Rabi frequency

It is proportional to the transition dipole moment of the two levels and to the amplitude (not intensity) of the electromagnetic field.

Population transfer between the levels of such a 2-level system illuminated with light exactly resonant with the difference in energy between the two levels will occur at the Rabi frequency; when the incident light is detuned from this energy difference (detuned from resonance) then the population transfer occurs at the generalized Rabi frequency.

The Rabi frequency is a semiclassical concept since it treats the atom as an object with quantized energy levels and the electromagnetic field as a continuous wave.

In the context of a nuclear magnetic resonance experiment, the Rabi frequency is the nutation frequency of a sample's net nuclear magnetization vector about a radio-frequency field.

Consider two energy eigenstates of a quantum system with Hamiltonian

Treating the potential as a perturbation, we can expect the eigenstates of the perturbed Hamiltonian to be some mixture of the eigenstates of the original Hamiltonian with time dependent coefficients:

Plugging this into the time dependent Schrödinger equation

The two terms in parentheses are dipole matrix elements dotted into the polarization vector of the electromagnetic field.

of the Hydrogen atom potential, the diagonal matrix elements go to zero, leaving us with

transition, whose squared amplitude represents the strength of the interaction between the electromagnetic field and the atom, and

is the vector electric field amplitude, which includes the polarization.

is the angle between the polarization of the light and the transition dipole moment.

, such that we can discard the high frequency oscillating terms, we have

is called the detuning between the laser and the atomic frequencies.

This is the probability as a function of detuning and time of the population of state

A plot as a function of detuning and ramping the time from 0 to

is commonly referred to as the "generalized Rabi frequency."

is the detuning, a measure of how far the light is off-resonance relative to the transition.

For instance, examining the above animation at an offset frequency of ±1.73, one can see that during the 1/2 Rabi cycle (at resonance) shown during the animation, the oscillation instead undergoes one full cycle, thus at twice the (normal) Rabi frequency

Also note that as the incident light frequency shifts further from the transition frequency, the amplitude of the Rabi oscillation decreases, as is illustrated by the dashed envelope in the above plot.

Coherent Rabi oscillations may also be driven by two-photon transitions.

In this case we consider a system with three atomic energy levels,

Instead, the atom absorbs two photons simultaneously and is promoted directly between the initial and final states.

The beat note of the two photons must be resonant with the two-photon transition (difference between initial and final state frequencies):

Delta determines the rate of scattering off of the intermediate state.

We may derive the two-photon Rabi frequency by returning to the equations

which now describe excitation between the ground and intermediate states.

is the generalized Rabi frequency for the transition from the initial to intermediate state.

Similarly for the intermediate to final state transition we have the equations

transitions, divided by the detuning from the intermediate state