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Physical Review A

Systematic description of laser phase by linearized Ito equations

Klaus M. Gheri, Daniel F. Walls, Monika A. M. Marte

Published Nov 1, 1992• 9 citations
Publisher

Abstract

We show that well above threshold the use of linearized Ito quantum stochastic differential equations allows a systematic description of all quantum stochastic properties of laser light including the phase properties. This is accomplished by the use of mixed field-atom operators which are phase independent and around whose finite mean values we may linearize. As a consequence we are able to derive an equation for an operator describing the phase properties of the laser light. We illustrate our method by applying it to three different models for sub-shot-noise lasers.

DOI
10.1103/physreva.46.6002

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We show that well above threshold the use of linearized Ito quantum stochastic differential equations allows a systematic description of all quantum stochastic properties of laser light including the phase properties. This is accomplished by the use of mixed field-atom operators which are phase independent and around whose finite mean values we may linearize. As a consequence we are able to derive an equation for an operator describing the phase properties of the laser light. We illustrate our method by applying it to three different models for sub-shot-noise lasers.
We derive the equations of motion of arbitrary momentum-dependent observables for a fully first-quantized, driven two-level atom in a linear potential under the presence of a stochastic laser phase. In particular, we show that such momentum-dependent observables evolve under stochastic differential equations (SDEs) in the form of geometrical Brownian motion and find that these SDEs reduce to ordinary differential equations (ODEs) for the expectation values under white laser phase noise. As an example, we solve these ODEs numerically and determine the effect of laser phase noise on Rabi oscillations including quantized center-of-mass degrees of freedom of the atom. Moreover, the SDE formalism we derive holds for any phase noise source and can also be applied to extended noise models such as color noise.
We propose a model, based on a quantum stochastic differential equation (QSDE), to describe the scattering of polarized laser light by an atomic gas. The gauge terms in the QSDE account for the direct scattering of the laser light into different field channels. Once the model has been set, we can rigorously derive quantum filtering equations for balanced polarimetry and homodyne detection experiments, study the statistics of output processes, and investigate a strong driving, weak coupling limit.
We show that the Langevin equation for a nonlinear-optical system may be obtained directly from the Heisenberg equation of motion for the annihilation operators, provided a certain linearization procedure is valid. We apply the technique to the parametric oscillator used to generate squeezed light and compare our results to those obtained from Fokker-Planck-type equations. We argue that, only when the Wigner, as opposed to the P or Q, representation of quantum optics is used, do we get a correct description of the underlying stochastic process. We show how the linearization procedure may be carried out to describe the operation of the parametric oscillator both below threshold, where a squeezed vacuum state results, and above threshold, where we find a squeezed coherent state. In the region of the threshold a heuristic extension of the method leads to a possible description of the system by means of a nonlinear Langevin equation.
We study the nonlinear quantum master equation describing a laser under the mean field approximation. The quantum system is formed by a single mode optical cavity and two level atoms, which interact with reservoirs. Namely, we establish the existence and uniqueness of the regular solution to the nonlinear operator equation under consideration, as well as we get a probabilistic representation for this solution in terms of a mean field stochastic Schrödinger equation. To this end, we find a regular solution for the nonautonomous linear quantum master equation in Gorini–Kossakowski–Sudarshan–Lindblad form, and we prove the uniqueness of the solution to the nonautonomous linear adjoint quantum master equation in Gorini–Kossakowski–Sudarshan–Lindblad form. Moreover, we obtain rigorously the Maxwell–Bloch equations from the mean field laser equation.
The quantum kinetic equation describing slow atomic motion in laser light is derived by an operational method which provides mathematical expressions with a transparent physical structure. The authors prove in a general way that the coefficients appearing in this equation, which is of a Fokker-Planck type, are simply related to the mean value and to the correlation functions of the Heisenberg radiative force of the semiclassical approach, where the atomic position is treated classically. They derive in particular a new theoretical expression for the damping force responsible for radiative cooling and they interpret it in terms of linear response theory. The authors also obtain a new crossed r-p derivative term, which does not appear in semiclassical treatments, but which they find to be very small in most situations. Finally, all the theoretical expressions derived are valid for any J g to J e transitions and are not restricted to two-level atoms.
A quantum theory for noise in the steady state of the laser oscillator above threshold is presented. The concept of operator noise source is introduced. A simple laser model operating in a single mode with resonant frequency tuned to the center of the material line is considered. The resonance linewidth of the inverted material is collision-broadened, and assumed to be much broader than the cold-cavity width. We restrict ourselves to one type of collision. The quantum description of this laser model leads to nonlinear operator equations. These are linearized, but this linearization is justified only for operation "sufficiently" above threshold. The first- and second-order Glauber functions and the expectation value of the field commutator are derived for the field inside the laser cavity. These results are compared with the predictions of a "semiclassical" theory in which the classical equations contain noise sources such that properties of the field below threshold are correctly predicted. Our results differ by certain terms from the results of this semiclassical theory. These terms are too small to have been detected experimentally.