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REVIEW 3 major objections 4 minor 43 references

Tomographic reconstruction of free-electron quantum states

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Tomographic reanalysis of free-electron data finds attosecond pulses near 224 as, almost three times shorter than the original 655 as estimate.

desk verdict A reanalysis of attosecond electron data that likely shortens the reported pulse duration, but the missing data-model consistency check leaves the headline number conditional. read the letter →

arxiv 2508.17594 v1 pith:DXL5EBYR submitted 2025-08-25 quant-ph

Hao Jeng , Claus Ropers This is my paper · ORCID
classification quant-ph
keywords quantumstatetomographyfree-electronopticsattosecondelectronpulsesmaximumlikelihoodestimationBayesianinversionneuralnetworkreconstructionWignerfunctiondensitymatrix
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to establish that the quantum state of a free-electron pulse after laser energy exchange and free-space propagation can be fully recovered from phase-resolved energy spectra, and that doing so with three independent algorithms on previously recorded data gives much shorter electron pulses than originally reported. The reconstructed density matrices put the attosecond pulse-train at 224 as (maximum likelihood), 245(39) as (Bayesian), and 203 as (neural network), versus the original 655 as full-width-at-half-maximum estimate, and predict a degree of coherence of about 36 percent for the radiation and excitations these electrons produce. A sympathetic reader would care because accurate state characterization decides what ultrafast electron microscopes can see and what quantum states of light they can generate.

What carries the argument

The load-bearing object is the spectrogram $S(\phi,N)$, built from the displacement-like unitary $U_\phi=\exp(g b^\dagger-\bar g b)$ acting on the electron energy ladder; reference [19] guarantees that this family of measurements identifies the state uniquely. On top of that, the paper defines a Wigner function on the phase space $\mathbb{Z}\times S^1$ (integer energy change and periodic temporal coordinate) so the reconstructed matrix can be visualized and its marginals read off directly. Three reconstruction schemes carry the argument: a fixed-point maximum-likelihood map, a Bayesian Markov-chain sampler with a curvature-informed preconditioned Crank-Nicholson proposal, and a deep feedforward network paired with a noise-filtering autoencoder; all three take the spectrogram to a physical density matrix.

What would settle it

Increase the Hilbert-space cutoff (say from 25 to 40 energy states) in the maximum-likelihood reconstruction of the same data; if the derived full-width-at-half-maximum moves by more than the Bayesian error bar, the truncated model is not stable. Alternatively, add a fitted constant background channel to the likelihood; if the 224 as value changes substantially, the data require more than the clean unitary model.

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Extended reading notes

Core claim

The central claim is that the phase-resolved energy spectrum $S(\phi,N)=\langle N|U_\phi \rho U_\phi^\dagger|N\rangle$, obtained by applying a laser interaction of fixed strength and varying phase before measuring electron energy, is tomographically complete for the discrete energy-state Hilbert space, so a density matrix can be reconstructed from it. Applying maximum-likelihood iteration, Bayesian inversion with a curvature-informed proposal, and a trained neural network to the same archival data, the paper obtains consistent attosecond pulse durations near 200-250 as and a degree of coherence $|\langle b\rangle| \approx 0.36$ (about 63 percent of the theoretical maximum for this class of pulses), concluding the original 655 as estimate is a substantial overestimate.

Load-bearing premise

If the measured phase-resolved energy spectrum contains any contributions that are not captured by the ideal laser-interaction model—background counts, detector response variations, or laser phase and amplitude jitter—the reconstructed density matrix and every derived number shift systematically.

Editorial extensions

If this is right

  • If the reconstructed density matrices are correct, the original 655 as full-width-at-half-maximum is an overestimate by roughly a factor of three, meaning the same apparatus produced much shorter pulses than was claimed.
  • The predicted degree of coherence of about 36 percent, or 63 percent of the theoretical maximum, gives a concrete benchmark for coherent cathodoluminescence and coherent excitation experiments with these pulse trains.
  • The Bayesian error bar of 245(39) as turns the pulse-duration claim into an uncertainty-quantified statement rather than a single number.
  • Because the neural network returns a state in about 10 milliseconds, the same reconstruction could support real-time feedback and online state characterization in electron-beam experiments.
  • The machinery transfers to qubit and qudit superpositions of electron energy states and, in principle, to joint electron-photon tomography of entangled systems.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If independent datasets confirm the shorter duration, the resolution ceiling for attosecond electron microscopy may be lower than the literature currently assumes for the same laser and electron-source parameters.
  • The three methods agree, but the neural network's lower fidelity (0.66 against the Bayesian mean) suggests the true value may sit closer to the Bayesian 245 as until the network is trained on more and broader data.
  • A direct experimental test of the coherence prediction would be to measure the interference visibility of cathodoluminescence from these pulses against an external local oscillator and compare it with the predicted 36 percent.
  • Applying the same pipeline to other archived free-electron datasets recorded at different interaction strengths would show whether the shortening effect is general or specific to this dataset.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The manuscript develops three tomographic reconstruction approaches for the discrete energy-state density matrix of swift electrons interacting with a laser field: an iterative maximum-likelihood estimator, a Bayesian MCMC method using a curvature-informed preconditioned Crank-Nicolson proposal, and a deep neural network trained on simulated spectrograms. The forward model is the phase-dependent displaced unitary of Eqs. (3)-(4), with measurements consisting of energy spectra S(phi,N). The algorithms are applied to archival data of Priebe et al. for attosecond electron pulse trains; the authors report FWHM pulse durations of 224 as (MLE), 245(39) as (Bayesian), and 203 as (neural network), and predict a cathodoluminescence coherence |<b>| of about 36%, i.e. about 63% of a theoretical maximum of 0.58. They conclude that the originally reported 655 as FWHM is a substantial overestimate, with the pulses being almost three times shorter.

Significance. If the central quantitative result survives scrutiny, the paper would provide both practical reconstruction tools and a significant reinterpretation of a published attosecond electron pulse-train experiment; shorter pulses and quantified coherence are directly relevant to free-electron quantum optics and ultrafast electron microscopy. The methodological strengths are the physical-state-preserving MLE iteration, the Bayesian treatment with four MCMC chains and convergence diagnostics (Gelman-Rubin statistic and autocorrelation), and the demonstration of a fast neural-network reconstruction with validation on random states. The paper also gives a useful visual representation via a discrete Wigner function. However, the dramatic factor-of-three claim is conditional on an ideal unitary measurement model whose agreement with the archival data is never demonstrated; the credible contributions at this stage are the algorithms and the convergence machinery, not yet the revised pulse duration.

major comments (3)
  1. [§2, Eqs. (4)–(5); §3, Fig. 2] The central numerical claim is not supported by any data-model consistency check. The paper never displays the measured spectrogram S(phi,N), the reconstructed model predictions, residuals, or a goodness-of-fit statistic; the only quantitative validation, the fidelity of 0.837 in Section 3, compares the Bayesian posterior mean with a separately fitted forward model, not the forward model with the data. Because the MLE, Bayesian, and neural-network reconstructions all invert the same ideal unitary model of Eq. (4), their mutual agreement is expected even if the model is systematically wrong. Systematic errors from background counts, detector response, laser phase/amplitude fluctuations, or the 25-state truncation could shift the reconstructed density matrix and all derived durations and coherences; the quoted 245(39) as is therefore a conditional statistical uncertainty, and the factor-of-three shortening relative to 655 as is not yet established for the archival data.
  2. [§3, Fig. 2(c)] The forward-model comparison is partly circular. The model parameters (g between 3.73 and 4.52, free-space propagation distance 1.7 mm, 6.4% phase noise) are obtained by minimizing the Frobenius norm between the forward model and the Bayesian posterior mean, so the 'excellent agreement' between the forward-model temporal profile and the Bayesian result is an in-sample fit. The phase-noise level is also an assumed decoherence source rather than a quantity estimated from the raw data; this does not independently certify the reconstruction or the predicted 36% coherence.
  3. [§2, Eq. (5); §4, Fig. 5] The likelihood model and the quoted uncertainties omit a noise model. Eq. (5) treats S(phi,N) as if it were the exact expectation value of the model, with no description of Poisson counting statistics, background, per-phase normalization, or detector response, and the paper does not state the count totals of the archival data. The ML and NN pulse durations (224 as and 203 as) are quoted without uncertainty estimates, and the NN reconstruction has fidelity 0.66 with respect to the Bayesian mean and an energy spread about four times |g|; hence the claimed agreement among the three methods is not quantitatively established and cannot replace a direct residual test against the measured spectrogram.
minor comments (4)
  1. [Eq. (17)] The summation limits in Eq. (17) appear to be misprinted ('∞∑_{n=∞}'); the lower limit should be n = -∞, since negative photon-exchange orders are included in the Bessel sum.
  2. [§3, first paragraph] The statement that the Bayesian result is in 'good agreement' with the maximum-likelihood result would be more informative if the ML estimate had an uncertainty; please report credible intervals for the ML duration as well.
  3. [Introduction and Acknowledgements] The Introduction says 'it is now believed' that the Priebe et al. duration is a conservative estimate, but the only support mentioned is an unpublished master thesis in the Acknowledgements; please provide a citation or state the basis of this claim explicitly.
  4. [Data availability] The manuscript would benefit from a data-availability statement for the archival Priebe et al. spectrogram and for the neural-network training set; without these, the numerical results cannot be reproduced or independently checked.

Circularity Check

1 steps flagged · score 1.0 of 10

Central reconstruction chain is not circular; only a minor in-sample forward-model comparison is self-referential.

  1. fitted input called prediction [Section 3 (Bayesian inversion), after Fig. 2: 'We have fitted the posterior mean density matrix...' through '...agreement with that obtained using Bayesian estimation.']
    "We have fitted the posterior mean density matrix with a forward model that accounts for various sources of decoherence, by minimising the Frobenius norm between density matrices. Despite the indirect optimisation to attosecond pulses, the temporal profile of the forward model shows excellent agreement with that obtained using Bayesian estimation."

    The forward model is optimized directly against the Bayesian posterior-mean density matrix, and the temporal profile is a deterministic functional of that same matrix. Hence the reported 'excellent agreement' is an in-sample fit, not an independent prediction; the agreement is statistically forced once the density matrix is chosen. This is only mildly circular because the main pulse-duration and coherence claims are computed directly from the reconstructed density matrix, not from this fitted model, and the fitted forward-model parameters (g range, propagation distance, phase noise) are not the target outputs.

full rationale

The paper's central derivation is self-contained with respect to circularity: the archival spectrogram enters the likelihood (Eq. 5), and MLE, Bayesian, and neural-network routes each invert the same forward model (Eqs. 3-4) to produce a density matrix; pulse durations (224/245/203 as) and coherence (about 36%) are then read off as functionals of that reconstructed matrix rather than being fitted targets. The tomographic-completeness premise is cited to Ref. [19], a published inverse-problems theorem; although one of the current authors is a coauthor there, this is a citation to an external mathematical result, not a parameter fitted in this paper, so it does not by itself create a circular derivation. The theoretical maximum coherence 0.58 comes from the external Ref. [36]. The one self-referential element is the forward-model fit used to summarize the posterior mean: because that model is fitted directly to the posterior-mean density matrix, its temporal-profile and coherence agreement is an in-sample consistency check rather than a genuine prediction. Also, the paper does not report a data-model residual check, so systematic errors in the ideal model (Eq. 4) are a real validation risk; but that is a correctness/robustness concern, not a circularity of the derivation. Overall the central claims are not equivalent to their inputs by construction; the circularity is at most minor.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The central reconstruction rests on the discrete energy-ladder model and the unitarity of the laser interaction, on the tomographic completeness result from Ref [19], and on a product likelihood with an implicit counting noise model. The forward-model comparison adds several fitted physical parameters. No new physical entities are introduced.

free parameters (5)
  • Forward-model interaction strength g range = 3.73 to 4.52, assumed uniformly distributed
    Fitted by minimizing Frobenius norm between forward model and Bayesian posterior mean density matrix (Section 3).
  • Free-space propagation distance = 1.7 mm
    Fitted as part of the forward model comparison in Section 3.
  • Phase noise level = 6.4 percent normally distributed phase noise, normalized to optical period
    Fitted as part of the forward model comparison in Section 3.
  • MCMC proposal step beta = 0.02
    Adjusted to yield acceptance probability 0.45 and sufficiently small autocorrelation (Section 3, Fig. 3d). Tuning parameter, not physical.
  • Energy-state truncation dimension = Unstated for MLE and Bayesian; 25 for neural network output
    The infinite energy basis must be truncated for any numerical reconstruction, and the chosen truncation level is not specified for two of the three methods.
assumptions (6)
  • domain assumption The Hilbert space for swift electrons interacting with light is spanned by discrete energy states |N> with shift operators satisfying [b,b^dagger]=0.
    Section 2, Eqs. (1)-(3); this is the standard discrete energy ladder model for free-electron-light interaction.
  • domain assumption The interaction with the intense laser is the unitary U = exp(g b^dagger - \bar g b) with g proportional to the complex laser amplitude.
    Section 2, Eq. (3); the phase-dependent spectrogram is generated by this unitary.
  • standard math The spectrogram S(phi,N) is tomographically complete: it contains all information needed to deduce rho.
    Section 2, after Eq. (4); the paper invokes Ref [19] for uniqueness and stability of the inverse problem.
  • domain assumption The likelihood factorizes as a product over spectrogram values S(phi,N) used as exponents in Eq. (5), implying independent counting statistics.
    Section 2, Eq. (5); no explicit noise model or normalization for the measured counts is stated.
  • domain assumption A uniform prior over physical density matrices is used in the Bayesian inversion.
    Section 3, after Eq. (14); the prior choice influences posterior uncertainty estimates.
  • standard math Fourier analysis on the circle and integer lattice (Z x S1) supports the Wigner function definition.
    Section 2, Eqs. (8)-(10), cites Ref [23] for Fourier analysis on bounded domains.

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Cite this review

Pith. "Pith review of Tomographic reconstruction of free-electron quantum states." pith.science (2026). https://pith.science/paper/DXL5EBYR

@misc{pith2026250817594,
  author       = {Pith},
  title        = {Pith review of: Tomographic reconstruction of free-electron quantum states},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DXL5EBYR}},
  note         = {Machine review of arXiv:2508.17594}
}
read the original abstract

We give several algorithms for reconstructing quantum states of swift electrons, using maximum likelihood estimation, Bayesian inversion, and deep learning. We apply these algorithms to data previously recorded for an attosecond electron pulse-train to retrieve the density matrix and to analyse its physical properties. Based on the reconstructed quantum state, we obtain pulse-durations of about 245as and predict a degree of coherence of 36 per cent for radiations and excitations produced by these electrons.

Figures

Figures reproduced from arXiv: 2508.17594 by the authors.

Figure 1
Figure 1. Maximum likelihood reconstruction of attosecond electron pulse-trains. Main figure shows the Wigner function calculated from the reconstructed density matrix using Eq. 8, with the periodic temporal coordinate on the X axis and the discrete energy states on the Y axis. The temporal coordinates have been normalised to the wavelength of the laser light used (800nm), while the energies are in units of the photon energy.… view at source ↗
Figure 2
Figure 2. Bayesian inversion. a & b Real and imaginary parts of the reconstructed density matrix, with the mean value of each entry represented by the bar height and uncertainties by the colour. Labels on the X and Y axes indicate changes in the electron energy in units of photon energy. c Temporal density, showing mean values for the Bayesian reconstruction with 1σ uncertainty to either side, and a forward model fitted to th… view at source ↗
Figure 3
Figure 3. Markov chain Monte Carlo. a Dependence of the log-posterior on the iteration number for each of the four chains, labelled by colour, with close-up views of the first 20,000 samples and of those from 100,000 onwards. b Distributions of the occupation probability of the zero-loss peak for each of the four chains, labelled by colour. c & d The Gelman-Rubin statistic and autocorrelation of chains calculated with the dat… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Structure of neural network. Red layers denote the primary multilayer feedforward network used for recognising quantum states, and blue layers denote an auxiliary autoencoder network used to filter noise. Detailed descriptions of each component contained in the main te…
Figure 5
Figure 5. Figure 5: Quantum state reconstruction with the neural network. a Training and validation error. b Recon￾struction fidelities for randomly generated density matrices of various dimensions (1000 samples for each dimension) visualised using a box plot. c Reconstruction fidelities …

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Works this paper leans on

43 extracted references · 40 canonical work pages

  1. [1]

    Photon-induced near- field electron microscopy

    Brett Barwick, David J. Flannigan, and Ahmed H. Zewail. “Photon-induced near- field electron microscopy”. Nature 462, 902 (2009)

    work page 2009

  2. [2]

    Quantum coher- ent optical phase modulation in an ultrafast transmission electron microscope

    Armin Feist, Katharina E. Echternkamp, Jakob Schauss, Sergey V. Yalunin, Sascha Schäfer, and Claus Ropers. “Quantum coher- ent optical phase modulation in an ultrafast transmission electron microscope”. Nature 521, 200 (2015)

    work page 2015

  3. [3]

    Optical excitations in electron microscopy

    F. J. García de Abajo. “Optical excitations in electron microscopy”. Rev. Mod. Phys. 82, 209–275 (2010)

    work page 2010

  4. [4]

    Cavity- mediated electron-photon pairs

    Armin Feist, Guanhao Huang, Germaine Arend, Yujia Yang, Jan-Wilke Henke, Arslan Sajid Raja, F. Jasmin Kappert, Rui Ning Wang, Hugo Lourenço-Martins, Zheru Qiu, Junqiu Liu, Ofer Kfir, Tobias J. Kippenberg, and Claus Ropers. “Cavity- mediated electron-photon pairs”. Science 377, 777–780 (2022)

    work page 2022

  5. [5]

    Elec- trons herald non-classical light

    Germaine Arend, Guanhao Huang, Armin Feist, Yujia Yang, Jan-Wilke Henke, Zheru Qiu, Hao Jeng, Arslan Sajid Raja, Rudolf Haindl, Rui Ning Wang, Tobias J. Kip- penberg, and Claus Ropers. “Elec- trons herald non-classical light” (2024). arXiv:2409.11300

    work page arXiv 2024

  6. [6]

    Entanglements of elec- trons and cavity photons in the strong- coupling regime

    Ofer Kfir. “Entanglements of elec- trons and cavity photons in the strong- coupling regime”. Phys. Rev. Lett. 123, 103602 (2019)

    work page 2019

  7. [7]

    Observation of quan- tum entanglement between free electrons and photons

    Jan-Wilke Henke, Hao Jeng, Murat Sivis, and Claus Ropers. “Observation of quan- tum entanglement between free electrons and photons” (2025). arXiv:2504.13047

    arXiv 2025

  8. [8]

    Experimental verification of electron-photon entanglement

    Alexander Preimesberger, Sergei Bogdanov, IsobelC.Bicket, PhilaRembold, andPhilipp Haslinger. “Experimental verification of electron-photon entanglement” (2025). arXiv:2504.13163

    arXiv 2025

Show all 43 references
  1. [9]

    Attosecond electron pulse trains and quan- tum state reconstruction in ultrafast trans- mission electron microscopy

    Katharina E. Priebe, Christopher Rathje, Sergey V. Yalunin, Thorsten Hohage, Armin Feist, Sascha Schäfer, and Claus Ropers. “Attosecond electron pulse trains and quan- tum state reconstruction in ultrafast trans- mission electron microscopy”. Nature Pho- tonics 11, 793–797 (2017)

    2017

  2. [10]

    Diffrac- tion and microscopy with attosecond elec- tron pulse trains

    Yuya Morimoto and Peter Baum. “Diffrac- tion and microscopy with attosecond elec- tron pulse trains”. Nature Physics14, 252– 256 (2018)

    2018

  3. [11]

    Ponderomotive generation and 10 detection of attosecond free-electron pulse trains

    M. Kozák, N. Schönenberger, and P. Hom- melhoff. “Ponderomotive generation and 10 detection of attosecond free-electron pulse trains”. Phys. Rev. Lett.120, 103203 (2018)

    2018

  4. [12]

    Attosecond electron microscopy by free-electron homodyne detection

    John H. Gaida, Hugo Lourenço-Martins, Murat Sivis, Thomas Rittmann, Armin Feist, F. Javier García de Abajo, and Claus Ropers. “Attosecond electron microscopy by free-electron homodyne detection”. Nature Photonics 18, 509–515 (2024)

    2024

  5. [13]

    Attosec- ond electron microscopy of sub-cycle optical dynamics

    David Nabben, Joel Kuttruff, Levin Stolz, Andrey Ryabov, and Peter Baum. “Attosec- ond electron microscopy of sub-cycle optical dynamics”. Nature 619, 63–67 (2023)

    2023

  6. [14]

    Electron-beam spectroscopy for nanophotonics

    Albert Polman, Mathieu Kociak, and F. Javier García de Abajo. “Electron-beam spectroscopy for nanophotonics”. Nature Materials 18, 1158–1171 (2019)

    2019

  7. [15]

    Creation of optical cat and gkp states using shaped free electrons

    Raphael Dahan, Gefen Baranes, Alexey Gor- lach, Ron Ruimy, Nicholas Rivera, and Ido Kaminer. “Creation of optical cat and gkp states using shaped free electrons”. Phys. Rev. X 13, 031001 (2023)

    2023

  8. [16]

    Tunable quantum light by modulated free electrons

    Valerio Di Giulio, Rudolf Haindl, and Claus Ropers. “Tunable quantum light by modulated free electrons” (2025). arXiv:2501.16771

    2025 arXiv

  9. [17]

    Probing electron-photon entanglement using a quantum eraser

    Jan-Wilke Henke, Hao Jeng, and Claus Rop- ers. “Probing electron-photon entanglement using a quantum eraser”. Phys. Rev. A111, 012610 (2025)

    2025

  10. [18]

    Designs for a quantum elec- tron microscope

    P. Kruit, R.G. Hobbs, C-S. Kim, Y. Yang, V.R. Manfrinato, J. Hammer, S. Thomas, P. Weber, B. Klopfer, C. Kohstall, T. Juff- mann, M.A. Kasevich, P. Hommelhoff, and K.K. Berggren. “Designs for a quantum elec- tron microscope”. Ultramicroscopy164, 31– 45 (2016)

    2016

  11. [19]

    Density matrix reconstructions in ultrafast transmission electron microscopy: uniqueness, stability, andconvergencerates

    Cong Shi, Claus Ropers, and Thorsten Ho- hage. “Density matrix reconstructions in ultrafast transmission electron microscopy: uniqueness, stability, andconvergencerates”. Inverse Problems 36, 025005 (2020)

    2020

  12. [20]

    Iterative maximum-likelihood reconstruction in quantum homodyne to- mography

    A I Lvovsky. “Iterative maximum-likelihood reconstruction in quantum homodyne to- mography”. Journal of Optics B: Quantum and Semiclassical Optics6, S556 (2004)

    2004

  13. [21]

    Semi-classical mechanics in phase space: A study of wigner’s function

    Michael Victor Berry and John Michael Zi- man. “Semi-classical mechanics in phase space: A study of wigner’s function”. Philo- sophical Transactions of the Royal Society of London. Series A, Mathematical and Physi- cal Sciences 287, 237–271 (1977)

    1977

  14. [22]

    Wigner distribution for angle coordinates in quantum mechanics

    N. Mukunda. “Wigner distribution for angle coordinates in quantum mechanics”. Ameri- can Journal of Physics47, 182–187 (1979)

    1979

  15. [23]

    Fourier analysis: An introduction

    Elias M. Stein and Rami Shakarchi. “Fourier analysis: An introduction”. Princeton University Press. (2003). url: https: //press.princeton.edu/books/ebook/ 9781400831234/fourier-analysis-pdf

    2003

  16. [24]

    Tailored high-contrast attosec- ond electron pulses for coherent excita- tion and scattering

    Sergey V. Yalunin, Armin Feist, and Claus Ropers. “Tailored high-contrast attosec- ond electron pulses for coherent excita- tion and scattering”. Phys. Rev. Res. 3, L032036 (2021)

    2021

  17. [25]

    Phase and angle variables in quan- tum mechanics

    Peter Carruthers and Michael Martin Ni- eto. “Phase and angle variables in quan- tum mechanics”. Rev. Mod. Phys.40, 411– 440 (1968)

    1968

  18. [26]

    Optimal, reliable es- timation of quantum states

    Robin Blume-Kohout. “Optimal, reliable es- timation of quantum states”. New Journal of Physics 12, 043034 (2010)

    2010

  19. [27]

    A practical and efficient approach for bayesian quantum state estimation

    Joseph M Lukens, Kody J H Law, Ajay Jasra, and Pavel Lougovski. “A practical and efficient approach for bayesian quantum state estimation”. New Journal of Physics 22, 063038 (2020)

    2020

  20. [28]

    Mcmc methods for functions: Modifying old algo- rithms to make them faster

    Simon L. Cotter, Gareth O. Roberts, An- drew M. Stuart, and David White. “Mcmc methods for functions: Modifying old algo- rithms to make them faster”. Statistical Sci- ence 28, 424–446 (2013)

    2013

  21. [29]

    hippylib-muq: A bayesianinferencesoftwareframeworkforin- tegration of data with complex predictive models under uncertainty

    Ki-Tae Kim, Umberto Villa, Matthew Parno, Youssef Marzouk, Omar Ghattas, and Noemi Petra. “hippylib-muq: A bayesianinferencesoftwareframeworkforin- tegration of data with complex predictive models under uncertainty”. ACM Trans. Math. Softw.49 (2023)

    2023

  22. [30]

    A simple automatic deriva- tive evaluation program

    R. E. Wengert. “A simple automatic deriva- tive evaluation program”. Commun. ACM7, 463–464 (1964)

    1964

  23. [31]

    Bayesian homodyne and hetero- dyne tomography

    Joseph C. Chapman, Joseph M. Lukens, Bing Qi, Raphael C. Pooser, and Nicholas A. Peters. “Bayesian homodyne and hetero- dyne tomography”. Opt. Express30, 15184– 15200 (2022). 11

    2022

  24. [32]

    Algorithms for kullback–leibler approximation of probability measures in in- finite dimensions

    F. J. Pinski, G. Simpson, A. M. Stuart, and H. Weber. “Algorithms for kullback–leibler approximation of probability measures in in- finite dimensions”. SIAM Journal on Scien- tific Computing 37, A2733–A2757 (2015)

    2015

  25. [33]

    Optical co- herence transfer mediated by free electrons

    Ofer Kfir, Valerio Di Giulio, F. Javier García de Abajo, and Claus Ropers. “Optical co- herence transfer mediated by free electrons”. Science Advances7, eabf6380 (2021)

    2021

  26. [34]

    Free- electron–bound-electron resonant interac- tion

    Avraham Gover and Amnon Yariv. “Free- electron–bound-electron resonant interac- tion”. Phys. Rev. Lett.124, 064801 (2020)

    2020

  27. [35]

    Quantum entanglement and modu- lation enhancement of free-electron–bound- electron interaction

    Zhexin Zhao, Xiao-Qi Sun, and Shanhui Fan. “Quantum entanglement and modu- lation enhancement of free-electron–bound- electron interaction”. Phys. Rev. Lett.126, 233402 (2021)

    2021

  28. [36]

    Modula- tion of cathodoluminescence emission by in- terference with external light

    Valerio Di Giulio, Ofer Kfir, Claus Ropers, and F. Javier García de Abajo. “Modula- tion of cathodoluminescence emission by in- terference with external light”. ACS Nano 15, 7290–7304 (2021)

    2021

  29. [37]

    Multilayer feedforward net- works are universal approximators

    Kurt Hornik, Maxwell Stinchcombe, and Halbert White. “Multilayer feedforward net- works are universal approximators”. Neural Networks 2, 359–366 (1989)

    1989

  30. [38]

    Real-time quantum feedback prepares and stabilizes photon number states

    Clément Sayrin, Igor Dotsenko, Xingxing Zhou, Bruno Peaudecerf, Théo Rybarczyk, Sébastien Gleyzes, Pierre Rouchon, Maz- yar Mirrahimi, Hadis Amini, Michel Brune, Jean-Michel Raimond, and Serge Haroche. “Real-time quantum feedback prepares and stabilizes photon number states”. ...

    2011

  31. [39]

    Spatiotemporal electron beam fo- cusing through parallel interactions with shaped optical fields

    F. Javier García de Abajo and Claus Rop- ers. “Spatiotemporal electron beam fo- cusing through parallel interactions with shaped optical fields”. Phys. Rev. Lett.130, 246901 (2023)

    2023

  32. [40]

    Free-electron qubits and maximum-contrast attosecond pulses via temporal talbot re- vivals

    M. V. Tsarev, A. Ryabov, and P. Baum. “Free-electron qubits and maximum-contrast attosecond pulses via temporal talbot re- vivals”. Phys. Rev. Res.3, 043033 (2021)

    2021

  33. [41]

    Free-electronqubits

    OriReinhardt, ChenMechel, MorganLynch, andIdoKaminer. “Free-electronqubits”. An- nalen der Physik533, 2000254 (2021)

    2021

  34. [42]

    Coulomb- correlated electron number states in a trans- mission electron microscope beam

    Rudolf Haindl, Armin Feist, Till Domröse, Marcel Möller, John H. Gaida, Sergey V. Yalunin, and Claus Ropers. “Coulomb- correlated electron number states in a trans- mission electron microscope beam”. Nature Physics 19, 1410–1417 (2023)

    2023

  35. [43]

    Few-electron correlations af- ter ultrafast photoemission from nanomet- ric needle tips

    Stefan Meier, Jonas Heimerl, and Peter Hommelhoff. “Few-electron correlations af- ter ultrafast photoemission from nanomet- ric needle tips”. Nature Physics 19, 1402– 1409 (2023). 12

    2023

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.