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Statistical characterization of valley coupling in Si/SiGe quantum dots via $g$-factor measurements near a valley vortex

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

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Pith's one-line read Valley splitting data alone cannot determine valley-coupling statistics in disordered Si/SiGe quantum wells; g-factor loops around valley vortices can.

desk verdict Solid statistical caution, but the g-factor phase extraction has a sign ambiguity that leaves the central claim unsupported as stated. read the letter →

arxiv 2507.05160 v1 pith:EUF3FZER submitted 2025-07-07 cond-mat.mes-hall quant-ph

Benjamin D. Woods , Merritt P. Losert , Nasir R. Elston , M. A. Eriksson , S. N. Coppersmith , Robert Joynt , Mark Friesen This is my paper · ORCID
classification cond-mat.mes-hallquant-ph
keywords valleysplittingphaseSi/SiGequantumwellsspinqubitsg-factoralloydisordermaximumlikelihoodestimationvortex
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 in disorder-dominated Si/SiGe quantum wells, valley splitting data alone cannot determine the two statistical parameters that set qubit failure rates: the deterministic valley coupling magnitude $|\Delta_0|$ and the random-alloy disorder strength $\sigma_\Delta$. Maximum-likelihood fits to splitting distributions systematically overestimate $|\Delta_0|$, misclassifying disordered wells as deterministically enhanced even when the data set contains a thousand dots. The paper shows that adding the valley phase $\varphi_v$ — the argument of the complex valley coupling — makes the inference accurate. Because the phase is hard to measure directly, it proposes to extract it from the electron g-factor, whose measured value depends on $\cos \varphi_v$, and to calibrate that dependence by moving the dot in a loop around a valley vortex, where the phase winds by $\pm 2\pi$. If the protocol works, it gives an experimentally practical way to certify quantum wells for large-scale spin qubit arrays.

What carries the argument

The load-bearing object is the relation between the measurable g-factor and the valley phase, $g = g_0 - g_\tau \cos \varphi_v \sin(2\theta_B)$, together with the complex valley coupling $\Delta = \Delta_0 + \Delta_\delta$ whose argument defines $\varphi_v$. A valley vortex is a point where $|\Delta| = 0$: the valley splitting vanishes there, and the valley phase winds by $\pm 2\pi$ around it. Measuring $g$ along a loop that winds once around a vortex guarantees that $g$ visits both extrema $g_0 \pm g_\tau$, which fixes the calibration constant $g_\tau$ with certainty. The statistical analysis then uses the Rician distribution of $E_v$ and the complex-normal distribution of $\Delta$ to show that phase information, not just splitting information, resolves the inference problem.

What would settle it

Map the valley splitting and the g-factor over the same region of a Si/SiGe well while shuttling a single electron, and locate a valley vortex. If the density of points where $E_v = 0$ disagrees with $(0.184/\ell^2)\exp(-|\Delta_0|^2/\sigma_\Delta^2)$, the statistical model of valley coupling is wrong; if a loop with winding number one does not send the measured g through both extrema $g_0 \pm g_\tau$, or if the g-inferred valley phase contradicts the splitting landscape, Eq. (4) is falsified.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that valley splitting statistics alone are unreliable precisely in the regime that matters for large quantum computers, and that the missing observable is the valley phase. The splitting $E_v = 2|\Delta|$ follows a Rician (magnitude-of-complex-Gaussian) distribution whose shape depends only on the ratio $|\Delta_0|/\sigma_\Delta$, so a disorder-dominated well and a deterministically enhanced well can have nearly identical splitting distributions and the same average splitting; maximum likelihood therefore often reports a large deterministic coupling when the true deterministic coupling is zero. In the disorder-dominated regime, $|\Delta_0|/\sigma_\Delta \lesssim \sqrt{\pi}/2$, realistic samples of 20 dots misclassify the well, and even $N=1000$ dots leave more than half of the fits with a spuriously large deterministic ratio. Measuring the valley phase breaks the degeneracy, and the g-factor provides a practical channel for that information: $g = g_0 - g_\tau \cos \varphi_v \sin(2\theta_B)$. The paper also derives the density of valley vortices, points where $E_v = 0$, as $\rho_{VV} = (0.184/\ell^2)\exp(-|\Delta_0|^2/\sigma_\Delta^2)$, which keeps the calibration loop practical in the disordered regime.

Load-bearing premise

The load-bearing premise is that the measured electron g-factor tracks the valley phase through a simple cosine law with a single fixed calibration constant, so that inverting a measured g gives the true valley phase; the authors validate that law only inside their own effective-mass model, and it recovers the phase only up to sign.

Editorial extensions

If this is right

  • Valley splitting surveys alone — whether across many dots or along a shuttled dot — cannot certify a quantum well as deterministically enhanced; reported high-splitting distributions may still be disorder-dominated.
  • Adding valley phase information restores reliable maximum-likelihood estimation of $|\Delta_0|$ and $\sigma_\Delta$ in the disorder-dominated regime, with no misclassification in the demonstrated examples.
  • The g-factor mapping protocol needs only standard qubit resonance measurements and dot position control, so it can be implemented in current experiments.
  • A loop enclosing a single valley vortex fixes the calibration constant $g_\tau$ with certainty, because the valley phase winding sends the measured g-factor through both of its extreme values.
  • The predicted vortex density implies the calibration step is practical in the disordered regime: a $100 \times 100$ nm$^2$ region is expected to contain about nine vortices for the parameters shown.

Reading between the lines

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

  • An implication the authors leave implicit is that Eq. (4) determines only $\cos \varphi_v$, so an experimental protocol must either resolve the sign of $\varphi_v$ or work with $|\varphi_v|$; the paper's own error statistics use absolute phases, suggesting the latter route.
  • The same g-factor maps could be used to extract spatial correlation lengths of alloy disorder, not just $|\Delta_0|$ and $\sigma_\Delta$, because the covariance structure of $\Delta$ shapes the landscapes — an extension the paper does not pursue.
  • A direct check of the model's random-field assumption would be to count valley vortices in a mapped valley-splitting landscape and compare the density with $\rho_{VV} = (0.184/\ell^2)\exp(-|\Delta_0|^2/\sigma_\Delta^2)$.
  • If $g_\tau$ itself fluctuates spatially, through the short-length-scale interface sharpness that generates the spin-orbit term, the phase extraction would inherit that noise; the paper's small-error result assumes a deterministic $g_\tau$.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. The paper studies statistical characterization of the complex intervalley coupling Δ = Δ0 + Δδ in Si/SiGe quantum dots, where Δδ is a complex Gaussian describing alloy disorder. Using maximum likelihood estimation on valley splitting Ev alone (a Rician distribution), the authors show that the deterministic component |Δ0| is strongly overestimated in the disorder-dominated regime, even for groups of 1000 dots. They then show, via simulations, that adding knowledge of the valley phase φv substantially improves the estimates. Since φv is not directly measurable, they propose to reconstruct it from g-factor measurements, using the relation g = g0 − gτ cos φv sin 2θB (Eq. (4)), with gτ calibrated by measuring g along a loop enclosing a valley vortex, where the phase winds by ±2π and g visits its extrema. The paper derives the vortex density ρVV = (0.184/ℓ^2) exp(−|Δ0|^2/σΔ^2) and validates the phase reconstruction numerically, reporting small errors. The stated outcome is an experimentally implementable protocol for characterizing valley coupling statistics.

Significance. If the central claim holds, the paper provides a practical characterization tool for Si/SiGe spin qubits, where disorder-dominated valley physics is a known obstacle. The MLE overestimation result is well supported by explicit simulations for group sizes up to N = 1000, and the vortex-density derivation in the Supplementary Materials is careful and consistent with the simulated landscape. The g-factor valley-phase mapping is an original and potentially useful proposal. However, the advertised statistical improvement is demonstrated for the full complex Δ, whereas the proposed g-factor protocol supplies only cos φv; the censored-likelihood analysis needed to close this gap is absent. Because that gap is directly load-bearing for the central practical claim, the paper needs additional work before the protocol can be regarded as established.

major comments (3)
  1. [Eq. (4) and Fig. 1] The observable supplied by the g-factor protocol is not the signed valley phase φv but its cosine: inverting Eq. (4) determines only |φv|, because gτ and g0 are real, so the sign of sin φv is undetermined. The salmon-color maximum-likelihood demonstration in Fig. 1 is generated from the full complex Δ distribution of Eq. (1), which encodes the signed phase. The likelihood appropriate to the actually measurable data, f_cens(|Δ|, cos φv) = Σ_{s=±1} f(|Δ|(cos φv + i s √(1−cos^2 φv)); |Δ0|, φ0, σΔ), is never analyzed in the main text or the Supplementary Materials. The paper should add a maximum-likelihood study using this censored observable and report whether |Δ0|/σΔ and φ0 remain identifiable; at minimum, the statement that g-factor measurements deliver the statistical improvement shown by the salmon data is currently unsupported.
  2. [Fig. 2(g) and Eq. (4) validation] The numerical validation of the g-to-phase inversion in Fig. 2(g) uses the same effective-mass model that produced Eq. (4) through Ref. [29], and it quantifies only the error in the absolute phase, |φv^(g)| − |φv|. This checks the size of higher-order disorder corrections but does not independently test the validity of Eq. (4) and does not address the sign ambiguity of φv. An independent benchmark, for example computing g and the valley phase from an sp3d5s* tight-binding model for the same disorder realizations, would materially strengthen the claim that Eq. (4) can be inverted in real devices.
  3. [Using valley vortices to determine gτ] The procedure for locating a valley vortex by following a curve of constant g is stated with the dichotomy that the curve either passes through a vortex or forms a closed loop. Since g depends on cos φv, its level sets can in principle have multiple components, saddle points, or open ends, so this dichotomy is not immediate from Eq. (4) alone. The claim is not load-bearing for the statistical argument, but a brief numerical demonstration would substantially clarify the proposed experimental protocol.
minor comments (3)
  1. [Eq. (2)] The sentence 'Since Ev = 2|Δ| is much easier to measure that Δ' contains a typo: 'that' should be 'than'.
  2. [Eq. (5)] The main text defines ℓ = √(ℏ/(mt ωt)) just above Eq. (5), while the Supplementary Materials use ℓt for the same quantity; the notation should be harmonized.
  3. [Supplementary Materials S.2] The statement that a non-zero φ0 can be removed by a global redefinition of the valley degrees of freedom is correct for the Δ distribution but should be reconciled with the g-factor protocol, where gτ is assumed real and positive; the phase convention used for φv extracted from g should be stated explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the statistical simulations are self-contained, and the g-factor–valley-phase relation is a derived model result rather than a fitted or self-referential input.

full rationale

The paper's statistical claims are generated by explicitly stated forward models: Ev samples are drawn from the Rician distribution in Eq. (2), and maximum-likelihood fits are then evaluated against those same generative distributions. This is a self-contained Monte Carlo analysis, not a case of a fitted parameter being renamed as a prediction. The salmon-color improvement uses the full complex Δ distribution in Eq. (1), which is a richer observable than Ev; that is a statistical consistency demonstration, and its information content follows from the model rather than being imported by circular reasoning. The central physical bridge, Eq. (4), is presented as a review of Ref. [29], but the supplement re-derives it from a stated effective-mass Hamiltonian via a second-order Schrieffer-Wolff transformation, so the paper does not rely solely on a self-citation for the relation. The numerical validation in Fig. 2(g) compares the inversion of Eq. (4) against a direct eigenvalue solution within the same model, which is a self-consistency check rather than an independent experimental benchmark; however, it is not a logical reduction of the prediction to its input. The valley-vortex calibration argument is topological and parameter-free, and the vortex density in Eq. (5) is derived analytically and then checked against a simulated landscape. A genuine caveat is that inverting Eq. (4) yields only cos ϕ_v, and the paper's own error metric uses |ϕ_v| values, whereas the Fig. 1 improvement is demonstrated for the full signed phase; this is a gap between the proposed observable and the demonstrated statistical benefit, but it is a missing analysis or correctness limitation, not circularity by construction. Overall, no claimed derivation is equivalent to its own inputs by definition or by fit.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

No new physical entities are introduced: the valley vortex is a zero of the existing valley splitting field, a topological feature rather than a new object. All assumptions are modeling choices about the valley coupling statistics and the effective Hamiltonian.

free parameters (1)
  • Representative device parameters = nbar=0.3, W=0.8 nm, Fz=5 mV/nm, hbar*omega_t=2 meV
    Chosen by hand for the numerical demonstration; quantitative predictions such as rho_VV and g_tau depend on them, while the qualitative claims do not.
assumptions (5)
  • domain assumption The complex inter-valley coupling Delta is a complex Gaussian random field with uncorrelated real and imaginary parts and covariance exp(-|r1-r2|^2/(2 l_t^2)) (Eq. (1) and SM S.11-S.12).
    Imported from Ref. [13] by the same group; it underlies the MLE analysis, the valley phase distribution, and the vortex density derivation.
  • domain assumption The effective Hamiltonian of Eq. (3) and the resulting relation g = g0 - g_tau cos(phi_v) sin(2 theta_B) (Eq. (4)) accurately describe the physics, with g_tau deterministic and spatially uniform.
    Cited to companion paper Ref. [29] by the same authors; the entire proposed g-factor mapping rests on this relation.
  • standard math The lines Re[Delta]=0 and Im[Delta]=0 are smooth and effectively straight on the scale of the small triangles used to derive the vortex density.
    Asymptotic small-size limit used in SM S.4; the numerical integral for the crossing probability Pint is evaluated for straight line segments.
  • domain assumption Sampled dots are statistically independent in the MLE analysis.
    The simulations sample 'a statistically independent set of quantum dots'; spatial correlations between dots are not considered.
  • domain assumption Higher-order alloy-disorder corrections to g_tau are negligible, entering only at third order in perturbation theory.
    Argued in SM S.6 and tested numerically in Fig. 2(g), but only within the same effective-mass model.

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Cite this review

Pith. "Pith review of Statistical characterization of valley coupling in Si/SiGe quantum dots via $g$-factor measurements near a valley vortex." pith.science (2026). https://pith.science/paper/EUF3FZER

@misc{pith2026250705160,
  author       = {Pith},
  title        = {Pith review of: Statistical characterization of valley coupling in Si/SiGe quantum dots via $g$-factor measurements near a valley vortex},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EUF3FZER}},
  note         = {Machine review of arXiv:2507.05160}
}
abstract

The presence of low-energy valley excitations in Si/SiGe heterostructures often causes spin qubits to fail. It is therefore important to develop robust protocols for characterizing the valley coupling. Here, we show that realistically sized samplings of valley energy distributions tend to dramatically overestimate the average valley coupling. But we find that knowledge of the valley phase, in addition to the valley splitting, can significantly improve our estimates. Motivated by this understanding, we propose a novel method to probe the valley phase landscape across the quantum well using simple $g$-factor measurements. An important calibration step in this procedure is to measure $g$ in a loop enclosing a valley vortex, where the valley phase winds by $\pm 2\pi$ around a zero of the valley splitting. This proposal establishes an important new tool for probing spin qubits, and it can be implemented in current experiments.

Figures

Figures reproduced from arXiv: 2507.05160 by the authors.

Figure 1
Figure 1. FIG. 1. Statistical characterization of valley splitting distributions. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Probing the valley phase using [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 3 Pith papers

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

  1. Complete measurement of tunnel- and valley-coupling parameters in a silicon double quantum dot

    quant-ph 2026-07 conditional novelty 7.0 of 10

    Researchers extracted the full set of tunnel-coupling magnitudes and valley phases in a Si/SiGe double quantum dot by measuring all four anticrossing gaps and fitting a four-level model.

  2. Mapping g-factors and complex intervalley coupling in Si/SiGe by conveyor-mode shuttling

    quant-ph 2026-03 conditional novelty 7.0 of 10

    Entangled-pair conveyor shuttling yields a 2D g-factor map of Si/SiGe quantum dots with two anticorrelated valley components, enabling the first lateral reconstruction of the complex intervalley coupling.

  3. Confinement drives valley splitting above 4K in buried silicon quantum wells

    cond-mat.mes-hall 2026-07 accept novelty 6.0 of 10

    Tight electrostatic confinement drives single-electron valley splitting above 4 K (up to 0.76 meV) in buried silicon quantum wells, with a linear coefficient ~0.22 versus orbital energy in three of four dots.

Reference graph

Works this paper leans on

36 extracted references · 31 canonical work pages · cited by 3 Pith papers

  1. [29]

    B. D. Woods, M. P. Losert, R. Joynt, and M. Friesen, g-factor theory of Si/SiGe quantum dots: spin-valley and giant renor- malization effects, arXiv:2412.19795 (2025)

    arXiv 2025

  2. [1]

    Burkard, T

    G. Burkard, T. D. Ladd, A. Pan, J. M. Nichol, and J. R. Petta, Semiconductor spin qubits, Rev. Mod. Phys.95, 025003 (2023)

    2023

  3. [2]

    X. Xue, M. Russ, N. Samkharadze, B. Undseth, A. Sammak, G. Scappucci, and L. M. K. Vandersypen, Quantum logic with spin qubits crossing the surface code threshold, Nature601, 343 (2022)

    2022

  4. [3]

    A. R. Mills, C. R. Guinn, M. J. Gullans, A. J. Sigillito, M. M. Feldman, E. Nielsen, and J. R. Petta, Two-qubit silicon quan- tum processor with operation fidelity exceeding 99%, Science Advances 8, eabn5130 (2022)

    work page 2022

  5. [4]

    Noiri, K

    A. Noiri, K. Takeda, T. Nakajima, T. Kobayashi, A. Sammak, G. Scappucci, and S. Tarucha, Fast universal quantum gate above the fault-tolerance threshold in silicon, Nature 601, 338 (2022)

    work page 2022

  6. [5]

    S. G. J. Philips, M. T. M ˛ adzik, S. V . Amitonov, S. L. de Snoo, M. Russ, N. Kalhor, C. V olk, W. I. L. Lawrie, D. Brousse, L. Tryputen, B. P. Wuetz, A. Sammak, M. Veldhorst, G. Scap- pucci, and L. M. K. Vandersypen, Universal control of a six- qubit quantum processor in silicon, Nature 609, 919 (2022)

    work page 2022

  7. [6]

    F. A. Zwanenburg, A. S. Dzurak, A. Morello, M. Y . Simmons, L. C. L. Hollenberg, G. Klimeck, S. Rogge, S. N. Coppersmith, and M. A. Eriksson, Silicon quantum electronics, Rev. Mod. Phys. 85, 961 (2013)

    work page 2013

  8. [7]

    Buterakos and S

    D. Buterakos and S. Das Sarma, Spin-valley qubit dynamics in exchange-coupled silicon quantum dots, PRX Quantum 2, 040358 (2021)

    work page 2021

Show all 36 references
  1. [8]

    J. D. Teske, F. Butt, P. Cerfontaine, G. Burkard, and H. Bluhm, Flopping-mode electron dipole spin resonance in the strong- driving regime, Phys. Rev. B107, 035302 (2023)

    2023

  2. [9]

    Langrock, J

    V . Langrock, J. A. Krzywda, N. Focke, I. Seidler, L. R. Schreiber, and L. Cywi ´nski, Blueprint of a scalable spin qubit shuttle device for coherent mid-range qubit transfer in disor- dered Si/SiGe/SiO2, PRX Quantum 4, 020305 (2023)

    2023

  3. [10]

    M. L. V . Tagliaferri, P. L. Bavdaz, W. Huang, A. S. Dzurak, D. Culcer, and M. Veldhorst, Impact of valley phase and split- ting on readout of silicon spin qubits, Phys. Rev. B 97, 245412 (2018)

    2018

  4. [11]

    A. J. Weinstein, M. D. Reed, A. M. Jones, R. W. Andrews, D. Barnes, J. Z. Blumoff, L. E. Euliss, K. Eng, B. H. Fong, S. D. Ha, D. R. Hulbert, C. A. C. Jackson, M. Jura, T. E. Keat- ing, J. Kerckhoff, A. A. Kiselev, J. Matten, G. Sabbir, A. Smith, J. Wright, M. T. Rakher, T. D....

    2023

  5. [12]

    B. P. Wuetz, M. P. Losert, S. Koelling, L. E. A. Stehouwer, A.-M. J. Zwerver, S. G. J. Philips, M. T. M ˛ adzik, X. Xue, G. Zheng, M. Lodari, S. V . Amitonov, N. Samkharadze, A. Sammak, L. M. K. Vandersypen, R. Rahman, S. N. Copper- smith, O. Moutanabbir, M. Friesen, and G. Sc...

    2022

  6. [13]

    M. P. Losert, M. A. Eriksson, R. Joynt, R. Rahman, G. Scap- pucci, S. N. Coppersmith, and M. Friesen, Practical strategies for enhancing the valley splitting in Si/SiGe quantum wells, Phys. Rev. B 108, 125405 (2023)

    2023

  7. [14]

    J. R. F. Lima and G. Burkard, Valley splitting depending on the size and location of a silicon quantum dot, Phys. Rev. Mater.8, 036202 (2024)

    2024

  8. [15]

    J. R. F. Lima and G. Burkard, Interface and electromagnetic effects in the valley splitting of Si quantum dots, Materials for Quantum Technology 3, 025004 (2023)

    2023

  9. [16]

    L. F. Peña, J. C. Koepke, J. H. Dycus, A. Mounce, A. D. Baczewski, N. T. Jacobson, and E. Bussmann, Modeling Si/SiGe quantum dot variability induced by interface disorder reconstructed from multiperspective microscopy, npj Quantum Inf 10, 33 (2024)

    2024

  10. [17]

    D. D. Esposti, L. E. A. Stehouwer, Önder Gül, N. Samkharadze, C. Déprez, M. Meyer, I. N. Meijer, L. Tryputen, S. Karwal, M. Botifoll, J. Arbiol, S. V . Amitonov, L. M. K. Vandersypen, A. Sammak, M. Veldhorst, and G. Scappucci, Low disorder and high valley splitting in silicon,...

    2024

  11. [18]

    V olmer, T

    M. V olmer, T. Struck, A. Sala, B. Chen, M. Oberländer, T. Of- fermann, R. Xue, L. Visser, J.-S. Tu, S. Trellenkamp, Łukasz Cywi´nski, H. Bluhm, and L. R. Schreiber, Mapping of valley- splitting by conveyor-mode spin-coherent electron shuttling, arXiv.2312.17694 (2023)

    2023 arXiv

  12. [19]

    Friesen, S

    M. Friesen, S. Chutia, C. Tahan, and S. N. Coppersmith, Valley splitting theory of SiGe/Si/SiGe quantum wells, Phys. Rev. B 75, 115318 (2007)

    2007

  13. [20]

    Tariq and X

    B. Tariq and X. Hu, Impact of the valley orbit coupling on ex- change gate for spin qubits in silicon, npj Quantum Information 8, 53 (2022)

    2022

  14. [21]

    Zhao and X

    X. Zhao and X. Hu, Measurement of tunnel coupling in a Si double quantum dot based on charge sensing, Phys. Rev. Appl. 17, 064043 (2022)

    2022

  15. [22]

    Borjans, X

    F. Borjans, X. Zhang, X. Mi, G. Cheng, N. Yao, C. Jackson, L. Edge, and J. Petta, Probing the variation of the intervalley tunnel coupling in a silicon triple quantum dot, PRX Quantum 2, 020309 (2021)

    2021

  16. [23]

    E. H. Chen, K. Raach, A. Pan, A. A. Kiselev, E. Acuna, J. Z. Blumoff, T. Brecht, M. D. Choi, W. Ha, D. R. Hulbert, M. P. Jura, T. E. Keating, R. Noah, B. Sun, B. J. Thomas, M. G. Borselli, C. Jackson, M. T. Rakher, and R. S. Ross, Detun- ing axis pulsed spectroscopy of valley-...

    2021

  17. [24]

    J. C. Marcks, E. Eagen, E. C. Brann, M. P. Losert, T. Oh, J. Reily, C. S. Wang, D. Keith, F. A. Mohiyaddin, F. Luthi, M. J. Curry, J. Zhang, F. J. Heremans, M. Friesen, and M. A. Eriks- son, Valley splitting correlations across a silicon quantum well, arXiv:2504.12455 (2025)

    2025

  18. [25]

    Struck, M

    T. Struck, M. V olmer, L. Visser, T. Offermann, R. Xue, J.-S. Tu, S. Trellenkamp, Łukasz Cywi ´nski, H. Bluhm, and L. R. Schreiber, Spin-EPR-pair separation by conveyor-mode single electron shuttling in Si/SiGe, Nature Communications15, 1325 (2024)

    2024

  19. [26]

    See Supplementary Materials for additional technical details and statistical results

  20. [27]

    We also observe an enhancement of counts with | ˆ∆0| = 0 , caused by | ˆ∆0| = 0 being an endpoint of the fitting procedure

  21. [28]

    Note that there is no enhancement of counts at | ˆ∆0| = 0 in this case, because this is no longer an endpoint in the fitting 6 procedure

  22. [30]

    More generally, gτ ∈ C; however, the zero of the valley phase can be redefined to give gτ ∈ R+, without loss of generality

  23. [31]

    B. D. Woods, M. A. Eriksson, R. Joynt, and M. Friesen, Spin- orbit enhancement in Si/SiGe heterostructures with oscillating Ge concentration, Phys. Rev. B 107, 035418 (2023)

    2023

  24. [32]

    2(b), such curves are obtained by starting at a random point and following a path that remains at the same color as the starting point

    In Fig. 2(b), such curves are obtained by starting at a random point and following a path that remains at the same color as the starting point. 7 Supplementary Materials: Statistical characterization of valley coupling in Si/SiGe quantum dots via g-factor measurements near a v...

  25. [34]

    = fR′ 1 (R′ 1)fR′ 2 (R′

  26. [35]

    (S.21) The integral in Eq

    ≈ 1 2πσ 2 ∆ 2ℓt s exp − (R′ 1)2 2σ2 ∆ exp − 2ℓ2 t s2 (R′ 2)2 2σ2 ∆ . (S.21) The integral in Eq. (S.17) then reduces to P R • − •(s) = 1 πσ 2 ∆ 2ℓt s ˆ ∞ 0 exp − 2ℓ2 t s2 (R′ 2)2 2σ2 ∆ ˆ − √ 2∆0+R′ 2 − √ 2∆0−R′ 2 exp − (R′ 1)2 2σ2 ∆ dR′ 1 ! dR′ 2 + O(s2). (S.22) 11 Here, the in...

  27. [36]

    (S.22) is strongly suppressed

    Only small values of R′ 2 are relevant in the limit s → 0, however, because the first exponential factor in Eq. (S.22) is strongly suppressed. The inner R′ 1 integral can therefore be expanded to first order in R′ 2, giving ˆ − √ 2∆0+R′ 2 − √ 2∆0−R′ 2 exp − (R′ 1)2 2σ2 ∆ dR′ 1...

  28. [110]

    crystallographic axis) to maximize the size of g-factor fluctuations. In Eqs. (3) and (4), we note that the amplitude and phase of the valley coupling vary in the plane of the quantum well due to alloy disorder, ∆ = ∆( x, y), yielding valley splitting landscapes like the one s...

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