Rotating Wave Approximation

In the Heisenberg picture, the time dependence for an operator Q can be written as

(3.54)

where the operator Q does not need to have explicit time dependence.

For the annihilation operator the Heisenberg equation becomes

(3.55)

which has the time dependant solution

(3.56)

In the same way, it can be shown that

(3.57)

We have already shown that the total Hamiltonian is given by

(3.58)

With the help of the time dependant solutions for the operators, the operator products look like

(3.59)

We consider the case of little detuning between the field frequency ω0 and the atoms transition frequency ω, i.e. ω0 ≈ ω. In this case the terms which contain (ω0 + ω) vary much faster than the other two. Furthermore, not all the terms conserve energy:

(3.60)

At this point we are making the RWA. This means we are going to drop the terms which do not obey conservation of energy to approximate the total Hamiltonian of this system. Our total Hamiltonian becomes

(3.61)

where

(3.62)

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