ShapeInvariant Solutions of the Quantum FokkerPlanck Equation for an Optical Oscillator
ShapeInvariant Solutions of the Quantum FokkerPlanck 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 timedependent Hermitian operator can be represented by a realvalued function of the form (t)=∫P(α,α,t)α><αdα, the socalled representation. The equation of motion for is the Fokker–Planck equation (FPE) =(αP)+(αP)+γP, where is the decay constant of the optical mode and denotes the mean number of quanta in the thermal reservoir.
^
ρ
α
^
ρ
P(α,α,t)
*
^
ρ
*
2
P
P(α,α,t)
*
∂P
∂t
γ
2
∂
∂α
∂
∂α
*
*
_
n
∂
2
∂α∂α
*
γ
_
n
The normalized stationary solution of this equation with the property =0 (steadystate solution) has the simple form (this Gaussian function means a thermal distribution with average value ), and a timedependent 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 shapeinvariant 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 , .
∂P
∂t
P(α,α)=
*
e

α
2
_
n
π
_
n
_
n
t=0
P(α,α,t)=
*
e
1e
α
2
_
n
γt
π(1e)
_
n
γt
But it seems difficult to find other analytical solutions starting at
t=0
α≠0
[0≤t<∞)
t∞
P(α,α,0)
*
x=0
y=1
x=1
y=0
x=Re(α)
y=Im(α)