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A silicon double quantum dot's complete low-energy Hamiltonian — including the complex phases that control tunneling between valley states — can be extracted from the four anticrossing gaps in a single delta-axis spectroscopy measurement.

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2026-08-02 07:28 UTC pith:TTQ2OG4H

load-bearing objection First simultaneous measurement of all four valley anticrossings in a Si/SiGe double dot, with careful fits and a real but acknowledged parameter ambiguity resolved by the authors' own disorder model. the 3 major comments →

arxiv 2607.09638 v2 pith:TTQ2OG4H submitted 2026-07-10 quant-ph cond-mat.mes-hall

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

Daniel J. King , Minyoung Kim , J. Reily , Jonathan C. Marcks , Mark Friesen , Benjamin D. Woods , M. A. Eriksson This is my paper
classification quant-ph cond-mat.mes-hall
keywords silicon quantum dotsvalley phasestunnel couplingintervalley couplingdelta-axis spectroscopySi/SiGe heterostructuredouble quantum dotalloy disorder
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.

In a silicon double quantum dot, tunneling between the two dots is not a single number: it depends on a valley-conserving tunnel coupling, intradot valley-mixing couplings, and an interdot-intervalley coupling, each with its own complex phase. The paper claims to measure all of these — including the phases — in one device by using delta-axis spectroscopy to resolve all four anticrossings between ground and excited valley states. This matters because prior work measured only three of the four anticrossings and typically set the interdot-intervalley coupling to zero; the paper finds it non-negligible and essential for fitting the data. If the extraction is correct, the anticrossing gaps themselves become a direct readout of the local atomic alloy disorder, and the valley phases can be tracked as gate voltages move the dots.

Core claim

The paper claims that in a Si/SiGe double quantum dot, the complete single-electron low-energy Hamiltonian — the valley-conserving tunnel coupling tc, the two intradot intervalley couplings ΔL and ΔR, and the interdot-intervalley coupling ΔRL, including the complex phases — can be extracted from a single set of delta-axis spectroscopy measurements. The essential experimental step is resolving all four anticrossings between ground and excited valley states; each anticrossing gap encodes a combination of the four effective tunnel couplings, and the four gaps together suffice to invert the model and obtain the magnitudes and phases. The paper also argues that ΔRL is non-negligible (|ΔRL|/tc ≈ 0

What carries the argument

The engine of the measurement is delta-axis spectroscopy (DAXS): square-wave voltage pulses along the δ axis shift both quantum dot potentials together, so an energy level crossing the reservoir Fermi level produces a detectable charge-sensor signal, mapping the energy dispersion directly. The central identity is the unitary transformation between the ±z-valley basis (with six parameters) and the ground/excited-valley basis (with four effective tunnel couplings and two valley splittings); the four anticrossing gaps E_gg, E_ge, E_eg, E_ee are approximately 2|t̃_μν|, and closed-form inversion equations recover tc, |ΔRL|, Δϕ, and ΔϕRL. The paper combines high-resolution scans at each anticrossi

Load-bearing premise

The results rest on assigning the measured spectra to the odd sign convention (s = -1) and assuming the ordinary tunnel coupling exceeds the interdot-intervalley coupling; the paper's own supplementary analysis shows the even-sign fits are almost equally good and that the two couplings can be interchanged by a gauge choice, so the assignment leans on the paper's alloy-disorder statistics model rather than on the spectra alone.

What would settle it

Take DAXS spectra over a detuning window several times wider than the ones used here: the s = +1 and s = -1 fits, nearly degenerate in the present window, predict visibly different spectra at large detuning, so the sign convention would be decided by data rather than by the disorder model. If the wider-window scan favors s = +1, or an independent measurement of a single anticrossing gap disagrees with the fitted value, the reported tc, |ΔRL| and valley phases would need revision.

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

If this is right

  • The four anticrossing gaps of any Si/SiGe double dot now have a known information content: their ratios determine the valley phase difference and the interdot-intervalley coupling ratio without full fitting.
  • Models that omit the interdot-intervalley coupling ΔRL will systematically mis-estimate tunnel couplings and gap ratios in devices like this one, where |ΔRL|/tc ≈ 0.2.
  • Valley phases are not fixed constants: they shift with screening-gate voltage (as dot positions move) and with barrier-gate voltage (as the barrier overlap region changes), so phase variations must be tracked device-by-device and gate-tuning-by-gate-tuning.
  • The paper's extraction provides a way to map local alloy disorder, since the phases depend on the atomic Ge distribution sampled by the dot wavefunctions.
  • The ratio distribution |ΔRL|/|Δi| is independent of disorder strength, so a single measured ratio already tests the wavefunction model.

Where Pith is reading between the lines

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

  • The same four-gap inversion could be applied to simulated spectra for other valley materials, such as Ge/SiGe or Si MOS, to test whether an interdot-intervalley coupling of this relative size is generic or specific to the alloy profile here.
  • A wider-detuning DAXS protocol could remove the sign-convention ambiguity entirely, making the parity sector a measured outcome rather than a statistical prior; the paper itself acknowledges that the two sign conventions fit nearly equally well in the current window.
  • Because g-factors and spin-orbit coupling depend on valley phases, a two-dimensional phase map across a quantum well could become a design tool for placing qubits at favorable disorder sites — the paper notes this as a next step but does not implement it.
  • The rapid variation of ΔϕRL with barrier-gate voltage implies that pulsed shuttling, which modulates the barrier dynamically, will also modulate the valley content of the tunnel coupling; this connection is not explored in the paper.

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 / 4 minor

Summary. The paper reports a set of DAXS (delta-axis spectroscopy) measurements on a Si/SiGe double quantum dot that resolve, for the first time, all four single-electron anticrossing gaps between the ground and first-excited valley states of the two dots. The authors fit the measured energy-level positions to a four-level Hamiltonian containing valley-conserving tunnel coupling tc, intradot intervalley couplings ΔL, ΔR, an interdot-intervalley coupling ΔRL, and two relative valley phases Δϕ and Δϕ_RL. They find that ΔRL is non-negligible over the whole measured range and that the extracted phases evolve with dot position and barrier-gate voltage. They support the interpretation with an effective-mass alloy-disorder model that predicts distributions of |ΔRL|/|Δi| and with simulated DAXS spectra that validate the fitting procedure on 128 disorder realizations. The central claim is that the complex valley phases and all tunnel- and valley-coupling parameters have been completely measured.

Significance. If the extracted parameters are indeed uniquely determined, this is a significant advance: valley phases have previously been difficult to access, and the paper provides a new experimental route to characterizing the full low-energy Hamiltonian of Si/SiGe double quantum dots, including the interdot-intervalley coupling ΔRL that is usually neglected. The experimental work is careful: 24 energy levels are extracted per tuning, each high-resolution scan is repeated 10 times, uncertainties are propagated by Monte Carlo, and the fitting method is validated against 128 simulated disorder realizations with median magnitude errors of order 2% and phase errors of order 0.1 rad. The paper also demonstrates, via the gap-ratio relations, that the four anticrossing gaps contain more information than previously exploited. The main weakness is that the central parameter set is not uniquely determined by the DAXS spectra alone; the authors resolve this degeneracy with a model prior, and the reported 'complete measurement' is therefore conditional on that prior.

major comments (3)
  1. [SM S1D, Eqs. (S16)–(S19), Fig. S1] The extracted parameter set is not uniquely determined by the data. SM S1D states that the two parity sectors s=+1 and s=−1 fit the DAXS spectra nearly equally well, and that gauge transformations produce four equivalent parameter sets, including sets with tc and |ΔRL| interchanged. The authors resolve these degeneracies by assuming s=−1 and tc>|ΔRL|, justified by comparing the resulting |ΔRL|/|Δi| ratios to distributions from their own effective-mass alloy-disorder model. This is a model prior, not a measurement. Fig. S1 shows that in the s=+1 sector tc and |ΔRL| are nearly equal and |ΔRL| is roughly twice the s=−1 value, so the reported phases and coupling magnitudes would change substantially under the alternative sector. Because the headline claim is a 'complete measurement,' the manuscript should either provide an independent experimental constraint on the sector/ordering or explici
  2. [Main text Fig. 3j and SM S1D] The consistency test in Fig. 3j is not fully independent. The same |ΔRL|/|Δi| distributions from the authors' effective-mass model are used in SM S1D to select the s=−1 parity sector and the tc>|ΔRL| ordering; Fig. 3j then shows that the experimental ratios lie near the mode of those same distributions. This agreement is partly by construction. Similarly, the gap-ratio comparisons in Figs. 2f and 2h compare quantities derived from the same four-level Hamiltonian fit; good agreement is expected from the algebraic identities (S7)–(S10) and does not provide an independent confirmation of the phase extraction. The authors should identify at least one prediction made before or independently of the fit that is tested against the data.
  3. [SM S4D, Eq. (S76)] The method uncertainty, which the authors state exceeds the measurement uncertainty by factors of 3–10, is obtained from noiseless simulations of the same effective-mass model used to break the parity-sector and tc/|ΔRL| degeneracies. Consequently, the reported error bars do not include uncertainty in the model's wavefunction shapes or disorder statistics. Since the central values themselves are selected using this model, the stated uncertainties are likely to be underestimated if the effective-mass model is inaccurate. At minimum, the model dependence of the central values should be listed as a separate systematic uncertainty, and the sensitivity of the reported phases to reasonable variations of the model parameters should be quantified.
minor comments (4)
  1. [Main text, 'SYSTEM EVOLUTION WITH BARRIER-GATE VOLTAGE'] The sentence 'has a large affect on the wavefunction overlap' should read 'has a large effect'.
  2. [Fig. 2h caption] The quantity sqrt(F−/F+) is defined only in the main text; providing the definition in the caption would improve readability.
  3. [SM S1D, Eq. (S18)] The four parameter sets are useful, but the notation '2Δϕ_RL − Δϕ' is easy to misread; a brief derivation or a table with the sign assignments would clarify this ambiguity.
  4. [Abstract and Conclusion] The phrase 'complete measurement' should be tempered in light of the degeneracy discussion in SM S1D; the main text should explicitly state that the reported parameters are obtained within a chosen parity sector and under the tc>|ΔRL| assumption, rather than leaving this caveat only in the Supplementary Materials.

Circularity Check

2 steps flagged

Central parameter extraction is benchmarked and not circular; however the gap-ratio 'checks' (Figs. 2f,h) are algebraic identities of the same four-level fit, and the parity/ordering resolution is model-dependent rather than data-determined.

specific steps
  1. fitted input called prediction [Fig. 2f and main text 'SYSTEM EVOLUTION WITH SCREENING-GATE VOLTAGE...' (also SM S2C)]
    "In Fig. 2f, we plot both of these quantities, tan(∆ϕ/2) and (Ege + Eeg)/(Egg + Eee). Here, ∆ϕ is extracted from a rigorous fit to the four-level model, and Eµν are the gap ratios shown in Fig. 2d, multiplied by Egg. ... The remarkable agreement ... demonstrates that ∆ϕ is encoded in the four gaps."

    Both plotted quantities are computed from the same fitted Hamiltonian parameters: the gaps Eµν are obtained by diagonalizing the fitted four-level model (SM S2C), and ∆ϕ is the inversion of the same fitted ~tµν via Eqs. (S11)-(S14). In the low-tunneling limit Eµν≈2|~tµν|, and Eqs. (S7)-(S10) make the displayed ratio algebraically equal to tan(∆ϕ/2). The agreement is therefore an identity of the fit, not an independent test that the phases are 'encoded in the gaps'.

  2. fitted input called prediction [Fig. 2h and main text 'SYSTEM EVOLUTION WITH SCREENING-GATE VOLTAGE...']
    "In Fig. 2h, we plot fitting results for |∆RL|/tc and the dimensionless gap parameter p F−/F+. Similar to Fig. 2f, we again observe excellent agreement between the two quantities, with small differences arising from interactions between the anticrossings."

    The quantity |∆RL|/tc is obtained by inverting the fitted ~tµν through Eqs. (S11)-(S12), while the gap parameter sqrt(F−/F+) is built from the Eµν gaps calculated from the same fitted ~tµν. With Eµν≈2|~tµν|, the two expressions coincide by construction. Thus the displayed agreement is a self-consistency relation of the four-level fit, not an independent confirmation that |∆RL| is determined by the raw gap data.

full rationale

The core extraction is not circular: 24 energy levels sampled from repeated high-resolution DAXS scans are fit to a four-level Hamiltonian, and the six reported parameters are then obtained by closed-form inversion. SM S4 benchmarks this fitting on noiseless device simulations with known ground-truth parameters (Figs. S5-S6), which is an independent, parameter-free check that does not presuppose the experimental values. The main non-uniqueness is explicitly acknowledged in SM S1D: the s=±1 parity sectors give nearly equally good fits, and a gauge transformation swaps tc and |∆RL|; the paper resolves both with the authors' own effective-mass alloy-disorder model. That resolution is model-dependent (and makes the 'complete measurement' claim conditional on the model), but it is not circular, since the model's assumptions (Ge profile, envelope shapes, disorder statistics) are stated and do not contain the fitted results. The genuinely circular element is smaller: the Figs. 2f and 2h 'agreement' demonstrations compare two derived quantities of the same fitted Hamiltonian, so they reduce to algebraic identities in the low-tunneling limit. These checks are not load-bearing for the parameter extraction itself, which is why the score is moderate rather than high.

Axiom & Free-Parameter Ledger

6 free parameters · 5 axioms · 0 invented entities

The model and fitting procedure rely on a four-level truncation, the mapping between DAXS signal and eigenenergies, an effective-mass disorder model used to pick the parity sector, and the gauge assumption tc>|ΔRL|. The six output parameters are fitted to data and are the measurement itself, not ad hoc constants.

free parameters (6)
  • tc (valley-conserving tunnel coupling) = ≈56–99 μeV (Fig. 3i)
    Fitted to DAXS high-resolution energy levels; central output, appears in Eqs. (S11)-(S14).
  • |ΔL| (left intradot intervalley coupling) = Varies by factor ~2 over VSG range (Fig. 2e)
    Fitted; determines left valley splitting Ev,L=2|ΔL|.
  • |ΔR| (right intradot intervalley coupling) = Comparable to |ΔL|, changes less with VSG (Fig. 2e)
    Fitted; determines right valley splitting.
  • |ΔRL| (interdot-intervalley coupling) = ≈0.17–0.25 tc (Figs. 2h, 3i)
    Fitted; new parameter central to the claim that ΔRL is essential.
  • Δϕ (=ϕL−ϕR) = ≈5π/16 to π/2 (Fig. 2g)
    Fitted valley phase difference; sign assumed positive by gauge convention.
  • ΔϕRL (=ϕRL−ϕR) = Varies significantly with VSG and VBC (Figs. 2g, 3h)
    Fitted interdot-intervalley phase; most affected by parity ambiguity.
axioms (5)
  • domain assumption The low-energy spectrum is well described by four levels: one orbital per dot times two valley states; higher orbital/valley states are omitted.
    Invoked throughout, Eq. (S1). Higher bands are visible in DAXS; their effect is bounded but not included directly in the fit (SM S4).
  • domain assumption DAXS peak positions map directly to the DQD eigenenergies as a function of detuning, with a small potential-shape deformation captured only as method uncertainty.
    Used to justify using 24 energy levels as data; SM S4 quantifies deviations.
  • domain assumption Alloy disorder statistics are described by an effective-mass Hamiltonian with random Ge occupancy; the real/imaginary parts of intervalley couplings are circular Gaussians with covariance given by envelope overlap integrals.
    Used for theoretical distributions in Figs. 3j/S1 and for method uncertainty; SM S3.
  • ad hoc to paper Gauge/parity sector s=-1 is the true sector, and tc>|ΔRL|.
    Spectra fit both s=±1 nearly equally well; s=-1 is chosen because s=1 puts measured ratios in low-probability tails of the authors' own disorder model (SM S1D, Fig. S1).
  • domain assumption With no magnetic field and weak spin-orbit coupling, spin can be ignored and the orbital envelopes can be chosen real, giving ΔRL=ΔLR.
    SM S4B; used to reduce the number of Hamiltonian parameters.

pith-pipeline@v1.3.0-alltime-deepseek · 30127 in / 13269 out tokens · 141893 ms · 2026-08-02T07:28:33.844254+00:00 · methodology

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read the original abstract

Tunneling is essential in the initialization, measurement, and control of quantum dot qubits. In silicon, such tunneling connects not only the qubit states but also valley minima in the conduction band on opposite sides of the Brillouin zone, with large consequences for the quantum dot behavior. Here we present a full characterization of the intravalley and intervalley tunnel couplings, including their complex phases -- the valley phases. These phases are shown to control measurable parameters, including the ratios of the gaps at anticrossings between quantum states of a double quantum dot. The valley phases themselves evolve as a function of the quantum dot gate voltages and depend on the underlying atomic structure of the quantum well. Knowledge of the valley phases completes the picture and fills a key gap in our understanding of sample-wide variations of valley couplings and the physical parameters that depend on them, including spin-orbit coupling, valley-orbit mixing, and Land\'e $g$-factors.

Figures

Figures reproduced from arXiv: 2607.09638 by Benjamin D. Woods, Daniel J. King, Jonathan C. Marcks, J. Reily, M. A. Eriksson, Mark Friesen, Minyoung Kim.

Figure 1
Figure 1. Figure 1: b. Here, a single spatial orbital (L/R) is considered in each dot. In addition, each spatial orbital has a valley degree of freedom (±z) due to the two-fold degenerate valley minima in the conduction band of the biaxially strained Si quantum well [36]. The various level cou￾plings are illustrated by arrows in Fig. 1b. These include valley-conserving tunnel coupling tc as well as intradot and interdot valle… view at source ↗
Figure 2
Figure 2. Figure 2: a the anticrossing gaps are labeled, while in Fig. 2b, the locations of the high-resolution DAXS measurements are indicated. Fig. 2c shows a typical high-resolution measurement, where the peaks are fit to derivatives of the Fermi-Dirac distribution. Following the procedure described above, for each of six values of VSG, we fit these peak locations in order to extract the Hamilto￾nian parameters, yielding t… view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

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