# Shape-Invariant Solutions of the Quantum Fokker-Planck Equation for an Optical Oscillator

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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.

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Contributed by: Reinhard Tiebel (July 2011)

Open content licensed under CC BY-NC-SA

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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).

References

[1] M. O. Scully and M. S. Zubairy,* Quantum Optics*, Cambridge: Cambridge University Press, 1997.

[2] P. Meystre and M. Sargent III, *Elements of Quantum Optics, *Berlin Heidelberg: Springer Verlag, 1991.

[3] J. Perina, *Coherence of Light, *Dordrecht-Boston-Lancaster: D. Reidel Publishing Company, 1985.

[4] M. Abramowitz and I. A. Stegun, *Pocketbook of Mathematical Functions,* Frankfurt: Verlag Harri Deutsch, 1984.

[5] W. I. Smirnow, *Lehrgang der Höheren Mathematik*, Teil 2: Frankfurt: Verlag Harri Deutsch, 2004.

[6] W. S. Wladimirow, *Gleichungen der mathematischen Physik, *Berlin: Deutscher Verlag der Wissenschaften, 1972.