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arXiv:2406.20082 · Physical Review X

Hybrid Josephson Rhombus: A Superconducting Element with Tailored Current-Phase Relation

Luca Banszerus, C. W. Andersson, W. Marshall, Tyler Lindemann, Michael J. Manfra, Charles M. Marcus, S. Vaitiekėnas

Published Feb 4, 2025• 14 citations
arXivHTMLPDF

Abstract

Controlling the current-phase relation (CPR) of Josephson elements is essential for tailoring the eigenstates of superconducting qubits, tuning the properties of parametric amplifiers, and designing nonreciprocal superconducting devices. Here, we introduce the hybrid Josephson rhombus, a highly tunable superconducting circuit containing four semiconductor-superconductor hybrid Josephson junctions embedded in a loop. Combining magnetic frustration with gate-voltage-controlled tuning of individual Josephson couplings provides deterministic control of the harmonic content of the rhombus CPR. We show that, for balanced Josephson couplings at full frustration, the hybrid rhombus displays a π -periodic cos ( 2 φ ) potential, indicating coherent charge- 4 e transport. Tuning away from the balanced configuration, we observe a superconducting diode effect with efficiency exceeding 25%. These results showcase the potential of hybrid Josephson rhombi as fundamental building blocks for noise-resilient qubits and quantum devices with custom transport properties.

DOI
10.1103/physrevx.15.011021

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Controlling the current-phase relation (CPR) of Josephson elements is essential for tailoring the eigenstates of superconducting qubits, tuning the properties of parametric amplifiers, and designing nonreciprocal superconducting devices. Here, we introduce the hybrid Josephson rhombus, a highly tunable superconducting circuit containing four semiconductor-superconductor hybrid Josephson junctions embedded in a loop. Combining magnetic frustration with gate-voltage-controlled tuning of individual Josephson couplings provides deterministic control of the harmonic content of the rhombus CPR. We show that, for balanced Josephson couplings at full frustration, the hybrid rhombus displays a π -periodic cos ( 2 φ ) potential, indicating coherent charge- 4 e transport. Tuning away from the balanced configuration, we observe a superconducting diode effect with efficiency exceeding 25%. These results showcase the potential of hybrid Josephson rhombi as fundamental building blocks for noise-resilient qubits and quantum devices with custom transport properties.
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Abstract Hybrid superconductor-semiconductor Josephson field-effect transistors (JoFETs) function as Josephson junctions with a gate-tunable critical current. Additionally, they can feature a non- sinusoidal current-phase relation (CPR) containing multiple harmonics of the superconducting phase difference, a so-far underutilized property. In this work, we exploit this multi-harmonicity to create a Josephson circuit element with an almost perfectly π-periodic CPR, indicative of a largely dominant charge-4e supercurrent transport. Such a Josephson element was recently proposed as the basic building block of a protected superconducting qubit. Here, it is realized using a superconducting quantum interference device (SQUID) with low-inductance aluminum arms and two nominally identical JoFETs. The latter are fabricated from a SiGe/Ge/SiGe quantum-well heterostructure embedding a high-mobility two-dimensional hole gas. By carefully adjusting the JoFET gate voltages and finely tuning the magnetic flux through the SQUID close to half a flux quantum, we achieve a regime where the sin(2φ) component accounts for more than 95 % of the total supercurrent. This result demonstrates a new promising route for the realization of superconducting qubits with enhanced coherence properties.
Coherence of superconducting qubits can be improved by implementing designs that protect the parity of Cooper pairs on superconducting islands. Here, we introduce a parity-protected qubit based on voltage-controlled semiconductor nanowire Josephson junctions, taking advantage of the higher harmonic content in the energy-phase relation of few-channel junctions. A symmetric interferometer formed by two such junctions, gate-tuned into balance and frustrated by a half-quantum of applied flux, yields a cos(2φ) Josephson element, reflecting coherent transport of pairs of Cooper pairs. We demonstrate that relaxation of the qubit can be suppressed tenfold by tuning into the protected regime.
Hybrid superconductor(S)-semiconductor(Sm) devices bring a range of functionalities into superconducting circuits by introducing Josephson field-effect transistors (JoFETs): Josephson junctions with a gate-tunable critical current. In particular, hybrid parity-protected qubits and Josephson diodes were recently proposed and experimentally demonstrated. Such devices leverage the multiharmonicity of the Josephson current-phase relation (CPR) in highly transparent S-Sm-S junctions, a so-far underutilized property. In this work we report an experimental study of superconducting quantum-interference devices (SQUIDs) embedding JoFETs fabricated from a SiGe/Ge/SiGe heterostructure grown at CEA-Grenoble. The single junction CPRs shows multiple harmonics with gate tunable amplitudes. In the presence of microwave irradiation, the appearance of integer and half integer Shapiro steps confirms this observation. A combination of magnetic-flux and gate-voltage control enables tuning the SQUID functionality from a nonreciprocal Josephson-diode regime with 27 % asymmetry to a π-periodic Josephson regime. Within a double SQUID device, we are able to quantitatively characterize a regime where the sin(2φ) component accounts for more than 95 % of the total supercurrent. Finally, this result is placed into perspective through a discussion of the implementation of a cos(2φ) parity-protected qubit within the Al-Ge/SiGe hybrid platform
Superconducting quantum circuits derive their nonlinearity from the Josephson energy-phase relation. Besides the fundamental cos⁡φ\cosφcosφ term, this relation can also contain higher Fourier harmonics cos⁡(kφ)\cos(kφ)cos(kφ) corresponding to correlated tunneling of kkk Cooper pairs. The parity of the dominant tunneling process, i.e.~whether an odd or even number of Cooper pairs tunnel, results in qualitatively different properties, and controlling this opens up a wide range of applications in superconducting technology. However, access to even-dominated regimes has remained challenging and has so far relied on complex multi-junction or all-hybrid architectures. Here, we demonstrate a simple "harmonic parity qubit" (HPQ); an element that combines two aluminum-oxide tunnel junctions in parallel to a gate-tunable InAs/Al nanowire junction forming a SQUID, and use spectroscopy versus flux to reconstruct its energy-phase relation at 85 gate voltage points. At half flux quantum, the odd harmonics of the Josephson potential can be suppressed by up to two orders of magnitude relative to the even harmonics, producing a double-well potential dominated by even harmonics with minima near ±π/2\pmπ/2±π/2. The ability to control harmonic parity enables supercurrent carried by pairs of Cooper pairs and provides a new building block for Fourier engineering in superconducting circuits.
Voltage-tunable superconductor-semiconductor devices offer a unique platform to realize dynamic tunability in superconducting quantum circuits. By galvanically connecting a gated InAs\mathrm{In}\mathrm{As}InAs-Al\mathrm{Al}Al Josephson junction to a coplanar waveguide resonator, we demonstrate the use of a superconducting element with wideband gate tunability. We show that the resonant frequency is controlled via a gate-tunable Josephson inductance and that the nonlinearity of the InAs\mathrm{In}\mathrm{As}InAs-Al\mathrm{Al}Al junction is nondissipative as is the case with conventional AlOx-Al{\mathrm{Al}\mathrm{O}}_{\mathrm{x}}\text{-Al}AlOx​-Al junctions. As the gate voltage is decreased, the inductive participation of the junction increases up to $44\mathrm{%}$, resulting in the resonant frequency being tuned by over 2GHz2\phantom{\rule{0.2em}{0ex}}\mathrm{GHz}2GHz. Utilizing the wide tunability of the device, we demonstrate that two resonant modes can be adjusted such that they strongly hybridize, exhibiting an avoided-level crossing with a coupling strength of 51MHz51\phantom{\rule{0.2em}{0ex}}\mathrm{MHz}51MHz. Implementing such voltage-tunable resonators is the first step toward realizing wafer-scale continuous voltage control in superconducting circuits for qubit-qubit coupling, quantum limited amplifiers, and quantum memory platforms.