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

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

Problem and Context
Hybrid semiconductor-superconductor devices offer a versatile platform for exploring the interplay between superconductivity, Coulomb interactions, and quantum confinement. While Josephson junctions (JJs) formed by quantum dots (QDs) coupled to superconducting reservoirs have been extensively studied in the single-level limit (where transport is governed by a single orbital) and the metallic island limit (large number of orbitals), the intermediate "multi-level regime" remains less understood. In this regime, several QD orbitals contribute simultaneously to superconducting transport. Previous experimental studies of this regime are scarce and have largely relied on direct current (DC) transport measurements. Consequently, there is a need for techniques that provide additional access to the inductive and dissipative response of the junction to fully characterize the ground state phase offset ($\phi_0$) and the nature of 0-$\pi$ transitions. Furthermore, while phase control has been demonstrated in nanowires and carbon nanotubes, implementations in planar material systems have largely relied on superconducting islands rather than gate-defined QDs.

Methodology
The authors investigate multi-level $\pi$-junctions formed in a proximitized Ge/SiGe quantum well (QW) heterostructure. The device fabrication utilizes a shallow 15 nm Ge/SiGe QW proximitized by an in-situ grown, nominally 8 nm thick Aluminum (Al) thin film. The heterostructure features asymmetric SiGe barriers (80% and 95% Ge content) and a thin top barrier (5 nm) designed to enhance the proximity effect.

The study employs a dual-approach methodology combining DC and Radio-frequency (RF) measurements:

1. DC-SQUID Setup: The Josephson junction is embedded in an asymmetric DC-SQUID. One arm contains a gate-defined channel ($\sim$200 nm wide, $\sim$70 nm long) tuned by a gate voltage ($V_{TG}$), while the other arm serves as a reference junction tuned by $V_{ref}$. This setup allows for the control of the superconducting phase drop via an external magnetic flux ($\phi_{ext}$).
2. RF-SQUID Setup: The JJ is embedded in an RF-SQUID coupled to a superconducting coplanar transmission line resonator (CPTR). This configuration enables the probing of the junction's inductance ($L_{JJ}$) and dissipative response via microwave transmission measurements ($S_{12}$). The resonance frequency ($f_r$) and loaded quality factor ($Q_l$) are extracted to infer the critical current and phase state.

Key Results
The experiments reveal the formation of a high-quality proximitized QD with an induced superconducting gap of $\Delta^* \simeq 240 \, \mu\text{eV}$, which is higher than previously reported for similar Ge-based heterostructures.

1. Identification of 0-$\pi$ Transitions: By sweeping the gate voltage ($V_{TG}$) and external flux, the authors observe transitions between the 0-phase (positive supercurrent) and $\pi$-phase (negative supercurrent) ground states.
2. Distinction of Transition Types: The study distinguishes between two qualitatively different types of 0-$\pi$ transitions:
   • Sharp Transitions: These occur when the occupation of the QD changes (e.g., crossing Coulomb diamonds). They are characterized by a sudden sign change in the critical current and a coexistence of 0 and $\pi$ states at the transition point, leading to a split in resonance frequency and a drop in the quality factor ($Q_l$).
   • Smooth Transitions: These occur at a fixed QD charge occupation but involve a change in the weights of co-tunneling paths through excited orbitals. These transitions are characterized by a gradual inversion of the supercurrent, a vanishing oscillation amplitude without a full state split, and a negligible change in $Q_l$.
3. Multi-level Regime Signatures: The observation of smooth transitions, where the phase offset changes without a change in ground state parity, serves as a fingerprint of the multi-level regime. In this regime, high-order co-tunneling processes involving multiple orbitals mediate spin permutations, leading to a non-zero probability of a $\pi$-shift for both odd and even occupations.
4. Microwave Response: The on-chip microwave resonator successfully detects these transitions through shifts in resonance frequency and phase. The time-trace measurements at the transition point reveal jumps between 0 and $\pi$ states, confirming the sensitivity of the microwave readout to the junction's ground state.

Significance and Claims
The paper establishes proximitized Ge as a robust platform for investigating hybrid superconductor/semiconductor physics. The primary significance of the work lies in the demonstration of gate-defined superconducting quantum devices integrated with high-quality on-chip microwave resonators within a planar Ge/SiGe system.

The authors claim that their results provide a clear picture of 0-$\pi$ transitions in a planar SC-QD-SC junction, revealing that these transitions are not solely determined by ground state parity but are also influenced by the weight shifts of co-tunneling processes via low-energy excited states. The combination of a large induced gap ($\Delta^* \simeq 240 \, \mu\text{eV}$) and the ability to resolve inductive and dissipative signatures via microwave techniques paves the way for exploring superconductivity, large spin-orbit interaction (SOI), and sizeable Zeeman energies in a single device at fast time-scales. The work suggests that this platform is well-suited for studying individual Andreev bound states, crossed Andreev reflection, and potentially emerging topological states.