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

​
t
0
In quantum optics an equation of motion for the density operator
ρ
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
α
. The time-dependent Hermitian operator
ρ
can be represented by a real-valued function
P(α,
*
α
,t)
of the form
ρ
(t)=∫P(α,
*
α
,t)α><α
2
d
α
, the so-called
P
-representation. The equation of motion for
P(α,
*
α
,t)
is the Fokker–Planck equation (FPE)
∂P
∂t
=
γ
2
∂
∂α
(αP)+
∂
∂
*
α
(
*
α
P)+γ
_
n
2
∂
∂α∂
*
α
P
, where
γ
is the decay constant of the optical mode and
_
n
denotes the mean number of quanta in the thermal reservoir.
The normalized stationary solution of this equation with the property
∂P
∂t
=0
(steady-state solution) has the simple form
P(α,
*
α
)=
-
α
2
|
n
e
π
_
n
(this Gaussian function means a thermal distribution with average value
n
), and a time-dependent solution with a singularity at
t=0
is​
P(α,
*
α
,t)=
-
α
2
|
_
n
1-
-γt
e
e
π
_
n
(1-
-γt
e
)
(this known function is not demonstrated here).But it seems difficult to find other analytical solutions starting at
t=0
with a maximum at
α≠0
. Here we show two of three completely shape-invariant solutions of FPE defined on the complete time interval
[0≤t<∞)
that tends for
t∞
to the stationary solution, and the initial functions
P(α,
*
α
,0)
have maximal values at
x=0
,
y=1
and
x=1
,
y=0
, where
x=Re(α)
,
y=Im(α)
.
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