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.
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.
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