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.
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.
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.
The quantum theory of two-level atom single-mode laser system of Sargent, Scully and Lamb is extended to two-mode laser in a three-level atomic system with common lower level, and master equation is obtained for the system with all the three levels being pumped. There appear four terms in the master equation that are absent in the single-mode case and those interpreted as 2-photon processes play an especial role in the present case. The master equation is represented by a probability flow diagram in two-dimensions with the photon numbers n1 and n2 of the two-mode as variables. The probability flow diagram can be extended to infinity, and each arrow of it represents a term of the right hand side of master equation. By taking sum over one of the variables n1 or n2, the master equation is reduced to two equations, each of them can be represented by an one-dimensional probability flow diagram. The equation of motion under steady state is obtained by considering the correspondence between the macroscopic balance and the microscopic balance.A parameter H is introduced to reduce the equation of motion and to facilitate the mathematical procedure and a formal solution is thus obtained. Using the equation of motion and its formal solution various operation characteristics are discussed. Among them are: the threshold condition for each mode; the condition of only one mode operation; the variation of photon statistics between single-mode operation and two-mode operation and the variation of photon statistics with lower level being pumped and not pumped.
The quantum theory of the three-energy-level two-mode laser in which all the levels are being pumped is investigated and the master equation of the laser operation is obtained. It contains two variables,
n1, the photon number of the first mode, and
n2, the photon number of the second mode. There appear four terms in the master equation which represent two-photon processes between the two modes, the absorption of a photon from one mode, and emission of a photon to the other. They are absent in the single-mode case and in the two-mode case with pumping only to the upper level. The master equation can be represented by a diagram of probability in two dimensions which can be extended to infinity, and each arrow of it represents a term on the right-hand side of the master equation. By summing each variable (
n1 or
n2) of the master equation, two equations can be obtained. Each equation contains only one variable, and can be expressed by a diagram of probability flow in one dimension. By considering the correspondence between macroscopic equilibrium and microscopic detailed balance the equations of motion in the steady state are obtained. By introducing a parameter
H, the equation of motion is simplified and a formal solution can be deduced which is dependent on
H. It appears that the parameter
H cannot be fully determined, but some properties of it can be deduced which prove that the introduction of
H is reasonable. The operation characteristics of the laser under different conditions are discussed on the basis of the formal solution: among them are the threshold condition, the condition for one-mode operation, and the change of the photon-statistical distribution. We especially discuss the laser output power curves under two-mode operation above threshold. The curve of one mode, which oscillates first, shows a bending-down phenomenon as the excitation increases when the other mode goes beyond its threshold as demonstrated experimentally by Otsuka. This phenomenon is explained qualitatively and is attributed to the two-photon processes between the two modes.