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Physical Review

Quantum Optics. II. Description of a Mode

I. R. Senitzky

Published Oct 25, 1968• 4 citations
Publisher

Abstract

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.

DOI
10.1103/physrev.174.1588

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Quantum Optics. I. Excitation of a Damped Radiation Mode by Weakly Coupled Sources

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Induced and Spontaneous Emission in a Coherent Field. IV

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Physical Review•Jul 1, 1958•66 citations
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.
The isolation usually encountered in optics between the part of a field that is of interest and its source motivates the consideration of a radiation mode weakly coupled to a quantum-mechanical source. After the introduction of some refinements into the quantum mechanics of a damped radiation mode, the field is expressed as the sum of two parts, one due to the source (the source field) and the other due to the loss mechanism (the "vacuum" field). The characteristic function for the field is calculated up to second order in perturbation theory. This function is then compared with the characteristic function for the field in the presence of a classical source. A method is exhibited by which a classical source can be found such that the two characteristic functions are identical when averaged over a half cycle. In particular, the two sources yield the same expectation values for the instantaneous amplitude and energy of the field. The description of the equivalent classical source must be given in statistical terms, in general, and requires only a knowledge of 〈S(0)(t)〉〈{S}^{(0)}(t)〉〈S(0)(t)〉 and 〈S(0)(t1)S(0)(t2)〉〈{S}^{(0)}({t}_{1}){S}^{(0)}({t}_{2})〉〈S(0)(t1​)S(0)(t2​)〉, where S(0){S}^{(0)}S(0) is the dipole-moment operator of the quantum-mechanical source unperturbed by the mode under consideration (but otherwise arbitrarily complex, with the possibility of strong coupling to other modes). The theory is illustrated by a consideration of several simple sources---a two-level system, a harmonic oscillator, and a blackbody---for which equivalent classical sources are found. The two-time correlation functions for the field obtained with the two types of sources are compared and are shown to be the same up to first order in $\ensuremath{\xi}\ensuremath{\tau}$, where $\ensuremath{\tau}$ is the difference between the two times and $\ensuremath{\xi}$ is the inverse of the field relaxation time; the physical meaning of the second-order difference in the correlation functions is discussed. A limiting process, in which both the coupling to the source and the damping become small, is suggested as a method of adapting the results to free fields, but it is pointed out that for discussion of a single mode, a free field is physically less satisfactory than a damped field. It is concluded that, within a reasonable approximation scheme, the source field may be described classically (the "vacuum" field furnishing all the necessary quantum-mechanical properties of the total field).
A large number of "molecules," which are taken to be angular momentum oscillators of frequency $\ensuremath{\nu}$, are considered to interact through their dipole moment with two electromagnetic cavity modes of frequencies ${\ensuremath{\omega}}_{1}$ and ${\ensuremath{\omega}}_{2}$, respectively, where $\ensuremath{\nu}={\ensuremath{\omega}}_{1}\ifmmode\pm\else\textpm\fi{}{\ensuremath{\omega}}_{2}$. The behavior of the combined system is analyzed up to fourth order in perturbation theory. Phenomena known as two-photon emission or absorption, parametric amplification, various types of Raman emission, and frequency conversion are exhibited. The analytical method used is applicable to both a quantum-mechanical and a classical description of the modes and molecules; it displays explicitly the resulting differences. Expressions for the rate of energy change of the modes and molecules indicate resonance in second and fourth order. The second-order results are the same for the classical and quantum-mechanical treatments, while the fourth-order results show certain differences, which are examined in detail. The conditions under which each of the above phenomena occurs are discussed in a self-consistent manner, both classically and quantum-mechanically. It is shown that the second-order expression and the most significant part of the fourth-order expression can be derived very simply from a resonance interaction Hamiltonian that consists of a sum of products of three variables---two referring to the two modes, and the third to the molecules. The relationship between this resonance Hamiltonian and the fundamental dipole-moment Hamiltonian is examined, and the approximations by which the resonance Hamiltonian may be derived are considered. Arguments concerning the applicability of perturbation theory to steady-state situations are presented; it is pointed out that the second-order expressions for the energy transfer between molecules and field may be applicable to such a situation, in which case the classical and quantum-mechanical treatments are equivalent.
Introduction In the preceding chapters we have been largely concerned with the simplest coherence effects of optical fields, namely those which depend on the correlation of the field variable at two space-time points ( r 1 , t 1 ) and ( r 2 , t 2 ). As we have seen, these effects include the most elementary coherence phenomena involving interference, diffraction and radiation from fluctuating sources. In this chapter we present an extension of the theory to cover more complicated situations, which have to be described by correlations of higher order, i.e. by correlations of the field variables at more than two space-time points or the expectation values involving various powers and products of the field variables. Situations of this kind have become of considerable importance since the development of the laser and of nonlinear optics. The basic difference between the statistical properties of thermal light and laser light can, in fact, only be understood by going beyond the elementary second-order correlation theory. Many of the higher-order coherence phenomena are most clearly manifest in the photoelectric detection process, which can only be adequately described by the quantum theory of detection or by taking into account the quantum features of the field, both of which will be studied in the succeeding chapters. However, because the field is still described classically in the semi-classical theory of light detection, and also because the classical field description provides a natural stepping stone to the quantum description of field correlations, we will now discuss the general description of field correlations of all orders on the basis of the classical theory of the fluctuating wavefield.
Introduction We have now prepared the way for a quantum treatment of problems that are of interest in optics. We have examined some basic properties of the quantized field and have explored the coherent-state representation and some of its virtues. Until now our discussions have been essentially formal; we have not concerned ourselves with what is actually measured in the laboratory, nor with questions concerning the interactions of the quantized field with other systems. In the following sections we shall turn our attention to some of the quantum field theoretic quantities that are of particular interest in quantum optics and are of significance in the measurement of the field. Our discussions in this chapter will still be somewhat formal, in that we shall not yet come to grips with the problem of the interaction of the field with the apparatus. That problem will be tackled in Chapter 14. For the moment we shall continue to oversimplify and treat the field as a free field. However, within the idealized context of the free field we shall examine some of the questions encountered in measurements in quantum optics. We shall see that normally ordered correlations play a dominant role and examine some of their properties. Anti-normally ordered correlations will be seen to arise in much less common and less useful measurements of the field. We shall also examine the sense in which a photon may be said to be localized in a measurement in which a photoelectric emission is registered.
In the previous articles of this series, dealing with the interaction between a number of molecules and the electromagnetic field in a resonant cavity, both the molecules and the field were treated by perturbation theory. The perturbation restriction on the field is removed in the present article, allowing large changes in the field, but the molecules are still assumed to undergo a small change during the time under consideration. The justification for this type of analysis, involving the generalization of the conventional concepts of induced and spontaneous emission, the applicability to a molecular amplifier during the buildup period, and the re-examination of a calculation by Serber and Townes concerning the fundamental limits of molecular amplification, is discussed.Two different molecular distributions are considered. In one (the resonant case) all molecules have the same frequency as the cavity, and in the other (the nonresonant case) there is a uniform frequency distribution. The molecules are assumed to be initially in an emissive state. Several types of driving fields are considered. Expressions are obtained for the field operators by the solution of a Volterra integral equation, and expectation values are obtained for the field strength and field energy.In the resonant case, both the coherent and incoherent fields increase exponentially after a sufficiently long time, no matter how small the initial gain is. Their ratio becomes constant and is equal to the number of photons in the driving field only in the absence of dissipation. An interesting related result is the fact that the signal-to-noise ratio for constant signal input power increases as the cavity dissipation increases. An estimate of the total time for which the theory is valid is obtained from a consideration of the energy emitted by the molecules. Contact is made with perturbation theory for sufficiently small gain and short time.In the nonresonant case the effect of the molecules is shown to be that of a negative dissipation. In contrast to the resonant case, the gain becomes exponential only if the negative dissipation exceeds, in absolute value, the true dissipation. The ratio of induced to spontaneous emission is, in this case also, equal to the number of photons in the driving field only in the absence of disipation. However, the signal-to-noise ratio for constant input power drops with increasing cavity dissipation.
The interaction between a coherently oscillating radiation field, such as is generally encountered in micro-wave experiments, and a number of similar atomic systems coupled to the field through their electric dipole moments is analyzed for the case of resonance between atomic system and field, with both the field and molecules treated quantum-mechanically. Expressions are derived for the expectation value of field strength and the energy, and comparison between the two makes it possible to distinguish between coherent and incoherent parts of the energy, the former having its counterpart in the expectation value of the field strength, while the latter has no such counterpart. Terms corresponding to induced and spontaneous emission are identified, and it is shown that the latter includes both coherent and incoherent components. Special situations related to masers and to the coherence of spontaneous radiation are discussed. The distinction between the behavior of correlated and uncorrelated states is examined, and it is shown that both the coherent emission from uncorrelated states and the incoherent emission from correlated states may be proportional to the square of the number of molecules. The phase of the field is predictable for the coherent emission but unpredictable for the incoherent emission.