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:
Write the Hamiltonian with two optical fields, one acting only on the
a→b transition, and acting only on the
b→c transition:
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: