the beat frequencies. The contribution arising from the interference of the free field and the source terms to antinormally ordered correlations is carefully evaluated. The nature of beats present in normally and antinormally ordered correlations is discussed, and their detection from the viewpoint of causing transitions in another atom is considered. The change in the nature of beats due to atomic collective effects is studied. Finally, it is shown that because of quantum effects, intensity correlations do not show any beats.