# Two-Level System

A two-level atomic system subject to an optical field is analyzed.

## Time-Dependent Solution and the Rabi Frequency

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.

We first neglect relaxation and solve for the dynamics of the system.

Solve the system of equations with DSolve. We see that the upper state has population given by with generalized Rabi frequency and amplitude A=Ω _{R}^{2}/(Ω_{R}^{2}+Δ^{2}). |

## Steady-State Solution and the Saturation Parameter

If we add relaxation to the system, we can find a steady-state solution.

There are two forms of relaxation present: intrinsic relaxation of the upper state due to spontaneous decay and "transit" relaxation due to atoms leaving the system. A "relaxation matrix" accounts for these processes.

The ground state is repopulated by the same two mechanisms: spontaneous decay transfers atoms from the upper state to the ground state, and ground-state atoms enter the system. This is taken into account by the "repopulation matrix":

Here are the evolution equations, including relaxation and repopulation. Since the density matrix elements are now explicitly time independent, we obtain time-independent equations for the steady state.

Create evolution equations with LiouvilleEquation. |

We can simplify these solutions by assuming that the transit rate is much slower than the natural width of the upper state, and by writing them in terms of the upper-state saturation parameter κ=/Γ^{2}.

We see that the upper-state population is a power-broadened Lorentzian in detuning. Here is a plot of the upper-state population as a function of detuning for various values of the saturation parameter.

Setting the detuning to zero, we see why κ is called the saturation parameter for this system: when κ is small, the upper-state population is linear in κ; when κ is increased to unity, the upper-state population begins to saturate, and approaches its limiting value of 1/2.