The Rabi frequency is the frequency at which the
probability amplitudes of two
atomic energy levels fluctuate in an oscillating electromagnetic field. It is proportional to the
Transition Dipole Moment
The transition dipole moment or transition moment, usually denoted \mathbf_ for a transition between an initial state, m, and a final state, n, is the electric dipole moment associated with the transition between the two states. In general the tra ...
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
A quantum mechanical system or particle that is bound—that is, confined spatially—can only take on certain discrete values of energy, called energy levels. This contrasts with classical particles, which can have any amount of energy. The t ...
and the electromagnetic field as a continuous wave.
In the context of a
nuclear magnetic resonance
Nuclear magnetic resonance (NMR) is a physical phenomenon in which nuclei in a strong constant magnetic field are perturbed by a weak oscillating magnetic field (in the near field) and respond by producing an electromagnetic signal with a ...
experiment, the Rabi frequency is the nutation frequency of a sample's net nuclear magnetization vector about a radio-frequency field. (Note that this is distinct from the
Larmor frequency
In physics, Larmor precession (named after Joseph Larmor) is the precession of the magnetic moment of an object about an external magnetic field. The phenomenon is conceptually similar to the precession of a tilted classical gyroscope in an extern ...
, which characterizes the precession of a transverse nuclear magnetization about a static magnetic field.)
Derivation
Consider two energy eigenstates of a quantum system with Hamiltonian
(for example, this could be the Hamiltonian of a particle in a
potential, like the Hydrogen atom or the Alkali atoms):
We want to consider the time dependent Hamiltonian
where
is the potential of the electromagnetic field. Treating the potential as a
perturbation
Perturbation or perturb may refer to:
* Perturbation theory, mathematical methods that give approximate solutions to problems that cannot be solved exactly
* Perturbation (geology), changes in the nature of alluvial deposits over time
* Perturbat ...
, 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 Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
taking the inner product with each of
and
, and using the orthogonality condition of eigenstates
, we arrive at two equations in the coefficients
and
:
where
. The two terms in parentheses are
dipole matrix elements dotted into the polarization vector of the electromagnetic field. In considering the spherically symmetric spatial eigenfunctions
of the
Hydrogen atom potential, the diagonal matrix elements go to zero, leaving us with
or
Here,
is the Rabi Frequency.
Intuition
In the numerator we have the
transition dipole moment
The transition dipole moment or transition moment, usually denoted \mathbf_ for a transition between an initial state, m, and a final state, n, is the electric dipole moment associated with the transition between the two states. In general the tra ...
for the
transition, whose squared amplitude represents the strength of the interaction between the electromagnetic field and the atom, and
is the
vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
electric field amplitude
The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
, which includes the
polarization. The numerator has dimensions of energy, so dividing by
gives an
angular frequency
In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit tim ...
.
By analogy with a
classical dipole
In physics, a dipole () is an electromagnetic phenomenon which occurs in two ways:
*An electric dipole deals with the separation of the positive and negative electric charges found in any electromagnetic system. A simple example of this system ...
, it is clear that an atom with a large dipole moment will be more susceptible to perturbation by an
electric
Electricity is the set of physical phenomena associated with the presence and motion of matter that has a property of electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as described by ...
field. The
dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alge ...
includes a factor of
, where
is the angle between the polarization of the light and the transition dipole moment. When they are parallel the interaction is strongest, when they are perpendicular there is no interaction at all.
If we rewrite the differential equations found above:
and apply the
Rotating-wave approximation, which assumes that
, such that we can discard the high frequency oscillating terms, we have
where
. We can solve these equations, assuming at time
the atom is in
(i.e.
) to find
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
gives:
![Animation of optical resonance, frequency domain](https://upload.wikimedia.org/wikipedia/commons/f/f8/Animation_of_optical_resonance%2C_frequency_domain.gif)
We see that for
the population will oscillate between the two states at the Rabi frequency.
Generalized Rabi frequency
The quantity
is commonly referred to as the "generalized Rabi frequency." For cases in which
,
Rabi flopping actually occurs at this frequency, where
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
, just as predicted by this equation. 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.
Two-Photon Rabi Frequency
Coherent Rabi oscillations may also be driven by
two-photon transitions. In this case we consider a system with three atomic energy levels,
,
, and
, where
represents a so-called intermediate state with corresponding frequency
, and an electromagnetic field with two frequency components:
Now,
may be much greater than both
and
, or
, as illustrated in the figure on the right.
![Two_Photon_Excitation_Schema](https://upload.wikimedia.org/wikipedia/commons/a/ac/Two_Photon_Excitation_Schema.png)
A two-photon transition is ''not'' the same as excitation from the ground to intermediate state, and then out of the intermediate state to the excited state. Instead, the atom absorbs two photons simultaneously and is promoted directly between the initial and final states. There are two necessary conditions for this two-photon process (also known as a
Raman process), to be the dominate model of the light-matter interaction:
In words, the sum of the frequencies of the two photons must be on resonance with the transition between the initial and final states, and the individual frequencies of the photons must be detuned from the intermediate state to initial and final state transitions. If the latter condition is not met and
, the dominate process will be one governed by rate equations in which the intermediate state is populated and stimulated and
Spontaneous emission events from that state prevent the possibility of driving coherent oscillations between the initial and final states.
We may derive the two-photon Rabi frequency by returning to the equations
which now describe excitation between the ground and intermediate states. We know we have the solution
where
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
Now we plug
into the above equation for
Such that, upon solving this equation, we find the coefficient to be proportional to:
This is the effective or two-photon Rabi frequency.
It is the product of the individual Rabi frequencies for the
and
transitions, divided by the detuning from the intermediate state
.
See also
*
Rabi cycle
In physics, the Rabi cycle (or Rabi flop) is the cyclic behaviour of a two-level quantum system in the presence of an oscillatory driving field. A great variety of physical processes belonging to the areas of quantum computing, condensed matter, ...
*
Vacuum Rabi oscillation
*
Rabi resonance method
References
{{reflist, 1
Quantum optics
Atomic physics
Atomic, molecular, and optical physics