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Annalen der Physik

Quantenmechanische Behandlung des Lasers mit nichtresonanter Rückkopplung. II Berechnung von Korrelationsfunktionen für das Strahlungsfeld

W. Brunner, Henryk Paul

Published Jan 1, 1969• 5 citations
Publisher

Abstract

In the model introduced in a preceding paper [1] (in which only the homogeneous broadening of the atomic levels is taken into account and the radiation field is assumed to be monochromatic) the first and second order correlation functions for the radiation field in one mode as well as for the total field are calculated explicitly by solving the HEISENBERG equations for the photon creation and annihilation operators b+λ, bλ and the operator \documentclass{article}\pagestyle{empty}\begin{document}$ Q \equiv \sum\limits_\lambda {b_\lambda ^ + } b_\lambda $\end{document} representing the total photon number. The linewidth of the radiation (which is the same for a single mode and the total radiation field) is found to be 2M‐times (M number of modes) larger than in the corresponding case of 1‐mode operation with the same total photon number.

DOI
10.1002/andp.19694780709

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In the model introduced in a preceding paper [1] (in which only the homogeneous broadening of the atomic levels is taken into account and the radiation field is assumed to be monochromatic) the first and second order correlation functions for the radiation field in one mode as well as for the total field are calculated explicitly by solving the HEISENBERG equations for the photon creation and annihilation operators b+λ, bλ and the operator \documentclass{article}\pagestyle{empty}\begin{document}Q≡∑λbλ+bλQ \equiv \sum\limits_\lambda {b_\lambda ^ + } b_\lambdaQ≡λ∑​bλ+​bλ​\end{document} representing the total photon number. The linewidth of the radiation (which is the same for a single mode and the total radiation field) is found to be 2M‐times (M number of modes) larger than in the corresponding case of 1‐mode operation with the same total photon number.
Abstract Extending the method developed in a previous paper [1] the correlation functions both for electric field strength and intensity are calculated. In the latter case the result is obtained quite easily by solving a differential equation. In the first case, on the contrary, a detailed knowledge of the photon creation operator b+ is necessary, especially of the different temporal behaviour with respect to phase and amplitude. In the main, the results are in agreement with those published by other authors. Of special interest is the result that the damping constant for intensity correlations tends to the value corresponding to the case of thermal radiation, in the region far from threshold. In addition, the ansatz b+ = (ro + ϱ) exp{i χ(t)} (ro c‐number) introduced by HAKEN is discussed from a quantum mechanical point of view.
Abstract Making use of the eigenfunctions of the total Hamiltonian, the correlation function 〈𝔈̂(−) (t1) 𝔈̂(+) (t2)〉 is calculated explicitly for a one‐mode radiation field interacting incessantly with an atom. In the frequency spectrum following from this correlation function there appear frequency shifts corresponding, just as in [7], in one case, to a change of the effective dielectric constant within the cavity due to the atom, and, in the other case, to an amplitude modulation due to the alternating absorption and emission of a photon by the atom.
Abstract Previous calculations [1] yielding the frequency spectrum of a one‐mode radiation field interacting with a single atom are extended to the case of M atoms. Under the assumption M „N (N mean photon number) the total Hamiltonian is diagonalized, and the correlation function 〈𝔈̂(t1 𝔈̂(t2)〉 revealing the frequency spectrum of the interacting field calculated quite generally. The following special initial conditions are discussed in detail: a) Any atom is either in the upper or the lower state, and b) the electric dipole moment of the ensemble of atoms is oscillating in phase or with a phase difference of π with respect to the electric field strength. In case a) the initial monochromatic line is broadened, and in case b) a frequency shift appears.
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.