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Physical Review A

Adiabatic expansion for the single-mode laser

Fritz Haake, Maciej Lewenstein

Published Feb 1, 1983• 35 citations
Publisher

Abstract

We carry out a systematic adiabatic elimination of the atomic degrees of freedom from the quantum-mechanical master equation for the single-mode laser. We represent the reduced density operator of the field mode by various quasiprobabilities and construct the respective equations of motion. The generators of infinitesimal time translations are obtained as series in powers of two parameters, the smallness of which defines the adiabatic limit. Our "adiabatic" expansion treats that part of the atom-field interaction as a zeroth-order effect which describes the action of the field on the atoms, while the reaction of the atoms is treated perturbatively. The adiabatic equilibrium thus assigned to the atomic variables at all times is a conditional one, contingent on the current state of the field mode. As a result, saturation effects in the atoms are fully accounted for in low orders of our expansion. In second order, especially, we obtain a Fokker-Planck equation for the Wigner function of the field mode which is valid for arbitrary pump strengths below, near, and above threshold. We compare our results with those of previous theories.

DOI
10.1103/physreva.27.1013

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We carry out a systematic adiabatic elimination of the atomic degrees of freedom from the quantum-mechanical master equation for the single-mode laser. We represent the reduced density operator of the field mode by various quasiprobabilities and construct the respective equations of motion. The generators of infinitesimal time translations are obtained as series in powers of two parameters, the smallness of which defines the adiabatic limit. Our "adiabatic" expansion treats that part of the atom-field interaction as a zeroth-order effect which describes the action of the field on the atoms, while the reaction of the atoms is treated perturbatively. The adiabatic equilibrium thus assigned to the atomic variables at all times is a conditional one, contingent on the current state of the field mode. As a result, saturation effects in the atoms are fully accounted for in low orders of our expansion. In second order, especially, we obtain a Fokker-Planck equation for the Wigner function of the field mode which is valid for arbitrary pump strengths below, near, and above threshold. We compare our results with those of previous theories.
The fully quantum-mechanical model of a single-mode laser is examined. A previously introduced method of solution where both the pump and loss reservoirs are treated simultaneously as a weak perturbation of the atom-light system is employed. This method is similar to perturbation theory for degenerate systems in that the reservoirs remove the nonuniqueness (degeneracy) of the steady state of the unperturbed atom-light system. In the limit of weak reservoir coupling, the exact steady-state solution of the master equation is derived. This solution still contains the atomic variables explicitly and therefore remains valid even in situations where the conventional adiabatic elimination of the atomic variables is no longer possible. A compact discrete representation of the master equation in terms of atomic occupation numbers and photon numbers is developed which allows one to perform all calculations in an elementary manner, requiring only appropriate summations over photon and occupation numbers. General exact relations are obtained for the steady-state probability distribution function p(n,m)p(n,m)p(n,m) to find nnn photons in the light field and mmm atoms in the excited state. Our solution constitutes the first steady-state laser distribution function that is valid for an arbitrary number of laser atoms and in the complete region from far below to far above threshold. It provides a quantitative criterion for the range of validity of approximate methods in laser theory.
Starting from the master equation for density operator of the atoms and the field mode and applying the operator method of adiabatic elimination of the atomio variables developed by Haake and Lewenstein, we derived the Fokker-Planck Equation for Wigner function of the lightfield and its steady-state solution. With a Gaussian approximation to the solution, analytical and numerical results on the photon statistics for various pumping strengths are obtained. Comparisons with previous works are made.
We present a regorous adiabatic elimination of the atomic variables in the framework of the quantum-statistical single-mode laser model. This procedure, which is systematic and similar to the one used in the semiclassical theory, is based on the hierarchy method proposed years ago by one of us. Using a rigorous definition of the adiabatic limit, we show that the hierarchy is exactly truncated by the limit itself. Thus we recover the Fokker-Planck equation previously derived by us using a nonsystematic procedure. We discuss the advantages of the present adiabatic elimination method over the ones previously used by various authors.
The single-mode laser model as formulated by Haken and co-workers is considered. Within the approximation which takes into account only the interaction (at all orders in the coupling constant) between the mode and the single atoms, this model leads to a linear time-evolution equation for the Glauber quasiprobability distribution $P(\ensuremath{\beta},{\ensuremath{\beta}}^{*},t)$ of the field mode, containing derivatives of all orders in $\ensuremath{\beta}$ and ${\ensuremath{\beta}}^{*}$. This equation is solved exactly in the stationary situation. Thus the quantum effects due to the terms with derivatives of order higher than the second one are fully taken into account. It is shown that these terms have a shrinking effect on the stationary distribution above threshold, counteracting the broadening effect due to the second-derivative term. The photon and photoelectron distributions are given. Exact relations are deduced, which allow the calculation of all the moments of the steady-state distribution from the first one. This distribution is compared with the stationary solutions of Risken's equation, of P. Mandel's equation, and with the stationary distribution deduced by Weidlich, Risken, and Haken. The connection with the single-mode laser theory of Scully and Lamb is analyzed in the high-intensity region. The complete Scully-Lamb master equation is translated into the Glauber representation, yielding a linear equation for $P(\ensuremath{\beta},{\ensuremath{\beta}}^{*},t)$ containing derivatives of all orders in $\ensuremath{\beta}$ and ${\ensuremath{\beta}}^{*}$. Suitably connecting the parameters AAA, BBB, and CCC of the Scully-Lamb equation with the parameters of the model of Haken and co-workers, it is shown that there is perfect agreement between the strong-signal Scully-Lamb steady-state distribution and that of Weidlich, Risken, and Haken for all values of the pump parameter.
An exact ccc-number generalized Fokker-Planck equation is derived for six noncommuting variables: the field variables, atomic populations, and atomic polarization of a laser model treated extensively by Lax. Langevin procedures yield the correct Fokker-Planck terms. Adiabatic eliminations are performed on the classical Markoff process given by the Fokker-Planck equation to compare with previous results. It is found that higher-order diffusion coefficients influence the second-order ones slightly during adiabatic elimination.
We present the complete analytical solution of the cavity QED of a two-level atom and a field mode at zero temperature. It includes both dissipation of the field due to a finite Q value of the cavity and incoherent decay mechanisms for the atom. This analytical solution is provided by a powerful method for treating general master equations that appear in quantum optics. As distinct from the usual approaches we first deal with that part of the master equation which describes the dissipative coupling of the field and the atom to their reservoirs. Rather than using number-state or dressed-state bases we expand the density operator into the eigenstates of the nonunitary parts of the master equation which model the dissipative part of the dynamics. The set of these eigenstates is the damping basis. With the aid of this expansion we find the eigenvalues and eigenstates of the total Liouville operator. The evolution of an arbitrary initial state is then known. We employ these results to give an exact solution of the dynamics of the photon field in realistic experiments with one-atom masers at very low temperatures. It includes detuning, cavity leakage effects, spontaneous decay mechanisms for the atoms, a Fizeau-type velocity distribution for the atomic beam, and a statistical parameter for the probability of the excitation of incoming atoms, covering the limits of Poissonian pumping and of regular pumping. On the same grounds one can treat the one-atom laser, consisting of a single atom which stays in permanent interaction with the field mode and which is continuously pumped by external heat baths.