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Temperature and Magnetic-Field Dependence of Energy Relaxation in a Fluxonium Qubit

T0 review · 2 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A low-frequency fluxonium qubit maps how flux noise and dielectric loss respond to temperature and magnetic field, giving microscopic noise theories concrete scaling targets.

desk verdict Careful fluxonium study with a solid AΦ∝T result and a convincing B-field effect; the T^3 charge-noise scaling is a fit-dependent claim that needs a direct bath-temperature test. read the letter →

arxiv 2507.01175 v1 pith:GLONRQGF submitted 2025-07-01 quant-ph cond-mat.mes-hall

Lamia Ateshian , Max Hays , David A. Rower , Helin Zhang , Kate Azar , Réouven Assouly , Leon Ding , Michael Gingras
show 9 more authors
Hannah Stickler Bethany M. Niedzielski Mollie E. Schwartz Terry P. Orlando Joel Î-j. Wang Simon Gustavsson Jeffrey A. Grover Kyle Serniak William D. Oliver
This is my paper
classification quant-phcond-mat.mes-hall
keywords fluxoniumqubitenergyrelaxationfluxnoisedielectriclosstwo-levelsystemstemperaturedependencemagnetic-fieldsuperconductingcoherence
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper reports temperature- and magnetic-field trends in the two dominant sources of energy decay in a low-frequency fluxonium qubit, and argues these trends should constrain microscopic theories of noise in superconducting circuits. Using a qubit whose transition frequency reaches 52 MHz, the authors measure $T_1$ as a function of flux bias while heating the mixing chamber to 100 mK and while applying in-plane fields up to 100 G. They find the low-frequency flux-noise magnitude scales roughly linearly with temperature, $A_\Phi \propto T$, and that the high-frequency charge noise from dielectric loss follows an approximate power law $T^3$ (fitted exponent $\beta_2 = 2.9$). They also find that weak in-plane magnetic fields increase the dielectric-loss contribution to relaxation, suggesting the charge-coupled defects respond to magnetic field. The paper presents a multi-level decoherence model—a rate matrix over six fluxonium levels—that captures parts of the data a two-level model misses, particularly at intermediate flux biases.

What carries the argument

The carrier of the argument is the fluxonium qubit itself, used as a spectrometer of its own noise environment: with minimum frequency $f_{01} = 52$ MHz and widely tunable $|\langle 0|\hat{\phi}|1\rangle|$ and $|\langle 0|\hat{n}|1\rangle|$ matrix elements, the device maps $T_1$ versus flux bias into the low-frequency flux-noise spectrum $S_\Phi(\omega) = A_\Phi(2\pi/\omega)^\alpha$ and the dielectric-loss spectrum $S_Q(\omega)$ via Fermi's golden rule. The second piece of machinery is the $N$-level rate-matrix decoherence model ($N = 6$), which accounts for heating transitions out of the computational subspace and is needed to capture $T_1$ at intermediate flux biases, especially for dielectric loss. The third is the phenomenological power-law model of Eq. 8, which jointly fits flux- and charge-noise terms with independent frequency and temperature exponents and yields the headline values $\alpha = 1.5$, $\beta_1 = 0.32$, $\gamma = 0.19$, $\beta_2 = 2.9$.

What would settle it

Repeat the temperature sweep while independently measuring the effective temperature of the lossy defect bath (for example, through qubit population thermometry or through a calibrated TLS thermometer at the same frequencies); if the bath temperature does not track $T_{MC}$, the fitted $\beta_2 \approx 2.9$ no longer describes the material. Alternatively, extend the sweep below 35 mK: if the $T^3$ power law flattens or saturates, the attribution to a thermal defect bath fails.

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Extended reading notes

Core claim

The central claim is that in a fluxonium qubit operated at low frequency, the intrinsic noise that limits $T_1$ has three empirically distinguishable behaviors: flux-noise amplitude grows approximately linearly with temperature ($A_\Phi \propto T$ for 35–100 mK, probed at 1–100 MHz through $T_1$ and echo dephasing); the dielectric-loss charge noise grows as roughly $T^3$, extracted from a phenomenological fit with fitted exponent $\beta_2 = 2.9$; and in-plane magnetic fields up to 100 G increase the inferred dielectric loss while also reshaping the flux-noise contribution, so that the dominant loss mechanism crosses over from flux noise to charge noise at high field. The authors argue these scaling laws, taken with the multi-level rate-matrix corrections, mean that fluxonium coherence models should not assume temperature-independent noise spectra, and that any microscopic theory of surface spins and charge TLS must reproduce the observed temperature and field trends.

Load-bearing premise

The central $T^3$ result rests on assuming the defect bath that causes the loss sits at the measured mixing-chamber temperature across the 35–100 mK sweep; the paper itself notes that an elevated bath temperature would alter or remove the $T^3$ conclusion.

Editorial extensions

If this is right

  • Fluxonium coherence simulations should incorporate transitions beyond the $|0\rangle \leftrightarrow |1\rangle$ manifold whenever transition energies approach $k_BT$; the $N$-level rate matrix materially changes predicted $T_1$ at intermediate flux biases, where the two-level model fails.
  • Microscopic theories of flux noise must allow for a temperature-dependent noise amplitude: $A_\Phi \propto T$ contradicts earlier observations of temperature-independent flux noise below about 100 mK in SQUIDs and flux qubits, so either device-specific or regime-specific mechanisms are required.
  • Dielectric-loss models for low-frequency qubits cannot assume a temperature-independent loss tangent: the $T^3$ growth implies existing two-level dielectric models underestimate how fast $T_1$ degrades above base temperature.
  • Magnetic-field studies of fluxonium coherence should treat the dielectric-loss channel as field-sensitive: the increase in inferred loss up to 100 G means charge-coupled defects, not only surface spins, respond to applied fields.
  • The reported scaling laws supply concrete benchmark targets—$A_\Phi(T)$, $\beta_2$, and the field-dependent crossover—against which future microscopic models of surface spins and charge TLS can be tested.

Reading between the lines

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

  • I would read the $T^3$ dielectric-loss term as a signature worth testing against a resonant-absorption mechanism in which the TLS lifetime itself scales as $T^\beta$; the paper notes this could produce a power-law qubit rate, and a direct measurement of the TLS lifetime temperature dependence on the same device would discriminate it from relaxation absorption.
  • A natural next experiment is to separate frequency and temperature scaling by measuring $T_1$ at fixed frequencies rather than along the flux-bias sweep; fitting the phenomenological model with independent temperature sweeps at, say, 100, 300, and 800 MHz would pin down whether $\beta_2$ is truly a material exponent or an artifact of the elevated-bath-temperature ambiguity the paper flags.
  • The apparent magnetic response of dielectric loss could be checked on a non-qubit device, such as a superconducting resonator without a flux-tunable element, at the same fields; if resonator loss rises with $B$, the effect is a property of the charge-TLS bath, while if only the qubit sees it, fit assumptions about fixed $\alpha$ and $\epsilon$ deserve scrutiny.
  • If $A_\Phi \propto T$ is confirmed as a general feature of low-frequency fluxonium, it may also explain part of the variation in reported flux-noise amplitudes across devices: measurements taken at different base temperatures would be compared only after rescaling to a common temperature.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The manuscript reports a study of energy relaxation in a low-frequency fluxonium qubit, with the aim of characterizing the temperature and in-plane magnetic-field dependence of flux and charge noise. T1 is measured as a function of flux bias at base temperature, over a mixing-chamber temperature range of 35–100 mK, and for in-plane fields up to 100 G. The paper's central claims are: (i) the low-frequency flux-noise amplitude AΦ increases approximately linearly with temperature; (ii) the high-frequency charge-noise contribution scales approximately as T^3 (β2 = 2.9 in Eq. 8); and (iii) weak in-plane magnetic fields increase the dielectric-loss contribution to relaxation. The analysis combines circuit-level loss models with a multi-level rate-matrix treatment, and the magnetic-field analysis is supported by bootstrapped confidence intervals and a sensitivity study over fixed noise exponents.

Significance. If correct, these observations would constitute new empirical constraints on microscopic models of flux and charge noise in superconducting circuits: the linear AΦ(T) trend differs from earlier reports of temperature-independent flux noise, and the T^3 charge-noise scaling lies outside simple two-level dielectric-loss predictions. The experimental work has notable strengths: data from three cooldowns are consistent; the AΦ(T) trend is cross-checked with independent spin-echo dephasing data; the magnetic-field section uses empirical bootstrapping and a sensitivity analysis; and the N-level model is tested for convergence and exponentiality. The principal weakness is the provenance of the T^3 exponent, which rests on a fit that assumes T_bath = TMC and reports no uncertainties.

major comments (2)
  1. [Section IV, Eq. (8)] The reported T^3 scaling of the high-frequency charge noise (β2 = 2.9) is not securely established because the fit sets T = TMC and reports no confidence intervals for the four fitted parameters. The physical dielectric-loss expression in Eq. (6) contains a coth(ℏω/2kBTeff) factor that is a strong function of temperature in the measured range (f ≈ 0.1–1.6 GHz, T ≈ 35–100 mK), whereas Eq. (8) folds this factor into a pure power law and attributes all remaining temperature dependence to the noise amplitude. The manuscript itself (Section IV and footnote 72) states that the N-level model residuals 'could also suggest an elevated bath temperature relative to the MC stage' and allows different effective temperatures for the low- and high-frequency defect baths. If the high-frequency bath temperature saturates or grows sublinearly with TMC, the fitted β2 would be biased upward. A re-analysis with independently measured effective temperatures, or at least a sensitivity analysis over plausible T_bath(TMC) curves with bootstrap errors on β2, is required before the T^3 claim can be supported.
  2. [Section IV, Eq. (8) and Appendix E] The same phenomenological fit returns β1 = 0.32 for the flux-noise term, which the text immediately discounts because the second term in Eq. (8) goes to zero as ω → 0. Since β2 is extracted from the same correlated four-parameter fit, the paper should demonstrate that β2 is not similarly contaminated by the low-frequency degeneracy—for example, by refitting with a physically motivated low-frequency model or by excluding low-frequency points—and should reconcile β1 = 0.32 with the separately claimed AΦ ∝ T from Appendix E. Without this, the reader cannot judge whether the T^3 exponent is a stable property of the high-frequency noise or an artifact of the chosen fitting form.
minor comments (5)
  1. [Section III and Section V] The parameter bookkeeping for the magnetic-field fits is ambiguous: the text first states that tanδ0C = 4 × 10−6 and α = 0.62 are used to constrain the B = 0 fit, then states that α and ϵ are held fixed while AΦ and tanδ0C are fitted at each field. Please state explicitly which parameters are fixed and which are free at each step.
  2. [Section IV] The exclusion of the stray TLS near 150 MHz is qualitative; please specify the frequency window excluded and the criterion used to identify it, since the high-frequency fit to Eq. (8) depends on this choice.
  3. [Abstract and Section IV] The abstract claims a 'power-law dependence of dielectric loss T^3' without the 'approximately' qualifier and without mentioning that the fitted exponent is β2 = 2.9 with no reported uncertainty; the abstract should be qualified to match the body.
  4. [Appendix E] For TMC > 35 mK, the echo-dephasing parameters are obtained by maximizing the R² of the linearity plot in Fig. 9b; please report the R² values and the sensitivity of the extracted AΦ to this procedure.
  5. [Figure 3] The averaging of points at equal frequencies is described only in the caption; please state how many points are averaged and whether the averaging is performed before or after the model comparison, because it affects the apparent scatter in Fig. 3.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central claims are empirical fits with independent cross-checks; the T^3 caveat is a fitting-assumption risk, not a circular derivation.

full rationale

This is an empirical characterization paper whose headline results are fits to measured T1 data cross-checked against independent noise spectroscopies, not quantities derived from previously fitted parameters. The T^3 charge-noise exponent is obtained from the phenomenological power-law fit in Eq. 8 ('Fitting Eq. 8 simultaneously to the full dataset with T = TMC ... we extract ... beta2 = 2.9'), and the paper explicitly frames it as a fit ('an approximately T^3 scaling fits the data best'), not as a prediction implied by the dielectric-loss model. The A_Phi versus T claim is supported by two routes: a temperature-dependent fit to T1 (Appendix E) and independent spin-echo dephasing measurements in a separate sweep, with the linear dependence overlaid only after both extractions; it is not inserted as the ansatz that generates the finding. The B-field dielectric-loss increase is obtained by refitting A_Phi and tan_delta0 at each field with fixed exponents, with bootstrap confidence intervals and a sensitivity analysis over alpha and epsilon, and the qualitative trend is reproduced in a second dataset (Appendix K). The N-level model is fitted at base temperature and then evaluated at higher temperatures without additional free parameters, which is an out-of-sample consistency check rather than a circular reduction. Self-citations (e.g., Refs. [36,66,84]) provide experimental context, typical flux-noise magnitudes, or suggestions for future work; none functions as an imported uniqueness theorem or a load-bearing ansatz. The paper itself flags the main caveat in Section IV and footnote [72] that the effective bath temperature may differ from T_MC, which affects the reliability of the fitted beta2; that is a fitting-assumption risk, not a circularity: nothing in the paper defines the T^3 scaling into existence by using it as an input. No equation in the paper equals its own input by construction, and no fitted parameter is renamed as a prediction.

Assumptions & free parameters 8 free parameters · 8 assumptions · 0 invented entities

The central claims rely on several empirical modeling choices, particularly the power-law forms of the flux and charge noise spectra and the fixed exponents in the B-field analysis. No new physical entities are postulated. The main free parameters are the noise amplitudes, exponents, and effective temperatures that enter the Fermi's-golden-rule fits.

free parameters (8)
  • Flux noise exponent alpha = 0.62 (main analysis), 1.0 (alternative)
    Fitted to spin-locking and echo dephasing data (Appendix D); alternative value 1.0 gives A_Phi=(3.7 uPhi0)^2. Used in all T1 models and the B-field analysis.
  • Flux noise amplitude A_Phi = (0.25 uPhi0)^2 at 1 Hz
    Extracted from echo dephasing with alpha=0.62 and also inferred from T1 near half-flux; the linear-in-T trend for A_Phi is the central temperature claim.
  • Dielectric loss tangent tan_delta0_C = ~4e-6
    Estimated upper bound at integer flux and refined by fitting T1 versus flux (Section III); central to the dielectric-loss contribution in all analyses.
  • Dielectric loss frequency exponent epsilon = 0.26 (baseline), 0.45 (temperature dataset), 0.31 (B-field)
    Phenomenological exponent in tan_delta_C(omega)=tan_delta0(omega/omega_ref)^epsilon; fitted separately across cooldowns.
  • Effective qubit temperature T_eff = ~50 mK
    Estimated from steady-state |0> and |1> populations (Appendix C); used in models at base temperature.
  • Phenomenological power-law exponents (Eq. 8) = alpha=1.5, beta1=0.32, gamma=0.19, beta2=2.9
    Fit simultaneously to the full temperature dataset; beta2 approx 3 is the central T^3 claim.
  • Resonator temperature T_res = 70 mK (upper bound)
    From echo dephasing T_E=71 us at the sweet spot, assuming shot-noise-limited dephasing; sets Purcell contribution.
  • B-field-dependent A_Phi and tan_delta0 = Varies with B (Fig. 4 and Appendix K)
    Fitted at each field with alpha and epsilon fixed; used to infer the dielectric-loss increase with field.
assumptions (8)
  • standard math Fermi's golden rule gives transition rates between fluxonium levels
    Eq. 2 and Appendix F use FGR to relate T1 to noise spectral densities.
  • domain assumption The noise sources (flux, charge/dielectric, Purcell, QP) are independent and additive
    Total decay rate is the sum of individual FGR rates (Eq. 2); assumes no interference between mechanisms.
  • ad hoc to paper Flux noise spectrum has power-law form S_Phi(omega)=A_Phi(2pi/omega)^alpha
    Empirical form used throughout; alpha is fitted. This is a modeling choice, not derived.
  • ad hoc to paper Dielectric loss tangent follows power law tan_delta_C(omega)=tan_delta0(omega/omega_ref)^epsilon
    Phenomenological model from prior works [50,54]; epsilon fitted per cooldown.
  • domain assumption The environment is in thermal equilibrium and detailed balance holds for transition rates
    Used in the N-level rate matrix (Appendix H) and in relating upward and downward rates.
  • domain assumption The qubit Hamiltonian parameters from spectroscopy are accurate and the N=6 truncation converges
    Matrix elements computed with scQubits using fitted parameters; convergence claimed for N=6.
  • ad hoc to paper The stray TLS at 150 MHz can be excluded from the analysis as an isolated feature
    Excluded to study ensemble properties; could affect fits if it has broad tails.
  • standard math Caldeira-Leggett impedance noise model for voltage and current noise
    Used for dielectric loss, Purcell decay, radiation, and QP models (Appendix F).

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Cite this review

Pith. "Pith review of Temperature and Magnetic-Field Dependence of Energy Relaxation in a Fluxonium Qubit." pith.science (2026). https://pith.science/paper/GLONRQGF

@misc{pith2026250701175,
  author       = {Pith},
  title        = {Pith review of: Temperature and Magnetic-Field Dependence of Energy Relaxation in a Fluxonium Qubit},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GLONRQGF}},
  note         = {Machine review of arXiv:2507.01175}
}
abstract

Noise from material defects at device interfaces is known to limit the coherence of superconducting circuits, yet our understanding of the defect origins and noise mechanisms remains incomplete. Here we investigate the temperature and in-plane magnetic-field dependence of energy relaxation in a low-frequency fluxonium qubit, where the sensitivity to flux noise and charge noise arising from dielectric loss can be tuned by applied flux. We observe an approximately linear scaling of flux noise with temperature $T$ and a power-law dependence of dielectric loss $T^3$ up to 100 mK. Additionally, we find that the dielectric-loss-limited $T_1$ decreases with weak in-plane magnetic fields, suggesting a potential magnetic-field response of the underlying charge-coupled defects. We implement a multi-level decoherence model in our analysis, motivated by the widely tunable matrix elements and transition energies approaching the thermal energy scale in our system. These findings offer insight for fluxonium coherence modeling and should inform microscopic theories of intrinsic noise in superconducting circuits.

Figures

Figures reproduced from arXiv: 2507.01175 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. c). The low-frequency parameters are impacted by the fact that the second term in Eq. 8 goes to zero as ω → 0, whereas a more realistic model comprising flux noise and dielectric loss has contributions from both mechanisms at low frequency. At high frequency, on the other hand, the contribution from flux noise falls off, and the remaining loss can be largely attributed to charge noise. This component displays a temp… view at source ↗
Figures from the paper (17 more)
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Wiring diagram of experimental setup. [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Qubit and resonator spectroscopy. (a) Qubit two-tone spectroscopy and fit to obtain Hamiltonian parameters. (b) [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Flux noise spectrum at [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Temperature dependence of spin-echo dephasing. (a) Echo pure dephasing time [PITH_FULL_IMAGE:figures/full_fig_p014_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Equivalent circuit used to model Purcell decay. [PITH_FULL_IMAGE:figures/full_fig_p017_10.png]
Figure 12
Figure 12. Figure 12: FIG. 12. Evolution of state population under transition ma [PITH_FULL_IMAGE:figures/full_fig_p018_12.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Comparison of temperature-dependence of models at [PITH_FULL_IMAGE:figures/full_fig_p018_11.png]
Figure 13
Figure 13. Figure 13: FIG. 13. Transition rates for dielectric loss. [PITH_FULL_IMAGE:figures/full_fig_p019_13.png]
Figure 15
Figure 15. Figure 15: FIG. 15. Eigenvectors of the [PITH_FULL_IMAGE:figures/full_fig_p020_15.png]
Figure 16
Figure 16. Figure 16: FIG. 16. Simulated effect of leakage state mis-assignment. [PITH_FULL_IMAGE:figures/full_fig_p021_16.png]
Figure 17
Figure 17. Figure 17: FIG. 17. Dependence of qubit parameters on magnetic [PITH_FULL_IMAGE:figures/full_fig_p023_17.png]
Figure 18
Figure 18. Figure 18: FIG. 18. Magnetic-field dependence for dataset B. [PITH_FULL_IMAGE:figures/full_fig_p023_18.png]
Figure 19
Figure 19. Figure 19: FIG. 19. Fit parameters for both magnetic field dependence [PITH_FULL_IMAGE:figures/full_fig_p024_19.png]
Figure 20
Figure 20. Figure 20: FIG. 20. Full [PITH_FULL_IMAGE:figures/full_fig_p025_20.png]
Figure 21
Figure 21. Figure 21: FIG. 21. Full [PITH_FULL_IMAGE:figures/full_fig_p025_21.png]

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Quasiparticle-induced transitions in a fluxonium qubit

    quant-ph 2026-07 conditional novelty 7.0 of 10

    Fluxonium decoherence under quasiparticle injection is accurately described only when the asymmetric superconducting gaps across the junctions are included, reconciling previously contradictory quasiparticle-density bounds.

Reference graph

Works this paper leans on

134 extracted references · 72 canonical work pages · cited by 1 Pith paper

  1. [72]

    L. B. Nguyen, Y. H. Lin, A. Somoroff, R. Mencia, N. Grabon, and V. E. Manucharyan, High-coherence fluxonium qubit, Physical Review X 9 (2019)

    work page 2019

  2. [1]

    Only one qubit (our device-under-test) had both charge and flux lines wired to the external package connectors

    Sample The sample comprised six uncoupled, floating fluxo- nium qubits with individual dispersively-coupled readout resonators coupled to a common transmission line. Only one qubit (our device-under-test) had both charge and flux lines wired to the external package connectors. The sample was designed, fabricated, and packaged at MIT Lincoln Laboratory usi...

  3. [2]

    Magnet The magnet used in our experiment is similar to the one discussed in detail in Ref. [36]. It comprised two NbTi superconducting coils (hand-wound, 627 turns for each coil) with an inner radius of 15 mm, smaller than that of the coils used in [36]. The sample package was mounted between the two coils such that the rectangular junction loop had arms ...

  4. [3]

    Datasets were collected after observing equilibration of repeated T1 measurements at half-flux, typically within 45 minutes after a change in the current setting

    T emperature dependence For the temperature dependence measurements, TMC was set with an applied heating current at the mixing chamber stage. Datasets were collected after observing equilibration of repeated T1 measurements at half-flux, typically within 45 minutes after a change in the current setting. We note that the DR underwent a thermal cycle to roo...

  5. [4]

    Accounting for α ̸= 1 in extracting AΦ from echo dephasing measurements. In the two-level approximation, the qubit is described by the Hamiltonian ˆH = ℏ(ω01 + δω(t))σz/2, where ω01 is the qubit transition frequency and δω(t) describes the stochastic frequency fluctuations induced by flux noise. A superposition state evolves with the phase ϕ(t) = ω01t + δ...

  6. [5]

    In our qubit, this is believed to be the dom- inant loss mechanism at half-flux, where the qubit fre- quency is f01 = 52 MHz

    1/f flux noise Low-frequency magnetic flux noise couples to the qubit phase operator through the inductive term of the Hamil- tonian, λ(t) = δΦext(t) and ˆDλ = (2πEL/Φ0) ˆϕ, yielding a depolarization rate Γ1/f 1 = 8 πEL ℏΦ0 2 | ⟨0| ˆϕ |1⟩ |2SΦ(ω01), (S9) where the flux noise power spectrum is defined as [6–8] SΦ(ω) = AΦ 2π ω α (S10) As mentioned, we obtai...

  7. [6]

    This results in a depolarization rate Γind 1 = 2EL ℏQL |⟨0| ˆϕ|1⟩|2 coth ℏω01 2kBTeff

    Inductive loss Inductive loss is related to current fluctuations in the inductor (implemented here by a Josephson-junction array superinductor) leading to bias-flux fluctuations, λ(t) = δΦext(t) and ˆDλ = (2πEL/Φ0) ˆϕ, with the power spectrum given by Johnson-Nyquist current noise: SΦ,ind(ω) = ℏΦ2 0 4π2ELQL coth ℏω 2kBTeff , (S11) where QL is the inductiv...

  8. [7]

    The loss rate can be written in terms of the capacitive loss tangent tan δC(ω) as Γdiel 1 = 16EC ℏ |⟨0|ˆn|1⟩|2 tan δC(ω01) coth ℏω01 2kBTeff

    Dielectric loss Dissipation through lossy dielectric materials in the de- vice is modeled as Johnson-Nyquist voltage noise across the capacitor, or equivalent noise in the gate charge λ(t) = δQg(t) = 2 eng(t), with ˆDλ = (4 EC/e)ˆn. The loss rate can be written in terms of the capacitive loss tangent tan δC(ω) as Γdiel 1 = 16EC ℏ |⟨0|ˆn|1⟩|2 tan δC(ω01) c...

Show all 134 references
  1. [8]

    QPs at the small junction For current noise from QPs tunneling across the small junction, we have λ(t) = δIQP(t) and ˆDλ = (Φ0/π) sin(ˆϕ/2), which results in a decay rate [54, 70, 105] ΓQP 1 = 2 Φ0 ℏπ 2 |⟨0| sin( ˆϕ/2)|1⟩|2SQP(ω01), (S16) where the spectral density SQP(ω) = ℏω...

  2. [9]

    The T1 at half-flux places an upper bound of xQPA ≈ 7 × 10−9 on QPs in the array

    QPs in the junction array QPs tunneling in the junction array are suspected to constitute a source of inductive loss in fluxonium qubits [106], with a relaxation rate described by ΓQPA 1 = 2 Φ0 ℏπ 2 EL|⟨0| ˆϕ 2 |1⟩|2 ¯SQP(ω01), (S20) where ¯SQP(ω) = SQP(ω)/EJ [70]. The T1 at h...

  3. [10]

    Charge line

    Radiation to the charge and flux lines Radiative decay to the charge (flux) lines is estimated by Johnson-Nyquist voltage (current) noise from a 50 Ω environment connected to the qubit via a coupling ca- pacitance (inductance) set by the control line geometry. Charge line. Vol...

  4. [11]

    Purcell decay through the readout resonator We model the fluxonium’s Purcell decay rate through the readout resonator by considering the equivalent- circuit model of the coupled qubit-resonator system, shown in Fig. 10. A λ/4 transmission-line resonator with characteristic imp...

  5. [12]

    N -Level Decoherence Model Consider a vector ⃗ p(t) = p0 p1 · · ·pN −1 T with length N , representing the populations of the first N lev- els of the fluxonium at time t. The dynamics of the level occupation probabilities pi, are described by the matrix rate equation ∂t⃗ p(t) =...

  6. [13]

    Multiple loss mechanisms To evaluate the population evolution under multiple loss mechanisms, we must evaluate a single rate matrix FIG. 14. Comparison of 2-level vs. N -level models. Gray shaded region indicates range of measured T1 data, showing that heating events impact on...

  7. [14]

    Evaluating exponential behavior Although in general the solution ⃗ p(t) contains multi- ple exponentially decaying terms, the resulting dynam- ics can often be approximately characterized by a single rate if there is a dominant exponentially decaying eigen- mode in the initial...

  8. [15]

    worst case

    Effect of leakage state mis-assignment on measured T1 Here we address the possibility that, under the N -level decoherence model, leakage states ( |n⟩ with n >1) may be populated but not distinguished in qubit readout. In- stead, they may be mis-assigned to|0⟩ or |1⟩. We consi...

  9. [16]

    The critical photon number nc is related to the TLS coherence times nc ∝ (τ1τ2)−1, so for large photon numbers the loss scales approximately as tan δres ∝ (τ1τ2)−1/2 tanh ℏω 2kB T

    Resonant absorption The contribution to the dielectric loss tangent from resonant TLS absorption is [19, 65] tan δres(ω, ¯n, T) = πP |p|2 3ϵ0ϵr tanh ℏω 2kB T p 1 + ¯n/nc (S39) where p is the electric dipole moment of the TLS,ϵr is the material relative permittivity, ¯n is the ...

  10. [17]

    Following Refs

    Relaxation absorption To calculate the relaxation contribution, one integrates the individual TLS contribution over the defect density of states (DOS). Following Refs. [19] and [65], similar to Ref. [82], this gives for the relaxation contribution to the loss tangent, tanδTLS,...

  11. [18]

    This is, of course, further complicated by the presence of higher levels in the system

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