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Physical Review A

Quantum effects in the single-mode laser

Federico Casagrande, Luigi A. Lugiato

Published Aug 1, 1976• 28 citations
Publisher

Abstract

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 $A$, $B$, and $C$ 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.

DOI
10.1103/physreva.14.778

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Annual Meeting Optical Society of America•Jan 1, 1985•0 citations
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(\ensuremathβ,\ensuremathβ∗,t)P(\ensuremath{\beta},{\ensuremath{\beta}}^{*},t)P(\ensuremathβ,\ensuremathβ∗,t) of the field mode, containing derivatives of all orders in \ensuremathβ\ensuremath{\beta}\ensuremathβ and \ensuremathβ∗{\ensuremath{\beta}}^{*}\ensuremathβ∗. 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(\ensuremathβ,\ensuremathβ∗,t)P(\ensuremath{\beta},{\ensuremath{\beta}}^{*},t)P(\ensuremathβ,\ensuremathβ∗,t) containing derivatives of all orders in \ensuremathβ\ensuremath{\beta}\ensuremathβ and \ensuremathβ∗{\ensuremath{\beta}}^{*}\ensuremathβ∗. 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.
We apply Lugiato's theory of open systems to the single-mode laser model of Haken and co-workers, in order to investigate the effect on laser dynamics of processes involving one photon and two atoms. A Fokker-Planck time-evolution differential equation is obtained for the field quasiprobability distribution $P(\ensuremath{\beta}, {\ensuremath{\beta}}^{*}, t)$, which includes terms proportional to the squared number of atoms N2{N}^{2}N2. A detailed analysis reveals a twofold effect of these terms. The main effect that becomes relevant well above threshold is a broadening of the stationary field-intensity distribution. This broadening is due to a "cooperative noise," whose source is a squared polarization created by the two-atom processes. These processes also give rise to a very weak loss mechanism, which affects the maximum of the stationary intensity distribution in the threshold region, slightly increasing the threshold for the laser emission.
The effect of counter-rotating terms in the interaction Hamiltonian on the photon statistics is investigated for a single-mode laser. The treatment is based on the Scully-Lamb model. A perturbation expansion is used to derive an equation of motion for the field-density matrix. The counter-rotating terms are found to lead to both transient and secular effects. With some reasonable approximations the steady-state photon-number distribution is derived and it is shown that the counter-rotating terms tend to raise the threshold of laser action and broaden the photon-number distribution. The relative size of these effects is found to be very small and depends on the square of the ratio of the atomic linewidth and the transition frequency.
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.
The quantum-statistical properties of the optical field of a two-mode laser with two coupled transitions have been studied by means of a generalization of the Scully and Lamb theory. The photon number distribution and the mode intensity distribution are obtained in the steady state for a system of homogeneously broadened atoms in resonance with the laser field. It is shown that the mode-coupling constant $\ensuremath{\xi}$ is unity and that near threshold the results of earlier treatments are recovered. Furthermore, it is shown that certain limit measures for the relative intensity fluctuations predicted by the semiclassical Fokker-Planck treatments based on the third-order theory are valid even in the limit of arbitrarily high gain levels.