---
title: "How dispersed phases form a photon echo"
description: "Track a static ensemble through a rephasing pathway, calculate echo timing and decay, then select the complex signal with phase cycling."
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  zh-CN: "https://sandboxphysics.com/zh/stories/multidimensional-spectroscopy/photon-echoes/"
last_modified: "2026-10-28"
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[Multidimensional spectroscopy](https://sandboxphysics.com/stories/multidimensional-spectroscopy/)

L2 · Chapter 4

# How dispersed phases form a photon echo

Sandbox Physics · 2026-10-28

Faster oscillations move ahead of slower ones. Later interactions can reverse the direction in which the response accumulates phase, allowing those differences to unwind. A static frequency distribution can then refocus into radiation after the third pulse.

## Before you begin

Read the timing and phase-matching discussion in chapter three. This chapter uses a two-level reference pathway. Chapter five restores excited-state absorption for a vibration.

[Review the preceding chapter](https://sandboxphysics.com/stories/multidimensional-spectroscopy/third-order-response/)

- Derive a Gaussian static-ensemble echo envelope.
- Distinguish two-pulse and three-pulse echo timing.
- Identify which phase spread an echo can refocus.

## Follow one molecular phase first

In a frame rotating at the mean resonance frequency, one molecule has offset $\delta\omega$. The first and last coherence intervals are $t_1$ and $t_3$. A rephasing contribution depends on their difference; a nonrephasing contribution depends on their sum.

The reversal belongs to the coherence pathway. Atoms do not travel backward, and information dispersed irreversibly into the surroundings does not automatically return. A frequency change between the intervals also prevents perfect refocusing.

$$
R_{\mathrm R}\propto e^{-i\delta\omega(t_3-t_1)},\qquad R_{\mathrm{NR}}\propto e^{-i\delta\omega(t_3+t_1)}
$$

An overall phase convention may reverse both signs. The difference between a sum and a difference of intervals determines refocusing.

Reading: Andrei Tokmakoff

## Add the ensemble of arrows

Average over a static Gaussian distribution with standard deviation $\sigma_\omega$ and add exponential coherence decay in each interval. Let $A(T)$ represent the specified population-pathway amplitude during the wait. Its form depends on the population model.

At $t_3=t_1$, the static Gaussian factor returns to one. The homogeneous decay remains. Echo amplitudes measured at several first-interval delays can therefore isolate the remaining homogeneous decay.

$$
R_{\mathrm R}(t_1,T,t_3)=A(T)e^{-\gamma(t_1+t_3)}e^{-\sigma_\omega^2(t_3-t_1)^2/2}
$$

A rotating-frame reference response with frequencies fixed during and between both coherence intervals.

$$
R_{\mathrm{NR}}(t_1,T,t_3)=A(T)e^{-\gamma(t_1+t_3)}e^{-\sigma_\omega^2(t_3+t_1)^2/2}
$$

The same distribution gives no positive-time refocusing centre for the nonrephasing pathway.

## Worked example timed from the third pulse

Choose $t_1=0.6\,\mathrm{ps}$ and $T=1.0\,\mathrm{ps}$. The pulse times are $0,\ 0.6,\ 1.6\,\mathrm{ps}$. Static phases refocus $0.6\,\mathrm{ps}$ after the third pulse, at laboratory time $2.2\,\mathrm{ps}$.

With $\gamma=0.5\,\mathrm{ps}^{-1}$, the homogeneous amplitude factor there is $e^{-0.6}\approx0.549$. Isolated signal intensity squares the field factor; a heterodyne cross term is linear in that field. Homogeneous decay can shift the envelope maximum slightly earlier than the static refocusing centre, so those two times need separate labels.

A two-pulse echo is often timed at twice the delay after the first pulse because the second pulse supplies two interactions. A three-pulse stimulated echo has a separate population interval. Doubling the total sequence duration would give the wrong answer.

## Select a phase component

Step each of the first two pulses through four quarter-turn phases, giving sixteen records. A rephasing field rotates oppositely to the first phase and with the second. Compensating complex weights align the selected component; constant background and single-field leakage cancel.

$$
\phi_n=\frac{n\pi}{2},\quad S_{\mathrm R}=\frac1{16}\sum_{n,m=0}^{3}Z(\phi_n,\phi_m)e^{i(\phi_n-\phi_m)}
$$

The record is the calibrated complex quadrature signal; its target phase is the second controlled phase minus the first. A finite cycle can alias other phase orders; experiments choose adequate cycles.

Reading: Park

## Finite pulses select a frequency range

A finite pulse has a finite spectrum and excites far-detuned molecules weakly. The reference experiment represents this with Gaussian spectral weighting and short, ordered interactions. Overlapping pulses require explicit envelope convolution and a treatment of multiple time orders.

The two-level reference isolates the echo mechanism. A vibration has a second excited level, whose absorption can cancel the nonlinear signal. Restoring that contribution is essential for understanding signed infrared spectra.

Reading: Andrei Tokmakoff

## Try the prediction

### [Drag a pulse and move the echo](https://sandboxphysics.com/labs/photon-echo-pulse-sequence/)

1. Predict how increasing the first interval should move the refocusing centre, then drag the delay carriage.
2. Compare echo widths for narrow and broad static frequency distributions.
3. Export a complex record with its parameters and seed. Identify which stored values retain phase information.

## Exercises and explanations

1. If the first interval is $0.4\,\mathrm{ps}$ and the third pulse arrives at $1.5\,\mathrm{ps}$, when is the static refocusing centre?

   Read the explanation

   At $1.9\,\mathrm{ps}$, which is $0.4\,\mathrm{ps}$ after the third pulse. The wait moves the third pulse but does not change the phase accumulated in the first interval.

2. Does increasing static frequency spread reduce the ideal refocusing factor at the echo centre?

   Read the explanation

   Not when frequencies stay static and pulse selection and total excitation are held fixed. The echo narrows, but the phases still agree at its centre. Finite bandwidth can also change the number of excited molecules and hence the total amplitude.

[Previous: How three pulses build a third-order response](https://sandboxphysics.com/stories/multidimensional-spectroscopy/third-order-response/) [Next: How two vibrations meet in a 2D spectrum](https://sandboxphysics.com/stories/multidimensional-spectroscopy/coupled-vibrations/)

## Sources and calculation scope

1. [Andrei Tokmakoff, Nonlinear and Two-Dimensional Spectroscopy (June 2011, v1)](https://knowledge.uchicago.edu/records/3tv7m-5ha50)

   Author lecture notes, nonlinear-spectroscopy section: density-matrix perturbation theory; printed pp. 23–25, Eqs. (51)–(55), for rephasing and nonrephasing; pp. 29–30 for the vibrational harmonic limit. The course examples are new teaching calculations.

2. [Park, Kwak and Fayer, Ultrafast 2D-IR vibrational echo spectroscopy (2007)](https://web.stanford.edu/group/fayer/articles/361.pdf)

   Laser Physics Letters 4, 704–718, DOI 10.1002/lapl.200710046. Experimental review: pulse sequence, local oscillator and absorptive phasing in Sections 2–3 and Fig. 3. Its center-line convention uses the inverse slope for the opposite slice direction; the course states its own axes before defining a slope.