Photon echo

Simulation of the photon-echo phenomenon in a gas of two-level atoms. Each dot on the canvas represents a group of atoms with a particular velocity (and therefore a different Doppler shift); its grayscale shade encodes the oscillation phase of its electric dipole, from white to black over a $2\pi$ cycle.

The "Pulse 1" and "Pulse 2" buttons in the "Light pulses" panel trigger each light pulse, with its area set by the θ₁/θ₂ sliders. The "Atomic system" panel adjusts the Doppler-distribution width (Δ_D), time resolution (Δt), and decoherence time (T₁₂). The "Graph" panel shows the squared polarization (the echo signal) over time.

The atomic gas is excited by a sequence of light pulses, each with an area defined by $$ \theta = \frac{\mu_{12}}{\hslash}\int_{-\infty}^{\infty} E_0(t)dt, $$ where $\mu_{12} = \left\langle 1 | e\hat{r} | 2 \right\rangle$ is the electronic-transition matrix element and $E_0$ is the light-pulse envelope. In the standard photon-echo configuration, the first pulse, with area $\pi/2$, leaves the gas with maximum coherence. Over time, decoherence caused by Doppler broadening reduces the polarization to zero over a half-life of $1/\Delta_D$. The second pulse, with area $\pi$, reverses the dephasing, causing the sample to emit a light pulse (the echo) after a time interval equal to that between the two incident pulses. The polarization is calculated as $$ P(t) = \eta\mu_{12}\int_{-\infty}^{\infty} \text{Re}\left[\rho_{12}(t)\right] g(\delta)\, d\delta, $$ where $\eta$ is the atomic density, $g$ is the Maxwell–Boltzmann velocity distribution, $\delta$ is the Doppler shift, and $\rho_{12}$ is an element of the density matrix, whose evolution is governed by the Liouville–von Neumann equation $$ \frac{\partial \hat{\rho}}{\partial t} = -\frac{i}{\hslash}\left[\hat{H}, \hat{\rho}\right]. $$ $\hat{H}$ is the Hamiltonian of a two-level system in the electric-dipole approximation, with rectangular-envelope pulses.

Marco P. M. de Souza