Rabi frequency
Encyclopedia
The Rabi frequency is the frequency of population oscillation for a given atomic transition in a given light field. It is associated with 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.)
where
is the transition dipole moment for the 
transition and 
is the vector electric field
amplitude
which includes the polarization. The numerator has dimensions of energy, so dividing by
gives an angular frequency
. Although
, one cannot assume that 
as 
may be complex, as in the case of circularly polarized light.
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
, where 
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.
In real systems, with more than two levels, the correct dipole transition matrix element for the relevant transition has to be calculated. When equating the Rabi frequency the correct Clebsch-Gordan coefficient has to be considered which is a common source of mistakes throughout scientific literature. There are also some publications that use a different definition for the Rabi frequency which is by a factor of two smaller than the one introduced here.
. Rabi flopping actually occurs at the generalized Rabi frequency.
where
is the detuning
, a measure of how far the light is off resonance with the transition.
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
atomic transition
Energy level
A quantum mechanical system or particle that is bound -- that is, confined spatially—can only take on certain discrete values of energy. This contrasts with classical particles, which can have any energy. These discrete values are called energy levels...
and a classical
Classical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...
light field
Electromagnetic field
An electromagnetic field is a physical field produced by moving electrically charged objects. It affects the behavior of charged objects in the vicinity of the field. The electromagnetic field extends indefinitely throughout space and describes the electromagnetic interaction...
.
In the context of a nuclear magnetic resonance
Nuclear magnetic resonance
Nuclear magnetic resonance is a physical phenomenon in which magnetic nuclei in a magnetic field absorb and re-emit electromagnetic radiation...
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
Larmor precession
In physics, Larmor precession is the precession of the magnetic moments of electrons, atomic nuclei, and atoms about an external magnetic field...
, which characterizes the precession of a transverse nuclear magnetization about a static magnetic field.)
Definition
The Rabi frequency is defined aswhere
is the transition dipole moment for the transition and is the vector electric fieldElectric field
In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...
amplitude
Amplitude
Amplitude is the magnitude of change in the oscillating variable with each oscillation within an oscillating system. For example, sound waves in air are oscillations in atmospheric pressure and their amplitudes are proportional to the change in pressure during one oscillation...
which includes the polarization. The numerator has dimensions of energy, so dividing by
gives an angular frequencyAngular frequency
In physics, angular frequency ω is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity...
. Although
, one cannot assume that as may be complex, as in the case of circularly polarized light.By analogy with a classical
Classical physics
What "classical physics" refers to depends on the context. When discussing special relativity, it refers to the Newtonian physics which preceded relativity, i.e. the branches of physics based on principles developed before the rise of relativity and quantum mechanics...
dipole
Dipole
In physics, there are several kinds of dipoles:*An electric dipole is a separation of positive and negative charges. The simplest example of this is a pair of electric charges of equal magnitude but opposite sign, separated by some distance. A permanent electric dipole is called an electret.*A...
, it is clear that an atom with a large dipole moment will be more susceptible to perturbation by electric
Electric field
In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...
and magnetic
Magnetic field
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...
fields. The dot product
Dot product
In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products...
includes a factor of
, where 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.In real systems, with more than two levels, the correct dipole transition matrix element for the relevant transition has to be calculated. When equating the Rabi frequency the correct Clebsch-Gordan coefficient has to be considered which is a common source of mistakes throughout scientific literature. There are also some publications that use a different definition for the Rabi frequency which is by a factor of two smaller than the one introduced here.
Generalized Rabi frequency
For light that is off resonance with a transition, it is common to define the generalized Rabi frequency. Rabi flopping actually occurs at the generalized Rabi frequency.where
is the detuningLaser detuning
In optical physics, laser detuning is the tuning of a laser to a frequency that is slightly off from a quantum system's resonant frequency. Lasers can be detuned in the lab frame so that they are Doppler shifted to the resonant frequency in a moving system, which allows lasers to affect only...
, a measure of how far the light is off resonance with the transition.