Pith. sign in

REVIEW 3 major objections 5 minor 57 references

A single superconducting transmon at the end of a semi-infinite transmission line, driven by two continuous microwave tones, acts as a cavity-free tunable frequency converter and frequency-comb generator whose peak spacing is the drive detu

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-04 07:05 UTC pith:HPK4QFFF

load-bearing objection Solid single-transmon frequency-conversion demo; the mode-locking and comb claims overreach the evidence. the 3 major comments →

arxiv 2510.26749 v1 pith:HPK4QFFF submitted 2025-10-30 quant-ph

Tunable frequency conversion and comb generation with a superconducting artificial atom

Fahad Aziz , Zhengqi Niu , Tzu-Yen Hsieh , Kuan Ting Lin , Yu-Huan Huang , Yen-Hsiang Lin , Ching-Yeh Chen , Yu-Ting Cheng
show 7 more authors
Kai-Min Hsieh Jeng-Chung Chen Anton Frisk Kockum Guin-Dar Lin Zhi-Rong Lin Ping-Yi Wen Io-Chun Hoi
This is my paper
classification quant-ph
keywords superconducting artificial atomtransmonfrequency conversionfrequency combwave mixingwaveguide quantum electrodynamicspower spectral densitymicrowave photonics
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper claims that one superconducting artificial atom—a transmon attached to the end of a semi-infinite transmission line—can act as both a frequency converter and a frequency-comb generator when driven by two continuous microwave tones. The atom's nonlinear energy ladder mixes the two drive frequencies, producing new spectral lines at equally spaced combinations such as 2ω2−ω1, 2ω1−ω2, 3ω2−2ω1, and higher orders. The spacing between adjacent peaks is set by the detuning Δω=ω2−ω1, and sweeping that detuning from about 10 MHz to 52 MHz produces combs with up to ten visible peaks. The authors report agreement between the measured spectra and a five-level theoretical model built from independently measured qubit parameters. A sympathetic reader would care because this offers a cavity-free, small-footprint, in-situ tunable microwave frequency source for on-chip quantum optics.

Core claim

The paper's central claim is that a single transmon at the end of a semi-infinite transmission line, pumped by two continuous rf fields, emits a power spectral density containing sidebands at frequencies (n+1)ω1−nω2 and (n+1)ω2−nω1 with constant spacing Δω=ω2−ω1. With equal input powers the sidebands appear symmetrically around the two carriers; increasing one drive power selects up-conversion or down-conversion. Sweeping ω2 while holding ω1 at the qubit transition tunes the converted peak continuously, and increasing drive power or decreasing detuning multiplies the number of equally spaced comb lines. The emitted spectrum is attributed to wave mixing mediated by the transmon's anharmonic l

What carries the argument

The central object is the transmon itself, modeled as a five-level artificial atom (M=5) with anharmonic ladder ωm,m−1 ≈ ω10 − (m−1)EC/¯h and decay rates Γm,m−1 = mΓ10. This uneven level spacing is what lets two input frequencies mix into new sum-and-difference frequencies. The theory writes the transmon's master equation in a frame rotating at the average of the two drive frequencies, leaving a term oscillating at the half-detuning δ=(ω1−ω2)/2; expanding the atomic operators as a Fourier series in δ turns the problem into equations for each harmonic order. The measured spectrum is then split into a coherent part (elastic scattering, Lorentzian broadened to match the analyzer resolution) and

Load-bearing premise

The load-bearing premise is that the sideband peaks are created by the artificial atom bending the microwave fields as they pass, not by ordinary mixing inside the cables, combiner, amplifiers, or analyzer; the paper subtracts traces with the drives off but does not show a control measurement with the atom absent or far off resonance.

What would settle it

Run the same two-tone experiment with the qubit transition tuned far away from both drive frequencies, or with the chip replaced by a short circuit or through-line; if the same sideband peaks appear at comparable strength, they were never produced by the atom.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Frequency conversion over tens of megahertz is achievable from a single transmon with no external cavity; the conversion side is selected simply by which drive is stronger.
  • The same device switches from conversion to multi-line comb generation by adjusting input powers and detuning, demonstrated with up to ten equally spaced peaks.
  • Comb spacing is set by the drive detuning, so the comb can be re-tuned in situ rather than being fixed by a resonator's free spectral range.
  • Because emission is collected in one direction by the terminated waveguide, the converter and comb generator occupy a single-qubit footprint and can be integrated with other superconducting circuits on one chip.
  • A five-level theoretical model using measured qubit parameters reproduces the observed peak positions and relative amplitudes, providing a predictive tool for choosing powers and frequencies.

Where Pith is reading between the lines

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

  • If the atom-mediated interpretation survives an off-resonant calibration, the same transmon should also convert fields at the single-photon level; the paper's drives are classical tones near −120 dBm, so quantum-level operation remains untested.
  • Driving with more than two tones is a natural next step: the harmonic-expansion model suggests the comb spacing and density could be engineered from the set of pairwise detunings.
  • The broad incoherent emission centered at ω10 between the carriers appears in the data but is not exploited; it could serve as an independent in-situ probe of drive-induced broadening or dephasing.
  • The self-locked, cavity-free comb implies phase coherence among the generated lines; a phase-sensitive two-tone or heterodyne measurement would make that prediction testable.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper reports power spectral density (PSD) measurements from a superconducting transmon coupled to the end of a semi-infinite transmission line and driven by two continuous microwave tones near 4.82 GHz. The measurements show sidebands at mixing frequencies such as 2ω1−ω2 and 3ω1−2ω2, and when the two-tone detuning is small, up to ten equally spaced peaks are observed, with spacing equal to the drive detuning. The authors develop an M-level (M=5) master-equation model, use a Fourier expansion in the half-detuning to obtain the coherent spectrum, and compare the model with data using a fitting function that includes background power, Lorentzian broadenings, and Rabi-frequency calibration factors. They claim this constitutes a cavity-free, tunable frequency converter and frequency-comb generator, with self-sustained mode locking without external synchronization. The strongest part is that the peak positions follow exactly from the two drive frequencies via the Fourier expansion, independent of fitting. The main weaknesses are the lack of a control against classical intermodulation in the measurement chain, the contradiction between the "self-sustained mode locking" claim and the externally synchronized sources, and the fitted nature of the amplitude/linewidth agreement.

Significance. If the atom-mediated origin of the sidebands is firmly established, this would be a compact, on-chip microwave frequency converter and comb generator with tunable spacing set by the drive detuning—potentially useful in quantum signal processing and waveguide QED. The theoretical framework in the Supplementary Material is standard but carefully executed, and the parameter-free prediction of comb-line positions is a clear strength. However, the significance hinges on the control experiment: the paper does not demonstrate that the transmon, rather than classical nonlinearities in the RF chain, generates the observed peaks. The mode-locking claim, if retained, would require direct phase-coherence or time-domain evidence.

major comments (3)
  1. [Sec. S1 and "Frequency up- and down-conversion"] No atom-free control is reported. The two drive tones are combined in an RF combiner and travel through a chain of attenuators, a circulator, and a HEMT amplifier; the only background subtraction is pump-on minus pump-off (Fig. S1b and the text after Fig. 1). Classical intermodulation in any of these components would generate peaks at exactly the same mixing frequencies (2ω1−ω2, etc.), independent of the transmon. The theoretical fits to peak amplitudes cannot distinguish these origins because the amplitudes are fitted. A control measurement with the transmon tuned far off resonance, or with the sample replaced by a linear termination, is essential to support the central claim that the peaks are "mediated by the artificial atom."
  2. [Introduction (last paragraph) and Sec. S1] The claim of "self-sustained mode locking without the need for external synchronization" is directly contradicted by Sec. S1, which states that both RF sources are synchronized to a reference signal from the spectrum analyzer. The equal spacing and phase coherence of the sidebands are therefore inherited from the externally synchronized drives. The PSD alone is also insensitive to the relative phases of the comb lines, so it cannot demonstrate mode locking. This claim and the comparison to actively mode-locked cavity combs should be removed or replaced with a statement that the comb is externally synchronized.
  3. [Sec. S4 C, Eqs. (S21) and (S24)] The quantitative agreement between theory and experiment is weakened by the fitting procedure. The coherent spectrum is obtained by replacing the delta functions of Eq. (S20) with Lorentzians of ad hoc width ε_l, and the final fitting function depends on Poff, ε_i, k1, and k2. Thus, while peak positions are parameter-free, the peak heights and widths are not predicted from first principles; they are matched by construction. The paper should report the fitted values, their uncertainties, and the number of fit parameters, and ideally show error bars or multiple traces to substantiate the "good agreement" claimed in Figs. 2–4.
minor comments (5)
  1. [Fig. 4(a) caption and main text] The qubit frequency is given as "ω10/2π = 4.82 MHz" in the Fig. 4(a) caption; this should be GHz.
  2. [Figs. 2–4] The experimental traces are shown as single curves without error bars or measurement repetitions. State how many averages were taken and whether the traces are representative.
  3. [Sec. S4 C, Eqs. (S21)–(S24)] The notation for the broadening parameter switches from ε_l in Eq. (S21) to ε_i in Eq. (S24). Clarify whether the Lorentzian widths are fit individually for each peak and whether they are expected to be equal.
  4. [Introduction and Abstract] The phrase "across a relatively wide frequency range (tens of MHz, exceeding the linewidth)" is marginal because Γ10/2π = 44.2 MHz and the demonstrated detunings are tens of MHz. Suggest quantifying the total comb span and comparing it explicitly to the linewidth.
  5. [Title] The title in the LaTeX source is typeset with a space: "T unable frequency conversion...". Correct the typo.

Circularity Check

1 steps flagged

Comb line positions are genuine predictions, but the quantitative 'agreement' is partly calibrated by fitting Poff, εi, k1, k2 to the same spectra; the advertised self-sustained mode-locking claim additionally conflicts with the stated external synchronization of the RF sources.

specific steps
  1. fitted input called prediction [Supplementary Sec. S4 C, Eq. (S24) and following paragraph; main-text Figs. 2–4 captions]
    "The fitting parameters for PSDn include Poff, εi (width of the peak, where i denotes the peak index), k1, and k2 (the scaling factors connecting input power and Rabi frequency of the driving fields)."

    The main text repeatedly calls the solid curves 'theoretical predictions' (e.g., Fig. 2: 'solid black curves depict theoretical predictions, showing good agreement'). But the plotted F(ω) in Eq. (S24) is normalized by a fitted background Poff, broadened by fitted widths εi, and scaled by fitted input-power-to-Rabi factors k1,k2, all fitted to the same PSDn data. Thus the amplitude and width agreement is a calibration rather than an independent first-principles prediction. The comb line positions are not fitted—they follow from ωs + lδ (Eq. S20) with δ=(ω1−ω2)/2—so the circularity is partial.

full rationale

The derivation chain for the line positions is self-contained: the Hamiltonian and Born–Markov master equation (Secs. S4 A/B) contain only the measured atom parameters (ω10, Γ10, γ10, EC) and the two drive frequencies; the Floquet expansion in Eq. (S8) puts the coherent sidebands at ωs + lδ, which is exactly the observed equal spacing. No uniqueness theorem or load-bearing self-citation is used to force the result. The score is above zero only because the quantitative spectral curves are not parameter-free: Poff, εi, k1, and k2 are fitted to the same PSDn data in Eq. (S24), so the repeated 'good agreement' between theory and experiment partly restates the fit. Separately (and not counted as circularity), the Introduction's claim of 'self-sustained mode locking without the need for external synchronization' is contradicted by Sec. S1, which states that 'Both RF sources are synchronized to a reference signal provided by the spectrum analyzer'; this is a support/correctness gap, not an equation-level circularity. Also, no atom-free calibration is reported to exclude classical intermodulation in the RF chain, which is an external-validity concern rather than a circularity of the derivation.

Axiom & Free-Parameter Ledger

3 free parameters · 5 axioms · 0 invented entities

No new entities are introduced. The theory is standard waveguide-QED; the only ad hoc elements are the 5-level truncation and the fitted broadening, noise, and Rabi parameters used to compare PSD spectra. These shift the burden of agreement from pure prediction to calibration against the same data.

free parameters (3)
  • k1, k2 (power-to-Rabi conversion factors) = not reported in paper
    In Eq. S24, k1 and k2 connect input powers to Rabi frequencies Ω1 and Ω2; they are fitted to the PSD spectra, so theoretical amplitudes are not parameter-free.
  • Poff (background noise power) = not reported in paper
    Poff is treated as a fitting parameter in the PSDn fitting function F(ω) = 1 + [Sinco + Sco]Δω_RBW / Poff (Eq. S24).
  • ε_i (Lorentzian widths of coherent peaks) = not reported in paper
    The delta functions in Sco (Eq. S20) are replaced by Lorentzians with ad hoc widths ε_i (Eq. S21), which are fitted per peak.
axioms (5)
  • domain assumption The transmon is accurately described by a five-level system with relaxation rates Γm,m−1 = mΓ10 and pure dephasing m²Γφ.
    Supp. S4 uses M = 5 and states the sixth-level population is negligible; no explicit verification at the highest drive powers is provided.
  • domain assumption Born–Markov master equation and input–output formalism apply; all emitted field is collected at the single port.
    Eqs. S1–S15 rely on the standard Born–Markov tracing of waveguide modes and the semi-infinite-line boundary condition.
  • domain assumption The two drive tones are the only inputs, and no significant classical nonlinear mixing occurs in the measurement chain.
    Required to attribute the observed sidebands to transmon-mediated wave mixing; no atom-free calibration is reported.
  • standard math Fourier/Floquet expansion and the quantum regression theorem give the steady-state spectrum.
    Standard technique for periodically driven quantum systems (Ficek-Freedhoff, Ref. S14); not re-derived in the paper.
  • ad hoc to paper The coherent spectrum can be represented as Lorentzians with ad hoc widths ε_l instead of delta functions.
    Eq. S21: 'we introduce a decaying parameter ε_l into the exponential' to match finite resolution; ε_l are fitted, not derived from physical linewidths.

pith-pipeline@v1.3.0-alltime-deepseek · 17934 in / 14984 out tokens · 136407 ms · 2026-08-04T07:05:10.702190+00:00 · methodology

0 comments
read the original abstract

We investigate the power spectral density emitted by a superconducting artificial atom coupled to the end of a semi-infinite transmission line and driven by two continuous radio-frequency fields. In this setup, we observe the generation of multiple frequency peaks and the formation of frequency combs with equal detuning between those peaks. The frequency peaks originate from wave mixing of the drive fields, mediated by the artificial atom, highlighting the potential of this system as both a frequency converter and a frequency-comb generator. We demonstrate precise control and tunability in generating these frequency features, aligning well with theoretical predictions, across a relatively wide frequency range (tens of MHz, exceeding the linewidth of the artificial atom). The extensive and simple tunability of this frequency converter and comb generator, combined with its small physical footprint, makes it promising for quantum optics on chips and other applications in quantum technology.

Figures

Figures reproduced from arXiv: 2510.26749 by Anton Frisk Kockum, Ching-Yeh Chen, Fahad Aziz, Guin-Dar Lin, Io-Chun Hoi, Jeng-Chung Chen, Kai-Min Hsieh, Kuan Ting Lin, Ping-Yi Wen, Tzu-Yen Hsieh, Yen-Hsiang Lin, Yu-Huan Huang, Yu-Ting Cheng, Zhengqi Niu, Zhi-Rong Lin.

Figure 1
Figure 1. Figure 1: FIG. 1. Experimental scheme, sample, and theoretical model. (a) [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Frequency up- and down-conversion. Two continuous fields at frequencies [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Tunable frequency conversion. (a) Measured PSD [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Generation of frequency combs. (a) Measured PSD [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

discussion (0)

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

Reference graph

Works this paper leans on

57 extracted references · 27 canonical work pages

  1. [1]

    Blais , author A

    author author A. Blais , author A. L. \ Grimsmo , author S. M. \ Girvin , \ and\ author A. Wallraff ,\ title title Circuit quantum electrodynamics , \ 10.1103/RevModPhys.93.025005 journal journal Reviews of Modern Physics \ volume 93 ,\ pages 025005 ( year 2021 ) NoStop

    doi:10.1103/revmodphys.93.025005 2021

  2. [2]

    Roy , author C

    author author D. Roy , author C. M. \ Wilson , \ and\ author O. Firstenberg ,\ title title Colloquium: Strongly interacting photons in one-dimensional continuum , \ 10.1103/RevModPhys.89.021001 journal journal Reviews of Modern Physics \ volume 89 ,\ pages 021001 ( year 2017 ) NoStop

    doi:10.1103/revmodphys.89.021001 2017

  3. [3]

    Gu , author A

    author author X. Gu , author A. F. \ Kockum , author A. Miranowicz , author Y.-x. \ Liu , \ and\ author F. Nori ,\ title title Microwave photonics with superconducting quantum circuits , \ 10.1016/j.physrep.2017.10.002 journal journal Physics Reports \ volume 718--719 ,\ pages 1 ( year 2017 ) NoStop

    doi:10.1016/j.physrep.2017.10.002 2017

  4. [4]

    author author A. S. \ Sheremet , author M. I. \ Petrov , author I. V. \ Iorsh , author A. V. \ Poshakinskiy , \ and\ author A. N. \ Poddubny ,\ title title Waveguide quantum electrodynamics: Collective radiance and photon-photon correlations , \ 10.1103/RevModPhys.95.015002 journal journal Reviews of Modern Physics \ volume 95 ,\ pages 015002 ( year 2023 ) NoStop

    doi:10.1103/revmodphys.95.015002 2023

  5. [6]

    author author O. V. \ Astafiev , author A. A. \ Abdumalikov , author A. M. \ Zagoskin , author Y. A. \ Pashkin , author Y. Nakamura , \ and\ author J. S. \ Tsai ,\ title title Ultimate On-Chip Quantum Amplifier , \ 10.1103/PhysRevLett.104.183603 journal journal Physical Review Letters \ volume 104 ,\ pages 183603 ( year 2010 b ) NoStop

    work page doi:10.1103/physrevlett.104.183603 2010

  6. [7]

    \ Hoi , author A

    author author I.-C. \ Hoi , author A. F. \ Kockum , author T. Palomaki , author T. M. \ Stace , author B. Fan , author L. Tornberg , author S. R. \ Sathyamoorthy , author G. Johansson , author P. Delsing , \ and\ author C. M. \ Wilson ,\ title title Giant Cross-Kerr Effect for Propagating Microwaves Induced by an Artificial Atom , \ 10.1103/PhysRevLett.11...

    work page doi:10.1103/physrevlett.111.053601 2013

  7. [8]

    author author A. F. \ van Loo , author A. Fedorov , author K. Lalumiere , author B. C. \ Sanders , author A. Blais , \ and\ author A. Wallraff ,\ title title Photon-Mediated Interactions Between Distant Artificial Atoms , \ 10.1126/science.1244324 journal journal Science \ volume 342 ,\ pages 1494 ( year 2013 ) NoStop

    doi:10.1126/science.1244324 2013

  8. [10]

    author author P. Y. \ Wen , author K.-T. \ Lin , author A. F. \ Kockum , author B. Suri , author H. Ian , author J. C. \ Chen , author S. Y. \ Mao , author C. C. \ Chiu , author P. Delsing , author F. Nori , author G.-D. \ Lin , \ and\ author I.-C. \ Hoi ,\ title title Large Collective Lamb Shift of Two Distant Superconducting Artificial Atoms , \ 10.1103...

    work page doi:10.1103/physrevlett.123.233602 2019

  9. [11]

    Mirhosseini , author E

    author author M. Mirhosseini , author E. Kim , author X. Zhang , author A. Sipahigil , author P. B. \ Dieterle , author A. J. \ Keller , author A. Asenjo-Garcia , author D. E. \ Chang , \ and\ author O. Painter ,\ title title Cavity quantum electrodynamics with atom-like mirrors , \ 10.1038/s41586-019-1196-1 journal journal Nature \ volume 569 ,\ pages 69...

    doi:10.1038/s41586-019-1196-1 2019

  10. [12]

    u ller , author D. K. \ Kim , author M. Kjaergaard , author P. Krantz , author A. Melville , author B. M. \ Niedzielski , author A. Veps\

    author author B. Kannan , author M. J. \ Ruckriegel , author D. L. \ Campbell , author A. Frisk Kockum , author J. Braum \"u ller , author D. K. \ Kim , author M. Kjaergaard , author P. Krantz , author A. Melville , author B. M. \ Niedzielski , author A. Veps\"al\"ainen , author R. Winik , author J. L. \ Yoder , author F. Nori , author T. P. \ Orlando , a...

    2020

  11. [13]

    author author P. Y. \ Wen , author O. V. \ Ivakhnenko , author M. A. \ Nakonechnyi , author B. Suri , author J.-J. \ Lin , author W.-J. \ Lin , author J. C. \ Chen , author S. N. \ Shevchenko , author F. Nori , \ and\ author I.-C. \ Hoi ,\ title title Landau--Zener--St\"uckelberg--Majorana interferometry of a superconducting qubit in front of a mirror , \...

    work page doi:10.1103/physrevb.102.075448 2020

  12. [14]

    Lu , author A

    author author Y. Lu , author A. Bengtsson , author J. J. \ Burnett , author B. Suri , author S. R. \ Sathyamoorthy , author H. R. \ Nilsson , author M. Scigliuzzo , author J. Bylander , author G. Johansson , \ and\ author P. Delsing ,\ title title Quantum efficiency, purity and stability of a tunable, narrowband microwave single-photon source , \ 10.1038/...

    work page doi:10.1038/s41534-021-00480-5 2021

  13. [15]

    author author A. F. \ Kockum ,\ title title Quantum Optics with Giant Atoms - the First Five Years , \ in\ 10.1007/978-981-15-5191-8_12 booktitle International Symposium on Mathematics, Quantum Theory, and Cryptography (Mathematics for Industry, vol 33) \ ( publisher Springer ,\ year 2021 )\ pp.\ pages 125--146 NoStop

    work page doi:10.1007/978-981-15-5191-8_12 2021

  14. [16]

    Wiegand , author P.-Y

    author author E. Wiegand , author P.-Y. \ Wen , author P. Delsing , author I.-C. \ Hoi , \ and\ author A. F. \ Kockum ,\ title title Ultimate quantum limit for amplification: a single atom in front of a mirror , \ https://dx.doi.org/10.1088/1367-2630/abf1d8 journal journal New Journal of Physics \ volume 23 ,\ pages 043048 ( year 2021 ) NoStop

    work page doi:10.1088/1367-2630/abf1d8 2021

  15. [17]

    \ Lin , author Y

    author author W.-J. \ Lin , author Y. Lu , author P. Y. \ Wen , author Y.-T. \ Cheng , author C.-P. \ Lee , author K. T. \ Lin , author K. H. \ Chiang , author M. C. \ Hsieh , author C.-Y. \ Chen , author C.-H. \ Chien , author J. J. \ Lin , author J.-C. \ Chen , author Y. H. \ Lin , author C.-S. \ Chuu , author F. Nori , author A. F. \ Kockum , author G....

    2022

  16. [18]

    Lu , author N

    author author Y. Lu , author N. Lambert , author A. F. \ Kockum , author K. Funo , author A. Bengtsson , author S. Gasparinetti , author F. Nori , \ and\ author P. Delsing ,\ title title Steady-State Heat Transport and Work With a Single Artificial Atom Coupled to a Waveguide: Emission Without External Driving , \ 10.1103/PRXQuantum.3.020305 journal journ...

    work page doi:10.1103/prxquantum.3.020305 2022

  17. [19]

    \ Cheng , author K.-M

    author author Y.-T. \ Cheng , author K.-M. \ Hsieh , author B.-Y. \ Wu , author Z. Q. \ Niu , author F. Aziz , author Y.-H. \ Huang , author P. Y. \ Wen , author K.-T. \ Lin , author Y.-H. \ Lin , author J. C. \ Chen , author A. F. \ Kockum , author G.-D. \ Lin , author Z.-R. \ Lin , author Y. Lu , \ and\ author I.-C. \ Hoi ,\ title title Group Delay Cont...

    work page doi:10.1103/fkzb-fxv4 2025

  18. [20]

    Aziz , author K.-T

    author author F. Aziz , author K.-T. \ Lin , author P.-Y. \ Wen , author Samina , author Y.-C. \ Lin , author E. Weigand , author C.-P. \ Lee , author Y.-T. \ Cheng , author Y. Lu , author C.-Y. \ Chen , author C.-H. \ Chien , author K.-M. \ Hsieh , author Y.-H. \ Huang , author H.-T. \ Huang , author H. Ian , author J.-C. \ Chen , author Y.-H. \ Lin , au...

    2025

  19. [21]

    \ Hoi , author C

    author author I.-C. \ Hoi , author C. M. \ Wilson , author G. Johansson , author T. Palomaki , author B. Peropadre , \ and\ author P. Delsing ,\ title title Demonstration of a Single-Photon Router in the Microwave Regime , \ 10.1103/PhysRevLett.107.073601 journal journal Physical Review Letters \ volume 107 ,\ pages 073601 ( year 2011 ) NoStop

    doi:10.1103/physrevlett.107.073601 2011

  20. [22]

    Kono \ and\ author Y

    author author S. Kono \ and\ author Y. Nakamura ,\ @noop title Microwave photon control device, microwave photon transmitter, microwave photon receiver, microwave photon repeater, and quantum computer , \ ( year 2024 ),\ note US Patent 12,184,339 NoStop

    2024

  21. [23]

    Chen \ and\ author A

    author author G. Chen \ and\ author A. F. \ Kockum ,\ title title Simulating open quantum systems with giant atoms , \ 10.1088/2058-9565/adb2bd journal journal Quantum Science and Technology \ volume 10 ,\ pages 025028 ( year 2025 ) NoStop

    work page doi:10.1088/2058-9565/adb2bd 2058

  22. [24]

    Almanakly , author B

    author author A. Almanakly , author B. Yankelevich , author M. Hays , author B. Kannan , author R. Assouly , author A. Greene , author M. Gingras , author B. M. \ Niedzielski , author H. Stickler , author M. E. \ Schwartz , author K. Serniak , author J. \^ I .-j. \ Wang , author T. P. \ Orlando , author S. Gustavsson , author J. A. \ Grover , \ and\ autho...

    doi:10.1038/s41567-025-02811-1 2025

  23. [25]

    \ Huang , author K.-M

    author author Y.-H. \ Huang , author K.-M. \ Hsieh , author F. Aziz , author Z. Q. \ Niu , author P. Y. \ Wen , author Y.-T. \ Cheng , author Y.-S. \ Tsai , author J. C. \ Chen , author X. Wang , author A. F. \ Kockum , author Z.-R. \ Lin , author Y.-H. \ Lin , \ and\ author I.-C. \ Hoi ,\ title title Tunable coherent microwave beam splitter and combiner ...

    work page doi:10.1063/5.0272687 2025

  24. [26]

    Kumar ,\ title title Quantum frequency conversion , \ 10.1364/ol.15.001476 journal journal Optics Letters \ volume 15 ,\ pages 1476 ( year 1990 ) NoStop

    author author P. Kumar ,\ title title Quantum frequency conversion , \ 10.1364/ol.15.001476 journal journal Optics Letters \ volume 15 ,\ pages 1476 ( year 1990 ) NoStop

    doi:10.1364/ol.15.001476 1990

  25. [27]

    Han , author W

    author author X. Han , author W. Fu , author C.-L. \ Zou , author L. Jiang , \ and\ author H. X. \ Tang ,\ title title Microwave-optical quantum frequency conversion , \ 10.1364/OPTICA.425414 journal journal Optica \ volume 8 ,\ pages 1050 ( year 2021 ) NoStop

    doi:10.1364/optica.425414 2021

  26. [28]

    Huang \ and\ author P

    author author J. Huang \ and\ author P. Kumar ,\ title title Observation of quantum frequency conversion , \ 10.1103/PhysRevLett.68.2153 journal journal Phys. Rev. Lett. \ volume 68 ,\ pages 2153--2156 ( year 1992 ) NoStop

    work page doi:10.1103/physrevlett.68.2153 1992

  27. [29]

    Yamazaki , author T

    author author T. Yamazaki , author T. Arizono , author T. Kobayashi , author R. Ikuta , \ and\ author T. Yamamoto ,\ title title Linear Optical Quantum Computation with Frequency-Comb Qubits and Passive Devices , \ 10.1103/PhysRevLett.130.200602 journal journal Physical Review Letters \ volume 130 ,\ pages 200602 ( year 2023 ) NoStop

    work page doi:10.1103/physrevlett.130.200602 2023

  28. [30]

    Chang , author S

    author author L. Chang , author S. Liu , \ and\ author J. E. \ Bowers ,\ title title Integrated optical frequency comb technologies , \ 10.1038/s41566-021-00945-1 journal journal Nature Photonics \ volume 16 ,\ pages 95 ( year 2022 ) NoStop

    doi:10.1038/s41566-021-00945-1 2022

  29. [31]

    author author S. W. \ Jolin , author G. Andersson , author J. R. \ Hern \'a ndez , author I. Strandberg , author F. Quijandr \' a , author J. Aumentado , author R. Borgani , author M. O. \ Thol \'e n , \ and\ author D. B. \ Haviland ,\ title title Multipartite entanglement in a microwave frequency comb , \ 10.1103/PhysRevLett.130.120601 journal journal Ph...

    doi:10.1103/physrevlett.130.120601 2023

  30. [32]

    Rivera Hern \'a ndez , author F

    author author J. Rivera Hern \'a ndez , author F. Lingua , author S. W. \ Jolin , \ and\ author D. B. \ Haviland ,\ title title Control of multi-modal scattering in a microwave frequency comb , \ 10.1063/5.0203426 journal journal APL Quantum \ volume 1 ,\ pages 036101 ( year 2024 ) NoStop

    doi:10.1063/5.0203426 2024

  31. [33]

    author author A. D. \ Ludlow , author M. M. \ Boyd , author J. Ye , author E. Peik , \ and\ author P. O. \ Schmidt ,\ title title Optical atomic clocks , \ 10.1103/RevModPhys.87.637 journal journal Reviews of Modern Physics \ volume 87 ,\ pages 637 ( year 2015 ) NoStop

    doi:10.1103/revmodphys.87.637 2015

  32. [34]

    author author S. T. \ Cundiff \ and\ author J. Ye ,\ title title Colloquium: Femtosecond optical frequency combs , \ 10.1103/RevModPhys.75.325 journal journal Reviews of Modern Physics \ volume 75 ,\ pages 325 ( year 2003 ) NoStop

    doi:10.1103/revmodphys.75.325 2003

  33. [35]

    author author A. Y. \ Dmitriev , author R. Shaikhaidarov , author V. Antonov , author T. H \"o nigl-Decrinis , \ and\ author O. Astafiev ,\ title title Quantum wave mixing and visualisation of coherent and superposed photonic states in a waveguide , \ 10.1038/s41467-017-01471-x journal journal Nature Communications \ volume 8 ,\ pages 1352 ( year 2017 ) NoStop

    doi:10.1038/s41467-017-01471-x 2017

  34. [36]

    H\"onigl-Decrinis , author I

    author author T. H\"onigl-Decrinis , author I. V. \ Antonov , author R. Shaikhaidarov , author V. N. \ Antonov , author A. Y. \ Dmitriev , \ and\ author O. V. \ Astafiev ,\ title title Mixing of coherent waves in a single three-level artificial atom , \ 10.1103/PhysRevA.98.041801 journal journal Physical Review A \ volume 98 ,\ pages 041801 ( year 2018 ) NoStop

    doi:10.1103/physreva.98.041801 2018

  35. [37]

    author author A. Y. \ Dmitriev , author R. Shaikhaidarov , author T. H\"onigl-Decrinis , author S. E. \ de Graaf , author V. N. \ Antonov , \ and\ author O. V. \ Astafiev ,\ title title Probing photon statistics of coherent states by continuous wave mixing on a two-level system , \ 10.1103/PhysRevA.100.013808 journal journal Physical Review A \ volume 100...

    work page doi:10.1103/physreva.100.013808 2019

  36. [38]

    author author P. Y. \ Wen , author A. F. \ Kockum , author H. Ian , author J. C. \ Chen , author F. Nori , \ and\ author I.-C. \ Hoi ,\ title title Reflective Amplification without Population Inversion from a Strongly Driven Superconducting Qubit , \ 10.1103/PhysRevLett.120.063603 journal journal Physical Review Letters \ volume 120 ,\ pages 063603 ( year...

    work page doi:10.1103/physrevlett.120.063603 2018

  37. [39]

    @noop note Supplementary Material NoStop

  38. [41]

    Cheng , author M

    author author R. Cheng , author M. Yu , author A. Shams-Ansari , author Y. Hu , author C. Reimer , author M. Zhang , \ and\ author M. Lon c ar ,\ title title Frequency comb generation via synchronous pumped (3) resonator on thin-film lithium niobate , \ 10.1038/s41467-024-48222-3 journal journal Nature Communications \ volume 15 ,\ pages 3921 ( year 2024 ...

    work page doi:10.1038/s41467-024-48222-3 2024

  39. [42]

    Gao , author H

    author author X. Gao , author H. He , author S. Sobolewski , author A. Cerjan , \ and\ author C. W. \ Hsu ,\ title title Dynamic gain and frequency comb formation in exceptional-point lasers , \ 10.1038/s41467-024-52957-4 journal journal Nature Communications \ volume 15 ,\ pages 8618 ( year 2024 ) NoStop

    work page doi:10.1038/s41467-024-52957-4 2024

  40. [43]

    Szabados , author B

    author author J. Szabados , author B. Sturman , \ and\ author I. Breunig ,\ title title Frequency comb generation threshold via second-harmonic excitation in (2) optical microresonators , \ 10.1063/5.0021424 journal journal APL Photonics \ volume 5 ,\ pages 116102 ( year 2020 ) NoStop

    work page doi:10.1063/5.0021424 2020

  41. [44]

    Wang , author M

    author author Y. Wang , author M. Zhang , author Z. Shen , author G.-T. \ Xu , author R. Niu , author F.-W. \ Sun , author G.-C. \ Guo , \ and\ author C.-H. \ Dong ,\ title title Optomechanical frequency comb based on multiple nonlinear dynamics , \ 10.1103/PhysRevLett.132.163603 journal journal Physical Review Letters \ volume 132 ,\ pages 163603 ( year ...

    work page doi:10.1103/physrevlett.132.163603 2024

  42. [45]

    author author M. P. \ Mrozowski , author J. Jeffers , \ and\ author J. D. \ Pritchard ,\ title title High-efficiency coupled-cavity optical frequency comb generation , \ 10.1364/OPTCON.481563 journal journal Optics Continuum \ volume 2 ,\ pages 894 ( year 2023 ) NoStop

    work page doi:10.1364/optcon.481563 2023

  43. [46]

    Udem , author R

    author author T. Udem , author R. Holzwarth , \ and\ author T. W. \ H \"a nsch ,\ title title Optical frequency metrology , \ 10.1038/416233a journal journal Nature \ volume 416 ,\ pages 233 ( year 2002 ) NoStop

    doi:10.1038/416233a 2002

  44. [47]

    \ Hoi , author A

    author author I.-C. \ Hoi , author A. F. \ Kockum , author L. Tornberg , author A. Pourkabirian , author G. Johansson , author P. Delsing , \ and\ author C. M. \ Wilson ,\ title title Probing the quantum vacuum with an artificial atom in front of a mirror , \ 10.1038/nphys3484 journal journal Nature Physics \ volume 11 ,\ pages 1045--1049 ( year 2015 ) NoStop

    work page doi:10.1038/nphys3484 2015

  45. [48]

    Astafiev , author A

    author author O. Astafiev , author A. M. \ Zagoskin , author A. Abdumalikov Jr , author Y. A. \ Pashkin , author T. Yamamoto , author K. Inomata , author Y. Nakamura , \ and\ author J. S. \ Tsai ,\ title title Resonance fluorescence of a single artificial atom , \ 10.1126/science.1181918 journal journal Science \ volume 327 ,\ pages 840--843 ( year 2010 ) NoStop

    doi:10.1126/science.1181918 2010

  46. [49]

    \ Cheng , author C.-H

    author author Y.-T. \ Cheng , author C.-H. \ Chien , author K.-M. \ Hsieh , author Y.-H. \ Huang , author P. Y. \ Wen , author W.-J. \ Lin , author Y. Lu , author F. Aziz , author C.-P. \ Lee , author K.-T. \ Lin , author C.-Y. \ Chen , author J. C. \ Chen , author C.-S. \ Chuu , author A. F. \ Kockum , author G.-D. \ Lin , author Y.-H. \ Lin , \ and\ aut...

    work page doi:10.1103/physreva.109.023705 2024

  47. [50]

    Probst , author F

    author author S. Probst , author F. B. \ Song , author P. A. \ Bushev , author A. V. \ Ustinov , \ and\ author M. Weides ,\ title title Efficient and robust analysis of complex scattering data under noise in microwave resonators , \ 10.1063/1.4907935 journal journal Review of Scientific Instruments \ volume 86 ,\ pages 024706 ( year 2015 ) NoStop

    doi:10.1063/1.4907935 2015

  48. [51]

    Lu , author A

    author author Y. Lu , author A. Bengtsson , author J. J. \ Burnett , author E. Wiegand , author B. Suri , author P. Krantz , author A. F. \ Roudsari , author A. F. \ Kockum , author S. Gasparinetti , author G. Johansson , \ and\ author P. Delsing ,\ title title Characterizing decoherence rates of a superconducting qubit by direct microwave scattering , \ ...

    doi:10.1038/s41534-021-00367-5 2021

  49. [52]

    @noop note The nonradiative relaxation rate is absorbed into _ . Stop

  50. [53]

    Koch , author T

    author author J. Koch , author T. M. \ Yu , author J. Gambetta , author A. A. \ Houck , author D. I. \ Schuster , author J. Majer , author A. Blais , author M. H. \ Devoret , author S. M. \ Girvin , \ and\ author R. J. \ Schoelkopf ,\ title title Charge-insensitive qubit design derived from the Cooper pair box , \ 10.1103/physreva.76.042319 journal journa...

    doi:10.1103/physreva.76.042319 2007

  51. [54]

    \ Lin , author T

    author author K.-T. \ Lin , author T. Hsu , author C.-Y. \ Lee , author I.-C. \ Hoi , \ and\ author G.-D. \ Lin ,\ title title Scalable collective Lamb shift of a 1D superconducting qubit array in front of a mirror , \ 10.1038/s41598-019-55545-5 journal journal Scientific Reports \ volume 9 ,\ pages 19175 ( year 2019 ) NoStop

    work page doi:10.1038/s41598-019-55545-5 2019

  52. [55]

    \ Lin , author T

    author author K.-T. \ Lin , author T. Hsu , author F. Aziz , author Y.-C. \ Lin , author P.-Y. \ Wen , author I.-C. \ Hoi , \ and\ author G.-D. \ Lin ,\ title title Single-atom amplification assisted by multiple sideband interference in waveguide QED systems , \ 10.1088/1367-2630/add7fe journal journal New Journal of Physics \ volume 27 ,\ pages 064108 ( ...

    work page doi:10.1088/1367-2630/add7fe 2025

  53. [56]

    author author A. F. \ Kockum , author M. Sandberg , author M. R. \ Vissers , author J. Gao , author G. Johansson , \ and\ author D. P. \ Pappas ,\ title title Detailed modelling of the susceptibility of a thermally populated, strongly driven circuit-QED system , \ 10.1088/0953-4075/46/22/224014 journal journal Journal of Physics B: Atomic, Molecular and O...

    work page doi:10.1088/0953-4075/46/22/224014 2013

  54. [57]

    author author R. H. \ Lehmberg ,\ title title Radiation from an N -Atom System. I. General Formalism , \ 10.1103/PhysRevA.2.883 journal journal Physical Review A \ volume 2 ,\ pages 883--888 ( year 1970 ) NoStop

    doi:10.1103/physreva.2.883 1970

  55. [58]

    Ficek \ and\ author H

    author author Z. Ficek \ and\ author H. S. \ Freedhoff ,\ title title Resonance-fluorescence and absorption spectra of a two-level atom driven by a strong bichromatic field , \ 10.1103/PhysRevA.48.3092 journal journal Physical Review A \ volume 48 ,\ pages 3092--3104 ( year 1993 ) NoStop

    work page doi:10.1103/physreva.48.3092 1993

  56. [59]

    Peropadre , author J

    author author B. Peropadre , author J. Lindkvist , author I.-C. \ Hoi , author C. M. \ Wilson , author J. J. \ Garcia-Ripoll , author P. Delsing , \ and\ author G. Johansson ,\ title title Scattering of coherent states on a single artificial atom , \ 10.1088/1367-2630/15/3/035009 journal journal New Journal of Physics \ volume 15 ,\ pages 035009 ( year 20...

    work page doi:10.1088/1367-2630/15/3/035009 2013

  57. [60]

    author author M. O. \ Scully \ and\ author M. S. \ Zubairy ,\ @noop title Quantum Optics \ ( publisher Cambridge University Press ,\ year 1997 ) NoStop

    1997