A quantum theory of a two-photon laser in a three-level atomic system of the cascade type is presented starting from an original microscopic Hamiltonian. The master equation of the laser operation is derived by using the method of Scully and Lamb [Phys. Rev. A 159, 208 (1967)] for a single-mode laser containing a two-level medium. The photon-number distribution of the laser is studied in detail. The curve of the distribution may have a two-peak structure. One of the peaks located at the origin comes from the single-photon process and reflects the thermal property of the laser, while another comes from the two-photon process and reflects the coherent property of the laser. We cannot find an exact criterion for the threshold of the two-photon laser. The two-photon laser can approximately be considered as a coherent state for certain values of A/C, which can even be much smaller than the threshold for the nonresonant single-mode laser in a two-level system, but it is much larger than the threshold for the resonant one.
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.
This paper presents new results on aspects of the two-photon laser. A semiclassical investigation of threshold behaviour reveals that the atom-field coupling is already so strong at threshold that a non-perturbative treatment of the dynamics is necessary. A master equation is derived on the basis of an effective Hamiltonian generalized in that it includes the effect of an overall two-photon detuning, the effect of ionization of the upper atomic level and the effect of dynamic Stark shifts. From the diagonal elements of the master equation, a Fokker-Planck equation is obtained and shown to yield a general analytical expression for the photon distribution which is found to simplify considerably in special cases. The agreement with the result obtained through numerical solution of the master equation itself is very good. It is further shown that a detuning from two-photon resonance and a significant Stark shift lead to sub-Poissonian statistics. Finally, time-dependent solutions for the photon distribution under an injected coherent state are presented.