Define a two-level system.
Find the Hamiltonian with states coupled by an optical field with angular frequency
.
Apply the rotating-wave approximation, specifying that the field of frequency
couples the lower state with label 1 to the upper state with label 2.
Specifying the detuning
is equivalent to making the replacement
→Energy[2]+.
For this system,
RotatingWaveApproximation can guess that state 1 is the lower state and state 2 is the upper state, so it is not necessary to specify the transition.
Here is an example with three levels and two fields.
Define a three level system.
Find the Hamiltonian with two optical fields. We make the assumption that the field
_{1} only interacts with the
a→b transition, while the field
_{2} only interacts with the
b→c transition, by setting the additional terms to zero.
Here we apply the RWA assuming that
Energy[a]<Energy[b]<Energy[c].
The two-photon transition between states
a and
c is resonant when the energies of the two states in the rotating-frame Hamiltonian are equal; in this case, when
0=Energy[c]-_{1}-_{2}, or
_{1}+_{2}=Energy[c]. This corresponds to the level diagram for the system:
If, instead, we assume that
Energy[a]<Energy[c]<Energy[b], we obtain the rotating-frame Hamiltonian:
Here, the two-photon resonance condition is
_{1}-_{2}=Energy[c], corresponding to the level diagram: