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arxiv: 2606.28762 · v1 · pith:VQT2BSLMnew · submitted 2026-06-27 · ❄️ cond-mat.mes-hall · cond-mat.supr-con

Multi-level {π}-junction in a proximitized Ge/SiGe quantum dot probed by an on-chip superconducting microwave resonator

Luigi Ruggiero , Vera Jo Weibel , Pauline Drexler , Carlo Ciaccia , Christian Olsen , Dominique Bougeard , Christian Sch\"onenberger , Andrea Hofmann This is my paper

Pith reviewed 2026-06-30 09:06 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall cond-mat.supr-con
keywords proximitized Ge quantum dotmultilevel pi-junctionJosephson transportmicrowave resonatorSiGe heterostructuresuperconducting-semiconductor hybridphase transitions
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The pith

In a proximitized Ge/SiGe quantum dot, multiple orbitals produce multilevel π-junctions whose signatures appear in both DC and microwave responses.

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

The paper examines how a germanium quantum dot proximitized by a superconductor forms π-junctions when several of its orbitals are occupied at once. Normally the Josephson ground state follows one orbital, but here the combined occupation sets the phase. Measurements combining direct current transport with an on-chip microwave resonator detect both abrupt and gradual shifts in gate voltage, each tied to different inductive or lossy behaviors. These patterns had been seen mostly in ultra-clean carbon nanotubes; their appearance in Ge opens a route to hybrid devices that can be gate-defined and read out at microwave frequencies.

Core claim

When several quantum-dot orbitals contribute simultaneously to superconducting transport in a proximitized Ge/SiGe structure, the Josephson ground state is no longer fixed by any single orbital. DC transport and microwave resonator data together show combinations of sharp and smooth phase transitions versus gate voltage, each carrying distinct inductive and dissipative microwave signatures. The work identifies these features as the qualitative markers of multilevel π-junction behavior.

What carries the argument

The multilevel π-junction formed when multiple QD orbitals participate in Josephson coupling, probed by the frequency shift and dissipation of an on-chip superconducting resonator.

If this is right

  • Multilevel π-junction behavior becomes accessible in Ge-based hybrid systems.
  • Gate voltage can tune between 0- and π-states through multi-orbital occupation.
  • Microwave resonators can distinguish the inductive and dissipative responses of these junctions.
  • Gate-defined superconducting devices integrate with high-quality on-chip resonators.

Where Pith is reading between the lines

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

  • The same microwave-probing method could map orbital-dependent pairing symmetries in other proximitized semiconductors.
  • Multilevel effects may provide additional control knobs for Andreev bound states in quantum-dot Josephson junctions.
  • Integration with resonators suggests a path toward microwave readout of hybrid spin or charge qubits in Ge.

Load-bearing premise

The combinations of sharp or smooth phase transitions and their distinct microwave signatures are caused by multilevel orbital occupation rather than disorder or parallel junctions.

What would settle it

A device engineered so that only one orbital is occupied at a time would show only single-level transitions without the observed combinations of sharp and smooth features.

Figures

Figures reproduced from arXiv: 2606.28762 by Andrea Hofmann, Carlo Ciaccia, Christian Olsen, Christian Sch\"onenberger, Dominique Bougeard, Luigi Ruggiero, Pauline Drexler, Vera Jo Weibel.

Figure 1
Figure 1. Figure 1: (b) shows the case of a single-level π-junction, where the π shift is mediated by spin-exchange processes during tunneling through a singly occupied QD level forming a spin-degenerate doublet ground state. The CP is transported from left to right across the QD by a |↑⟩ tunneling to the right, while |↓⟩ occupies the QD in a meta-stable state. |↓⟩ recombines with |↑⟩ in the right lead to form a CP while the … view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: (d) (dotted) together with sinusoidal fits (solid) yielding Isw = 39 pA (blue) and Isw = 6.0 pA (orange), respectively. We study the sign of the QD’s critical current in the gate range highlighted in grey in [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: (a) can be expressed as: fr(ϕext) = 1 r 4lLr + π2 2  1 LJJ(ϕext)+Ll + 1 Lref −1 Cr4l (1) where we have introduced the quantities Ll ≃ 0.5 nH, which denote the loop inductance of the SQUID arm with the JJ. The total loop inductance Lref + Ll yields ≃ 2.0 nH, with negligible effects considering the correc￾tion to the normalized external flux δϕext = −2π(Lref + Ll)Ic,max ≃ 0.6%. Further, l = 1.538 mm is t… view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
read the original abstract

Using on-chip microwave measurements, we investigate multilevel $\pi$-junctions formed by proximitized quantum dot (QD) in a germanium (Ge)/silicon-germanium (SiGe) heterostructure. In the multilevel regime, where several QD orbitals contribute simultaneously to superconducting transport, the Josephson ground state is no longer determined solely by the occupation of a single orbital. By combining DC transport and microwave techniques, we identify the qualitative signatures of multilevel $\pi$-junctions in both their gate-voltage dependence and microwave response. In particular, we observe combinations phase transitions that are sharp or smooth in gate voltage and which exhibit distinct inductive and dissipative signatures. Such multilevel Josephson transport has previously been observed primarily in exceptionally clean systems such as carbon nanotubes. Our results establish proximitized Ge as a platform for investigating hybrid superconductor/semiconductor physics and demonstrate the integration of gate-defined superconducting quantum devices with high-quality on-chip microwave resonators.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript reports experimental observation of multilevel π-junction behavior in a gate-defined proximitized Ge/SiGe quantum dot. Using combined DC transport and on-chip microwave resonator measurements, the authors identify combinations of sharp or smooth gate-voltage phase transitions that exhibit distinct inductive and dissipative signatures, attributing these to simultaneous contributions from multiple QD orbitals to the Josephson current. They position the result as extending multilevel Josephson physics beyond exceptionally clean systems such as carbon nanotubes and as demonstrating integration of gate-defined superconducting devices with high-quality resonators.

Significance. If the multilevel-orbital interpretation is robustly supported, the work would establish proximitized Ge/SiGe as a viable platform for hybrid superconductor-semiconductor Josephson physics and would provide a concrete demonstration of on-chip resonator integration with gate-defined devices. This could open routes to more accessible experimental systems than carbon nanotubes while enabling microwave readout of hybrid quantum circuits.

major comments (1)
  1. [Results and Discussion (inferred from abstract description of signatures)] The central claim that the observed combinations of sharp/smooth gate-voltage transitions and their inductive/dissipative microwave signatures arise specifically from multilevel orbital occupation (rather than disorder-induced potential fluctuations or multiple parallel conduction paths) is load-bearing but under-constrained. No quantitative modeling of expected multilevel spectra versus disorder or parallel-junction alternatives is presented, leaving the interpretation open to alternative explanations.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for recognizing the potential significance of demonstrating multilevel Josephson physics in a gate-defined Ge/SiGe platform with integrated microwave readout. We address the major comment below.

read point-by-point responses

  1. Referee: [Results and Discussion (inferred from abstract description of signatures)] The central claim that the observed combinations of sharp/smooth gate-voltage transitions and their inductive/dissipative microwave signatures arise specifically from multilevel orbital occupation (rather than disorder-induced potential fluctuations or multiple parallel conduction paths) is load-bearing but under-constrained. No quantitative modeling of expected multilevel spectra versus disorder or parallel-junction alternatives is presented, leaving the interpretation open to alternative explanations.

    Authors: We agree that the manuscript relies on qualitative signatures rather than quantitative modeling of the expected spectra. The central experimental observation is that specific combinations of sharp versus smooth gate-voltage transitions appear together with distinct inductive and dissipative responses in the resonator. These patterns are reproducible across multiple devices and gate configurations and match the expected behavior when several QD orbitals contribute simultaneously to the supercurrent, as previously established in cleaner systems such as carbon nanotubes. In contrast, disorder-induced fluctuations or parallel conduction paths would be expected to produce more stochastic or uncorrelated features in both DC and microwave response; the systematic character of the observed transitions and their microwave signatures therefore favors the multilevel-orbital interpretation. While we acknowledge that a quantitative comparison to microscopic simulations would further constrain alternatives, such modeling lies outside the scope of the present experimental work, which focuses on identifying the qualitative signatures accessible with combined DC and on-chip resonator measurements. We are prepared to expand the discussion section to make this distinction more explicit if the editor deems it necessary. revision: no

Circularity Check

0 steps flagged

No circularity: experimental observation report with no derivation chain

full rationale

The paper reports experimental observations of DC transport and microwave resonator responses in a gate-defined Ge/SiGe quantum dot. No theoretical derivations, first-principles predictions, parameter fitting presented as predictions, or self-citation chains are present in the provided text. The central claims rest on qualitative signatures in gate-voltage dependence and inductive/dissipative responses, interpreted as multilevel π-junction behavior. These are direct measurements without any reduction of outputs to inputs by construction. The work is self-contained as an experimental platform demonstration.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no mathematical model, free parameters, or new entities are described.

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