Optical Bloch equations
For an electron in a two-level system with time-independent states
(ground) and (excited),
consider the following general solution
to the time-dependent Schrödinger equation:
Perturbing this system with
an electromagnetic wave
introduces a time-dependent sinusoidal term to the Hamiltonian.
In the electric dipole approximation,
is given by:
Where is an electric field,
and is the dipole moment operator.
From Rabi oscillation,
we know that the time-varying coefficients and
can then be described by:
Where is the resonance frequency.
We want to rearrange these equations a bit,
so we split the field as follows,
where the amplitudes and
may be slowly-varying with respect to the carrier wave :
Since is real, .
Similarly, we define the transition dipole moment :
With these, the equations for and can be rewritten as shown below.
Note that and include the driving plane wave, and the
rotating wave approximation is still made:
For as defined above,
the corresponding pure density operator
is as follows:
We take the -derivative of the matrix elements,
and insert the equations for and :
Recognizing the density matrix elements allows us
to reduce these equations to:
These equations are correct if nothing else is affecting .
But in practice, these quantities decay due to various processes,
e.g. spontaneous emission.
Suppose decays with rate .
Because the total probability , we have:
Meanwhile, for whatever reason,
let decay into with rate :
And finally, let the diagonal (perpendicular) matrix elements
both decay with rate :
Putting everything together,
we arrive at the optical Bloch equations governing ,
which are the basis of the
and by extension all laser theory:
Some authors simplify these equations a bit by choosing
- F. Kärtner,
Ultrafast optics: lecture notes,
2005, Massachusetts Institute of Technology.
- H.J. Metcalf, P. van der Straten,
Laser cooling and trapping,