Photoelectron Metrology¶

Paul Hockett

National Research Council of Canada

Web: femtolab.ca

Books: phockett.github.io/Quantum-Metrology-with-Photoelectrons-Vol3

Slides: http://dx/doi/org/10.6084/m9.figshare.27468942

Abstract¶

Seminar: Photoelectron Metrology

In this seminar I will give an overview of recent work in photoelectron metrology (a.k.a. quantitative photoelectron spectroscopy), in which the main aim is the retrieval of photoionization matrix elements from experimental measurements. The work is enabled by various tools on both the theory and experimental side; recent work has focussed on the development of a suite of open-source python tools - the Photoelectron Metrology Toolkit - and application to various problems, and this will form the main topic of the talk. On the experimental side, time- and angle-resolved photoionization studies of aligned molecules forms the main methodology [1]. New results will be presented, including valence photoionization studies of OCS [2] and N2O. In a similar vein, electronic wavepacket reconstruction has also recently been explored, and will be discussed if time permits [3,4].

For a more comprehensive introduction and review, please see Quantum Metrology with Photoelectrons Vol. 3 (https://phockett.github.io/Quantum-Metrology-with-Photoelectrons-Vol3), an open-source book which introduces and explores the topic, including the code-base and full computational examples.

[1] Hockett, P. and Makhija, V. (2023) ‘Topical Review: Extracting molecular frame photoionization dynamics from experimental data’, Journal of Physics B: Atomic, Molecular and Optical Physics, 56(11), p. 112001. Available at: https://doi.org/10.1088/1361-6455/acd03e. [2] Woodhouse, J. et al. (2024) ‘Quantitative analysis of aligned molecule photoelectron angular distributions’, In press, Phys. Rev. A. (Data available at https://figshare.com/projects/OCS_alignment_and_photoionization_project_/194090) [3] Gregory, M. et al. (2022) ‘A laboratory frame density matrix for ultrafast quantum molecular dynamics’, The Journal of Chemical Physics, 157(16), p. 164301. Available at: https://doi.org/10.1063/5.0109607. [4] Morrigan, L. et al. (2023) ‘Ultrafast Molecular Frame Quantum Tomography’, Physical Review Letters, 131(19), p. 193001. Available at: https://doi.org/10.1103/PhysRevLett.131.193001.

Outline¶

  1. Defining photoelectron metrology, a.k.a. quantitative photoelectron spectroscopy, a.k.a. quantum metrology with photoelectrons
  1. Overview of the Photoelectron Metrology Toolkit - open-source python platform for data analysis and matrix element retrieval
  1. Retrieving matrix elements (making use of time- and angle-resolved data)
  1. Some recent work making use of the tools...
    1. Molecular alignment + HH probing of OCS
    2. Molecular alignment + HH probing of N2O
  1. Related work

Books...¶

Vol. 3 now available open source, HTML version https://phockett.github.io/Quantum-Metrology-with-Photoelectrons-Vol3

QMbooks

1. Photoelectron metrology¶

  1. Photoelectron spectroscopy with matrix element reconstruction

→ Quantitative photoelectron spectroscopy, full reconstruction of the ionization continuum (including phases).

  1. 1 + molecular (wavepacket) dynamics

→ Full reconstruction of the ionization continuum + wavepacket properties.

  1. Quantum - yes it is!

→ Reconstruct density matrix, correlation & coherence functions etc. (As well as the underlying physics of course.)

Conceptual introduction¶

Photoionization is an interferometric process, in which multiple paths can contribute to the final continuum photoelectron wavefunction.

At the simplest level, interferences between different final angular momentum states are manifest in the energy and angle resolved photoelectron spectra: metrology schemes making use of these interferograms are thus phase-sensitive, and provide a powerful route to detailed understanding of photoionization. In these cases, the continuum wavefunction (and underlying scattering dynamics) can be characterised.

At a more complex level, such measurements can also provide a powerful probe for other processes of interest, leading to a more general class of quantum metrology built on phase-sensitive photoelectron imaging.

Conceptual diagram for matrix retrieval using time and angle-resolved photoelectron measurements with impulsive molecular alignment.

For further introductory materials...

  • Quantum Metrology with Photoelectrons Vol. 3 (HTML version)
  • Hockett, Paul, and Varun Makhija. 2023. Topical Review: Extracting Molecular Frame Photoionization Dynamics from Experimental Data. Journal of Physics B: Atomic, Molecular and Optical Physics 56 (11): 112001. Also arXiv 2209.04301. And HTML with interactive figures on Authorea.
  • Phase-sensitive Photoelectron Metrology (presentation at DAMOP 2017), video presentation and slides.

2. Photoelectron metrology toolkit (PEMtk) & platform¶

  • The main aim of the platform is to provide a unifying data layer, and analysis routines, for photoelectron metrology, including new methods and tools, as well as a unifying bridge between these and existing tools.
  • Currently simulation of observables (MF, AF and LF PADs) and matrix-element fitting routines are implemented, along with some basic utility functions.

  • The tools are built in python, and available on Github or via Pypi. Docker builds including the Quantum Metrology with Photoelectrons Vol. 3 source are also available via Docker Hub.

  • ePSproc: https://github.com/phockett/ePSproc

  • PEMtk: https://github.com/phockett/PEMtk

Basic examples¶

Examples are available online as part of the package documentation and QM3.

These demonstrate some of the basic functionality, often making use of:

  • Photoionization matrix elements from ePolyScat (thanks Robert!).
  • Alignment distribution moments from Varun Makhija (expansion in spherical harmonics or Wigner D).
  • Example below follows Basic plotting + AF-$\beta_{LM}$ and AF-PAD calculations with ePSproc. (plus some recent plotter updates, may need dev branch for this).
  • PEMtk time-resolved data example
  • AF-BLM example with heatmaps (NO2 case)

(Note that the computational slides following may not be formatted correctly in the HTML version, please see the source Jupyter notebook or links given above instead.)

In [1]:
# Load base class
from epsproc.classes.base import ePSbase

# Instantiate class object.
# Minimally this needs just the dataPath, if verbose = 1 is set then some useful output will also be printed.
dataPath = '/home/jovyan/github/ePSproc/data/photoionization/n2_multiorb'
data = ePSbase(dataPath, verbose = 1)

# ScanFiles() - this will look for data files on the path provided, and read from them.
data.scanFiles()

OMP: Info #276: omp_set_nested routine deprecated, please use omp_set_max_active_levels instead.

* sparse not found, sparse matrix forms not available. 
* natsort not found, some sorting functions not available. 

* Setting plotter defaults with epsproc.basicPlotters.setPlotters(). Run directly to modify, or change options in local env.

* Set Holoviews with bokeh.
* pyevtk not found, VTK export not available. 

*** Job orb6 details
Key: orb6
Dir /home/jovyan/github/ePSproc/data/photoionization/n2_multiorb, 1 file(s).
{   'batch': 'ePS n2, batch n2_1pu_0.1-50.1eV, orbital A2',
    'event': ' N2 A-state (1piu-1)',
    'orbE': -17.09691397835426,
    'orbLabel': '1piu-1'}

*** Job orb5 details
Key: orb5
Dir /home/jovyan/github/ePSproc/data/photoionization/n2_multiorb, 1 file(s).
{   'batch': 'ePS n2, batch n2_3sg_0.1-50.1eV, orbital A2',
    'event': ' N2 X-state (3sg-1)',
    'orbE': -17.34181645456815,
    'orbLabel': '3sg-1'}
In [2]:
# Molecular info
# Note that this is currently assumed to be the same for all jobs in the data dir.
data.molSummary()

*** Molecular structure

*** Molecular orbital list (from ePS output file)
EH = Energy (Hartrees), E = Energy (eV), NOrbGrp, OrbGrp, GrpDegen = degeneracies and corresponding orbital numbering by group in ePS, NormInt = single centre expansion convergence (should be ~1.0).
props Sym EH Occ E NOrbGrp OrbGrp GrpDegen NormInt
orb
1 SG -15.6719 2.0 -426.454124 1.0 1.0 1.0 0.999532
2 SU -15.6676 2.0 -426.337115 1.0 2.0 1.0 0.999458
3 SG -1.4948 2.0 -40.675580 1.0 3.0 1.0 0.999979
4 SU -0.7687 2.0 -20.917393 1.0 4.0 1.0 0.999979
5 SG -0.6373 2.0 -17.341816 1.0 5.0 1.0 1.000000
6 PU -0.6283 2.0 -17.096914 1.0 6.0 2.0 1.000000
7 PU -0.6283 2.0 -17.096914 2.0 6.0 2.0 1.000000
In [3]:
# Basic plot of ePS cross-section results (from GetCro command)
data.plotGetCroComp()
In [4]:
# Compute MF-BLM from matrix elements (default options, all data)
data.MFBLM()

Calculating MF-BLMs for job key: orb6

Calculating MF-BLMs for job key: orb5
In [5]:
# Plot BLMs using Holoviews
# Note plotter defaults are set for AFBLM case
data.BLMplot(dataType='MFBLM', 
             thres = 1e-2,     # Passing a threshold value here will remove any spurious BLM parameters.
             backend='hv', hvType = 'line',
             width=800, renderPlot=False) 

BLMplot set data and plots to self.plots['BLMplot']
Formatted BLMplot added to self.plots['BLMplot']['hvPlotOut']
Out[5]:
True
In [7]:
# For slide rendering (cell outputs only)
data.plots['BLMplot']['hvPlotOut']
Out[7]:
In [9]:
# Plot some MF-PADs
# Use Plotly backend for interactive plots
data.padPlot(keys = 'orb5', Erange = [5, 10, 4], dataType='MFBLM',
            backend='pl')

Using default sph betas.
Summing over dims: set()
Plotting from self.data[orb5][MFBLM], facetDims=['Eke', 'Labels'], pType=a with backend=pl.
*** WARNING: plot dataset has min value < 0, min = (-2.9550010004271866e-06-1.8947143685522024e-18j). This may be unphysical and/or result in plotting issues.

*** WARNING: plot dataset has min value < 0, min = (-1.432562167896377e-05+6.4676612278544425e-18j). This may be unphysical and/or result in plotting issues.

3. Retrieving matrix elements¶

  • In general a difficult problem with high dimensionality.
  • In PEMtk, use tensor formulation to separate ND basis function and matrix elements; can then fit efficiently using standard fitting libraries.
$$ \beta_{L,M}^{u}(\epsilon,t)=\sum A_{L,M}^{u}(t) I(\epsilon) $$

(Simplified version, see here for full expansions.)

  • Make use of a bootstrapping methodology for separable parts of the problem, and work up in complexity.
  • In general, methods can be applied to any problem with a sufficient number of measurements (information content) in which only geometric parameters are changed, e.g. molecular alignment, rotational state, polarization geometry etc.
  • Currently the focus has been on time-resolved measurements making use of rotational wavepackets, to provide sufficient datasets. Retrieval from MF-PADs has also been tested.

$N_2$ example from book and fitting manuscript.

4. Recent experimental work¶

  • Experiments led by Jo Woodhouse (University of Southampton, UK), with various collaborators, at CLF/RAL/SFTC laser facilities (UK), Artemis monochromated high-harmonic beamline.
  • Alignment simulation and reconstuction by Varun Makhija and coworkers. (University Mary Washington, Fredericksburg, VA)
  • Photoionization simulation with ePolyScat + PEMtk.
  • Additional theory comparisons with R-matrix by Zdeněk Mašín and Jakub Benda (Charles University, Prague, Czech Republic)

E.g. Mašín, Z. et al. (2019) ‘UKRmol+: A suite for modelling electronic processes in molecules interacting with electrons, positrons and photons using the R-matrix method’, Computer Physics Communications, p. 107092. Available at: https://doi.org/10.1016/j.cpc.2019.107092.

  • Matrix element retrieval (ongoing) with PEMtk.

OCS¶

Recent theory + experimental work.

  • Impulsively aligned sample (rotational wavepacket).
  • Valence ionization with isolated high-harmonics.
  • VMI detection.

Manuscript in press: Woodhouse, J.; Benda, J.; Chapman, R. T.; Hockett, P.; Makhija, V.; Masin, Z.; Minns, R. S.; Reid, K.; Springate, E.; Thompson, J. O. F.; Wyatt, A. S.; Zhang, Y. Quantitative Analysis of Aligned Molecule Photoelectron Angular Distributions. In Press, Phys. Rev. A. 2024

Comparison of theory results - ePS and R-matrix at different levels of theory (Zdeněk Mašín and Jakub Benda) - and literature results (Carlson, Thomas A., Manfred O. Krause, and Frederick A. Grimm. 1982. “Angle Resolved Photoelectron Spectroscopy of CS2 and COS Measured as a Function of Photon Energy from 21 to 70 eV.” The Journal of Chemical Physics 77 (4): 1701–9. https://doi.org/10.1063/1.444067).

  • Hartree-Fock: this is single-channel model which includes exact exchange and only the static potential of the ion exactly. The HF orbitals were those of the neutral molecule.
  • Coupled-Hartree-Fock: this is a multi-channel model which still uses HF orbitals and a small active space of configurations to describe the ionic states, the neutral state and the polarization/correlation effects.
  • Complete Active Space: this is the most sophisticated model where all states are described using a CAS generated from CASSCF orbitals and we included 200 states of the ion.

ePolyScat vs. bond length extensions (OCS X-state).

Literature results: Carlson, Thomas A., Manfred O. Krause, and Frederick A. Grimm. 1982. “Angle Resolved Photoelectron Spectroscopy of CS2 and COS Measured as a Function of Photon Energy from 21 to 70 eV.” The Journal of Chemical Physics 77 (4): 1701–9. https://doi.org/10.1063/1.444067

OCS matrix element retrieval - initial work in QM3.

Experimental retrieval in progress!

OCS summary¶

  • Generally good comparison with theory and experiment (with some caveats).
  • Indicates good matrix elements from the ePS calculations, including sensitivity to shape resonance (X-state). (No surprises here!)
  • Some states may require multi-electron treatment for accurate matrix elements.
  • Matrix element retrieval promising, but currently limited by symmetry ($C_{\infty v} > D_{\infty h}$). Future work will explore this by, e.g., different polarization geometries.

N2O¶

  • Same team and method as OCS, but with additional tomographic measurement techniques (D. Townsend and coworkers, Herriot-Watt University, Edinburgh, UK) to explore polarization geometry and non-cylindrically symmetric VMI detection.
  • Aim to expand calculations to include variable polarization states (e.g. elliptically polarized light).
  • Aim to explore reconstruction techniques further.

ePS + ePSproc aligned-frame PADs ($\phi$-summed).

ePS + ePSproc aligned-frame PADs

Also tested AF theory results for polarization dependence, since it was not clear how clean polarization rotation would be experimentally (potentially up to ~20% ellipticity expected).

Experimental images

Experimental tomographic scan - rotated polarization and record VMI images.

5. Related work¶

  • MF-PAD reconstruction with matrix inversion method (no matrix elements): Gregory, Margaret, Paul Hockett, Albert Stolow, and Varun Makhija. 2021. “Towards Molecular Frame Photoelectron Angular Distributions in Polyatomic Molecules from Lab Frame Coherent Rotational Wavepacket Evolution.” Journal of Physics B: Atomic, Molecular and Optical Physics 54 (14): 145601. https://doi.org/10.1088/1361-6455/ac135f.
  • Lab-frame density matrix reconstruction, e.g. for electronic dynamics: Morrigan, Luna; Neville, S. P.; Gregory, M.; Boguslavskiy, A. E.; Forbes, R.; Wilkinson, I.; Lausten, R.; Stolow, A.; Schuurman, M. S.; Hockett, P.; Makhija, V. Ultrafast Molecular Frame Quantum Tomography. Phys. Rev. Lett. 2023, 131 (19), 193001. https://doi.org/10.1103/PhysRevLett.131.193001.

Summary & Conclusions¶

  • Phase-sensitive photoelectron metrology is - in general - possible with time- and angle-resolved photoelectron measurements.
  • General methods are implemented in the PEMtk toolkit. Photoionization matrix element retrieval (equivalently continuum density matrices) can be reconstructed in favourable cases, e.g. N2.
  • Scaling to larger/less symmetric systems remains challenging, both due to size of the retrieval problem, and symmetry limitations in any given case. In general, some additional experimental measurements may be required above a standard align-probe technique, e.g. polarization-geometry, orientated molecules, etc. (For general discussion see information content in QM3
  • Recent experimental work shows good agreement with theory and measurement in general, although full matrix element retrieval efforts are ongoing.
  • Open-source software and books now available to the community for those interested in testing, using and/or developing these techniques further.