# The AC Stark Effect

A formula for the energy shifts due to an optical field is obtained for a two-level atomic system.

We define an atomic system consisting of two states (a ground state labeled 1 and upper state labeled 2). This is a "toy" system that neglects angular momentum (J and M are not defined). We apply a light field detuned from resonance by a frequency Δ.

The energy eigenvalues are found by diagonalizing the Hamiltonian.

To find the Stark shifts, we subtract the eigenenergies for zero applied field. This gives one shift that is always positive, and one that is negative. (This is because the eigenvalue ordering changes as Δ goes through zero.) To obtain the desired physical behavior, in which the energy shifts flip sign as Δ goes through zero, we multiply by Δ/ (equivalent to Sign[Δ]).

Note that these solutions lose validity as . This is because we have neglected the natural line width of the upper state. When the detuning is on the order of or smaller than the natural line width, the formula must be modified. To analyze the general case, we can model the effect of the upper-state line width Γ by making the upper-state energy complex.

The energies are the real parts of the eigenvalues of the modified Hamiltonian. In order to find compact expressions for the real parts, it helps to do a little manipulation by hand.

To lowest order in the Rabi frequency the shifts are given by dispersive Lorentzian functions.