Resolving Andreev spin qubits in germanium-based Josephson junctions

Charles Tahan; Silas Hoffman

REVIEW 3 major objections 5 minor 1 cited by

Resolving Andreev spin qubits in germanium-based Josephson junctions

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A germanium Josephson junction's geometry can suppress its Andreev spin-qubit frequency below the thermal limit, and a narrower channel with higher filling can make it resolvable.

desk verdict A useful, honest theory paper explaining a null result in Ge Andreev spin qubits, with caveats about the assumed pairing symmetry. read the letter →

arxiv 2506.13988 v1 pith:DNJU7SRU submitted 2025-06-16 cond-mat.mes-hall quant-ph

Silas Hoffman , Charles Tahan This is my paper · ORCID

classification cond-mat.mes-hallquant-ph

keywords AndreevspinqubitgermaniumholegasJosephsonjunctionheavy-holebandFermivelocityproximity-inducedsuperconductivityfrequencytight-bindingmodel

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 argues that the apparent absence of spin-resolved Andreev states in germanium-based Josephson junctions may be a geometric artifact rather than a fundamental obstacle. For the 350 x 150 x 25 nm junction used in recent experiments, the model predicts a spin-qubit frequency of about 150 MHz at phase difference pi/2, below the 30 mK thermal scale of about 625 MHz, so the two Andreev levels cannot be resolved. Using a one-dimensional Luttinger-Kohn model and a tight-binding simulation, the authors show that narrowing the channel to Ly = 55 nm and raising the hole filling to 0.66 of the heavy-hole-light-hole gap raises the frequency to about 1.3 GHz, a resolvable value. The practical consequence is design guidance: maximize the Fermi-velocity difference between the two heavy-hole bands, choose the transverse confinement near the characteristic electric-field length, keep the superconducting length near the coherence length, minimize strain, and maximize interface transparency.

What carries the argument

The load-bearing object is the quasi-one-dimensional heavy-hole band structure of a germanium two-dimensional hole gas, confined by widths $L_y$ and $L_z$, with Rashba spin-orbit coupling. In the regime where only the heavy-hole band is filled, the model yields two right- and two left-moving bands with different Fermi velocities $v_1$ and $v_2$; this velocity difference is the origin of the nondegenerate Andreev levels. The paper defines the Andreev bound-state energy $\epsilon_j = \Delta \cos(\phi/2)\big/(1 + (L\Delta/v_{Fj})\sin(\phi/2))$ and the qubit frequency $\omega_q = |\epsilon_1 - \epsilon_2|$. The analytical result is supplemented by a tight-binding Hamiltonian that includes the full spin-3/2 Luttinger-Kohn model, Birs-Pikus strain, Zeeman coupling, and interface transparency, and it is this numerical model that produces the geometry- and filling-dependence of $\omega_q$.

What would settle it

Measure the tunnelling spectrum of a germanium Josephson junction with $L_y \approx 55\,\mathrm{nm}$, $L_z \approx 25\,\mathrm{nm}$, and filling $\bar{\mu} \approx 0.66\Delta_{HL}$ at $\phi = \pi/2$, and look for two spin-split Andreev levels separated by at least 1 GHz; a null result would contradict the model's core prediction.

Watch

Extended reading notes

Core claim

The central claim is that the Andreev spin qubit frequency in a germanium Josephson junction is controlled by the junction geometry through the difference in Fermi velocities of the two heavy-hole bands. The authors derive an analytic formula, $\epsilon_j = \Delta \cos(\phi/2)\big/(1 + (L\Delta/v_{Fj})\sin(\phi/2))$, showing that the two Andreev states are split only when the inner and outer heavy-hole bands have different velocities; the qubit frequency is $\omega_q = |\epsilon_1 - \epsilon_2|$. For an experimentally realized $350 \times 150 \times 25\,\mathrm{nm}^3$ junction with $\Delta = 50\,\mu\mathrm{eV}$ and transparency $r = 0.995$, this frequency is about 150 MHz at $\phi = \pi/2$, below the 30 mK thermal limit of about 625 MHz, explaining why spin-resolved Andreev states were not seen. The tight-binding calculation, which includes strain, magnetic field, and imperfect interfaces, predicts that a narrower channel ($L_y = 55\,\mathrm{nm}$) and higher filling ($\bar{\mu} = 0.66 \Delta_{HL}$) raise $\omega_q$ to about 1.3 GHz, making the qubit resolvable.

Load-bearing premise

The calculation assumes the superconductivity induced in germanium is a spin-3/2 singlet that pairs each heavy-hole or light-hole band with its time-reversed partner; if the real proximity pairing has a different spin structure or a heavy/light-hole asymmetry, the predicted qubit frequencies and the explanation of the null result would change.

Editorial extensions

If this is right

If the model is right, the null result of Ref. [17] is explained by geometry: at $L_x = 350\,\mathrm{nm}$, $L_y = 150\,\mathrm{nm}$, and $\Delta = 50\,\mu\mathrm{eV}$, the qubit frequency falls below the 30 mK thermal limit and cannot be resolved.
Design rules follow: $L_y$ should sit near the characteristic electric-field length (roughly 7 to 55 nm depending on field), $L_x$ near the coherence length (about 200 nm to 1 $\mu$m), filling near $0.66$ to $0.8\Delta_{HL}$, strain minimized, and interface transparency maximized.
A resolvable germanium Andreev spin qubit with $\omega_q \approx 1.3\,\mathrm{GHz}$ should be achievable without changing materials, using the experimentally observed gap and transparency of Ref. [17].
Because the mechanism is generic to two Fermi velocities, the same geometric optimization should apply to planar InAs junctions, with higher confined subbands playing the role of the light-hole band.
The predicted $g$-factor anisotropy, with $g_y \approx g_z$ near $L_y \approx L_z$ and $g_x$ small, suggests magnetic sweet spots where decoherence from nuclear noise could be minimized.

Reading between the lines

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

The paper's mechanism implies that earlier null searches for spin-resolved Andreev states may have been blinded by geometry; re-analyzing existing devices with $L_y \ll L_z$ could reveal the qubit without new materials work.
The strong geometric dependence of $\omega_q$ also means that the qubit frequency of a germanium Andreev spin qubit could serve as a sensitive local probe of confinement, strain, and filling, useful for device metrology.
The singlet-pairing assumption is the main uncertainty: if the proximity-induced pairing is not a spin-3/2 singlet, the two-level basis and the reported frequencies could shift, so a microscopic theory of the Ge/Al interface pairing would sharpen or overturn the design guidance.

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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

Colleague, here's the short version: Hoffman and Tahan argue that the reason nobody has seen Andreev spin qubits in Ge is simpler than anyone thought — the junction geometry used in the recent experiment (Ref. 17) puts the qubit frequency at ~150 MHz, below the thermal limit of a 30 mK fridge. They then show that shrinking Ly to ~55 nm and raising the filling to 0.66 of the HH-LH gap would bring it up to ~1.3 GHz. That is the new and practically useful result. The paper deserves a serious referee.

What is actually new: the application of the known two-velocity Andreev formula (Park & Yeyati) to planar Ge 2DHGs, a concrete tight-binding model built from the Luttinger-Kohn Hamiltonian, and a genuinely comprehensive scan over geometry, filling, strain, and magnetic field. The comparison to the Hinderling experiment is the best part — they reproduce the null result by fixing the transparency to r=0.995, and then give specific, actionable changes. The paper is also honest: it flags the singlet pairing assumption in Sec. IV and notes that the qubit frequency is sensitive to it.

Where it is soft: the pairing assumption is the load-bearing wall. They assume a spin-3/2 singlet in the Ge, which reduces the 8x8 Nambu problem to four 2x2 blocks and gives every reported omega_q. The authors admit this is not unique and that HH-LH pairing differences can shift or even destroy the gap. That makes the quantitative claims — 150 MHz, 1.3 GHz — conditional on a pairing structure not yet established from experiment or microscopic theory. The qualitative message that geometry can suppress the qubit frequency is probably robust, but the numbers should be read as indicative, not a roadmap. There is also a typo in Eq. (13) where four identical sine terms appear in B; the prefactor makes it look like different phases were intended. Minor, but worth fixing. No code or data is shipped, which is a pity but not disqualifying for a theory paper.

Overall: the paper does what it claims to do, within the stated assumptions. It would be valuable to experimentalists working on Ge Josephson junctions and to theorists modeling hole-based hybrids. It deserves peer review; I'd send it to a referee with instructions to press on the pairing robustness and to ask for a corrected equation. I would not cite the numbers as firm predictions, but I would cite the framework as a useful first pass. Bring it to reading group if your group works on superconducting-semiconducting hybrids; otherwise skim Sec. III.C.

Referee Report

3 major / 5 minor

Summary. The manuscript presents a theoretical study of Andreev spin qubits (ASQs) in Ge-based Josephson junctions. Starting from the Luttinger-Kohn Hamiltonian with transverse confinement, the authors construct a one-dimensional multiband model and a tight-binding model of a superconductor-normal-superconductor junction, including Rashba spin-orbit coupling, magnetic field, strain, and interface transparency. They derive an effective two-level Andreev Hamiltonian and study the qubit frequency as a function of junction geometry (Ly, Lz, Lx), filling, strain, and magnetic field. They use the model to argue that the failure to resolve spin-resolved Andreev states in a recent Ge experiment (Ref. 17) can be explained by a qubit frequency (~150 MHz) below the thermal energy (~625 MHz), and they propose design changes (Ly~55 nm, filling ~0.66 Delta_HL) that would raise the frequency to ~1.3 GHz.

Significance. If the central model is accepted, the paper provides a concrete and testable explanation for a puzzling null result and gives design guidance for Ge-based ASQs. The tight-binding model is described in sufficient detail to be reproduced, it reproduces expected qualitative features (different Fermi velocities, HH-LH gap, linear dispersion near phi=pi), and the parametric dependence on geometry and filling is computed directly from the model rather than fitted. The main quantitative conclusions, however, rest on a specific assumption about the spin structure of the proximity-induced pairing, which the authors themselves identify as non-unique; this makes the numerical values conditional rather than definitive. The paper is nevertheless valuable as a systematic study of the geometry dependence of ASQ frequencies in a hole-gas platform.

major comments (3)

[Sec. II.B and Sec. IV] The reduction of the 8x8 problem to four 2x2 blocks and the resulting Andreev energies in Eq. (11) rely on the spin-3/2 singlet pairing H_SC = i tau_y exp(i pi J_y) Delta cos(phi) - i tau_x exp(i pi J_y) Delta sin(phi), which pairs each particle solution Phi^+_{nu,e} with a specific hole solution Phi^-_{nu,h}. This pairing choice is not unique for a hole gas, and the authors concede in Sec. IV that it depends on the microscopic coupling of heavy and light holes to the superconductor and that small HH-LH pairing differences can alter the qubit frequency while large ones can destroy the gap. Since every reported omega_q, including the 150 MHz and 1.3 GHz values in Sec. III.C, is a direct output of this assumption, the central claims are conditional on a pairing structure not established by experiment or first principles. I request either a microscopic estimate of the proximity pairing matrix along the lines of Refs. 25 and 26, or an explicit sensitivity analysis over the HH/LH pairing amplitude ratio.
[Sec. III.C] The comparison to the experiment of Ref. 17 fixes the interface transparency parameter r=0.995 so that the Andreev level energies match experiment, and no sensitivity scan in r is shown. The conclusion that the null result is explained by omega_q ~ 150 MHz being below 625 MHz is partially robust because the r=1 limit with bulk Delta also gives only ~300 MHz, but the quantitative design recommendation of omega_q ~ 1.3 GHz is a single point in parameter space and needs a sensitivity analysis in both r and the pairing amplitude. Please provide omega_q as a function of r and state how the inferred resolvability threshold depends on this fit parameter.
[Eq. (13)] In Eq. (13), the expression for B contains four identical sine terms [3 sin(theta_1+theta_2+pi/3) + sin(theta_1+theta_2+pi/3) + sin(theta_1+theta_2+pi/3) + sin(theta_1+theta_2+pi/3)]/4, which reduces to (3/2) sin(theta_1+theta_2+pi/3) and makes the displayed magnetic-field matrix element incorrect as written. This affects the analytic model used for the magnetic-field dependence in Fig. 2(b). Although the numerical tight-binding results in Sec. III do not use Eq. (13), the analytical derivation is presented as supporting intuition and should be corrected.

minor comments (5)

[Eq. (6)] As printed, Eq. (6) appears to contain a product (k^2/m_s) tilde_J_x^2 tilde_J_y^2 and a bare -omega_z, whereas consistency with Eq. (3) requires separate terms -(k^2/m_s) tilde_J_x^2, -omega_z tilde_J_y^2, and -omega_y tilde_J_z^2.
[Sec. II.B] There are small grammatical errors: "two and two-by-two Hamiltonians" should be "two-by-two Hamiltonians", and "are are separated" should be "are separated".
[Fig. 5 and Fig. 7] The symbol E_s is used for strain in Sec. III.A and Fig. 7, but Fig. 5's inset appears to use "Es" for the characteristic electric-field length; these should use distinct notation to avoid confusion.
[Sec. III.A and Sec. III.C] The coherence length is estimated as xi ~ 200 nm in Sec. III.A, but Sec. III.C recommends Lx ~ xi ~ 1 micron; please reconcile these two estimates.
[Eq. (15)] The tight-binding superconducting term in Eq. (15) uses a bare J without defining it; it should be explicit that this is the same exp(i pi J_y) factor used in the analytical Hamiltonian, or a separate definition should be given.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central qubit-frequency predictions are direct outputs of a Luttinger-Kohn/tight-binding model; the experimental comparison calibrates one transparency parameter to the measured Andreev energy but does not fit the predicted splitting.

full rationale

The derivation chain is self-contained. The analytical Andreev spectrum uses Eq. (11), taken from the independent Ref. [22], and evaluates it with Fermi velocities obtained from the Luttinger-Kohn Hamiltonian (Eqs. (1)-(4)); the qubit frequency omega_q = |epsilon_1 - epsilon_2| is then a direct output of the band-structure input. The numerical tight-binding model (Eqs. (14)-(17)) is a direct simulation with Luttinger parameters, confinement lengths, and an explicit singlet-pairing assumption, and yields the geometry/filling dependence in Secs. III.A-III.B. In the experimental comparison (Sec. III.C), the single free parameter r = 0.995 is calibrated to match the measured Andreev-state energy (~3 GHz); the reported omega_q ~ 150 MHz is a distinct output of the calibrated model, not the fitted quantity, so the explanation of the null result is a postdiction rather than a reduction to the input. The authors' Sec. IV caveat that the spin-3/2 singlet pairing assumption 'is not unique' is an acknowledged modeling limitation, not a circular step. The only self-citation (Ref. 15) is used for qualitative phase/transparency behavior and nuclear-decoupling expectations, and is not load-bearing for the central geometry/filling predictions.

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

The calculation rests on a standard band-structure model for Ge, a strong infinite-square-well confinement idealization, and a specific singlet-pairing assumption for the induced superconductivity. The only parameter fitted to experimental data is the interface transparency r, used in the comparison to Ref. 17. No new physical entities are introduced.

free parameters (1)

Interface transparency parameter r = 0.995
In Sec. III.C (Comparison to experiment), r is fixed to 0.995 so that the calculated Andreev state energies match the experimental values of Ref. 17. This is a fit to data that affects the computed qubit frequency in that comparison. The central design guidance also depends on r in general, though the qualitative geometry dependence is robust.

assumptions (5)

standard math Bulk Ge is described by the Luttinger-Kohn Hamiltonian with Luttinger parameters gamma1 = 13.35, gamma2 = 4.25, gamma3 = 5.69 (Eq. 1).
Background material model from Ref. 18, used as the starting point for the band structure.
domain assumption Confinement along the y and z axes is an infinite square well of widths Ly and Lz (Eq. 3 and tight-binding setup).
This approximation determines the subband energies and wavefunctions used to project the 3D Hamiltonian to 1D; real confinement potentials are softer and may modify Fermi velocities.
domain assumption The induced superconductivity is singlet pairing in spin-3/2 space with a step-function phase profile (Eq. 9 and HSC).
The authors state in Sec. IV that 'we have assumed a singlet type of superconducting pairing... this choice is not unique and ultimately depends on the coupling of the heavy and light holes to the bulk superconductor.' This assumption directly sets the two-level Andreev structure that the qubit frequency is read from.
domain assumption Only the heavy-hole band is filled; light holes are neglected in the Andreev derivation.
Stated in Sec. II A: 'we focus on the regime when only the HH band is filled.' The authors note in footnote 23 that the calculation proceeds analogously for LH bands, so this is a regime restriction.
domain assumption Magnetic field and barrier transparency are treated perturbatively in the analytical model (Sec. II.B).
The effective Hamiltonian Eq. (12) is obtained by first-order perturbation theory in HZ and HU; the tight-binding numerics do not rely on this assumption but the analytical intuition does.

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Cite this review

Pith. "Pith review of Resolving Andreev spin qubits in germanium-based Josephson junctions." pith.science (2026). https://pith.science/paper/DNJU7SRU

@misc{pith2026250613988,
  author       = {Pith},
  title        = {Pith review of: Resolving Andreev spin qubits in germanium-based Josephson junctions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DNJU7SRU}},
  note         = {Machine review of arXiv:2506.13988}
}

read the original abstract

Andreev spin qubits (ASQs) are a promising platform for quantum information processing which benefit from both the small footprint of semiconducting spin qubits and the long range connectivity of superconducting qubits. While state-of-the-art experiments have developed ASQs in InAs nanowires, these realizations are coherence-time limited by nuclear magnetic noise which cannot be removed by isotopic purification. In Ge-based Josephson junctions, which can be isotopically purified, Andreev states have been experimentally observed but spin-resolved Andreev states remain elusive. Here, we theoretically demonstrate that the geometry of the Josephson junction can limit the qubit frequency to values below typical experimental temperatures and render the ASQ effectively invisible. ASQs could be experimentally resolved by judiciously choosing the geometry of the junction and filling of the underlying Ge. Our comprehensive study of ASQ frequency on in situ and ex situ experimentally controllable parameters provides design guidance of Ge-based Josephson junctions and paves the way towards realization of high-coherence ASQs.

Figures

Figures reproduced from arXiv: 2506.13988 by the authors.

**Figure 2.** FIG. 2. (a) Analytical band structures for translationally [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG. 3. Spectrum of the infinite semiconductor in the absence [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 5.** Figure 5: FIG. 5. Qubit frequency as a function of [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

**Figure 4.** Figure 4: FIG. 4. Typical spectrum (a), qubit frequency (b), spin (c), [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 8.** Figure 8: FIG. 8. Spectrum (a), qubit frequency (b), and spin (c) as [PITH_FULL_IMAGE:figures/full_fig_p007_8.png]

**Figure 9.** Figure 9: FIG. 9. Frequency as a function of magnetic field along the [PITH_FULL_IMAGE:figures/full_fig_p008_9.png]

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