The Optical Bloch Equations
The Fermi Golden Rule is valid for sufficiently small value of V and E0. If we are interested in following the dynamics when V and E0 are large we derive terms for the coefficients c1(t) and c2(t) in terms of atomic (or molecular) populations and coherences. By doing this we define the density matrix:
r
11 = |c1|2 = N1/Nr
22 = |c2|2 = N2/Nr
12 = c1c2*r
21 = c2c1*The diagonal elements are real and satisfy r11 + r22 = 1. The off-diagonal elements are complex and satisfy r21 = r12*. The equations of motion are readily found from the equations of motion of the coefficients.
with substitutions from
it is easily shown that
We can replace cos(
wt) with
and then apply the rotating wave approximation to obtain
These are the optical Bloch equations. They are similar to equations by Bloch to describe the motion of a spin in a magnetic field. The quantum mechanics of the two level atom or molecule described here is formally identical to that of a spin 1/2 system.
Rabi Oscillations
The above equations represent coupled first order differential equations. They can solved using the following trial solutions:
Substitution of these trial solutions into the above equations yields
The possible values of
l are the roots of the 4x4 matrix. The determinant of the coefficients yields the quartic equation:
The distinct roots are
The most general solutions for the density matrix elements have the form:
Together with the initial conditions at t=0 these equations provide sufficient information to determine the coefficients.
For example, for the initial conditions
r22 = 0 and r12 = 0:
The detuning is the mismatch between the transition frequency and the applied electric field frequency,
w0 - w. For example, if the detuning is zero and w0 = w, then
The oscillations occur between the ground |1> and excited |2> states as shown below. The oscillations are called Rabi oscillations and |V| is called the Rabi frequency.
Notice that the greater the detuning the higher the frequency of oscillation and the lower the amplitude. The lower amplitude means that less of the excited state |2> is being formed.