Evaluate
Evaluate Cell
Cut
Copy
Copy as
Plain Text
Input Text
Image
Paste
Convert to
InputForm
StandardForm
TraditionalForm
Image
Clear Formatting
Formatting...
Divide Cell
Merge Cells
Publish/Share Cell
Embed...
Email Cell to
Me
Others...
Email Cell...
Deploy Cell as
Instant API
Instant Web Computation
Instant Web Form
Embeddable Code
Advanced
Show Cell Expression...
Cell Update History...
WOLFRAM|DEMONSTRATIONS PROJECT

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|d
2
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
*
)=
e
-
|α|
2
_
n
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)=
e
-
|α|
2
_
n
1-e
-γt

π
_
n
(1-e
-γt
)
(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)
.