---
title: "Photon Echo Pulse Sequence Interactive Lab | Sandbox Physics"
description: "How can three light pulses make a dephased ensemble emit an echo? The rephasing pathway reverses the first interval of phase accumulation."
canonical_url: "https://sandboxphysics.com/labs/photon-echo-pulse-sequence/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/photon-echo-pulse-sequence/"
---

FR16 · Recover a fading optical coherence

# Photon Echo Pulse Sequence

**How can three light pulses make a dephased ensemble emit an echo?** The rephasing pathway reverses the first interval of phase accumulation. Fixed frequency differences then cancel after an equal second interval, producing an echo whose strength still decays through irreversible dephasing.

Move an optical delay and watch three pulses meet in a sample. Recover the echo field with phase cycling, then transform the saved quadrature samples into a spectrum.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

Model assumptions

This is a weak third-order two-level saturation response with static Gaussian frequency disorder, exponential homogeneous dephasing and a three-picosecond population lifetime. Pulse widths enter through Gaussian spectral selection; pulse overlap and propagation are omitted. The carrier follows the ensemble mean; the receiver offset changes only the rotating frame. A calibrated complex heterodyne receiver adds independent Gaussian noise. The optical paths and delay travel are schematically magnified. The sample dipoles display a magnified rotating-frame coherence with the optical carrier removed. All recordings are synthetic. Vibrational excited-state absorption and its harmonic-limit cancellation are treated in the coupled-vibration Lab.

Physics tutorial

## Recovering a phase memory

**Background** A short light pulse prepares many dipoles with a common phase. Their resonance frequencies differ, so their arrows spread around a circle and their summed field fades.

**Why it matters** A broad line can come from a spread of fixed frequencies or from rapid loss of coherence. A photon echo separates their effects in time.

Start with the essentials

- **Focus question**: How can three light pulses make a dephased ensemble emit an echo?
- **One-sentence intuition**: The rephasing pathway reverses the sign of the first phase accumulation, so static frequency differences cancel after equal coherence times.

### Core mathematical model

#### Opposite phase signs

$$
P_{\mathrm R}(t_3;t_1,T)=A e^{-\Gamma(t_1+t_3)-T/T_1}e^{-\frac12\Delta_{\mathrm{sel}}^2(t_3-t_1)^2}e^{-i\delta_0(t_3-t_1)}
$$

Times are in picoseconds. Gamma is a coherence decay rate; Delta is an angular frequency standard deviation. The static phase spread cancels at equal coherence times. The peak of the decaying envelope can occur slightly before that refocusing time.

#### Finite bandwidth selects the ensemble

$$
A=\left(1+3\Delta^2\sigma_t^2\right)^{-1/2},\qquad\Delta_{\mathrm{sel}}=A\Delta
$$

Three identical Gaussian field envelopes weight the static distribution. The pulse carrier tracks its mean. This spectral-selection approximation omits overlap dynamics and is used only for well-separated pulse centres.

#### Extract the signal from sixteen phase steps

$$
\widehat P_{\mathrm R}(t_3)=\frac1{16}\sum_{m,n=0}^3 E_{mn}(t_3)e^{i(\phi_m-\phi_n)},\qquad\phi_m=\frac{m\pi}2
$$

Each complex sample is phase-rotated before averaging. The rephasing phase survives, whereas a constant background and fields depending on only one pulse phase cancel. The nonrephasing pathway uses the conjugate weight.

#### Transform the retained observations

$$
\widetilde P(f)=\Delta t\sum_j\widehat P(t_j)e^{+2\pi i f t_j}
$$

The spectrum is a direct discrete Fourier sum of the saved complex observations, with no analytic replacement. Frequency is in cycles per picosecond, equal to terahertz. The finite window broadens features and the sample interval sets the Nyquist limit.

### Common difficulties

#### An echo restores only reversible phase spreading

**Typical misconception** The echo means the sample has recovered all the coherence it initially had.

**Better mental model** Frequency differences that stay fixed during the sequence can cancel. Exponential homogeneous dephasing and population loss continue to reduce the recovered field.

### Run the experiment

1. 01

   #### Move the refocusing time

   Choose Recover an echo and drag the glowing delay carriage along its rail.

   **What to observe:** The recorded field appears later when the first delay is longer. The ensemble arrows align when the two coherence times match.

2. 02

   #### Remove the phase reversal

   Choose No phase reversal, then compare the recorded field with Recover an echo.

   **What to observe:** In the nonrephasing pathway, static frequency differences accumulate with the same sign and the delayed echo disappears.

3. 03

   #### Separate two sources of broadening

   Choose Mostly homogeneous. Increase the homogeneous dephasing rate, then restore Recover an echo and increase the static spread.

   **What to observe:** Homogeneous dephasing reduces the field even at refocusing. Static spread mainly narrows the echo when the pulses remain broadband.

4. 04

   #### Keep the actual samples

   Open Phase cycling & saved samples, save a comparison and export the raw CSV. Change the noise seed, then restore the saved run.

   **What to observe:** Restoring brings back the same raw phase steps and Fourier spectrum. A new seed changes the noise realization.

Physical conclusion

Static disorder can rephase. Homogeneous dephasing continues to attenuate the echo during both coherence periods.

## Related experiments and articles

Multidimensional spectroscopy · Reading path

### Vibrations, echoes and frequency memory

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **Why a spectral line broadens**](https://sandboxphysics.com/stories/multidimensional-spectroscopy/coherence-and-linewidth/)
2. [Article **How three pulses build a third-order response**](https://sandboxphysics.com/stories/multidimensional-spectroscopy/third-order-response/)
3. [Article **How dispersed phases form a photon echo**](https://sandboxphysics.com/stories/multidimensional-spectroscopy/photon-echoes/)

Quantum & fields · Starter path

## Start from a single measurement

Predict an outcome, then count what the measurements actually give. The path starts with one qubit and moves on to wave packets, bound states and two-photon interference.

[Start here → **Quantum State & Born Measurement** Predict a probability, turn the analyzer, then compare your prediction with the measured frequency over many shots. Observe · 15 min](https://sandboxphysics.com/labs/qubit-state/)

1. [1 Quantum State & Born Measurement](https://sandboxphysics.com/labs/qubit-state/)
2. [2 Stern–Gerlach Sequence](https://sandboxphysics.com/labs/stern-gerlach-sequence/)
3. [3 Wave Packet & Uncertainty](https://sandboxphysics.com/labs/wave-packet-uncertainty/)
4. [4 Quantum Harmonic Oscillator](https://sandboxphysics.com/labs/quantum-harmonic-oscillator/)
5. [5 Which Path & Quantum Eraser](https://sandboxphysics.com/labs/quantum-eraser/)
6. [6 Hong–Ou–Mandel Two-Photon Interference](https://sandboxphysics.com/labs/hong-ou-mandel/)

[View the full path →](https://sandboxphysics.com/topics/quantum/#starter-path)