A number of $n$-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.
A number of n-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.
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.
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 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 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 chapter introduces and applies the method of second quantization, a technique that underpins the formulation of quantum many-particle theories. Second quantization formalism treats systems of bosons or fermions with a fixed or variable number of particles in a unified way. The chapter reviews the key aspects of this formalism. The formalism of second quantization treats systems of bosons with a fixed or variable number of particles in a unified manner. In many physical processes, the particle number does change. Different quantum operators are expressed most naturally in different representations, which make basis changes an essential issue in quantum physics. Quantum field theory is a universal tool that has applications ranging from atomic, molecular, and particle physics to condensed matter and statistical physics as well as modern quantum chemistry. The fundamental difference between classical and quantum mechanics relates to the concept of indistinguishability of identical particles.
The dynamics of the co-operative. spontaneous emission· from many· two-level atoms is investigated by introducing a specific form of quasi-probability distribution which apparently excludes the vacuum fluctuation of the radiation field but includes the ·fluctuation of the atomic dipole moment. It is shown that the equation of motion for the random variables associated with the atom-field coupled system can be written in a quite classical form. As a result, the formula is derived for computing the multi-time correlation functions of the radi ation field-essential to its coherence properties-which generalizes Bonifacio, Schwendimann and Haake's adiabatic corresponding theorem. § 1. Introduction. The co-operative spontaneous emission (so-called superradiance) from an as sembly of many two-level atoms first discussed by Dicke in 1954n has been studied by many authors with different methods, roughly speaking, classicall/> and quantum mechanically. 3>-5> The classical approach, though simple, has a special significance for the dynamics of a large number of atoms participating in phase, and has succeeded except when the population of the atoms is completely inverted. The failure of the simple classical treatment in the case of complete inversion is due to the fact that the effect of spontaneous emission is not considered at all, so that emission is impossible from such a state as with no dipole moment.2>.s> Therefore, if one takes into account adequately the quantum fluctuation of the tdtal dipole moment which exists even when the atoms are fully excited, one should be able to describe the spontaneous emission from a classical point of view. This idea was suggested by Senitzky6> in a rather intuitive argument. In the present paper an attempt is made to formulate the above problem from a quasi-probability point of view. To this end we make here a particular choice of the representation of the quantum density operators (or quasi-probability. distri bution functions abbreviated by QPDF hereafter) among others7l-Io> by which the quantum mechanical equation of motion may be transformed into a form in an . associated c-number phase space, i.e., the yvell-known P-representation 9> of Glauber IS taken for the field variables and the so-~alled Q~representation based on the
Conceptual problems in quantum optics are investigated by a study of the physical significance of various types of description of the field of a single mode. The descriptions are examined through the effect of the field on matter. Both undamped and damped modes are considered, the descriptions being those of various degrees of precision in classical and quantum-mechanical formalisms. The matter consists of a large number of "molecules," and the interaction is studied by means of low-order perturbation theory. Formal expressions are obtained for the energy interchange between field and molecules in terms of the zeroth-order (uncoupled) operators, with the effect of the commutation properties of the field variables explicitly separated from that of the excitation of the field, in second order. The significance of taking expectation values of the operator results is discussed, and it is concluded that this procedure is justified only when the experiment under consideration involves an averaging process; thus, the molecular variables should be averaged, but not necessarily those of the field. Conditions under which the averaging process is justified are considered, and it is reasoned that, when such a process is not justified, a statistical description of the field can result only in a statistical description of the interaction process. Results of various field descriptions are compared, and it is shown that for induced emission and for spontaneous emission from coherently oscillating molecules, classical and quantum-mechanical fields are equivalent. However, for a self-consistent theory of spontaneous emission from molecules that are not oscillating coherently, the quantum-mechanical commutation properties of the field are essential if the molecules are treated quantum-mecahnically, and the field must be treated classically if the molecules are treated classically; the type of description used for the excitation of the field is inconsequential. Photoelectric emission is examined, and, within the approximation framework used, shown to be accurately described by a classical treatment of the field. It is pointed out that there does not necessarily exist a connection between "photon statistics," that is, the statistics associated with an ensemble of fields, and photocurrent fluctuations in one or more detectors coupled to the same field.