Shape-Invariant Solutions of the Quantum Fokker-Planck Equation for an Optical Oscillator
In quantum optics an equation of motion for the density operator of an optical harmonic oscillator damped by a thermal bath of oscillators (reservoir) is derived. The optical mode may be described by the complex amplitude . The time-dependent Hermitian operator can be represented by a real-valued function of the form , the so-called -representation. The equation of motion for is the Fokker–Planck equation (FPE) , where is the decay constant of the optical mode and denotes the mean number of quanta in the thermal reservoir.[more]
The normalized stationary solution of this equation with the property (steady-state solution) has the simple form (this Gaussian function means a thermal distribution with average value ), and a time-dependent solution with a singularity at is (this known function is not demonstrated here). But it seems difficult to find other analytical solutions starting at with a maximum at . Here we show two of three completely shape-invariant solutions of FPE defined on the complete time interval that tends for to the stationary solution, and the initial functions have maximal values at , and , , where , .[less]
In their original meaning, the Fokker–Planck equations are partial differential equations for conditional probabilities in the theory of Markov processes, a special class of stochastic processes. In the quantum theory of a damped optical oscillator, Markovian properties also lead to a quantum FPE, valid for the quasi-probability density .
The above functions were found as follows.
In the real variables , , the FPE has the form . The reduction to a one-dimensional equation of motion gives a simplified FPE, more complicated than the heat conduction equation:
. From the method of separation of variables in and , the formal solution is (special case: ):
, where are Hermite polynomials of order . The coefficients follow from the initial condition : . The problem is that to obtain an analytical solution for , either the integration or the summation cannot be carried out for many known elementary initial functions . In the case chosen here, when that the initial condition is the normalized function , then both operations (integration and summation) are possible; the result is . Returning to the original two-dimensional problem is easy: is one of three quasi-probability densities normalized for all (this Demonstration shows two of three calculated functions).
 M. O. Scully and M. S. Zubairy, Quantum Optics, Cambridge: Cambridge University Press, 1997.
 P. Meystre and M. Sargent III, Elements of Quantum Optics, Berlin Heidelberg: Springer Verlag, 1991.
 J. Perina, Coherence of Light, Dordrecht-Boston-Lancaster: D. Reidel Publishing Company, 1985.
 M. Abramowitz and I. A. Stegun, Pocketbook of Mathematical Functions, Frankfurt: Verlag Harri Deutsch, 1984.
 W. I. Smirnow, Lehrgang der Höheren Mathematik, Teil 2: Frankfurt: Verlag Harri Deutsch, 2004.
 W. S. Wladimirow, Gleichungen der mathematischen Physik, Berlin: Deutscher Verlag der Wissenschaften, 1972.