Superinductor-based ultrastrong coupling in a superconducting circuit

Alba Torras-Coloma; Ariadna G\'omez-del-Pulgar-Mart\'inez; Elia Bertoldo; Luca Cozzolino; P. Forn-D\'iaz

REVIEW 3 major objections 6 minor 46 references

Superinductor-based ultrastrong coupling in a superconducting circuit

T0 review · 3 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A granular-aluminum superinductor galvanically couples a flux qubit to a resonator with g/ω_r ≈ 0.13, placing the system in the perturbative ultrastrong coupling regime.

desk verdict A novel and promising superinductor coupling scheme, but the paper's own quantitative checks (inductance estimate and Bloch-Siegert shift) don't line up with the fitted g, so the USC claim needs a consistent re-analysis. read the letter →

arxiv 2507.09339 v1 pith:ZEJ3FN4L submitted 2025-07-12 quant-ph

Alba Torras-Coloma , Luca Cozzolino , Ariadna Gómez-del-Pulgar-Martínez , Elia Bertoldo , P. Forn-Díaz This is my paper · ORCID

classification quant-ph

keywords ultrastrongcouplingsuperinductorgranularaluminumfluxqubitBloch-SiegertshiftquantumRabimodelkineticinductancesuperconductingcircuit

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

The paper claims that a superinductor, a large-inductance, low-loss circuit element, can replace the Josephson junction usually used to couple a flux qubit to a resonator and still push the system into the ultrastrong coupling regime. Spectroscopy of the coupled circuit fits the Quantum Rabi model with g/ω_r ≈ 0.13, above the 0.1 threshold, and shows a 23 MHz Bloch-Siegert shift, the hallmark of counter-rotating terms. An independent measurement of the coupler inductance from low-temperature resistance is reported as compatible with the strong coupling. If correct, this relaxes the design constraints for ultrastrong coupling circuits, allowing smaller qubit loops and lower persistent currents, which should improve coherence.

What carries the argument

The central object is the granular-aluminum (grAl) superinductor used as the shared coupling inductor L_c of a three-junction flux qubit and a lumped-element LC resonator. Granular aluminum is a disordered superconductor whose kinetic inductance provides a large surface inductance in a compact wire, giving a linear, low-loss coupler without a Josephson junction. The coupling strength, g ≈ ξ_R (L_eff I_p I_rms,R)/ħ, is derived from the circuit, and the resulting Quantum Rabi Hamiltonian is used to fit the spectra; the large L_c also renormalizes the resonator frequency, ω_r ≈ [(L_R + L_c)/C_R]^{-1/2}. The Bloch-Siegert shift, ω_BS = $g^{2}$/(ω_r + ω_q), serves as an independent check of the coupling strength.

What would settle it

A direct time-domain measurement of vacuum Rabi oscillations (or a resolved vacuum Rabi splitting) at the sweet spot that returns a coupling rate g/2π below about 0.45 GHz would put g/ω_r under 0.1 and would falsify the claim that the device is in the USC regime.

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Extended reading notes

Core claim

The central experimental result is a transmission spectrum of a galvanically coupled flux-qubit–resonator system fitted to the Quantum Rabi Hamiltonian with parameters I_p = (11.619 ± 0.004) nA, Δ/h = (5.707 ± 0.002) GHz, ω_r/2π = (4.463 ± 0.001) GHz, and g/2π = (0.578 ± 0.001) GHz, giving a coupling fraction g/ω_r ≈ 0.13. The difference between the QRM and Jaynes-Cummings spectra at the sweet spot is 23 MHz, attributed to the Bloch-Siegert shift, and this shift independently yields g/2π ≈ 0.48 GHz, consistent with the fit. The coupling inductor is a granular-aluminum wire with an independently estimated inductance L_c = (0.74 ± 0.14) nH obtained from low-temperature resistance and the Mattis-Bardeen formula, a value reported as compatible with g/ω ≳ 0.1. The paper concludes that superinductors can bring qubit-resonator systems into the USC regime while keeping persistent currents low and qubit loops small.

Load-bearing premise

The USC claim stands or falls on the spectral fit's value of g (g/ω_r ≈ 0.13); using the independently measured L_c with the fitted persistent current gives a sub-threshold coupling, so the fit, not the inductance measurement, carries the claim.

Editorial extensions

If this is right

Ultrastrong coupling can be reached with a linear superinductor instead of a shared Josephson junction, avoiding junction losses and stray nonlinearities.
Design constraints are relaxed: smaller qubit loops and lower persistent currents become compatible with g/ω_r > 0.1, which should translate into longer coherence times.
The coupling fraction can be pushed to g/ω_r > 0.3 by increasing the superinductance, while keeping the qubit loop small.
The fabrication flow is modular, so the superinductor material can be replaced by other high-kinetic-inductance materials such as nitrides.
The Bloch-Siegert shift of 23 MHz provides a clear spectral signature of the counter-rotating terms in the perturbative USC regime.

Reading between the lines

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

The same galvanic superinductor coupling scheme could be applied to other qubit types, potentially relaxing impedance-matching constraints for USC in transmon-like circuits.
Because the superinductor is linear, the approach may extend to ultrastrong coupling between two resonators or to multi-qubit USC networks without adding junction nonlinearities.
A time-domain measurement of vacuum Rabi oscillations in this device would directly verify the fitted coupling rate and also probe USC corrections to the decay dynamics.
The close agreement between the Bloch-Siegert shift and the QRM fit suggests that this shift could be used as a precision calibration tool for coupling strengths in other superconducting circuits.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

What you should know: this is the first demonstration I've seen of a superinductor—granular aluminum—used as the galvanic coupling element between a flux qubit and a resonator, with spectroscopy showing a transition spectrum that fits the Quantum Rabi model with g/omega_r ~ 0.13. That's a genuinely new device idea, and the fabrication and measurement work looks careful: calibration data for the grAl, raw transmission in the appendix, and a two-tone spectrum that shows the qubit-resonator anticrossing.

What's less solid is the quantitative validation. The paper claims the independently measured L_c=(0.74±0.14) nH is compatible with g/omega >= 0.1. That check uses Eq. (3) with the design persistent current I_p≈19.6 nA, which gives g/2π≈0.61 GHz. But the QRM fit returns I_p=(11.62±0.01) nA. Plugging that into the same equation gives g/2π≈0.36 GHz, i.e. g/omega_r≈0.08, below the USC threshold. That's a factor of 1.6 in g, and it's not within the error bars. The paper doesn't flag this.

The Bloch-Siegert shift is similarly inconsistent. The paper reports 23 MHz at the sweet spot and says this is consistent with the fitted g=0.578 GHz. Using their own formula omega_BS=g^2/(omega_r+omega_q) with omega_r/2π=4.463 GHz and Delta/h=5.707 GHz gives about 33 MHz. The measured 23 MHz would correspond to g/2π≈0.48 GHz, g/omega_r≈0.11. That's still USC, but it's 17% below the fitted value, and the manuscript states they are consistent without explaining the difference.

There's also a reproducibility issue: Eq. (3), the key link between L_c and g, is deferred to an unpublished manuscript (ref. 33). Appendix A derives a simpler limiting case, but the general form with xi_R is not derived here. That makes it hard for a reader to assess whether the linear scaling with I_p is even the right model for this circuit.

So where does this leave the central claim? The QRM fit itself is a reasonable reading of the spectroscopy, and the device clearly shows strong coupling features. But the two independent checks the paper presents to support the USC classification don't hold up quantitatively. The truth might be g/omega_r≈0.11 from the BS shift, which still enters USC, or it might be ~0.08 if the inductance estimate is right, which doesn't. The paper needs to reconcile these numbers before the headline is solid.

Who is this for? People working on USC circuit QED and superinductor materials. The demonstration is novel enough to deserve peer review, but the referees should ask for a consistent cross-check among I_p, L_c, g, and the Bloch-Siegert shift—and ideally a full derivation of the coupling formula.

Referee Report

3 major / 6 minor

Summary. The paper reports a flux qubit galvanically coupled to a lumped-element LC resonator through a granular aluminum (grAl) superinductor. Single-tone and two-tone spectroscopy are measured as a function of flux, and the observed transitions are fitted to the quantum Rabi model, yielding I_p = (11.619±0.004) nA, Δ/h = (5.707±0.002) GHz, ω_r/2π = (4.463±0.001) GHz, and g/2π = (0.578±0.001) GHz, i.e., g/ω_r ≈ 0.13, which is claimed to enter the perturbative ultrastrong coupling (USC) regime. The coupler inductance is independently estimated from room-temperature and 4 K resistance measurements via the Mattis-Bardeen formula, giving L_c = (0.74±0.14) nH. A Bloch-Siegert shift of 23 MHz is reported for the ω_01 transition at the sweet spot. The paper claims that both the inductance estimate and the Bloch-Siegert shift are consistent with the fitted coupling, thereby validating the USC regime.

Significance. If the central claim holds, the work demonstrates a promising alternative to Josephson-junction-based USC couplers: a linear superinductor made of granular aluminum can provide large coupling while keeping the qubit persistent current and loop area small, potentially improving coherence. The direct spectral fit to multiple transitions over a range of flux biases is a robust method for extracting the QRM parameters, and the fabrication and material characterization are detailed and reproducible. However, the two independent checks presented in support of the USC claim are internally inconsistent under the manuscript's own equations: Eq. (3) evaluated with the fitted persistent current gives a coupling below the USC threshold, and the Bloch-Siegert formula with the fitted parameters does not reproduce the reported 23 MHz shift. These quantitative tensions are load-bearing because the abstract and conclusions use them to claim validation of the USC regime. The central claim may still be correct, but it is not yet settled by the evidence presented.

major comments (3)

[Main text, Section 'Using Eq. (3)' (p. 2-3)] Equation (3) is linear in the persistent current I_p, and the predicted g/2π ≈ 0.61 GHz is evaluated with the design value I_p ≈ 19.6 nA. The QRM fit, however, returns I_p = (11.619±0.004) nA. Re-evaluating Eq. (3) with the fitted I_p and L_c = 0.74 nH gives g/2π ≈ 0.61 × (11.619/19.6) ≈ 0.36 GHz, corresponding to g/ω_r ≈ 0.08, which is below the USC threshold g/ω_r > 0.1 used in the paper. Therefore the statement that the independent L_c estimate is 'consistent with' the USC claim is not supported by the manuscript's own equations. The authors should re-evaluate Eq. (3) at the fitted current and discuss the resulting discrepancy, or justify why the design current should be used instead of the fitted one.
[Main text, Bloch-Siegert shift (p. 3)] The paper reports ω_BS = 23 MHz for the ω_01 transition at the sweet spot and gives the formula ω_BS = g²/(ω_r + ω_q). Substituting the fitted parameters g/2π = 0.578 GHz, ω_r/2π = 4.463 GHz, and Δ/h = 5.707 GHz yields ω_BS ≈ 33 MHz, not 23 MHz. Conversely, the observed 23 MHz implies, through the same formula, g/2π ≈ 0.48 GHz, about 17% below the fitted g/2π = 0.578 GHz. The manuscript states that these are 'consistent' without quantifying or explaining the discrepancy. The authors should clarify whether the analytic formula is expected to be accurate at g/ω_r ≈ 0.13, how the measured shift was extracted from the data, and why the full QRM fit and the Bloch-Siegert estimate differ.
[Abstract] The abstract states that the independently measured L_c = (0.74±0.14) nH is 'compatible with g/ω ≳ 0.1.' This claim depends on evaluating Eq. (3) with the design persistent current rather than the fitted current; with the fitted I_p, Eq. (3) gives g/ω_r ≈ 0.08, below the USC boundary. The abstract's compatibility claim therefore needs to be revised or the analysis underlying it must be corrected.

minor comments (6)

[Fig. 2 and Fig. 3 captions] The caption reads 'transmission magnitude |S_21| vs. flux bias (Φ_ext)' with inconsistent spacing; please add units for Φ_ext and ensure consistent formatting of subscripts.
[Eq. (15)] The term 'ˆp2 + 1 γ ˆp2 3' contains an orphan '1' and unclear grouping; the expression should be typeset as (1/γ) p̂_3² or with explicit parentheses.
[Appendix B] The quantity J_c = (0.66±0.03) µA/µm² is introduced without definition; please define it (presumably a critical current density).
[Footnote 34] The formula for ξ_R² contains a stray 'r' and unclear notation such as 'ω_2^4 = 1/L_eff C_tot'; the expression should be simplified and all symbols defined.
[Notation] The abstract uses 'g/ω' while the main text uses 'g/ω_r'; please use one notation consistently.
[Reference [33]] Equation (3) is central to the validation, but its derivation is deferred to a paper 'in preparation.' Since the consistency of the inductance estimate with the fitted coupling is a key point, the derivation should be summarized or the reference should be replaced by an available source.

Circularity Check

2 steps flagged · score 4.0 of 10

Secondary validations of the USC claim are self-referential: the Bloch-Siegert shift is computed from the fitted g and then used to re-derive g, and the L_c compatibility rests on an unpublished self-cited formula evaluated at a non-fitted current.

self definitional [Main text after fit parameters; Appendix C 1 'Quantum Rabi model and Jaynes-Cummings model']
"The fitted parameters can be used to calculate the spectrum using the Jaynes-Cummings (JC) Hamiltonian. The difference between the QRM and JC curves in the perturbative USC regime is the Bloch-Siegert shift, ω_BS = g^2/(ωr + ωq)... ω_BS is maximum at the sweetspot with a value of 23 MHz for the ω01 transition... ω_BS can be used to estimate the coupling coefficient g/2π≃0.48GHz and the fraction g/ω_r≃0.11 which is consistent with the fitted parameters."

The 23 MHz value is not an independent observable: Appendix C1 states 'We calculate the JC spectrum using the fitting parameters extracted from the QRM fit', so the QRM–JC difference is a function of the same fitted g. Solving ω_BS = g^2/(ωr+ωq) for g is therefore an algebraic inversion of the fitted input, not an independent cross-check. The claimed consistency is vacuous by construction and numerically fails (23 MHz implies g/2π≈0.48 GHz, not the fitted 0.578 GHz).
ansatz smuggled in via citation [Eq. (3); main text 'Using Eq. (3), we estimate the coupling to be g/2π≃0.61GHz...'; Appendix A 'The general case of an arbitrary L_c that leads to Eq. (3) will be the subject of future work33']
"the qubit-resonator coupling strength can be estimated using33 g≃ξ_R L_eff I_p I_rms,R / ħ ... where ... ξ_R is a coefficient that approaches 1 in our circuit... The details of the derivation will be provided ref.33. ... The coupling achieved is consistent with our estimate of L_k with Eq. (3)."

The paper's independent validation of USC via the measured L_c depends entirely on Eq. (3) to translate L_c into g. Eq. (3) is attributed to ref. [33], the authors' own unpublished 'in preparation' work, and its general form is explicitly deferred to that future paper; only the restrictive L_R >> L_c limit is derived in the appendix. Thus the 'independent' inductance check is not self-contained: it imports the coupling formula from a self-citation rather than deriving it here. The check is further compromised by evaluating Eq. (3) with the design I_p≈19.6 nA instead of the fitted I_p=(11.619±0.004) nA, so the claimed compatibility does not actually test the fitted parameters.

full rationale

The primary derivation of g/ω_r≈0.13 is a direct fit of the measured spectroscopy to the Quantum Rabi Hamiltonian (Eqs. 1–2), which is a legitimate, non-circular parameter extraction from external data. However, two supporting validations are circular or self-referential. First, the Bloch-Siegert shift of 23 MHz is obtained by comparing QRM and JC spectra computed with the same fitted parameters; using that shift to 'estimate' the coupling g is an inversion of the fitted input and cannot serve as independent confirmation. Second, the compatibility of the independently measured L_c=(0.74±0.14) nH with USC is established through Eq. (3), a formula attributed to the authors' own unpublished work (ref. 33) and evaluated with the design current (~19.6 nA) rather than the fitted current (~11.6 nA). With the fitted current, Eq. (3) would give g/2π≈0.36 GHz, i.e. g/ω_r≈0.08, below the USC threshold; the paper does not address this. These are correctness risks in addition to the circularity of the BS-shift check. The central claim still has independent content from the spectral fit, so the circularity score is moderate (4), not extreme.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the validity of the Quantum Rabi model as an effective description of the galvanically coupled circuit (which requires adiabatic elimination of the extra mode and a small phase drop across the coupler), on the Mattis-Bardeen formula for the grAl kinetic inductance, and on a four-parameter spectral fit. The free parameters are the fit outputs I_p, Delta, omega_r, and g.

free parameters (4)

I_p (persistent current) = 11.619 ± 0.004 nA
Fitted from the flux dispersion of the single-tone and two-tone spectra using the Quantum Rabi model.
Delta (qubit gap) = 5.707 ± 0.002 GHz
Fitted from the spectra; the authors note it contains the renormalization effect of the coupling inductor.
omega_r (resonator frequency) = 4.463 ± 0.001 GHz
Fitted resonator frequency, stated to be compatible with a high-power saturation measurement.
g (coupling strength) = 0.578 ± 0.001 GHz
Fitted coupling strength; the central USC claim hinges on this value.

assumptions (4)

domain assumption The circuit can be described by the Quantum Rabi Hamiltonian after adiabatic elimination of the extra oscillator mode formed by L_c and the qubit capacitances.
Assumed in the derivation in Appendix A; requires the extra mode to be at much higher frequency than the qubit and resonator, and relies on the quasi-steady-state condition for φ_4.
domain assumption The phase drop φ_4 across the coupling inductor is small, justifying a second-order expansion of the Josephson cosine term and the neglect of kinetic terms associated with L_c.
Expansion used in Appendix A to obtain the effective Hamiltonian; if φ_4 is not small, the pure Rabi model is not exact.
domain assumption The kinetic inductance of the grAl coupler is given by the Mattis-Bardeen formula L_k = 0.18 ℏ R_4K / (k_B T_c) in the local dirty limit at low frequency and low temperature.
Used to translate normal-state resistance and critical temperature into a coupler inductance; the formula is taken from ref. [36] and assumes the dirty limit.
domain assumption The two-level approximation for the flux qubit is valid, and the coupling operator I_q can be replaced by I_p σ_z.
Used to derive Eq. (21) and the final Rabi model; valid when the qubit nonlinearity is large compared to relevant energies.

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Cite this review

Pith. "Pith review of Superinductor-based ultrastrong coupling in a superconducting circuit." pith.science (2026). https://pith.science/paper/ZEJ3FN4L

@misc{pith2026250709339,
  author       = {Pith},
  title        = {Pith review of: Superinductor-based ultrastrong coupling in a superconducting circuit},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZEJ3FN4L}},
  note         = {Machine review of arXiv:2507.09339}
}

abstract

We present an ultrastrong superinductor-based coupling consisting of a flux qubit galvanically coupled to a resonator. The coupling inductor is fabricated in granular Aluminum, a superinductor material able to provide large surface inductances. Spectroscopy measurements on the qubit-resonator system reveal a Bloch-Siegert shift of \SI{23}{\mega\hertz} and a coupling fraction of $g/\omega_r \simeq 0.13$, entering the perturbative ultrastrong coupling (USC) regime. We estimate the inductance of the coupler independently by low-temperature resistance measurements providing $L_c = (0.74\pm0.14)\,\mathrm{nH}$, which is compatible with $g/\omega \gtrsim 0.1$. Our results show that superinductors are a promising tool to study USC physics in high-coherence circuits using flux qubits with small loop areas and low persistent currents.

Figures

Figures reproduced from arXiv: 2507.09339 by the authors.

**Figure 2.** FIG. 2. Normalized transmission magnitude [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG. 3. Normalized transmission magnitude vs. flux bias [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (7 more)

**Figure 4.** Figure 4: FIG. 4. Circuit design consisting of a 3-junction flux qubit galvanically coupled to an LC oscillator. [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: FIG. 5. Simulated spectrum of the system. Left panel: spectrum obtained with the design parameters of the circuit. Right panel: spectrum [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

**Figure 6.** Figure 6: FIG. 6. Magnitude of the raw transmission obtained via single-tone spectroscopy through the resonator (left) and qubit (right) feedline. [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]

**Figure 7.** Figure 7: FIG. 7. Magnitude of the raw transmission obtained via single-tone spectroscopy through the resonator (left) and qubit (right) feedline. [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]

**Figure 8.** Figure 8: FIG. 8. Zoom in around the resonator-like transition for the single-tone spectroscopy dataset presented in Fig. 2. The black dashed lines are [PITH_FULL_IMAGE:figures/full_fig_p010_8.png]

**Figure 9.** Figure 9: FIG. 9. Sheet resistance of grAl for different oxygen flow deposition. Each sample is measured right after fabrication (before bake) and after [PITH_FULL_IMAGE:figures/full_fig_p011_9.png]

**Figure 10.** Figure 10: FIG. 10. Resistance versus temperature curve for the grAl test structure. The dashed line indicates the [PITH_FULL_IMAGE:figures/full_fig_p011_10.png]

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author author M. Bell , author I. Sadovskyy , author L. Ioffe , author A. Y. \ Kitaev ,\ and\ author M. Gershenson ,\ @noop journal journal Physical review letters \ volume 109 ,\ pages 137003 ( year 2012 ) NoStop

2012
[24]

author author V. E. \ Manucharyan , author J. Koch , author L. I. \ Glazman ,\ and\ author M. H. \ Devoret ,\ @noop journal journal Science \ volume 326 ,\ pages 113 ( year 2009 ) NoStop

2009
[25]

Fraudet , author I

author author D. Fraudet , author I. Snyman , author D. M. \ Basko , author S. L \'e ger , author T. S \'e pulcre , author A. Ranadive , author G. Le Gal , author A. Torras-Coloma , author W. Guichard , author S. Florens , et al. ,\ @noop journal journal Physical Review Letter...

2025
[26]

Niepce , author J

author author D. Niepce , author J. Burnett ,\ and\ author J. Bylander ,\ @noop journal journal Physical Review Applied \ volume 11 ,\ pages 044014 ( year 2019 ) NoStop

2019
[27]

Torras-Coloma , author L

author author A. Torras-Coloma , author L. M. \ de Olcoz , author E. C \'e spedes , author E. Bertoldo , author D. L \'o pez-N \'u \ n ez , author S. Paul , author W. Wernsdorfer , author G. Rius ,\ and\ author P. Forn-D \' az ,\ @noop journal journal Superconductor Science an...

2024
[28]

author author H. G. \ Leduc , author B. Bumble , author P. K. \ Day , author B. H. \ Eom , author J. Gao , author S. Golwala , author B. A. \ Mazin , author S. McHugh , author A. Merrill , author D. C. \ Moore , et al. ,\ @noop journal journal Applied Physics Letters \ volume ...

2010
[29]

author author M. R. \ Vissers , author J. Gao , author D. S. \ Wisbey , author D. A. \ Hite , author C. C. \ Tsuei , author A. D. \ Corcoles , author M. Steffen ,\ and\ author D. P. \ Pappas ,\ @noop journal journal Applied Physics Letters \ volume 97 ,\ pages 232509 ( year 20...

2010
[30]

Gr \"u nhaupt , author M

author author L. Gr \"u nhaupt , author M. Spiecker , author D. Gusenkova , author N. Maleeva , author S. T. \ Skacel , author I. Takmakov , author F. Valenti , author P. Winkel , author H. Rotzinger , author W. Wernsdorfer , et al. ,\ @noop journal journal Nature materials \ ...

2019
[31]

Kamenov , author W.-S

author author P. Kamenov , author W.-S. \ Lu , author K. Kalashnikov , author T. DiNapoli , author M. T. \ Bell ,\ and\ author M. E. \ Gershenson ,\ @noop journal journal Physical Review Applied \ volume 13 ,\ pages 054051 ( year 2020 ) NoStop

2020
[32]

Gupta , author P

author author V. Gupta , author P. Winkel , author N. Thakur , author P. van Vlaanderen , author Y. Wang , author S. Ganjam , author L. Frunzio ,\ and\ author R. J. \ Schoelkopf ,\ @noop journal journal Physical Review Applied \ volume 23 ,\ pages 054067 ( year 2025 ) NoStop

2025
[33]

u ller , author L. Ding , author A. P. \ Veps \

author author J. Braum \"u ller , author L. Ding , author A. P. \ Veps \"a l \"a inen , author Y. Sung , author M. Kjaergaard , author T. Menke , author R. Winik , author D. Kim , author B. M. \ Niedzielski , author A. Melville , et al. ,\ @noop journal journal Physical Review...

2020
[34]

Abeles , author R

author author B. Abeles , author R. W. \ Cohen ,\ and\ author G. Cullen ,\ @noop journal journal Physical Review Letters \ volume 17 ,\ pages 632 ( year 1966 ) NoStop

1966
[35]

Rabi ,\ @noop journal journal Physical Review \ volume 49 ,\ pages 324 ( year 1936 ) NoStop

author author I. Rabi ,\ @noop journal journal Physical Review \ volume 49 ,\ pages 324 ( year 1936 ) NoStop

1936
[36]

Torras-Coloma et al., in preparation NoStop

@noop ( year 2025 ),\ note a. Torras-Coloma et al., in preparation NoStop

2025
[37]

The details of the derivation will be provided ref

note The coefficient takes the form: _ R ^2 = _R / _A with _A^2 = _R^2 + _4^2 2 - ( ^2 2 )^2 + g ^4 , _4^2 = 1 /L_ eff C_ tot , ^2 = _R^2 - _4^2 , g ^2 = _R / L_R C_ tot , and C_ tot = C_J+C_ sh . The details of the derivation will be provided ref. newarticle . Stop
[38]

Magazz \`u , author P

author author L. Magazz \`u , author P. Forn-D \' az ,\ and\ author M. Grifoni ,\ @noop journal journal Physical Review A \ volume 104 ,\ pages 053711 ( year 2021 ) NoStop

2021
[39]

Rotzinger , author S

author author H. Rotzinger , author S. Skacel , author M. Pfirrmann , author J. Voss , author J. M \"u nzberg , author S. Probst , author P. Bushev , author M. Weides , author A. Ustinov ,\ and\ author J. Mooij ,\ @noop journal journal Superconductor Science and Technology \ v...

2016
[40]

author author M. D. \ Reed , author L. DiCarlo , author B. Johnson , author L. Sun , author D. Schuster , author L. Frunzio ,\ and\ author R. Schoelkopf ,\ @noop journal journal Physical review letters \ volume 105 ,\ pages 173601 ( year 2010 ) NoStop

2010
[41]

author author E. T. \ Jaynes \ and\ author F. W. \ Cummings ,\ @noop journal journal PROCEEDINGS OF THE IEEE \ volume 51 ( year 1963 ) NoStop

1963
[42]

author author D. P. \ DiVincenzo , author F. Brito ,\ and\ author R. H. \ Koch ,\ @noop journal journal Physical Review B—Condensed Matter and Materials Physics \ volume 74 ,\ pages 014514 ( year 2006 ) NoStop

2006
[43]

Kafri , author C

author author D. Kafri , author C. Quintana , author Y. Chen , author A. Shabani , author J. M. \ Martinis ,\ and\ author H. Neven ,\ @noop journal journal Physical Review A \ volume 95 ,\ pages 052333 ( year 2017 ) NoStop

2017
[44]

Baust , author E

author author A. Baust , author E. Hoffmann , author M. Haeberlein , author M. Schwarz , author P. Eder , author J. Goetz , author F. Wulschner , author E. Xie , author L. Zhong , author F. Quijandr \' a , et al. ,\ @noop journal journal Physical Review B \ volume 93 ,\ pages ...

2016
[45]

merlin.mbs apsrev4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked

FUNCTION id.bst "merlin.mbs apsrev4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked" ENTRY address archive archivePrefix author bookaddress booktitle chapter collaboration doi edition editor eid eprint howpublished institution isbn issn journal key language month note number orga...

2010
[46]

merlin.mbs apsrmp4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked

FUNCTION id.bst "merlin.mbs apsrmp4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked" ENTRY address archive archivePrefix author bookaddress booktitle chapter collaboration doi edition editor eid eprint howpublished institution isbn issn journal key language month note number orga...

2010

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

merlin.mbs aapmrev4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked

FUNCTION id.bst "merlin.mbs aapmrev4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked" ENTRY address archive archivePrefix author bookaddress booktitle chapter collaboration doi edition editor eid eprint howpublished institution isbn issn journal key language month note number organization pages primaryClass publisher school SLACcitation series title translat...

2010

[2] [2]

merlin.mbs aipauth4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked

FUNCTION id.bst "merlin.mbs aipauth4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked" ENTRY address archive archivePrefix author bookaddress booktitle chapter collaboration doi edition editor eid eprint howpublished institution isbn issn journal key language month note number organization pages primaryClass publisher school SLACcitation series title translat...

2010

[3] [3]

merlin.mbs aipnum4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked

FUNCTION id.bst "merlin.mbs aipnum4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked" ENTRY address archive archivePrefix author bookaddress booktitle chapter collaboration doi edition editor eid eprint howpublished institution isbn issn journal key language month note number organization pages primaryClass publisher school SLACcitation series title translati...

2010

[4] [4]

Blais , author R.-S

author author A. Blais , author R.-S. \ Huang , author A. Wallraff , author S. M. \ Girvin ,\ and\ author R. J. \ Schoelkopf ,\ @noop journal journal Physical Review A—Atomic, Molecular, and Optical Physics \ volume 69 ,\ pages 062320 ( year 2004 ) NoStop

work page 2004

[5] [5]

Wallraff , author D

author author A. Wallraff , author D. I. \ Schuster , author A. Blais , author L. Frunzio , author R.-S. \ Huang , author J. Majer , author S. Kumar , author S. M. \ Girvin ,\ and\ author R. J. \ Schoelkopf ,\ @noop journal journal Nature \ volume 431 ,\ pages 162 ( year 2004 ) NoStop

work page 2004

[6] [6]

Stassi , author M

author author R. Stassi , author M. Cirio ,\ and\ author F. Nori ,\ @noop journal journal npj Quantum Information \ volume 6 ,\ pages 67 ( year 2020 ) NoStop

work page 2020

[7] [7]

author author C. L. \ Degen , author F. Reinhard ,\ and\ author P. Cappellaro ,\ @noop journal journal Reviews of modern physics \ volume 89 ,\ pages 035002 ( year 2017 ) NoStop

work page 2017

[8] [8]

Yoshihara , author T

author author F. Yoshihara , author T. Fuse , author S. Ashhab , author K. Kakuyanagi , author S. Saito ,\ and\ author K. Semba ,\ @noop journal journal Nature Physics \ volume 13 ,\ pages 44 ( year 2017 a ) NoStop

work page 2017