In spectroscopy, the Autler–Townes effect (also known as AC Stark effect), is a dynamical Stark effect corresponding to the case when an oscillating electric field (e.g., that of a laser) is tuned in resonance (or close) to the transition frequency of a given spectral line, and resulting in a change of the shape of the absorption/emission spectra of that spectral line. The AC Stark effect was discovered in 1955 by American physicists Stanley Autler and Charles Townes.
It is the AC equivalent of the static Stark effect which splits the spectral lines of atoms and molecules in a constant electric field. Compared to its DC counterpart, the AC Stark effect is computationally more complex.
While generally referring to atomic spectral shifts due to AC fields at any (single) frequency, the effect is more pronounced when the field frequency is close to that of a natural atomic or molecular dipole transition. In this case, the alternating field has the effect of splitting the two bare transition states into doublets or "dressed states" that are separated by the Rabi frequency. Alternatively, this can be described as a Rabi oscillation between the bare states which are no longer eigenstates of the atom–field Hamiltonian. The resulting fluorescence spectrum is known as a Mollow triplet.
The AC Stark splitting is integral to several phenomena in quantum optics, such as electromagnetically induced transparency and Sisyphus cooling. Vacuum Rabi oscillations have also been described as a manifestation of the AC Stark effect from atomic coupling to the vacuum field.
History
The AC Stark effect was discovered in 1955 by American physicists Stanley Autler and Charles Townes while at Columbia University and Lincoln Labs at the Massachusetts Institute of Technology. Before the availability of lasers, the AC Stark effect was observed with radio frequency sources. Autler and Townes' original observation of the effect used a radio frequency source tuned to 12.78 and 38.28 MHz, corresponding to the separation between two doublet microwave absorption lines of OCS.
The notion of quasi-energy in treating the general AC Stark effect was later developed by Nikishov and Ritis in 1964 and onward. This more general method of approaching the problem developed into the "dressed atom" model describing the interaction between lasers and atoms.
Prior to the 1970s there were various conflicting predictions concerning the fluorescence spectra of atoms due to the AC Stark effect at optical frequencies. In 1974 the observation of Mollow triplets verified the form of the AC Stark effect using visible light.
General semiclassical approach
In a semiclassical model where the electromagnetic field is treated classically, a system of charges in a monochromatic electromagnetic field has a Hamiltonian that can be written as:
where , , and are respectively the position, momentum, mass, and charge of the -th particle, and is the speed of light. The vector potential of the field, , satisfies
.
The Hamiltonian is thus also
periodic:
Now, the Schrödinger equation, under a periodic Hamiltonian is a linear homogeneous differential equation with periodic coefficients,
where here represents all coordinates. Floquet's theorem guarantees that the solutions to an equation of this form can be written as
Here, is the "bare" energy for no coupling to the electromagnetic field, and has the same time-periodicity as the Hamiltonian,
or
with the angular frequency of the field.
Because of its periodicity, it is often further useful to expand in a Fourier series, obtaining
or
where is the frequency of the laser field.
The solution for the joint particle-field system is, therefore, a linear combination of stationary states of energy , which is known as a quasi-energy state and the new set of energies are called the spectrum of quasi-harmonics.
Unlike the DC Stark effect, where perturbation theory is useful in a general case of atoms with infinite bound states, obtaining even a limited spectrum of shifted energies for the AC Stark effect is difficult in all but simple models, although calculations for systems such as the hydrogen atom have been done.
Examples
General expressions for AC Stark shifts must usually be calculated numerically and tend to provide little insight. However, there are important individual examples of the effect that are informative. Analytical solutions in these specific cases are usually obtained assuming the detuning is small compared to a characteristic frequency of the radiating system.