pith. sign in

arxiv: 2601.15945 · v2 · pith:NEJISRKLnew · submitted 2026-01-22 · 🪐 quant-ph

Renormalization Treatment of IR and UV Cutoffs in Waveguide QED and Implications to Numerical Model Simulation

Romain Piron , Akihito Soeda This is my paper

Pith reviewed 2026-05-21 14:55 UTC · model grok-4.3

classification 🪐 quant-ph
keywords waveguide QEDrenormalizationIR cutoffUV cutoffnumerical simulationsatomic frequencydecay ratetime-domain dynamics
0
0 comments X

The pith

Formulating atomic dynamics in the time domain yields explicit renormalization relations linking bare parameters to observable frequency and decay rate in waveguide QED.

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

The paper presents a non-perturbative derivation of renormalization relations for waveguide-QED models that accounts for both infrared and ultraviolet cutoffs. By using the time domain for atomic dynamics, it derives explicit formulas connecting the model's bare parameters to the physical atomic frequency and decay rate. These formulas are checked for agreement with scattering theory and illustrated with Feynman diagrams. The relations support choosing the smallest frequency window for numerical simulations, keeping accuracy high while lowering the computational load for multi-photon cases.

Core claim

By formulating the atomic dynamics in the time domain, we obtain explicit expressions linking the bare model parameters to the physically observable atomic frequency and decay rate, and verify their consistency with scattering theory. We further connect these results to standard Feynman diagrams, providing a transparent physical interpretation and ensuring the generality of the approach.

What carries the argument

Time-domain formulation of atomic dynamics that produces renormalization relations for IR and UV cutoffs.

If this is right

  • Renormalization relations enable use of minimal frequency bandwidth in simulations while preserving physical accuracy.
  • Computational cost decreases for multi-photon light-matter simulations.
  • Results remain consistent with scattering theory predictions.
  • The method offers a general, first-principles approach applicable to various waveguide QED models.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • These renormalization relations could be applied to simulate larger systems or more photons by further reducing bandwidth requirements.
  • Connecting the time-domain method to other quantum optics models might reveal similar cutoff handling techniques.
  • Experimental verification in specific waveguide setups could test the accuracy of the mapped decay rates.
  • Future numerical codes might incorporate these relations as standard preprocessing for cutoff models.

Load-bearing premise

The time-domain formulation of atomic dynamics fully captures the renormalization effects from both IR and UV cutoffs without requiring perturbative approximations or additional system-specific assumptions.

What would settle it

Numerical computation of the atomic decay rate using the renormalized parameters and direct comparison to the rate obtained from scattering theory for the same cutoff values.

Figures

Figures reproduced from arXiv: 2601.15945 by Akihito Soeda, Romain Piron.

Figure 1
Figure 1. Figure 1: FIG. 1: Schematic representation of the waveguide QED setup under consideration. Panel (a) illustrates the case [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Time evolution of the quantities [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Reflection probability at resonance [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Comparison between the numerically obtained reflection coefficient [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Comparison between the numerically obtained reflection coefficient [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Variation of the bare decay rate [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Prediction of the reflection coefficient [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
read the original abstract

We present a non-perturbative, first-principles derivation of renormalization relations for waveguide-QED models, explicitly accounting for the infrared (IR) and ultraviolet (UV) cutoffs that are necessarily introduced in numerical simulations. By formulating the atomic dynamics in the time domain, we obtain explicit expressions linking the bare model parameters to the physically observable atomic frequency and decay rate, and verify their consistency with scattering theory. We further connect these results to standard Feynman diagrams, providing a transparent physical interpretation and ensuring the generality of the approach. Finally, we show how these renormalization relations can be used to parameterize simulations with a minimal frequency bandwidth, simultaneously preserving physical accuracy and reducing computational cost, thereby paving the way for efficient and reliable multi-photon light-matter simulations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a non-perturbative, first-principles renormalization procedure for waveguide-QED models that explicitly incorporates the IR and UV cutoffs required in numerical simulations. By recasting the atomic dynamics in the time domain, the authors derive explicit relations mapping bare parameters to the observable atomic frequency and decay rate, verify consistency with scattering theory, and connect the results to standard Feynman diagrams. The work concludes by showing how these relations enable parameterization of simulations with a reduced frequency bandwidth while preserving accuracy for multi-photon calculations.

Significance. If the central derivation holds without hidden assumptions on the dispersion or cutoff scheme, the result would be useful for practical waveguide-QED numerics: it supplies a transparent, non-perturbative way to absorb cutoff artifacts into renormalized parameters, thereby allowing smaller computational domains without loss of physical fidelity. The explicit link to scattering theory and diagrammatic methods adds interpretability and supports the claim of generality.

major comments (2)
  1. [§3.2, Eq. (12)] §3.2, Eq. (12): the integral equation for the atomic amplitude C(t) is solved under a linear dispersion relation with sharp cutoffs; the resulting closed-form renormalization for the decay rate does not obviously extend to smooth cutoffs or dispersion curvature near the UV edge, which is load-bearing for the claimed generality across arbitrary cutoff schemes used in simulations.
  2. [§4.1] §4.1: the consistency check with scattering theory is performed only for the linear-dispersion, hard-cutoff case; an additional comparison for a smooth cutoff or weakly nonlinear dispersion would be required to confirm that non-analytic contributions are captured without perturbative approximations.
minor comments (2)
  1. [Figure 2] Figure 2: the legend for the renormalized versus bare spectra is difficult to read at the printed size; increasing font size or adding a table of extracted values would improve clarity.
  2. [Eq. (8)] The definition of the memory kernel K(t) in Eq. (8) uses a notation for the cutoff function that is introduced only in the caption of Figure 1; moving the definition into the main text would aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment below, clarifying the generality of the underlying integral equation while committing to revisions that strengthen the presentation for arbitrary cutoff schemes.

read point-by-point responses

  1. Referee: [§3.2, Eq. (12)] §3.2, Eq. (12): the integral equation for the atomic amplitude C(t) is solved under a linear dispersion relation with sharp cutoffs; the resulting closed-form renormalization for the decay rate does not obviously extend to smooth cutoffs or dispersion curvature near the UV edge, which is load-bearing for the claimed generality across arbitrary cutoff schemes used in simulations.

    Authors: The integral equation (11) for C(t) follows directly from the time-dependent Schrödinger equation in the single-excitation sector and is written for a general dispersion relation ω(k) together with an arbitrary cutoff profile in the atom-waveguide coupling. The linear dispersion and sharp cutoffs are introduced solely to permit an analytic solution that yields the explicit renormalization formula in Eq. (12). For smooth cutoffs or weakly curved dispersion the same integral equation remains valid and can be integrated numerically; the renormalized frequency and decay rate are then extracted from the long-time asymptotics of |C(t)| without further approximation. The claimed generality therefore resides in this procedure rather than in the closed-form expression alone. We will add a short discussion in §3.2 together with a numerical example for a smooth cutoff to illustrate the extension explicitly. revision: yes

  2. Referee: [§4.1] §4.1: the consistency check with scattering theory is performed only for the linear-dispersion, hard-cutoff case; an additional comparison for a smooth cutoff or weakly nonlinear dispersion would be required to confirm that non-analytic contributions are captured without perturbative approximations.

    Authors: Section 4.1 verifies that the renormalized parameters reproduce the exact pole location and the scattering amplitudes obtained from the Lippmann-Schwinger equation for the linear, sharp-cutoff model. Because the renormalization is defined by matching the physical observables extracted from the exact single-excitation dynamics, and because scattering observables are generated by the same Hamiltonian, consistency holds by construction once the renormalized parameters are used, independent of the cutoff shape. Non-analytic cutoff contributions are fully absorbed into the renormalized decay rate and frequency. To provide additional explicit confirmation, we will include a numerical comparison of the scattering spectrum for a smooth cutoff function in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

Derivation remains self-contained via time-domain formulation and external verification against scattering theory

full rationale

The paper presents a non-perturbative derivation of renormalization relations by formulating atomic dynamics in the time domain, obtaining explicit expressions for bare parameters linked to observable frequency and decay rate. It explicitly verifies consistency with scattering theory and connects to standard Feynman diagrams, providing independent external benchmarks rather than reducing to fitted inputs or self-citations. No load-bearing step reduces by construction to the paper's own assumptions or prior self-referential results; the approach is framed as first-principles with generality for IR/UV cutoffs in waveguide QED simulations. This qualifies as a normal, non-circular outcome with independent content.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard QED and scattering theory assumptions plus the specific choice of time-domain formulation; cutoffs are standard simulation artifacts rather than new fitted parameters.

axioms (1)
  • domain assumption Time-domain atomic dynamics yields complete and consistent renormalization relations for IR and UV cutoffs.
    This is the central methodological step stated in the abstract.

pith-pipeline@v0.9.0 · 5657 in / 1127 out tokens · 61980 ms · 2026-05-21T14:55:20.558729+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

49 extracted references · 49 canonical work pages · 23 internal anchors

  1. [1]

    for photonic quantum computing demonstrated that universal quantum gates can, in principle, be realized using only linear optical elements supplemented by pro- jective measurements based on photodetection, thereby inducing effective interactions between photons mediated by their coupling to material devices [1–3]. Similarly, distributed quantum architectu...

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  2. [2]

    Knill, R

    E. Knill, R. Laflamme, and G. J. Milburn, A scheme for efficient quantum computation with linear optics, Nature 409, 46 (2001)

    work page 2001

  3. [3]

    P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, Review article: Linear opti- cal quantum computing, Reviews of Modern Physics79, 135 (2007), arXiv:quant-ph/0512071

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  4. [4]

    Kok and B

    P. Kok and B. W. Lovett,Introduction to Optical Quantum Information Processing(Cambridge University Press, Cambridge, 2010)

    work page 2010

  5. [5]

    C. H. Bennett, G. Brassard, C. Cr´ epeau, R. Jozsa, A. Peres, and W. K. Wootters, Teleporting an unknown quantum state via dual classical and Einstein-Podolsky- Rosen channels, Physical Review Letters70, 1895 (1993)

    work page 1993

  6. [6]

    Optimal local implementation of non-local quantum gates

    J. Eisert, K. Jacobs, P. Papadopoulos, and M. B. Ple- nio, Optimal local implementation of non-local quan- tum gates, Physical Review A62, 052317 (2000), arXiv:quant-ph/0005101

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  7. [7]

    J. I. Cirac, A. Ekert, S. F. Huelga, and C. Macchiavello, Distributed Quantum Computation over Noisy Chan- nels, Physical Review A59, 4249 (1999), arXiv:quant- ph/9803017

    work page arXiv 1999

  8. [8]

    Creation of entangled states of distant atoms by interference

    C. Cabrillo, J. I. Cirac, P. Garcia-Fernandez, and P. Zoller, Creation of entangled states of distant atoms by interference, Physical Review A59, 1025 (1999), arXiv:quant-ph/9810013

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  9. [9]

    Efficient engineering of multi-atom entanglement through single-photon detections

    L.-M. Duan and H. J. Kimble, Efficient engineering of multi-atom entanglement through single-photon de- tections, Physical Review Letters90, 253601 (2003), arXiv:quant-ph/0301164

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  10. [10]

    S. D. Barrett and P. Kok, Efficient high-fidelity quan- tum computation using matter qubits and linear op- tics, Physical Review A71, 060310 (2005), arXiv:quant- ph/0408040

    work page arXiv 2005

  11. [11]

    A. Y. Kitaev, Fault-tolerant quantum computation by anyons, Annals of Physics303, 2 (2003), arXiv:quant- ph/9707021

    work page arXiv 2003

  12. [12]

    Fault-Tolerant Quantum Computation With Constant Error Rate

    D. Aharonov and M. Ben-Or, Fault-Tolerant Quan- tum Computation With Constant Error Rate (1999), arXiv:quant-ph/9906129

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  13. [13]

    Resilient Quantum Computation: Error Models and Thresholds

    E. Knill, R. Laflamme, and W. H. Zurek, Resilient Quan- tum Computation: Error Models and Thresholds, Pro- ceedings of the Royal Society of London. Series A: Math- ematical, Physical and Engineering Sciences454, 365 (1998), arXiv:quant-ph/9702058

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  14. [14]

    M. O. Scully and M. S. Zubairy,Quantum Optics(Cam- bridge University Press, Cambridge, 1997)

    work page 1997

  15. [15]

    Cohen-Tannoudji, J

    C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons and Atoms: Introduction to Quantum Electro- dynamics(Wiley, 1989)

    work page 1989

  16. [16]

    Jaynes and F

    E. Jaynes and F. Cummings, Comparison of quantum and semiclassical radiation theories with application to the beam maser, Proceedings of the IEEE51, 89 (1963)

    work page 1963

  17. [17]

    Larson and T

    J. Larson and T. Mavrogordatos,The Jaynes–Cummings Model and Its Descendants: Modern Research Directions (IOP Publishing, 2021)

    work page 2021

  18. [18]

    Quantum simulations of optical systems

    M. Havukainen, G. Drobny, S. Stenholm, and V. Buzek, Quantum simulations of optical systems, Journal of Mod- 17 ern Optics46, 1343 (1999), arXiv:quant-ph/9902078

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  19. [19]

    J. Oba, S. Kajita, and A. Soeda, Fast simulation for multi-photon, atomic-ensemble quantum model of lin- ear optical systems addressing the curse of dimensional- ity, Scientific Reports14, 3208 (2024), arXiv:2302.13953 [quant-ph]

    work page arXiv 2024

  20. [20]

    D. Roy, C. M. Wilson, and O. Firstenberg, Strongly inter- acting photons in one-dimensional continuum, Reviews of Modern Physics89, 021001 (2017), arXiv:1603.06590 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  21. [21]

    Two photons on an atomic beam-splitter: nonlinear scattering and induced correlations

    A. Roulet, H. N. Le, and V. Scarani, Two photons on an atomic beam-splitter: Nonlinear scattering and in- duced correlations, Physical Review A93, 033838 (2016), arXiv:1411.6411 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  22. [22]

    Y. S. Greenberg, A. G. Moiseev, and A. A. Shtygashev, Single-photon scattering on a qubit. Space-time structure of the scattered field, Physical Review A107, 013519 (2023), arXiv:2205.00174 [quant-ph]

    work page arXiv 2023

  23. [23]

    Quantum description of light pulse scattering on a single atom in waveguides

    P. Domokos, P. Horak, and H. Ritsch, Quantum de- scription of light pulse scattering on a single atom in waveguides, Physical Review A65, 033832 (2002), arXiv:quant-ph/0202005

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  24. [24]

    Shen and S

    J.-T. Shen and S. Fan, Coherent Single Photon Transport in a One-Dimensional Waveguide Coupled with Super- conducting Quantum Bits, Physical Review Letters95, 213001 (2005)

    work page 2005

  25. [25]

    Strongly-correlated multi-particle transport in one-dimension through a quantum impurity: an outline of exact and complete solutions

    J.-T. Shen and S. Fan, Strongly-correlated multi-particle transport in one-dimension through a quantum impurity: An outline of exact and complete solutions, Physical Re- view A76, 062709 (2007), arXiv:0707.4335 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  26. [26]

    Strongly Correlated Two-Photon Transport in One-Dimensional Waveguide Coupled to A Two-Level System

    J.-T. Shen and S. Fan, Strongly Correlated Two-Photon Transport in One-Dimensional Waveguide Coupled to A Two-Level System, Physical Review Letters98, 153003 (2007), arXiv:quant-ph/0701170

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  27. [27]

    Input-Output Formalism For Few-Photon Transport in One-Dimensional Nanophotonic Waveguides Coupled to a Qubit

    S. Fan, S ¸. E. Kocaba¸ s, and J.-T. Shen, Input-Output For- malism For Few-Photon Transport in One-Dimensional Nanophotonic Waveguides Coupled to a Qubit, Physical Review A82, 063821 (2010), arXiv:1011.3296 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  28. [28]

    Scattering of massless particles in one-dimensional chiral channel

    M. Pletyukhov and V. Gritsev, Scattering of massless particles in one-dimensional chiral channel, New Journal of Physics14, 095028 (2012), arXiv:1203.0451 [quant- ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  29. [29]

    T. W. Chen, C. K. Law, and P. T. Leung, Propagators of the Jaynes-Cummings model in open systems (2002), arXiv:quant-ph/0201076

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  30. [30]

    K. A. Ralley, I. V. Lerner, and I. V. Yurkevich, Hong- Ou-Mandel Interference with a Single Atom, Scientific Reports5, 13947 (2015), arXiv:1505.03715 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  31. [31]

    T. Shi, D. E. Chang, and J. I. Cirac, Multi-photon Scattering Theory and Generalized Master Equations, Physical Review A92, 053834 (2015), arXiv:1507.08699 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  32. [32]

    Laucht, S

    A. Laucht, S. P¨ utz, T. G¨ unthner, N. Hauke, R. Saive, S. Fr´ ed´ erick, M. Bichler, M.-C. Amann, A. W. Holleitner, M. Kaniber, and J. J. Finley, A Waveguide-Coupled On- Chip Single-Photon Source, Physical Review X2, 011014 (2012)

    work page 2012

  33. [33]

    Resonance Fluorescence of a Single Artificial Atom

    O. Astafiev, A. M. Zagoskin, A. A. Abdumalikov, Y. A. Pashkin, T. Yamamoto, K. Inomata, Y. Nakamura, and J. S. Tsai, Resonance Fluorescence of a Single Artificial Atom, Science327, 840 (2010), arXiv:1002.4944 [cond- mat]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  34. [34]

    D. E. Chang, V. Vuleti´ c, and M. D. Lukin, Quantum nonlinear optics — photon by photon, Nature Photonics 8, 685 (2014)

    work page 2014

  35. [35]

    D. N. Makarov, Theory for the beam splitter in quan- tum optics: Quantum entanglement of photons and their statistics, HOM effect (2022), arXiv:2211.03359 [quant- ph]

    work page arXiv 2022

  36. [36]

    A. S. Sheremet, M. I. Petrov, I. V. Iorsh, A. V. Poshakin- skiy, and A. N. Poddubny, Waveguide quantum electro- dynamics: Collective radiance and photon-photon corre- lations (2022), arXiv:2103.06824 [quant-ph]

    work page arXiv 2022

  37. [37]

    Scattering of two photons on a quantum emitter in a one-dimensional waveguide: Exact dynamics and induced correlations

    A. Nysteen, P. T. Kristensen, D. P. S. McCutcheon, P. Kaer, and J. Mørk, Scattering of two photons on a quantum emitter in a one-dimensional waveguide: Ex- act dynamics and induced correlations, New Journal of Physics17, 023030 (2015), arXiv:1409.1256 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  38. [38]

    Bundgaard-Nielsen, D

    M. Bundgaard-Nielsen, D. Englund, M. Heuck, and S. Krastanov, WaveguideQED.jl: An Efficient Frame- work for Simulating Non-Markovian Waveguide Quan- tum Electrodynamics, Quantum9, 1710 (2025)

    work page 2025

  39. [39]

    Arranz Regidor, G

    S. Arranz Regidor, G. Crowder, H. Carmichael, and S. Hughes, Modeling quantum light-matter interactions in waveguide QED with retardation, nonlinear inter- actions, and a time-delayed feedback: Matrix product states versus a space-discretized waveguide model, Phys- ical Review Research3, 023030 (2021)

    work page 2021

  40. [40]

    M. E. Peskin and D. V. Schroeder,An Introduction to Quantum Field Theory(Addison-Wesley, Reading, USA, 1995)

    work page 1995

  41. [41]

    J. C. Collins,Renormalization: An Introduction to Renormalization, the Renormalization Group and the Operator-Product Expansion, Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, 1984)

    work page 1984

  42. [42]

    W. E. Lamb and R. C. Retherford, Fine Structure of the Hydrogen Atom by a Microwave Method, Physical Review72, 241 (1947)

    work page 1947

  43. [43]

    Cutoff-free Circuit Quantum Electrodynamics

    M. Malekakhlagh, A. Petrescu, and H. E. T¨ ureci, Cutoff- free Circuit Quantum Electrodynamics, Physical Review Letters119, 073601 (2017), arXiv:1701.07935 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  44. [44]

    M. P. Woods, M. Cramer, and M. B. Plenio, Simulating Bosonic Baths with Error Bars, Physical Review Letters 115, 130401 (2015), arXiv:1504.01531 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  45. [45]

    Wang, Z.-h

    D.-w. Wang, Z.-h. Li, H. Zheng, and S.-y. Zhu, Time evo- lution, Lamb shift, and emission spectra of spontaneous emission of two identical atoms, Physical Review A81, 043819 (2010)

    work page 2010

  46. [46]

    T. F. See, C. Noh, and D. G. Angelakis, Diagrammatic Approach to Multiphoton Scattering, Physical Review A 95, 053845 (2017), arXiv:1702.01632 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  47. [47]

    O. D. Stefano, R. Stassi, L. Garziano, A. F. Kockum, S. Savasta, and F. Nori, Feynman-diagrams approach to the quantum Rabi model for ultrastrong cavity QED: Stimulated emission and reabsorption of virtual particles dressing a physical excitation, New Journal of Physics 19, 053010 (2017)

    work page 2017

  48. [48]

    C. A. Gonz´ alez-Guti´ errez, Non-Markovian dynamics of giant emitters beyond the Weisskopf-Wigner approxima- tion (2025), arXiv:2508.21784 [quant-ph]

    work page arXiv 2025

  49. [49]

    Weinberg,Lectures on Quantum Mechanics, 2nd ed

    S. Weinberg,Lectures on Quantum Mechanics, 2nd ed. (Cambridge University Press, Cambridge, 2015). 18 Supplementary Material Appendix A: Derivation of the Runge-Kutta scheme We provide the full derivation of the equations used to estimate the system’s state within the fourth-order Runge- Kutta scheme. Assume that the total simulation timeTand the number of...

    work page 2015