---
title: "Time ARPES: Pulses & Relaxation Interactive Lab | Sandbox Physics"
description: "Does a decay measured in time-resolved ARPES show the true lifetime or the length of the pulses? Both are in it."
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last_modified: "2026-10-27"
---

E55 · Photoelectron spectroscopy

# Time ARPES: Pulses & Relaxation

**Does a decay measured in time-resolved ARPES show the true lifetime or the length of the pulses?** Both are in it. The measured decay is the true relaxation smeared out by the finite length of the pump and probe pulses. A lifetime is believable only if the fit includes that calibrated time response, and even then it depends on the decay model you assume.

Pump a synthetic surface, scan the probe delay and sum the counts in a fixed energy–momentum window of each recorded spectrum. Balance pulse duration, spectral bandwidth and electrons per pulse, then fit a relaxation time with the measured time response included.

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

Model assumptions

A synthetic ground band and a pumped Gaussian band show a perturbation that repeats the same way with every pulse. The excited population decays as one assumed 600 fs exponential that starts only after the pump, with amplitude 0.12 times the fluence in mJ per square centimetre. This is not a microscopic material calculation, and the excitation model does not conserve particle number. The pump and probe have Gaussian intensity profiles, so their widths add in quadrature to give the cross-correlation width. The probe is a transform-limited Gaussian with a bandwidth-duration product of 1.824951 eV fs; chirp is left out. A phenomenological space-charge model adds a 0.01 eV shift and 0.012 eV of FWHM for each emitted electron per pulse; there is no Coulomb solver and no universal instrument calibration. The repetition rate changes the count rate without changing the charge per pulse, and cumulative heating is left out. Even the longest allowed interval is far shorter than the pulse period, and the sample is assumed to recover fully between pulses. The grounded work functions are 4.5 eV. The spectra get a Gaussian energy and angular response on a finite grid, with the edges clamped. Every delay scan records nine independent spectra, and the spectrum at the selected delay has its own independent noise. A fixed gate of 0.08–0.38 eV and ±0.2 inverse angstrom is summed from the stored pixels. The lifetime, a nonnegative amplitude and a baseline are fitted to those sums, optionally convolved with a supplied response width. Only the counts enter the fit; the known population is shown as a separate reference. Profile limits use a conditional chi-square increase of one; they do not guarantee coverage or include systematic uncertainty. If the fit hits a boundary, the signal is weak, the residual is too large or the profile is too wide, no lifetime is reported. Stroboscopic acquisition cannot follow an irreversible event. The optical delay length is schematic and is not derived from the drawn path; the animation does not advance the physical delay or record frames. The detector gain is logarithmic, the angular imaging is magnified, and the flux and the Gaussian-equivalent analyzer width are assumed. Real dispersion with perpendicular momentum, coherent phonons, pump matrix elements, multiphoton artifacts and damage are left out. Counts are Poisson below a mean of 40 and rounded Gaussian above.

Physics tutorial

## An ultrafast clock has a finite response

**Background** An infrared pump disturbs a surface state, and the disturbance repeats the same way with every pulse. A delayed VUV probe then ejects photoelectrons, and the analyzer records their energy and angle.

**Why it matters** A measured decay mixes the material's relaxation with the pump–probe cross-correlation. A shorter probe sharpens the time response but broadens its spectrum.

Start with the essentials

- **Focus question**: Does the measured decay show the material's lifetime, or just the probe pulse?
- **One-sentence intuition**: Fit the gate sums from the delay scan with a decay that starts at the pump, convolved with the calibrated response. A higher repetition rate raises the counts without crowding more electrons into each pulse.

### Core mathematical model

#### Gaussian transform limit

$$
\Delta E_{pr}\,\Delta t_{pr}=4\ln 2\,\hbar
$$

Both widths are intensity FWHM, in eV and fs. The relation holds for a transform-limited pulse with no chirp.

#### Pump–probe cross-correlation

$$
\Delta t_{cc}=\sqrt{\Delta t_p^2+\Delta t_{pr}^2}
$$

For Gaussian intensity pulses, the widths add in quadrature.

#### Model fitted to the gate counts

$$
\begin{aligned}C(\Delta t)&=B+A\,q(\Delta t)\\q&=f*g_\sigma\\f(t)&=e^{-t/\tau}\Theta(t)\end{aligned}
$$

The step function makes the decay start only after the pump, and the decay is convolved with the supplied Gaussian response. The fit estimates the baseline, amplitude and lifetime from nine stored spectra.

#### Mean emitted electron rate

$$
R=f_{rep}n_e
$$

Two settings with the same mean rate can put different numbers of electrons in each pulse, and so give different space-charge broadening.

### Common difficulties

#### A fitted lifetime depends on the response

**Typical misconception** Fitting the decay directly always gives the material lifetime.

**Better mental model** A broad time response changes the counts at early delays. Fit the convolved model and check the residuals.

#### Charge per pulse is not the mean rate

**Typical misconception** The same average electron rate means the same space charge.

**Better mental model** Space charge depends on how many electrons share one pulse. To keep the counts, raise the repetition rate and lower the charge per pulse.

### Run the experiment

1. 01

   #### Predict the pulse-length tradeoff

   Compare Recover the relaxation time with Short pulse broadens energy.

   **What to observe:** The short pulse sharpens the time response but broadens the energy spectrum.

2. 02

   #### Fit the recorded gate counts

   Turn on Include the time response in the fit and press Check target. Then move the Selected pump–probe delay.

   **What to observe:** The image at the selected delay changes, but the separately acquired delay scan and the fitted lifetime stay the same.

3. 03

   #### Test the response and the charge per pulse

   Choose Ignore a broad time response, then turn the response back on. Compare with Same average rate, crowded pulses.

   **What to observe:** Including the response changes the fit to the same counts. Two settings with the same mean rate can give different energy shifts and widths.

Physical conclusion

A fitted lifetime is only as good as the response and kinetic model behind it. Stroboscopic data need an excitation that repeats exactly and a sample that recovers between pulses.

Fields & matter · Starter path

## From a field to an electron microscope

Map an electric field and steer a charged particle. Then find the electron's wavelength and use electron beams to see a crystal and a surface.

[Start here → **Make an Electric Field Visible** Move the probe around a dipole and check that the field always points downhill on the potential landscape. Observe · 8 min](https://sandboxphysics.com/labs/electric-field/)

1. [1 Make an Electric Field Visible](https://sandboxphysics.com/labs/electric-field/)
2. [2 A Charge Bending through Fields](https://sandboxphysics.com/labs/charged-particle/)
3. [3 Electron Wavelength & Voltage](https://sandboxphysics.com/labs/electron-wavelength/)
4. [4 Electron Diffraction & Reciprocal Space](https://sandboxphysics.com/labs/electron-diffraction/)
5. [5 TEM Bright Field, Dark Field & SAED](https://sandboxphysics.com/labs/tem-mode-selector/)
6. [6 SEM Secondary-Electron Topography](https://sandboxphysics.com/labs/sem-topography/)

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