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.

Fitted relaxation time $\text{—}$

Conditional profile limits $\text{—}$

Combined energy FWHM $\text{—}$

Pump–probe cross-correlation FWHM $\text{—}$

More measurements and target

Recorded selected-delay counts $\text{—}$

Recorded counts in all nine gates $\text{—}$

Count-scaled fit residual $\text{—}$

Assumed space-charge energy shift $\text{—}$

Known teaching-model lifetime $\text{—}$

Total scan time · nine delays $\text{—}$

Experiment target $\text{—}$

Simulated examples; each plot identifies its data source.

Include the time response in the fitRefits the same nine recorded gate sums.Selected pump–probe delay

\Delta t

100\,\mathrm{fs}

Pump intensity FWHM

\Delta t_p

80\,\mathrm{fs}

Emitted electrons per probe pulse

n_e

5

Advanced settings

Probe photon energy

h\nu

21.2\,\mathrm{eV}

Probe intensity FWHM

\Delta t_{pr}

30\,\mathrm{fs}

Pump fluence

F

1\,\mathrm{\frac{mJ}{cm^2}}

Repetition frequency

f

0.2\,\mathrm{MHz}

Analyzer pass energy

E_p

30\,\mathrm{eV}

Entrance and exit slit width

s

0.5\,\mathrm{mm}

Exposure per delay

\tau_{acq}

1\,\mathrm{s}

Supplied response-width multiplier

q

1

Learning goal and model limits

A short pulse has a broad spectrum

First predict how pulse width trades time resolution against energy resolution. Then fit the nine recorded gate counts with the time response included. Recover the 600 fs relaxation time within 60 fs, with an energy width below 0.05 eV, a temporal width below 200 fs and at most one emitted electron per pulse. Finally, compare the same mean rate packed into fewer pulses.

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.

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 misconceptionFitting the decay directly always gives the material lifetime.

Better mental modelA 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 misconceptionThe same average electron rate means the same space charge.

Better mental modelSpace 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

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.
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.
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.