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The AC Stark effect (also known as the dynamic(al) Stark effect or Autler-Towns effect) refers to the shift of energy levels for an atom or molecule in an alternating electric field. It is the AC equivalent of the Stark effect which splits the spectral lines of atoms and molecules in a constant electric field. Compared to its DC counterpart though, the fields in the AC effect are typically much larger, and the effects are much harder to predict.

While generally referring to atomic spectral shifts due to AC fields at any (single) frequency, the effect is dramatically enhanced when the field is tuned to the frequency of a natural two level 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. This is commonly achieved by a laser tuned to (or near) the desired transition.

This splitting results in a Rabi cycle or Rabi oscillation between bare states which are no longer energy eigenstates of the atom-field Hamiltonian. The resulting fluorescence spectrum of an atom is known as a Mollow triplet. The AC stark splitting is integral to several other phenomena in quantum optics, such as Electromagnetically induced transparency and Sisyphus cooling. Vacuum Rabi oscillations have also been described as manifestation of the AC Stark effect from atomic coupling to the vacuum field.

History
Often also referred to as the Autler-Townes effect, the AC stark effect was discovered in 1955 by the eponymous American physicists Stanley Autler and Charles Townes while at Columbia University and Lincoln Labs at the Massachusetts Institute of Technology. Before the availability of shorter wavelength coherent sources (namely, 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 celebrated "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:

$$ H=\sum_i \frac{1}{2m_i}\left[\mathbf{p}_i-\frac{q_i}{c}\mathbf{A(\mathbf{r}_i, t)}\right]^2 +V(\mathbf{r}_i), \ \ \ \ \ \ \ \ \ \ \ $$

where $$\mathbf{r}_i \,$$, $$\mathbf{p}_i \,$$, $$m_i \,$$ and $$q_i \,$$ are respectively the position, momentum, mass, and charge of the $$i\,$$-th particle, and $$c \,$$ is the speed of light. The vector potential of the field, $$\mathbf{A}$$, satisfies

$$\mathbf{A}(t+\tau)=\mathbf{A}(t)$$.

The Hamiltonian is thus also periodic:

$$ H(t+\tau) = H(t). $$

Now, the Schrödinger equation, under a periodic Hamiltonian is a linear homogeneous differential equation with periodic coefficients,



i\hbar \frac{\partial}{\partial t} \psi(\mathbf{\xi},t) = H(t)\psi(\mathbf{\xi},t), $$

where $$\xi$$ here represents all coördinates. Floquet's theorem guarantees that the solutions to an equation of this form can be written as $$ \psi(\mathbf{\xi},t) = \exp[-iE_bt/\hbar]\phi(\mathbf{\xi},t). $$

Here, $$E_b$$ is the "bare" energy for no coupling to the electromagnetic field, and $$\phi\,$$ has the same time-periodicity as the Hamiltonian,

$$ \phi(\mathbf{\xi},t+\tau) = \phi(\mathbf{\xi},t) $$

or

$$ \phi(\mathbf{\xi},t+2\pi/\omega) = \phi(\mathbf{\xi},t) $$

with $$\omega=2\pi/\tau$$ the angular frequency of the field.

Because of it's periodicity, it is often further useful to expand $$\phi(\mathbf{\xi},t)$$ in a Fourier series, obtaining

\psi(\mathbf{\xi},t) = \exp[-iE_bt/\hbar] \sum_{k=-\infty}^{\infty}C_k(\mathbf{\xi})\exp[-ik\omega t] $$

or

$$ \psi(\mathbf{\xi},t) = \sum_{k=-\infty}^{\infty}C_k(\mathbf{\xi})\exp[-i(E_b+k\omega)t/\hbar] \ \ \ \ \ \ \ \ \ \ \ $$

where $$\omega =2\pi/T\,$$ 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 $$E_b+k\omega$$, 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.

Two level atom dressing
An atom driven by an electric field with frequency $$\omega_0$$ close to an atomic transition (consider detuning $$\Delta<<\omega_0$$) can be approximated as a two level quantum system since the off resonance states have low occupation probability. The Hamiltonian can be divided into the bare atom term plus a term for the interaction with the field as:

$$ \hat{H} = \hat{H}_{A} + \hat{H}_{int}. $$

In an appropriate rotating frame, and making the rotating wave approximation, $$\hat{H}$$ reduces to

$$ \hat{H}=-\hbar\Delta|e\rangle\langle e|+\frac{\hbar\Omega}{2}(|e\rangle\langle g|+|g\rangle \langle e|). $$

Where $$\Omega$$ is the Rabi frequency, and $$|g\rangle, |e\rangle$$ are the strongly coupled bare atom states. The energy eigenvalues are

$$ E_{\pm}=\frac{-\hbar\Delta}{2}\pm\frac{\hbar\sqrt{\Omega^2+\Delta^2}}{2} $$

and for small detuning,

$$ E_{\pm}\approx\pm\frac{\hbar\Omega}{2}. $$

The eigenstates of the atom-field system or dressed states are dubbed $$|+\rangle$$ and $$|-\rangle$$.

The result of the AC field on the atom is thus to shift the strongly coupled bare atom energy eigenstates states into two states $$|+\rangle$$ and $$|-\rangle$$ which are now separated by $$\hbar\Omega$$. Evidence of this shift is apparent in the atom's absorption spectrum, which shows two peaks around the bare transition frequency, seperated by $$\Omega$$ (Autler-Townes splitting). The modified absorption spectrum can be obtained by a pump-probe experiment, wherein a strong pump laser drives the bare transition while a weaker probe laser sweeps for a second transition between a third atomic state and the dressed states.

Another consequence of the AC Stark splitting here is the appearance of Mollow triplets, a triple peaked fluorescence profile. Historically an important confirmation of Rabi flopping, they were first predicted by Mollow in 1969 and confirmed in the 1970s experimentally.

Adiabatic elimination
In quantum system with three (or more) states, where a transition from one level, $$|g\rangle$$ to another $$|e\rangle$$ can be driven by an AC field, but $$|e\rangle$$ only decays to states other than $$|g\rangle$$, the dissipative influence of the spontaneous decay can be eliminated. This is achieved by increasing the AC Stark shift on $$|g\rangle$$ through large detuning and raising intensity of the driving field. Adiabatic elimination has been used to create comparatively stable effective two level systems in Rydberg atoms, which are of interest for qubit manipulations in quantum computing.

Electromagnetically induced transparency
Electromagnetically induced transparency (EIT), which gives some materials a small transparent area within an absorption line, can be thought of as a combination of Autler-Townes splitting and Fano interference, although the distinction may be difficult to determine experimentally. While both Autler-Towns splitting and EIT can produce a transparent window in an absorption band, EIT refers to a window that maintains transparency in a weak pump field, and thus requires Fano interference. Because Autler-Townes splitting will wash out Fano interference at stronger fields, a smooth transition between the two effects is evident in materials exhibiting EIT.