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# Rabi frequency

The Rabi frequency for a given atomic transition in a given light field gives the strength of the coupling between the light and the transition. Rabi flopping between the levels of a 2-level system illuminated with resonant light, will occur at the Rabi frequency. The Rabi frequency is a semiclassical concept as it is based on a quantum atomic transition and a classical light field.

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 radiofrequency field. (Note that this is distinct from the Larmor frequency, which characterizes the precession of a transverse nuclear magnetization about a static magnetic field.)

## Definition

$\chi_{i,j} = {\vec{d}_{i,j}.\vec{E}_0 \over \hbar}$

where $\scriptstyle{\vec{d}_{i,j}}$ is the transition dipole moment for the $\scriptstyle{i \rightarrow j}$ transition and $\scriptstyle{\vec{E}_0 = \hat{\epsilon}E_0}$ is the vector electric field amplitude which includes the polarization.

The numerator has dimensions of energy, dividing by $\scriptstyle{\hbar}$ gives an angular frequency. Although $\scriptstyle{\vec{d}_{i,j} = \vec{d}_{j,i}^*}$, one cannot assume that $\scriptstyle{\chi_{i,j} = \chi_{j,i}^*}$ as $\scriptstyle{\hat{\epsilon}E_0}$ may be complex, as in the case of circularly polarized light.[1]

By analogy with a classical dipole, it is clear that an atom with a large dipole moment will be more susceptible to perturbation by electric and magnetic fields. The dot product includes a factor of cosθ, where θ is the angle between the polarization of the light and the transition dipole moment. When they are parallel or antiparallel the interaction is strongest, when they are perpendicular there is no interaction at all. The vector electric field amplitude defines both the intensity and the polarization of the light.

## Generalized Rabi frequency

For light that is off resonance with a transition, it is common to define the generalized Rabi frequency Ωi,j. Rabi flopping actually occurs at the generalized Rabi frequency.

$\Omega_{i,j} = \sqrt{|\chi_{i,j}|^2 + \Delta^2}$

where Δ = ωlight − ωtransition is the detuning, a measure of how far the light is off resonance with the transition.