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Physical Review

Quantum Theory of Laser Radiation. I. Many-Atom Effects

J. A. Fleck

Published Sep 9, 1966• 47 citations
Publisher

Abstract

The interaction is considered between $N$ stationary two-level atoms and a radiation field described by a single cavity mode. The state vector for the complete system of atoms plus radiation is expressed as a linear superposition of states constructed from a product of photon states in the $n$ representation and products of Pauli spin eigenstates describing all combinations of atoms in the lower and upper energy levels. Equations for the corresponding probability amplitudes are derived by substituting this superposition into the Schr\"odinger equation. The resulting equations are combined into bilinear form and phenomenological damping contributions are added. After the neglect of certain of the bilinear quantities, a master equation is derived which governs the probability ${{P}_{m}}^{n}$ of having $m$ atoms in the lower level and $n$ photons in the mode. This master equation takes account of multiple single-quantum absorption and emission processes but not of simultaneous multiple processes involving two or more atoms at a time. The equation which governs the expected number of photons $〈n〉$ derived from the master equation bears a close resemblance to a rate equation. The effect of radiation loss from the cavity is incorporated into the master equation. Numerical calculations for a $Q$-spoiled laser show that the statistics of the number of photons in the mode bear a qualitative resemblance to Poisson statistics.

DOI
10.1103/physrev.149.309

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The interaction is considered between NNN stationary two-level atoms and a radiation field described by a single cavity mode. The state vector for the complete system of atoms plus radiation is expressed as a linear superposition of states constructed from a product of photon states in the nnn representation and products of Pauli spin eigenstates describing all combinations of atoms in the lower and upper energy levels. Equations for the corresponding probability amplitudes are derived by substituting this superposition into the Schr"odinger equation. The resulting equations are combined into bilinear form and phenomenological damping contributions are added. After the neglect of certain of the bilinear quantities, a master equation is derived which governs the probability Pmn{{P}_{m}}^{n}Pm​n of having mmm atoms in the lower level and nnn photons in the mode. This master equation takes account of multiple single-quantum absorption and emission processes but not of simultaneous multiple processes involving two or more atoms at a time. The equation which governs the expected number of photons 〈n〉〈n〉〈n〉 derived from the master equation bears a close resemblance to a rate equation. The effect of radiation loss from the cavity is incorporated into the master equation. Numerical calculations for a QQQ-spoiled laser show that the statistics of the number of photons in the mode bear a qualitative resemblance to Poisson statistics.
We have solved a master equation to obtain the diagonal elements of the density matrix for laser light. The master equation represents a generalization of a master equation derived earlier for a single radiation mode interacting with NNN stationary two-level atoms. The generalization takes into account the pumping scheme which characterizes a three-level laser. For positive temperatures and a lossless cavity, the equilibrium solution of the master equation gives the correct statistical-mechanical description of the atoms and radiation. For negative temperatures, the distribution function for the number of photons in the mode is either exponential or peaked, depending on whether the laser is operating below or above threshold. The calculated intensity fluctuations are in good agreement with semiclassical results for lasers operating slightly above threshold.
We have solved a master equation to obtain the diagonal elements of the density matrix for laser light. The master equation represents a generalization of a master equation derived earlier for a single radiation mode interacting with NNN stationary two-level atoms. The generalization takes into account the pumping scheme which characterizes a three-level laser. For positive temperatures and a lossless cavity, the equilibrium solution of the master equation gives the correct statistical-mechanical description of the atoms and radiation. For negative temperatures, the distribution function for the number of photons in the mode is either exponential or peaked, depending on whether the laser is operating below or above threshold. The calculated intensity fluctuations are in good agreement with semiclassical results for lasers operating slightly above threshold.
The steady-state intensity distribution for a laser operating somewhat above threshold is determined by solving a master equation. The master equation describes the interaction between a single radiation mode and NNN two-level atoms, it contains the effects of pumping and radiation loss from the cavity, and it is applicable to both homogeneous and Doppler broadening. Forms of the master equation applicable to both three- and four-level lasers are considered. Solutions are obtained under the assumption that the distribution of atoms and radiation intensity is a bivariate Gaussian. Fluctuations in the output of giant-pulse lasers are also considered.
In this paper we discuss the quantum features of cooperative radiation processes in a two-level system taking into account non-Markovian effects. If the initial condition is spatially uniform, the many-mode atom-field master equation which has been derived and semiclassically discussed in a previous paper, reduces to the well-known single-mode model laser master equation. Hence, using a suitable projection operator, we derive a generalized non-Markovian master equation which turns out to be correct already in the first Born approximation. This master equation incorporates the effects both of stimulated emission and absorption and of spontaneous emission. When stimulated processes can be neglected, it reduces to a previously known Markovian superradiance master equation, whereas in general it describes non-Markovian oscillations of the radiated intensity. Finally we derive a linear system of equations for the probability of occupation of the Dicke states which generalizes a previously discussed system, taking into account the contribution of stimulated processes.
We arrive at transition probabilities for a laserlike system by neglecting wave properties of matter and radiation, and treating the problem, in an analogy to chemical systems, entirely in a number representation. Thus, for example, stimulated emission is idealized as a collision between a photon and an excited atom and the transition rate is proportional to the amount of each species present in the cavity. In all, transition probabilities for stimulated emission, absorption of radiation, cavity loss, atomic deexcitation (other than stimulated emission), and pumping of atoms are included in the model. The resulting stochastic master equation analogous to that employed for the discussion of fluctuations in chemical systems is then used to obtain the deterministic behavior of atoms and field. This agrees with that of the quantum mechanical laser model of Casagrande and Lugiato; thus the rate parameters of the stochastic model may be related to the parameters of their model. Returning to the master equation, we project out the atomic variable and use adiabatic elimination to arrive at a closed, single-variable equation for P(n), the probability of having n photons in the cavity. The stationary solution of this equation is obtained and graphs of (n) and log[(n2)/(n)2] versus the pumping parameter show close agreement with the microscopic results of the Casa-grande-Lugiato model. A similar discussion of the fluctuations can also be given for a laser with saturable absorption. It shows a first-order phase transition. A comparison with the microscopic results of Lugiato et al. is outlined. Again a remarkable agreement of the stochastic approach via the master equation and the microscopic theory is obtained.
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{n}_{1}n1​, the photon number of the first mode, and n2{n}_{2}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{n}_{1}n1​ or n2{n}_{2}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 HHH, the equation of motion is simplified and a formal solution can be deduced which is dependent on HHH. It appears that the parameter HHH cannot be fully determined, but some properties of it can be deduced which prove that the introduction of HHH 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.