E55 · Photoelectron spectroscopy

Time ARPES: Pulses & Relaxation

Pump a synthetic surface, scan the probe delay and fit a fixed energy–momentum gate in the recorded spectra. Balance pulse duration, spectral bandwidth and electrons per pulse; recover a relaxation time with the measured response included.

Fitted relaxation time $\text{—}$

Conditional profile limits $\text{—}$

Combined energy FWHM $\text{—}$

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

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 $\text{—}$

A short pulse is a broad spectrum

Include the time response in the fitReprocess the same nine recorded gate sums.Probe photon energy

h\nu

21.2\,\mathrm{eV}

Selected pump–probe delay

\Delta t

100\,\mathrm{fs}

Pump intensity FWHM

\Delta t_p

80\,\mathrm{fs}

Probe intensity FWHM

\Delta t_{pr}

30\,\mathrm{fs}

Pump fluence

F

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

Emitted electrons per probe pulse

n_e

5

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

Predict the pulse-width tradeoff, then fit the nine acquired gate counts with the response included. Recover 600 fs within 60 fs, keep energy width below 0.05 eV, temporal width below 200 fs and emitted charge at most one electron per pulse. Compare the same mean rate concentrated into fewer pulses.

A synthetic ground band and pumped Gaussian band illustrate a repeatable perturbation. Excited population follows an assumed single 600 fs causal exponential, with amplitude 0.12 times fluence in mJ per square centimetre; it is not a microscopic material calculation or a particle-conserving excitation model. Gaussian pump/probe intensity pulses give a quadrature cross-correlation width. The transform-limited Gaussian probe obeys a 1.824951 eV fs bandwidth-duration product; chirp is omitted. A phenomenological space-charge model adds 0.01 eV shift and 0.012 eV FWHM per emitted electron per pulse; no Coulomb solver or universal instrument calibration. Repetition changes count rate independently of pulse charge; cumulative heating is omitted. The longest allowed interval remains vastly shorter than the pulse period, assuming full recovery. Grounded work functions are 4.5 eV. Spectra receive Gaussian energy/angular response on a finite grid with clamped boundary continuation. Every delay scan records nine independent spectra; the selected-delay spectrum has independent noise. A fixed 0.08–0.38 eV and ±0.2 inverse-angstrom gate is summed from stored pixels. Lifetime, nonnegative amplitude and baseline are fitted to those sums, optionally convolving a supplied response width. Counts alone enter the fit; known population is an explicitly separate reference. Profile limits use a conditional chi-square increase of one, not a coverage guarantee or systematic uncertainty. Boundary fits, weak signals, excessive residuals or wide profiles leave lifetime unresolved. Stroboscopic acquisition cannot follow an irreversible event. Optical delay length is schematic and not inferred from the drawn path; animation does not advance the physical delay or acquire frames. Log detector gain, magnified angular imaging, assumed flux and Gaussian-equivalent HDA width; no real kz bands, coherent phonons, pump matrix elements, multiphoton artifacts or damage. Poisson counts below mean 40, rounded Gaussian above.

Start with the essentials

Focus question: Does the decay reveal a lifetime—or the probe pulse?
One-sentence intuition: Fit gate sums from the acquired delay spectra with a causal decay convolved with the calibrated response. Increasing repetition can raise counts without crowding each pulse.

Core mathematical model

Gaussian transform limit

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

Intensity FWHM in eV and fs; a transform-limited pulse without chirp.

Pump–probe cross-correlation

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

Gaussian intensity pulse widths combine in quadrature.

Fit acquired 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}

A causal decay, selected by the step function, is convolved with the supplied Gaussian response. The fit estimates baseline, amplitude and lifetime from nine stored spectra.

Mean emitted electron rate

R=f_{rep}n_e

Equal mean rates can have different electrons per pulse and different space-charge broadening.

Common difficulties

Lifetime is conditional

Typical misconceptionA straight decay always reveals the material lifetime.

Better mental modelA broad time response changes early-delay counts; fit the convolved model and inspect residuals.

Pulse charge is different from mean rate

Typical misconceptionEqual average electron rate implies equal space charge.

Better mental modelSpace charge depends on pulse crowding. Increase repetition and reduce charge per pulse to retain counts.

Run the experiment

01
Predict the pulse tradeoff
Compare Recover the relaxation time and Short pulse broadens energy.
What to observe: The short pulse narrows the temporal response but broadens the energy spectrum.
02
Fit the actual gate counts
Check the target with the response enabled. Then move the selected delay.
What to observe: The selected image changes while the independently acquired scan and lifetime remain fixed.
03
Test response and pulse charge
Ignore a broad time response, then enable it. Compare the crowded pulse preset at the same mean rate.
What to observe: The response correction changes a fit of identical counts. Equal mean rates can give different energy shifts and widths.