The study of the appearance of quantum effects in an essentially classical experiment---involving the passage of an electron through a resonant cavity---as the frequency of the electromagnetic field increases, was begun in a previous paper, in which the electron velocity was examined. In the present paper, an analysis is made of the electric field in the same experiment. Both the expectation value of the field, and the expectation value of the square of the deviation from the expectation value (the dispersion) of the field are calculated to the lowest order in the electron charge. The expectation value of the field is seen to differ from the classical value by a small amount, the difference being due to the wave properties of the electron. The dispersion in the field is shown to consist of three parts: one due to the quantum-mechanical properties of the field alone, the second due to the wave properties of the electron, and the third due to the interaction between the quantum-mechanical properties of the field and the classical properties of the electron. The orders of magnitude of the three terms are shown to decrease in the order in which they are enumerated. A calculation is then made of the effect on the field of a number of electrons passing through the cavity, the interaction between the electrons being neglected. The number of electrons required in a "bunch" to produce a field comparable to the dispersion in the field is calculated, and, for a typical set of experimental conditions, it is seen that the highest frequency at which this requirement can be met is of the order of 3\ifmmode\times\else\texttimes\fi{}${10}^{12}$/sec.
The study of the appearance of quantum effects in an essentially classical experiment---involving the passage of an electron through a resonant cavity---as the frequency of the electromagnetic field increases, was begun in a previous paper, in which the electron velocity was examined. In the present paper, an analysis is made of the electric field in the same experiment. Both the expectation value of the field, and the expectation value of the square of the deviation from the expectation value (the dispersion) of the field are calculated to the lowest order in the electron charge. The expectation value of the field is seen to differ from the classical value by a small amount, the difference being due to the wave properties of the electron. The dispersion in the field is shown to consist of three parts: one due to the quantum-mechanical properties of the field alone, the second due to the wave properties of the electron, and the third due to the interaction between the quantum-mechanical properties of the field and the classical properties of the electron. The orders of magnitude of the three terms are shown to decrease in the order in which they are enumerated. A calculation is then made of the effect on the field of a number of electrons passing through the cavity, the interaction between the electrons being neglected. The number of electrons required in a "bunch" to produce a field comparable to the dispersion in the field is calculated, and, for a typical set of experimental conditions, it is seen that the highest frequency at which this requirement can be met is of the order of 3\ifmmode\times\else\texttimes\fi{}1012/sec.
The passage of an electron through a cavity resonator is considered for the purpose of studying the appearance of quantum mechanical effects in an essentially classical experiment as the frequency of the electromagnetic field becomes high. Both the electron and the field must be represented, initially, by wave packets---in space and momentum coordinates for the former, and in electric field and vector potential coordinates for the latter---in order to correspond to classically meaningful systems. The velocity of the electron at the output of the cavity is investigated. It is found that the expectation value of velocity has a quantum-mechanical correction term. A more significant effect, however, is the nonvanishing of the expectation value of the square of the deviation from the expectation value (mean square deviation from the mean) of the velocity. This deviation is a random phenomenon which will produce noise in an electron beam. An expression for the mean square deviation, in which the results of electron and field quantization are separately apparent, is derived, and its significance is discussed. An order-of-magnitude calculation is made for the ratio of minimum possible mean square deviation of velocity to velocity increment due to the field, and the frequency for which this ratio becomes unity is calculated for a particular set of conditions to be of the order of 3\ifmmode\times\else\texttimes\fi{}1012/sec.
An experiment carried out in order to obtain a direct indication of the quantum structhre of a cm wavelength electromagnetic field is described. The interaction between tbe electromagnetic field generated in a cavity resonntor and an electron beam passing through it was studied. According to the theoretical calculations on which this experiment is based the electrons leaving the cavity as a result of the interaction should have a directional scattering of given value. In the experiment, however, no directional scattering was observed or more precisely if there were any directionnl scattering it was not more than 10% of the expected value. (auth)
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.
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.
We develop a quantum field-theoretic method to describe the effect of frequency mixing using Josephson junctions or using an electron beam traversing a succession of microwave cavities. One of our purposes is to discuss the basic mechanism of current modulation and the subsequent radiation in the complete quantum mechanical context. We use the Heisenberg picture, so that the field operators depend on time and state vectors are fixed. It is essential to take quantum mechanical expectation values only for the quantities that are actually observed. Thus, in the case of frequency mixing, we do not evaluate the current but rather the radiated power, which is observable. An evaluation of the current would destroy correlations between Fourier components of the current produced in the first cavity or in the first junction. Leaving the current as an operator permits the correlations produced in the first cavity (junction) to affect the radiation in the second cavity (junction). Another purpose of the approach is to find the possible dependence of the output radiation on the quantum state of the incident radiation preexisting in cavities or in Josephson junctions. In the case of frequency mixing using two microwave cavities where the fields are treated as independent, we show that it is possible to distinguish the case where both cavities are in definite phase-certain coherent states from the phase-uncertain number states by the value of the wave function renormalization constant. However, if the relative phase of the two cavities cannot be defined, this distinction disappears. This indicates that the radiation occurs at the beat frequency even when the cavity fields are taken to be in energy eigenstates, so that no phase whatever can be associated with the fields in those cavities and thus a classical velocity modulation picture does not apply.
The theory developed in the first two articles of this series, dealing with the interaction between the electromagnetic field in a cavity resonator and a number of two-level molecules, is generalized to include a Gaussian spread in the molecular frequency. The center of the molecular frequency distribution coincides with the cavity resonant frequency. There is a coherent driving field in the cavity at the same frequency, and cavity loss is taken into account.Using the formalism previously developed for a quantum-mechanical field in a lossy cavity, expressions are obtained by means of second-order perturbation theory for the expectation values of the field strength and field energy in the cavity, and of the power loss by the molecules. It is shown that the parts of the field energy resulting from induced and spontaneous emission, respectively, initially increase as the square of the time and approach steady-state values after (different, in general) transient periods, each of which is determined by two time constants: cavity relaxation time and inverse molecular frequency spread.It is also shown that both the induced and spontaneous emission power radiated by the molecules increase initially linearly with the time and approach steady-state values after transient periods. For the induced emission power, the transient period is determined by only one time constant, the inverse molecular frequency spread, while for the spontaneous emission power it is determined both by the inverse molecular frequency spread and the cavity relaxation time. The ratio of induced to spontaneous emission is initially n, and approaches a steady-state value $n{[\mathrm{exp}({r}^{2})(1\ensuremath{-}\mathrm{erf}r)]}^{\ensuremath{-}1},$ where n is the driving field energy in units of the photon energy, and r is the ratio of the cavity resonance width to molecular frequency spread. The seeming inconsistency of this value with the classical value of the ratio of the Einstein coefficients is discussed.