The quantum electrodynamic formalism of Senitzky is developed so as to yield long-time differential equations for the time-average expectation values of electric field energy and excess molecular population in a system composed of one resonant cavity mode, two-level molecules, and a dissipation mechanism. The cavity-mode resonant frequency is essentially the mean molecular transition frequency of either a Gaussian or Lorentzian distribution of transition frequencies. The intermolecular and lattice T1 and T2 time constants are assumed to be much longer than the stimulated emission period. The field is coupled to either the electric or the magnetic dipole moments of the molecules. Senitzky's papers are summarized and the relevant expressions for the Heisenberg field and molecular operators are listed. Modified time-average expressions are obtained for the presence of a transient coherent driving field and perhaps an ``off-resonance'' molecular distribution of transition frequencies. Time averages of the derivatives of various terms in the expected field energy are compared to derivatives of time averages of those terms. Well-behaved differential equations are obtained for the time-average expectation values of field energy and molecular excess population, justified on a long-time basis by the application of an intermittent similarity transformation. A differential equation for the time-average dispersion (second moment) of electric field energy is obtained, which indicates that the relative dispersion tends to decrease during the pumping interval and increase back to the thermal value during the emission interval. Energy transfer from pumping field to molecules to the resonant field during the pumping interval is described qualitatively. The direct-product form for the density matrix ρ = ρ(field) × ρ(molecules) × ρ(dissipation mechanism) is justified by maximum-entropy inference. In conclusion, the equations of motion for the time-average expectation values of field energy and molecular population are interpreted so as to explain the envelope modulation of a solid-state laser beam during the emission period.
The quantum electrodynamic formalism of Senitzky is developed so as to yield long-time differential equations for the time-average expectation values of electric field energy and excess molecular population in a system composed of one resonant cavity mode, two-level molecules, and a dissipation mechanism. The cavity-mode resonant frequency is essentially the mean molecular transition frequency of either a Gaussian or Lorentzian distribution of transition frequencies. The intermolecular and lattice T1 and T2 time constants are assumed to be much longer than the stimulated emission period. The field is coupled to either the electric or the magnetic dipole moments of the molecules. Senitzky's papers are summarized and the relevant expressions for the Heisenberg field and molecular operators are listed. Modified time-average expressions are obtained for the presence of a transient coherent driving field and perhaps an ``off-resonance'' molecular distribution of transition frequencies. Time averages of the derivatives of various terms in the expected field energy are compared to derivatives of time averages of those terms. Well-behaved differential equations are obtained for the time-average expectation values of field energy and molecular excess population, justified on a long-time basis by the application of an intermittent similarity transformation. A differential equation for the time-average dispersion (second moment) of electric field energy is obtained, which indicates that the relative dispersion tends to decrease during the pumping interval and increase back to the thermal value during the emission interval. Energy transfer from pumping field to molecules to the resonant field during the pumping interval is described qualitatively. The direct-product form for the density matrix ρ = ρ(field) × ρ(molecules) × ρ(dissipation mechanism) is justified by maximum-entropy inference. In conclusion, the equations of motion for the time-average expectation values of field energy and molecular population are interpreted so as to explain the envelope modulation of a solid-state laser beam during the emission period.
The equations of motion for the resonant modes in a lossy cavity, which couples them through the nonresonant thermal loss mechanism described by Senitzky, are examined. The second-order perturbation treatment of dissipation is justified for long time intervals by reference to an intermittent similarity transformation. Self-consistent expressions for commutators such as 〈[Pn(t1), Pm(t2)]〉d, Pn being the operator-amplitude of the electric field in the nth mode, as averaged over the loss mechanism d with respect to its diagonal density matrix, are obtained from Laplace transform expressions for these operators. These averaged commutators describe correlations of fluctuations in different modes, in a way compatible with the ordinary commutation relations for lossless modes. The self-consistent anticommutator expressions, averaged over the loss mechanism, imply the Bose-Einstein formula for thermal energy in each mode at time t2 = t1. This is shown from direct examination of anticommutators and also from an integration of the correlation function. Commutators for the total electric and magnetic fields at two space-time points in the cavity have the proper causal properties. The dissipation terms in the field equations of motion cannot be derived from an equivalent Hamiltonian unless a modified analysis such as that of Kemeny is undertaken. Finally, the approximate equations of motion for the mode operator-amplitudes of a parametric amplifier are derived in the creation-annihilation operator framework and the spectral density of noise obtained at the signal frequency. The equilibrium noise present due to coupling with the idle modes is proportional to the number of photons in the idle frequency mode instead of that number plus one, in contrast to the conclusion of Wagner and Hellwarth. In this system the loss-averaged operators evolve from initial thermal equilibrium values to parametric equilibrium values according to nonlinear equations of motion containing time dependent commutators.
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.
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 the electromagnetic field and a number of identical atomic systems, individually characterized by an electric dipole moment and two energy levels, is analyzed for the case where the atomic systems are inside a lossy cavity and exposed to a coherent driving field, resonance being assumed between atomic system, cavity, and driving field.The problem of introducing loss into a quantum-mechanical formalism is treated first. Formal operator expressions are obtained for the field variables which include the absorption and the fluctuation (both thermal and quantum-mechanical) effects of the loss mechanism. Expectation values are then obtained for the field strength and the field energy which are valid for times short compared to the lifetime of the excited state. It is shown that the spontaneous-emission energy in the field increases initially as the square of the time and approaches a steady-state value after a transient period which is of the order of the cavity relaxation time. The induced emission contains two parts: incoherent emission induced by the thermal field, and coherent emission induced by the driving field. The incoherent induced-emission energy has the same time dependence as the spontaneous-emission energy, and the ratio of the former to the latter is the number of photons in the thermal (inducing) field. The coherent induced-emission energy does not approach a steady-state value, but, after a transient period, increases linearly with the time. The ratio of the coherent induced-emission energy to the spontaneous-emission energy is equal, initially, to the number of photons in the driving-field energy, but becomes $\ensuremath{\beta}t$ times as large after the transient period, where $\ensuremath{\beta}$ is the reciprocal of the cavity relaxation time.The expectation value of the rate of energy emission by the atomic systems is also obtained. It is shown that the ratio of the downward to the upward transition probabilities has the well-known value of n(n+1), where n is the field energy in units of the photon energy, only in the absence of a coherent field.
Abstract This present paper is an attempt to describe the operation of a laser by a method patterned closely after the classical theory of dispersion. There, the electromagnetic field is treated as a classical system interacting with a collection of classical harmonic oscillators which are at rest. In the present case the radiation field is still treated classically, but, in accordance with the correspondence principle, the oscillators are now virtual oscillators associated with the up-and-down transitions in effective two-state atoms which are described quantum mechanically. The atoms carrying these virtual oscillators are not at rest but move with different velocities. We show how the Lorentz averaging can be performed for such systems, and derive a closed set of equations linking the average electric field E, with the average polarization density P„ and the average population inversion density Mp associated with atoms moving with the velocity v. Next, we investigate the conditions for the existence of a steady state if pumping is present, and if the electric field is represented by a standing wave, or a travelling wave of a single frequency (single-mode case). It turns out that, notwithstanding the nonlinear nature of the equations, the steady-state conditions give a simple complex dispersion relation; moreover, the real and imaginary part of this dispersion relation are equivalent to the usual heuristic expressions which specify the operating frequency in terms of the index of refraction and cavity length, and which balance the gain against the losses. The single-mode case is analysed in detail without any smallness assumptions for the resultant intensity. The small-intensity case gives the usual results exhibiting the tuning dip. For high intensities the relative depth of the dip tends to zero. At the end of the paper we discuss proposed extensions and additional applications.
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.