Autler–Townes effect

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: : 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 coordinates. 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 its 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\hbar\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\hbar\omega, which is known as a quasi-energy state and the new set of energies are called the spectrum of quasi-harmonics. == Examples ==

General expressions for AC Stark shifts must usually be calculated numerically and tend to provide little insight. 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. and the light frequency is often far detuned from the atomic transition to avoid heating the atoms from the photon-atom scattering; in turn, the intensity of the light field (i.e. AC electric field) I(\mathbf{r}) is typically high to compensate for the large detuning. Typically, we have |\Delta| /\Gamma\gg I/I_{sat} \gg 1, where the atomic transition has a natural linewidth \Gamma and a saturation intensity: :I_\text{sat} = \frac{\pi}{3}{h c \Gamma \over \lambda_0^3}=\frac{\pi}{3}{h c \over {\lambda_0^3 \tau}}\,. Note the above expression for saturation intensity does not apply to all cases. For example, the above applies for the D2 line transition of Li-6, but not the D1 line, which obeys a different sum rule in calculating the oscillator strength. As a result, the D1 line has a saturation intensity 3 times larger than the D2 line. However, when the detuning from these two lines is much larger than the fine-structure splitting, the overall saturation intensity takes the value of the D2 line. In the case where the light's detuning is comparable to the fine-structure splitting but still much larger than the hyperfine splitting, the D2 line contributes twice as much dipole potential as the D1 line, as shown in Equation (19) of. : \begin{align} U_\text{dipole}(\mathbf{r})&=-\frac{\hbar \Omega^2}{4}\left(\frac{1}{\omega_0-\omega}+\frac{1}{\omega_0+\omega}\right)\\ &=-\frac{\hbar \Gamma^2}{8I_{sat}}\left(\frac{1}{\omega_0-\omega}+\frac{1}{\omega_0+\omega}\right)I(\mathbf{r})\\ &=-\frac{\text{Re}[\alpha(\omega)]}{2\varepsilon_0c}I(\mathbf{r})=-\frac{1}{2}\langle \mathbf{p\cdot E}\rangle. \end{align} Here, the Rabi frequency \Omega is related to the (dimensionless) saturation parameter s\equiv\frac{I(\mathbf{r})}{I_\text{sat}}=\frac{2\Omega^2}{\Gamma^2}, and \text{Re}[\alpha(\omega)] is the real part of the complex polarizability of the atom, the ODT light is so far detuned that counter-rotating term must be included in calculations, as well as contributions from adjacent atomic transitions with appreciable linewidth \Gamma. Note that the natural linewidth \Gamma here is in radians per second, and is the inverse of lifetime \tau. This is the principle of operation for Optical Dipole Trap (ODT, also known as Far Off Resonance Trap, FORT), in which case the light is red-detuned \Delta . When blue-detuned, the light beam provides a potential bump/barrier instead. The optical dipole potential is often expressed in terms of the recoil energy, which is the kinetic energy imparted in an atom initially at rest by "recoil" during the spontaneous emission of a photon: :\varepsilon_{recoil}=\frac{\hbar^2k^2}{2m}, where k is the wavevector of the ODT light (k\neq k_0 when detuned). The recoil energy, along with related recoil frequency \omega_{recoil}=\varepsilon_{recoil}/\hbar, are crucial parameters in understanding the dynamics of atoms in light fields, especially in the context of atom optics and momentum transfer. In applications that utilize the optical dipole force, it is common practice to use a far-off-resonance light frequency. This is because a smaller detuning would increase the photon-atom scattering rate much faster than it increases the dipole potential energy, leading to undesirable heating of the atoms. Quantitatively, the scattering rate is given by: 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-Townes 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. == See also ==