Shape-Invariant Solutions of the Quantum Fokker-Planck Equation for an Optical Oscillator
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 (t)=∫P(α,α,t)|α><α|dα, the so-called -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 (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 , .
∂P
∂t
P(α,α)=
*
e
-
|α|
2
_
n
π
_
n
_
n
t=0
P(α,α,t)=
*
e
-1-e
|α|
2
_
n
-γt
π(1-e)
_
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)