1
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Sestoft JE, Marnauza M, Olsteins D, Kanne T, Schlosser RD, Chen IJ, Grove-Rasmussen K, Nygård J. Shadowed versus Etched Superconductor-Semiconductor Junctions in Al/InAs Nanowires. NANO LETTERS 2024. [PMID: 38865258 DOI: 10.1021/acs.nanolett.4c02055] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/14/2024]
Abstract
Hybrid semiconductor-superconductor nanowires have emerged as a cornerstone in modern quantum devices. Integrating such nanowires into hybrid devices typically requires extensive postgrowth processing which may affect device performance unfavorably. Here, we present a technique for in situ shadowing superconductors on nanowires and compare the structural and electronic properties of Al junctions formed by shadowing versus etching. Based on transmission electron microscopy, we find that typical etching procedures lead to atomic-scale surface roughening. This surface perturbation may cause a reduction of the electron mobility as demonstrated in transport measurements. Further, we display advanced shadowing geometries aiding in the pursuit of bringing fabrication of hybrid devices in situ. Finally, we give examples of shadowed junctions exploited in various device geometries that exhibit high-quality quantum transport signatures.
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Affiliation(s)
- Joachim E Sestoft
- Center for Quantum Devices and Nano-science Center, Niels Bohr Institute, University of Copenhagen, 2100 Copenhagen, Denmark
| | - Mikelis Marnauza
- Center for Quantum Devices and Nano-science Center, Niels Bohr Institute, University of Copenhagen, 2100 Copenhagen, Denmark
| | - Dags Olsteins
- Center for Quantum Devices and Nano-science Center, Niels Bohr Institute, University of Copenhagen, 2100 Copenhagen, Denmark
| | - Thomas Kanne
- Center for Quantum Devices and Nano-science Center, Niels Bohr Institute, University of Copenhagen, 2100 Copenhagen, Denmark
| | - Rasmus D Schlosser
- Center for Quantum Devices and Nano-science Center, Niels Bohr Institute, University of Copenhagen, 2100 Copenhagen, Denmark
| | - I-Ju Chen
- Center for Quantum Devices and Nano-science Center, Niels Bohr Institute, University of Copenhagen, 2100 Copenhagen, Denmark
| | - Kasper Grove-Rasmussen
- Center for Quantum Devices and Nano-science Center, Niels Bohr Institute, University of Copenhagen, 2100 Copenhagen, Denmark
| | - Jesper Nygård
- Center for Quantum Devices and Nano-science Center, Niels Bohr Institute, University of Copenhagen, 2100 Copenhagen, Denmark
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2
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Zheng H, Cheung LY, Sangwan N, Kononov A, Haller R, Ridderbos J, Ciaccia C, Ungerer JH, Li A, Bakkers EPAM, Baumgartner A, Schönenberger C. Coherent Control of a Few-Channel Hole Type Gatemon Qubit. NANO LETTERS 2024. [PMID: 38848282 DOI: 10.1021/acs.nanolett.4c00770] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/09/2024]
Abstract
Gatemon qubits are the electrically tunable cousins of superconducting transmon qubits. In this work, we demonstrate the full coherent control of a gatemon qubit based on hole carriers in a Ge/Si core/shell nanowire, with the longest coherence times in group IV material gatemons to date. The key to these results is a high-quality Josephson junction obtained using a straightforward and reproducible annealing technique. We demonstrate that the transport through the narrow junction is dominated by only two quantum channels, with transparencies up to unity. This novel qubit platform holds great promise for quantum information applications, not only because it incorporates technologically relevant materials, but also because it provides new opportunities, like an ultrastrong spin-orbit coupling in the few-channel regime of Josephson junctions.
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Affiliation(s)
- Han Zheng
- Quantum- and Nanoelectronics Lab, Department of Physics, University of Basel, 4056 Basel, Switzerland
| | - Luk Yi Cheung
- Quantum- and Nanoelectronics Lab, Department of Physics, University of Basel, 4056 Basel, Switzerland
| | - Nikunj Sangwan
- Quantum- and Nanoelectronics Lab, Department of Physics, University of Basel, 4056 Basel, Switzerland
| | - Artem Kononov
- Quantum- and Nanoelectronics Lab, Department of Physics, University of Basel, 4056 Basel, Switzerland
| | - Roy Haller
- Quantum- and Nanoelectronics Lab, Department of Physics, University of Basel, 4056 Basel, Switzerland
| | - Joost Ridderbos
- MESA+ Institute for Nanotechnology University of Twente, 7500 AE Enschede, The Netherlands
| | - Carlo Ciaccia
- Quantum- and Nanoelectronics Lab, Department of Physics, University of Basel, 4056 Basel, Switzerland
| | - Jann Hinnerk Ungerer
- Quantum- and Nanoelectronics Lab, Department of Physics, University of Basel, 4056 Basel, Switzerland
| | - Ang Li
- Department of Applied Physics, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands
| | - Erik P A M Bakkers
- Department of Applied Physics, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands
| | - Andreas Baumgartner
- Quantum- and Nanoelectronics Lab, Department of Physics, University of Basel, 4056 Basel, Switzerland
- Swiss Nanoscience Institute, University of Basel, 4056 Basel, Switzerland
| | - Christian Schönenberger
- Quantum- and Nanoelectronics Lab, Department of Physics, University of Basel, 4056 Basel, Switzerland
- Swiss Nanoscience Institute, University of Basel, 4056 Basel, Switzerland
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3
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Ibabe Á, Steffensen GO, Casal I, Gómez M, Kanne T, Nygård J, Levy Yeyati A, Lee EJH. Heat Dissipation Mechanisms in Hybrid Superconductor-Semiconductor Devices Revealed by Joule Spectroscopy. NANO LETTERS 2024; 24:6488-6495. [PMID: 38771151 PMCID: PMC11157656 DOI: 10.1021/acs.nanolett.4c00574] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/02/2024] [Revised: 05/17/2024] [Accepted: 05/17/2024] [Indexed: 05/22/2024]
Abstract
Understanding heating and cooling mechanisms in mesoscopic superconductor-semiconductor devices is crucial for their application in quantum technologies. Owing to their poor thermal conductivity, heating effects can drive superconducting-to-normal transitions even at low bias, observed as sharp conductance dips through the loss of Andreev excess currents. Tracking such dips across magnetic field, cryostat temperature, and applied microwave power allows us to uncover cooling bottlenecks in different parts of a device. By applying this "Joule spectroscopy" technique, we analyze heat dissipation in devices based on InAs-Al nanowires and reveal that cooling of superconducting islands is limited by the rather inefficient electron-phonon coupling, as opposed to grounded superconductors that primarily cool by quasiparticle diffusion. We show that powers as low as 50-150 pW are able to suppress superconductivity on the islands. Applied microwaves lead to similar heating effects but are affected by the interplay of the microwave frequency and the effective electron-phonon relaxation time.
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Affiliation(s)
- Ángel Ibabe
- Departamento
de Física de la Materia Condensada, Universidad Autónoma de Madrid, E-28049 Madrid, Spain
- Condensed
Matter Physics Center (IFIMAC), Universidad
Autónoma de Madrid, E-28049 Madrid, Spain
| | - Gorm O. Steffensen
- Condensed
Matter Physics Center (IFIMAC), Universidad
Autónoma de Madrid, E-28049 Madrid, Spain
- Departamento
de Física Teórica de la Materia Condensada, Universidad Autónoma de Madrid, E-28049 Madrid, Spain
| | - Ignacio Casal
- Departamento
de Física de la Materia Condensada, Universidad Autónoma de Madrid, E-28049 Madrid, Spain
- Condensed
Matter Physics Center (IFIMAC), Universidad
Autónoma de Madrid, E-28049 Madrid, Spain
| | - Mario Gómez
- Departamento
de Física de la Materia Condensada, Universidad Autónoma de Madrid, E-28049 Madrid, Spain
- Condensed
Matter Physics Center (IFIMAC), Universidad
Autónoma de Madrid, E-28049 Madrid, Spain
| | - Thomas Kanne
- Center
for Quantum Devices, Niels Bohr Institute, University of Copenhagen, DK-2100 Copenhagen, Denmark
| | - Jesper Nygård
- Center
for Quantum Devices, Niels Bohr Institute, University of Copenhagen, DK-2100 Copenhagen, Denmark
| | - Alfredo Levy Yeyati
- Condensed
Matter Physics Center (IFIMAC), Universidad
Autónoma de Madrid, E-28049 Madrid, Spain
- Departamento
de Física Teórica de la Materia Condensada, Universidad Autónoma de Madrid, E-28049 Madrid, Spain
- Instituto
Nicolás Cabrera, Universidad Autónoma
de Madrid, E-28049 Madrid, Spain
| | - Eduardo J. H. Lee
- Departamento
de Física de la Materia Condensada, Universidad Autónoma de Madrid, E-28049 Madrid, Spain
- Condensed
Matter Physics Center (IFIMAC), Universidad
Autónoma de Madrid, E-28049 Madrid, Spain
- Instituto
Nicolás Cabrera, Universidad Autónoma
de Madrid, E-28049 Madrid, Spain
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4
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Levajac V, Barakov H, Mazur GP, van Loo N, Kouwenhoven LP, Nazarov YV, Wang JY. Supercurrent in the Presence of Direct Transmission and a Resonant Localized State. PHYSICAL REVIEW LETTERS 2024; 132:176304. [PMID: 38728734 DOI: 10.1103/physrevlett.132.176304] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/08/2023] [Revised: 01/25/2024] [Accepted: 04/02/2024] [Indexed: 05/12/2024]
Abstract
We study the current-phase relation (CPR) of an InSb-Al nanowire Josephson junction in parallel magnetic fields up to 700 mT. At high magnetic fields and in narrow voltage intervals of a gate under the junction, the CPR exhibits π shifts. The supercurrent declines within these gate intervals and shows asymmetric gate voltage dependence above and below them. We detect these features sometimes also at zero magnetic field. The observed CPR properties are reproduced by a theoretical model of supercurrent transport via interference between direct transmission and a resonant localized state.
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Affiliation(s)
- Vukan Levajac
- QuTech and Kavli Institute of Nanoscience, Delft University of Technology, 2600 GA Delft, The Netherlands
| | - Hristo Barakov
- Kavli Institute of Nanoscience, Delft University of Technology, 2628 CJ Delft, The Netherlands
| | - Grzegorz P Mazur
- QuTech and Kavli Institute of Nanoscience, Delft University of Technology, 2600 GA Delft, The Netherlands
| | - Nick van Loo
- QuTech and Kavli Institute of Nanoscience, Delft University of Technology, 2600 GA Delft, The Netherlands
| | - Leo P Kouwenhoven
- QuTech and Kavli Institute of Nanoscience, Delft University of Technology, 2600 GA Delft, The Netherlands
| | - Yuli V Nazarov
- Kavli Institute of Nanoscience, Delft University of Technology, 2628 CJ Delft, The Netherlands
| | - Ji-Yin Wang
- QuTech and Kavli Institute of Nanoscience, Delft University of Technology, 2600 GA Delft, The Netherlands
- Beijing Academy of Quantum Information Sciences, 100193 Beijing, China
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5
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Estrada Saldaña JC, Vekris A, Pavešič L, Žitko R, Grove-Rasmussen K, Nygård J. Correlation between two distant quasiparticles in separate superconducting islands mediated by a single spin. Nat Commun 2024; 15:3465. [PMID: 38658553 PMCID: PMC11043349 DOI: 10.1038/s41467-024-47694-7] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/20/2023] [Accepted: 04/05/2024] [Indexed: 04/26/2024] Open
Abstract
Controlled coupling between distant particles is a key requirement for the implementation of quantum information technologies. A promising platform are hybrid systems of semiconducting quantum dots coupled to superconducting islands, where the tunability of the dots is combined with the macroscopic coherence of the islands to produce states with non-local correlations, e.g. in Cooper pair splitters. Electrons in hybrid quantum dots are typically not amenable to long-distance spin alignment as they tend to be screened into a localized singlet state by bound superconducting quasiparticles. However, two quasiparticles coming from different superconductors can overscreen the quantum dot into a doublet state, leading to ferromagnetic correlations between the superconducting islands. We present experimental evidence of a stabilized overscreened state, implying correlated quasiparticles over a micrometer distance. We propose alternating chains of quantum dots and superconducting islands as a novel platform for controllable large-scale spin coupling.
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Affiliation(s)
| | - Alexandros Vekris
- Center for Quantum Devices, Niels Bohr Institute, University of Copenhagen, 2100, Copenhagen, Denmark
- Sino-Danish College (SDC), University of Chinese Academy of Sciences, Beijing, China
| | - Luka Pavešič
- Jožef Stefan Institute, Jamova 39, SI-1000, Ljubljana, Slovenia
- Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI-1000, Ljubljana, Slovenia
| | - Rok Žitko
- Jožef Stefan Institute, Jamova 39, SI-1000, Ljubljana, Slovenia.
- Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI-1000, Ljubljana, Slovenia.
| | - Kasper Grove-Rasmussen
- Center for Quantum Devices, Niels Bohr Institute, University of Copenhagen, 2100, Copenhagen, Denmark
| | - Jesper Nygård
- Center for Quantum Devices, Niels Bohr Institute, University of Copenhagen, 2100, Copenhagen, Denmark.
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6
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Valentini M, Sagi O, Baghumyan L, de Gijsel T, Jung J, Calcaterra S, Ballabio A, Aguilera Servin J, Aggarwal K, Janik M, Adletzberger T, Seoane Souto R, Leijnse M, Danon J, Schrade C, Bakkers E, Chrastina D, Isella G, Katsaros G. Parity-conserving Cooper-pair transport and ideal superconducting diode in planar germanium. Nat Commun 2024; 15:169. [PMID: 38167818 PMCID: PMC10762135 DOI: 10.1038/s41467-023-44114-0] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/14/2023] [Accepted: 11/30/2023] [Indexed: 01/05/2024] Open
Abstract
Superconductor/semiconductor hybrid devices have attracted increasing interest in the past years. Superconducting electronics aims to complement semiconductor technology, while hybrid architectures are at the forefront of new ideas such as topological superconductivity and protected qubits. In this work, we engineer the induced superconductivity in two-dimensional germanium hole gas by varying the distance between the quantum well and the aluminum. We demonstrate a hard superconducting gap and realize an electrically and flux tunable superconducting diode using a superconducting quantum interference device (SQUID). This allows to tune the current phase relation (CPR), to a regime where single Cooper pair tunneling is suppressed, creating a [Formula: see text] CPR. Shapiro experiments complement this interpretation and the microwave drive allows to create a diode with ≈ 100% efficiency. The reported results open up the path towards integration of spin qubit devices, microwave resonators and (protected) superconducting qubits on the same silicon technology compatible platform.
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Affiliation(s)
- Marco Valentini
- Institute of Science and Technology Austria, Klosterneuburg, Austria.
| | - Oliver Sagi
- Institute of Science and Technology Austria, Klosterneuburg, Austria
| | - Levon Baghumyan
- Institute of Science and Technology Austria, Klosterneuburg, Austria
| | - Thijs de Gijsel
- Institute of Science and Technology Austria, Klosterneuburg, Austria
- Department of Applied Physics, Eindhoven University of Technology, Eindhoven, The Netherlands
| | - Jason Jung
- Department of Applied Physics, Eindhoven University of Technology, Eindhoven, The Netherlands
| | | | - Andrea Ballabio
- L-NESS, Physics Department, Politecnico di Milano, Como, Italy
| | | | - Kushagra Aggarwal
- Institute of Science and Technology Austria, Klosterneuburg, Austria
- Department of Materials, University of Oxford, Oxford, UK
| | - Marian Janik
- Institute of Science and Technology Austria, Klosterneuburg, Austria
| | | | - Rubén Seoane Souto
- Center for Quantum Devices, Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark
- Instituto de Ciencia de Materiales de Madrid, Consejo Superior de Investigaciones Científicas (ICMM-CSIC), Madrid, Spain
| | - Martin Leijnse
- NanoLund and Solid State Physics, Lund University, Lund, Sweden
| | - Jeroen Danon
- Department of Physics, Norwegian University of Science and Technology, Trondheim, Norway
| | - Constantin Schrade
- Hearne Institute for Theoretical Physics, Department of Physics and Astronomy, Louisiana State University, Baton Rouge, USA
| | - Erik Bakkers
- Department of Applied Physics, Eindhoven University of Technology, Eindhoven, The Netherlands
| | | | - Giovanni Isella
- L-NESS, Physics Department, Politecnico di Milano, Como, Italy
| | - Georgios Katsaros
- Institute of Science and Technology Austria, Klosterneuburg, Austria.
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7
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Jin G, Kim SH, Han HJ. Synthesis and Future Electronic Applications of Topological Nanomaterials. Int J Mol Sci 2023; 25:400. [PMID: 38203574 PMCID: PMC10779379 DOI: 10.3390/ijms25010400] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/27/2023] [Revised: 12/19/2023] [Accepted: 12/22/2023] [Indexed: 01/12/2024] Open
Abstract
Over the last ten years, the discovery of topological materials has opened up new areas in condensed matter physics. These materials are noted for their distinctive electronic properties, unlike conventional insulators and metals. This discovery has not only spurred new research areas but also offered innovative approaches to electronic device design. A key aspect of these materials is now that transforming them into nanostructures enhances the presence of surface or edge states, which are the key components for their unique electronic properties. In this review, we focus on recent synthesis methods, including vapor-liquid-solid (VLS) growth, chemical vapor deposition (CVD), and chemical conversion techniques. Moreover, the scaling down of topological nanomaterials has revealed new electronic and magnetic properties due to quantum confinement. This review covers their synthesis methods and the outcomes of topological nanomaterials and applications, including quantum computing, spintronics, and interconnects. Finally, we address the materials and synthesis challenges that need to be resolved prior to the practical application of topological nanomaterials in advanced electronic devices.
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Affiliation(s)
- Gangtae Jin
- Department of Materials Science and Engineering, Cornell University, Ithaca, NY 14853, USA;
| | - Seo-Hyun Kim
- Department of Environment and Energy Engineering, Sungshin Women’s University, Seoul 01133, Republic of Korea;
| | - Hyeuk-Jin Han
- Department of Materials Science and Engineering, Cornell University, Ithaca, NY 14853, USA;
- Department of Environment and Energy Engineering, Sungshin Women’s University, Seoul 01133, Republic of Korea;
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8
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Matsuo S, Imoto T, Yokoyama T, Sato Y, Lindemann T, Gronin S, Gardner GC, Nakosai S, Tanaka Y, Manfra MJ, Tarucha S. Phase-dependent Andreev molecules and superconducting gap closing in coherently-coupled Josephson junctions. Nat Commun 2023; 14:8271. [PMID: 38092786 PMCID: PMC10719386 DOI: 10.1038/s41467-023-44111-3] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/02/2023] [Accepted: 11/30/2023] [Indexed: 12/17/2023] Open
Abstract
The Josephson junction (JJ) is an essential element of superconducting (SC) devices for both fundamental and applied physics. The short-range coherent coupling of two adjacent JJs forms Andreev molecule states (AMSs), which provide a new ingredient to engineer exotic SC phenomena such as topological SC states and Andreev qubits. Here we provide tunneling spectroscopy measurements on a device consisting of two electrically controllable planar JJs sharing a single SC electrode. We discover that Andreev spectra in the coupled JJ are highly modulated from those in the single JJs and possess phase-dependent AMS features reproduced in our numerical calculation. Notably, the SC gap closing due to the AMS formation is experimentally observed. Our results help in understanding SC transport derived from the AMS and promoting the use of AMS physics to engineer topological SC states and quantum information devices.
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Affiliation(s)
- Sadashige Matsuo
- Center for Emergent Matter Science, RIKEN, Saitama, 351-0198, Japan.
| | - Takaya Imoto
- Center for Emergent Matter Science, RIKEN, Saitama, 351-0198, Japan
- Department of Applied Physics, Tokyo University of Science, Tokyo, 162-8601, Japan
| | - Tomohiro Yokoyama
- Department of Materials Engineering Science, Osaka University, Osaka, 560-8531, Japan.
| | - Yosuke Sato
- Center for Emergent Matter Science, RIKEN, Saitama, 351-0198, Japan
| | - Tyler Lindemann
- Department of Physics and Astronomy, Purdue University, West Lafayette, Indiana, IN, 47907, USA
- Birck Nanotechnology Center, Purdue University, West Lafayette, Indiana, IN, 47907, USA
| | - Sergei Gronin
- Birck Nanotechnology Center, Purdue University, West Lafayette, Indiana, IN, 47907, USA
| | - Geoffrey C Gardner
- Birck Nanotechnology Center, Purdue University, West Lafayette, Indiana, IN, 47907, USA
| | - Sho Nakosai
- Department of Applied Physics, Nagoya University, Nagoya, 464-8603, Japan
| | - Yukio Tanaka
- Department of Applied Physics, Nagoya University, Nagoya, 464-8603, Japan
| | - Michael J Manfra
- Department of Physics and Astronomy, Purdue University, West Lafayette, Indiana, IN, 47907, USA
- Birck Nanotechnology Center, Purdue University, West Lafayette, Indiana, IN, 47907, USA
- School of Materials Engineering, Purdue University, West Lafayette, Indiana, IN, 47907, USA
- Elmore Family School of Electrical and Computer Engineering, Purdue University, West Lafayette, Indiana, IN, 47907, USA
| | - Seigo Tarucha
- Center for Emergent Matter Science, RIKEN, Saitama, 351-0198, Japan.
- RIKEN Center for Quantum Computing, RIKEN, Saitama, 351-0198, Japan.
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9
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Haxell DZ, Coraiola M, Sabonis D, Hinderling M, Ten Kate SC, Cheah E, Krizek F, Schott R, Wegscheider W, Belzig W, Cuevas JC, Nichele F. Microwave-induced conductance replicas in hybrid Josephson junctions without Floquet-Andreev states. Nat Commun 2023; 14:6798. [PMID: 37884490 PMCID: PMC10603169 DOI: 10.1038/s41467-023-42357-5] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/12/2023] [Accepted: 10/09/2023] [Indexed: 10/28/2023] Open
Abstract
Light-matter coupling allows control and engineering of complex quantum states. Here we investigate a hybrid superconducting-semiconducting Josephson junction subject to microwave irradiation by means of tunnelling spectroscopy of the Andreev bound state spectrum and measurements of the current-phase relation. For increasing microwave power, discrete levels in the tunnelling conductance develop into a series of equally spaced replicas, while the current-phase relation changes amplitude and skewness, and develops dips. Quantitative analysis of our results indicates that conductance replicas originate from photon assisted tunnelling of quasiparticles into Andreev bound states through the tunnelling barrier. Despite strong qualitative similarities with proposed signatures of Floquet-Andreev states, our study rules out this scenario. The distortion of the current-phase relation is explained by the interaction of Andreev bound states with microwave photons, including a non-equilibrium Andreev bound state occupation. The techniques outlined here establish a baseline to study light-matter coupling in hybrid nanostructures and distinguish photon assisted tunnelling from Floquet-Andreev states in mesoscopic devices.
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Affiliation(s)
| | - Marco Coraiola
- IBM Research Europe-Zurich, 8803, Rüschlikon, Switzerland
| | | | | | | | - Erik Cheah
- Laboratory for Solid State Physics, ETH Zürich, 8093, Zürich, Switzerland
| | - Filip Krizek
- IBM Research Europe-Zurich, 8803, Rüschlikon, Switzerland
- Laboratory for Solid State Physics, ETH Zürich, 8093, Zürich, Switzerland
| | - Rüdiger Schott
- Laboratory for Solid State Physics, ETH Zürich, 8093, Zürich, Switzerland
| | - Werner Wegscheider
- Laboratory for Solid State Physics, ETH Zürich, 8093, Zürich, Switzerland
| | - Wolfgang Belzig
- Fachbereich Physik, Universität Konstanz, D-78457, Konstanz, Germany
| | - Juan Carlos Cuevas
- Fachbereich Physik, Universität Konstanz, D-78457, Konstanz, Germany
- Departamento de Física Teórica de la Materia Condensada and Condensed Matter Physics Center (IFIMAC), Universidad Autónoma de Madrid, E-28049, Madrid, Spain
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10
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Siebrecht J, Huang H, Kot P, Drost R, Padurariu C, Kubala B, Ankerhold J, Cuevas JC, Ast CR. Microwave excitation of atomic scale superconducting bound states. Nat Commun 2023; 14:6794. [PMID: 37880208 PMCID: PMC10600199 DOI: 10.1038/s41467-023-42454-5] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/06/2023] [Accepted: 10/11/2023] [Indexed: 10/27/2023] Open
Abstract
Magnetic impurities on superconductors lead to bound states within the superconducting gap, so called Yu-Shiba-Rusinov (YSR) states. They are parity protected, which enhances their lifetime, but makes it more difficult to excite them. Here, we realize the excitation of YSR states by microwaves facilitated by the tunnel coupling to another superconducting electrode in a scanning tunneling microscope (STM). We identify the excitation process through a family of anomalous microwave-assisted tunneling peaks originating from a second-order resonant Andreev process, in which the microwave excites the YSR state triggering a tunneling event transferring a total of two charges. We vary the amplitude and the frequency of the microwave to identify the energy threshold and the evolution of this excitation process. Our work sets an experimental basis and proof-of-principle for the manipulation of YSR states using microwaves with an outlook towards YSR qubits.
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Affiliation(s)
- Janis Siebrecht
- Max-Planck-Institut für Festkörperforschung, Heisenbergstraße 1, 70569, Stuttgart, Germany
| | - Haonan Huang
- Max-Planck-Institut für Festkörperforschung, Heisenbergstraße 1, 70569, Stuttgart, Germany
| | - Piotr Kot
- Max-Planck-Institut für Festkörperforschung, Heisenbergstraße 1, 70569, Stuttgart, Germany
| | - Robert Drost
- Max-Planck-Institut für Festkörperforschung, Heisenbergstraße 1, 70569, Stuttgart, Germany
| | - Ciprian Padurariu
- Institut für Komplexe Quantensysteme and IQST, Universität Ulm, Albert-Einstein-Allee 11, 89069, Ulm, Germany
| | - Björn Kubala
- Institut für Komplexe Quantensysteme and IQST, Universität Ulm, Albert-Einstein-Allee 11, 89069, Ulm, Germany
- Institute for Quantum Technologies, German Aerospace Center (DLR), Wilhelm-Runge Straße 10, 89081, Ulm, Germany
| | - Joachim Ankerhold
- Institut für Komplexe Quantensysteme and IQST, Universität Ulm, Albert-Einstein-Allee 11, 89069, Ulm, Germany
| | - Juan Carlos Cuevas
- Departamento de Física Teórica de la Materia Condensada and Condensed Matter Physics Center (IFIMAC), Universidad Autónoma de Madrid, 28049, Madrid, Spain
| | - Christian R Ast
- Max-Planck-Institut für Festkörperforschung, Heisenbergstraße 1, 70569, Stuttgart, Germany.
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11
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Coraiola M, Haxell DZ, Sabonis D, Weisbrich H, Svetogorov AE, Hinderling M, Ten Kate SC, Cheah E, Krizek F, Schott R, Wegscheider W, Cuevas JC, Belzig W, Nichele F. Phase-engineering the Andreev band structure of a three-terminal Josephson junction. Nat Commun 2023; 14:6784. [PMID: 37880228 PMCID: PMC10600130 DOI: 10.1038/s41467-023-42356-6] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/02/2023] [Accepted: 10/09/2023] [Indexed: 10/27/2023] Open
Abstract
In hybrid Josephson junctions with three or more superconducting terminals coupled to a semiconducting region, Andreev bound states may form unconventional energy band structures, or Andreev matter, which are engineered by controlling superconducting phase differences. Here we report tunnelling spectroscopy measurements of three-terminal Josephson junctions realised in an InAs/Al heterostructure. The three terminals are connected to form two loops, enabling independent control over two phase differences and access to a synthetic Andreev band structure in the two-dimensional phase space. Our results demonstrate a phase-controlled Andreev molecule, originating from two discrete Andreev levels that spatially overlap and hybridise. Signatures of hybridisation are observed in the form of avoided crossings in the spectrum and band structure anisotropies in the phase space, all explained by a numerical model. Future extensions of this work could focus on addressing spin-resolved energy levels, ground state fermion parity transitions and Weyl bands in multiterminal geometries.
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Affiliation(s)
- Marco Coraiola
- IBM Research Europe-Zurich, 8803, Rüschlikon, Switzerland
| | | | | | - Hannes Weisbrich
- Fachbereich Physik, Universität Konstanz, D-78457, Konstanz, Germany
| | | | | | | | - Erik Cheah
- Laboratory for Solid State Physics, ETH Zürich, 8093, Zürich, Switzerland
| | - Filip Krizek
- IBM Research Europe-Zurich, 8803, Rüschlikon, Switzerland
- Laboratory for Solid State Physics, ETH Zürich, 8093, Zürich, Switzerland
- Institute of Physics, Czech Academy of Sciences, 162 00, Prague, Czech Republic
| | - Rüdiger Schott
- Laboratory for Solid State Physics, ETH Zürich, 8093, Zürich, Switzerland
| | - Werner Wegscheider
- Laboratory for Solid State Physics, ETH Zürich, 8093, Zürich, Switzerland
| | - Juan Carlos Cuevas
- Departamento de Física Teórica de la Materia Condensada and Condensed Matter Physics Center (IFIMAC), Universidad Autónoma de Madrid, E-28049, Madrid, Spain
| | - Wolfgang Belzig
- Fachbereich Physik, Universität Konstanz, D-78457, Konstanz, Germany
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12
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Haxell D, Coraiola M, Sabonis D, Hinderling M, ten Kate SC, Cheah E, Krizek F, Schott R, Wegscheider W, Nichele F. Zeeman- and Orbital-Driven Phase Shifts in Planar Josephson Junctions. ACS NANO 2023; 17:18139-18147. [PMID: 37694539 PMCID: PMC10540266 DOI: 10.1021/acsnano.3c04957] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/02/2023] [Accepted: 08/25/2023] [Indexed: 09/12/2023]
Abstract
We perform supercurrent and tunneling spectroscopy measurements on gate-tunable InAs/Al Josephson junctions (JJs) in an in-plane magnetic field and report on phase shifts in the current-phase relation measured with respect to an absolute phase reference. The impact of orbital effects is investigated by studying multiple devices with different superconducting lead sizes. At low fields, we observe gate-dependent phase shifts of up to φ0 = 0.5π, which are consistent with a Zeeman field coupling to highly transmissive Andreev bound states via Rashba spin-orbit interaction. A distinct phase shift emerges at larger fields, concomitant with a switching current minimum and the closing and reopening of the superconducting gap. These signatures of an induced phase transition, which might resemble a topological transition, scale with the superconducting lead size, demonstrating the crucial role of orbital effects. Our results elucidate the interplay of Zeeman, spin-orbit, and orbital effects in InAs/Al JJs, giving improved understanding of phase transitions in hybrid JJs and their applications in quantum computing and superconducting electronics.
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Affiliation(s)
| | - Marco Coraiola
- IBM
Research Europe−Zurich, 8803 Rüschlikon, Switzerland
| | | | | | | | - Erik Cheah
- Laboratory
for Solid State Physics, ETH Zürich, 8093 Zürich, Switzerland
| | - Filip Krizek
- IBM
Research Europe−Zurich, 8803 Rüschlikon, Switzerland
- Laboratory
for Solid State Physics, ETH Zürich, 8093 Zürich, Switzerland
- Institute
of Physics, Czech Academy of Sciences, 162 00 Prague, Czech Republic
| | - Rüdiger Schott
- Laboratory
for Solid State Physics, ETH Zürich, 8093 Zürich, Switzerland
| | - Werner Wegscheider
- Laboratory
for Solid State Physics, ETH Zürich, 8093 Zürich, Switzerland
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13
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Wesdorp JJ, Grünhaupt L, Vaartjes A, Pita-Vidal M, Bargerbos A, Splitthoff LJ, Krogstrup P, van Heck B, de Lange G. Dynamical Polarization of the Fermion Parity in a Nanowire Josephson Junction. PHYSICAL REVIEW LETTERS 2023; 131:117001. [PMID: 37774257 DOI: 10.1103/physrevlett.131.117001] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/25/2022] [Revised: 01/22/2023] [Accepted: 07/14/2023] [Indexed: 10/01/2023]
Abstract
Josephson junctions in InAs nanowires proximitized with an Al shell can host gate-tunable Andreev bound states. Depending on the bound state occupation, the fermion parity of the junction can be even or odd. Coherent control of Andreev bound states has recently been achieved within each parity sector, but it is impeded by incoherent parity switches due to excess quasiparticles in the superconducting environment. Here, we show that we can polarize the fermion parity dynamically using microwave pulses by embedding the junction in a superconducting LC resonator. We demonstrate polarization up to 94%±1% (89%±1%) for the even (odd) parity as verified by single shot parity readout. Finally, we apply this scheme to probe the flux-dependent transition spectrum of the even or odd parity sector selectively, without any postprocessing or heralding.
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Affiliation(s)
- J J Wesdorp
- QuTech and Kavli Institute of Nanoscience, Delft University of Technology, 2628 CJ, Delft, Netherlands
| | - L Grünhaupt
- QuTech and Kavli Institute of Nanoscience, Delft University of Technology, 2628 CJ, Delft, Netherlands
| | - A Vaartjes
- QuTech and Kavli Institute of Nanoscience, Delft University of Technology, 2628 CJ, Delft, Netherlands
| | - M Pita-Vidal
- QuTech and Kavli Institute of Nanoscience, Delft University of Technology, 2628 CJ, Delft, Netherlands
| | - A Bargerbos
- QuTech and Kavli Institute of Nanoscience, Delft University of Technology, 2628 CJ, Delft, Netherlands
| | - L J Splitthoff
- QuTech and Kavli Institute of Nanoscience, Delft University of Technology, 2628 CJ, Delft, Netherlands
| | - P Krogstrup
- NNF Quantum Computing Programme, Niels Bohr Institute, University of Copenhagen, Denmark
| | - B van Heck
- Microsoft Quantum Lab Delft, 2628 CJ, Delft, Netherlands
- Leiden Institute of Physics, Universiteit Leiden, Niels Bohrweg 2, 2333 CA Leiden, Netherlands
- Dipartimento di Fisica, Sapienza Università di Roma, P.le Aldo Moro 2, 00185 Roma, Italy
| | - G de Lange
- Microsoft Quantum Lab Delft, 2628 CJ, Delft, Netherlands
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14
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Bargerbos A, Pita-Vidal M, Žitko R, Splitthoff LJ, Grünhaupt L, Wesdorp JJ, Liu Y, Kouwenhoven LP, Aguado R, Andersen CK, Kou A, van Heck B. Spectroscopy of Spin-Split Andreev Levels in a Quantum Dot with Superconducting Leads. PHYSICAL REVIEW LETTERS 2023; 131:097001. [PMID: 37721843 DOI: 10.1103/physrevlett.131.097001] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/28/2022] [Accepted: 07/27/2023] [Indexed: 09/20/2023]
Abstract
We use a hybrid superconductor-semiconductor transmon device to perform spectroscopy of a quantum dot Josephson junction tuned to be in a spin-1/2 ground state with an unpaired quasiparticle. Because of spin-orbit coupling, we resolve two flux-sensitive branches in the transmon spectrum, depending on the spin of the quasiparticle. A finite magnetic field shifts the two branches in energy, favoring one spin state and resulting in the anomalous Josephson effect. We demonstrate the excitation of the direct spin-flip transition using all-electrical control. Manipulation and control of the spin-flip transition enable the future implementation of charging energy protected Andreev spin qubits.
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Affiliation(s)
- Arno Bargerbos
- QuTech and Kavli Institute of Nanoscience, Delft University of Technology, 2600 GA Delft, Netherlands
| | - Marta Pita-Vidal
- QuTech and Kavli Institute of Nanoscience, Delft University of Technology, 2600 GA Delft, Netherlands
| | - Rok Žitko
- Jožef Stefan Institute, Jamova 39, SI-1000 Ljubljana, Slovenia
- Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia
| | - Lukas J Splitthoff
- QuTech and Kavli Institute of Nanoscience, Delft University of Technology, 2600 GA Delft, Netherlands
| | - Lukas Grünhaupt
- QuTech and Kavli Institute of Nanoscience, Delft University of Technology, 2600 GA Delft, Netherlands
| | - Jaap J Wesdorp
- QuTech and Kavli Institute of Nanoscience, Delft University of Technology, 2600 GA Delft, Netherlands
| | - Yu Liu
- Center for Quantum Devices, Niels Bohr Institute, University of Copenhagen, 2100 Copenhagen, Denmark
| | - Leo P Kouwenhoven
- QuTech and Kavli Institute of Nanoscience, Delft University of Technology, 2600 GA Delft, Netherlands
| | - Ramón Aguado
- Instituto de Ciencia de Materiales de Madrid (ICMM), Consejo Superior de Investigaciones Cientificas (CSIC), Sor Juana Ines de la Cruz 3, 28049 Madrid, Spain
| | | | - Angela Kou
- Department of Physics and Frederick Seitz Materials Research Laboratory, University of Illinois Urbana-Champaign, Urbana, Illinois 61801, USA
| | - Bernard van Heck
- Leiden Institute of Physics, Leiden University, Niels Bohrweg 2, 2333 CA Leiden, Netherlands
- Dipartimento di Fisica, Università di Roma "La Sapienza", P.le Aldo Moro 5, 00185 Roma, Italy
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15
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Ibabe A, Gómez M, Steffensen GO, Kanne T, Nygård J, Yeyati AL, Lee EJH. Joule spectroscopy of hybrid superconductor-semiconductor nanodevices. Nat Commun 2023; 14:2873. [PMID: 37208316 DOI: 10.1038/s41467-023-38533-2] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/06/2023] [Accepted: 05/05/2023] [Indexed: 05/21/2023] Open
Abstract
Hybrid superconductor-semiconductor devices offer highly tunable platforms, potentially suitable for quantum technology applications, that have been intensively studied in the past decade. Here we establish that measurements of the superconductor-to-normal transition originating from Joule heating provide a powerful spectroscopical tool to characterize such hybrid devices. Concretely, we apply this technique to junctions in full-shell Al-InAs nanowires in the Little-Parks regime and obtain detailed information of each lead independently and in a single measurement, including differences in the superconducting coherence lengths of the leads, inhomogeneous covering of the epitaxial shell, and the inverse superconducting proximity effect; all-in-all constituting a unique fingerprint of each device with applications in the interpretation of low-bias data, the optimization of device geometries, and the uncovering of disorder in these systems. Besides the practical uses, our work also underscores the importance of heating in hybrid devices, an effect that is often overlooked.
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Affiliation(s)
- A Ibabe
- Departamento de Física de la Materia Condensada, Universidad Autónoma de Madrid, Madrid, Spain
- Condensed Matter Physics Center (IFIMAC), Universidad Autónoma de Madrid, Madrid, Spain
| | - M Gómez
- Departamento de Física de la Materia Condensada, Universidad Autónoma de Madrid, Madrid, Spain
- Condensed Matter Physics Center (IFIMAC), Universidad Autónoma de Madrid, Madrid, Spain
| | - G O Steffensen
- Condensed Matter Physics Center (IFIMAC), Universidad Autónoma de Madrid, Madrid, Spain
- Departamento de Física Teórica de la Materia Condensada, Universidad Autónoma de Madrid, Madrid, Spain
| | - T Kanne
- Center for Quantum Devices, Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark
| | - J Nygård
- Center for Quantum Devices, Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark
| | - A Levy Yeyati
- Condensed Matter Physics Center (IFIMAC), Universidad Autónoma de Madrid, Madrid, Spain
- Departamento de Física Teórica de la Materia Condensada, Universidad Autónoma de Madrid, Madrid, Spain
| | - E J H Lee
- Departamento de Física de la Materia Condensada, Universidad Autónoma de Madrid, Madrid, Spain.
- Condensed Matter Physics Center (IFIMAC), Universidad Autónoma de Madrid, Madrid, Spain.
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16
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Trivini S, Ortuzar J, Vaxevani K, Li J, Bergeret FS, Cazalilla MA, Pascual JI. Cooper Pair Excitation Mediated by a Molecular Quantum Spin on a Superconducting Proximitized Gold Film. PHYSICAL REVIEW LETTERS 2023; 130:136004. [PMID: 37067302 DOI: 10.1103/physrevlett.130.136004] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/01/2022] [Accepted: 02/17/2023] [Indexed: 06/19/2023]
Abstract
Breaking a correlated pair in a superconductor requires an even number of fermions providing at least twice the pairing energy Δ. Here, we show that a single tunneling electron can also excite a pair breaking excitation in a proximitized gold film in the presence of magnetic impurities. Combining scanning tunneling spectroscopy with theoretical modeling, we map the excitation spectrum of an Fe-porphyrin molecule on the Au/V(100) proximitized surface into a manifold of entangled Yu-Shiba-Rusinov and spin excitations. Pair excitations emerge in the tunneling spectra as peaks outside the spectral gap only in the strong coupling regime, where the presence of a bound quasiparticle in the ground state ensures the even fermion parity of the excitation. Our results unravel the quantum nature of magnetic impurities on superconductors and demonstrate that pair excitations unequivocally reveal the parity of the ground state.
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Affiliation(s)
| | - Jon Ortuzar
- CIC nanoGUNE-BRTA, 20018 Donostia-San Sebastián, Spain
| | | | - Jingchen Li
- CIC nanoGUNE-BRTA, 20018 Donostia-San Sebastián, Spain
- School of Physics, Sun Yat-sen University, Guangzhou 510275, China
| | - F Sebastian Bergeret
- Centro de Física de Materiales (CFM-MPC) Centro Mixto CSIC-UPV/EHU, E-20018 Donostia-San Sebastián, Spain
- Donostia International Physics Center (DIPC), 20018 Donostia-San Sebastian, Spain
| | - Miguel A Cazalilla
- Donostia International Physics Center (DIPC), 20018 Donostia-San Sebastian, Spain
- Ikerbasque, Basque Foundation for Science, 48013 Bilbao, Spain
| | - Jose Ignacio Pascual
- CIC nanoGUNE-BRTA, 20018 Donostia-San Sebastián, Spain
- Ikerbasque, Basque Foundation for Science, 48013 Bilbao, Spain
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17
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Haxell DZ, Cheah E, Křížek F, Schott R, Ritter MF, Hinderling M, Belzig W, Bruder C, Wegscheider W, Riel H, Nichele F. Measurements of Phase Dynamics in Planar Josephson Junctions and SQUIDs. PHYSICAL REVIEW LETTERS 2023; 130:087002. [PMID: 36898094 DOI: 10.1103/physrevlett.130.087002] [Citation(s) in RCA: 3] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/12/2022] [Revised: 09/15/2022] [Accepted: 01/19/2023] [Indexed: 06/18/2023]
Abstract
We experimentally investigate the stochastic phase dynamics of planar Josephson junctions (JJs) and superconducting quantum interference devices (SQUIDs) defined in epitaxial InAs/Al heterostructures, and characterized by a large ratio of Josephson energy to charging energy. We observe a crossover from a regime of macroscopic quantum tunneling to one of phase diffusion as a function of temperature, where the transition temperature T^{*} is gate-tunable. The switching probability distributions are shown to be consistent with a small shunt capacitance and moderate damping, resulting in a switching current which is a small fraction of the critical current. Phase locking between two JJs leads to a difference in switching current between that of a JJ measured in isolation and that of the same JJ measured in an asymmetric SQUID loop. In the case of the loop, T^{*} is also tuned by a magnetic flux.
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Affiliation(s)
- D Z Haxell
- IBM Research Europe-Zurich, Säumerstrasse 4, 8803 Rüschlikon, Switzerland
| | - E Cheah
- Solid State Physics Laboratory, ETH Zurich, 8093 Zurich, Switzerland
| | - F Křížek
- IBM Research Europe-Zurich, Säumerstrasse 4, 8803 Rüschlikon, Switzerland
- Solid State Physics Laboratory, ETH Zurich, 8093 Zurich, Switzerland
| | - R Schott
- Solid State Physics Laboratory, ETH Zurich, 8093 Zurich, Switzerland
| | - M F Ritter
- IBM Research Europe-Zurich, Säumerstrasse 4, 8803 Rüschlikon, Switzerland
| | - M Hinderling
- IBM Research Europe-Zurich, Säumerstrasse 4, 8803 Rüschlikon, Switzerland
| | - W Belzig
- Fachbereich Physik, Universität Konstanz, D-78457 Konstanz, Germany
| | - C Bruder
- Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland
| | - W Wegscheider
- Solid State Physics Laboratory, ETH Zurich, 8093 Zurich, Switzerland
| | - H Riel
- IBM Research Europe-Zurich, Säumerstrasse 4, 8803 Rüschlikon, Switzerland
| | - F Nichele
- IBM Research Europe-Zurich, Säumerstrasse 4, 8803 Rüschlikon, Switzerland
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18
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Rieger D, Günzler S, Spiecker M, Paluch P, Winkel P, Hahn L, Hohmann JK, Bacher A, Wernsdorfer W, Pop IM. Granular aluminium nanojunction fluxonium qubit. NATURE MATERIALS 2023; 22:194-199. [PMID: 36482206 DOI: 10.1038/s41563-022-01417-9] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/18/2022] [Accepted: 10/20/2022] [Indexed: 06/17/2023]
Abstract
Mesoscopic Josephson junctions, consisting of overlapping superconducting electrodes separated by a nanometre-thin oxide layer, provide a precious source of nonlinearity for superconducting quantum circuits. Here we show that in a fluxonium qubit, the role of the Josephson junction can also be played by a lithographically defined, self-structured granular aluminium nanojunction: a superconductor-insulator-superconductor Josephson junction obtained in a single-layer, zero-angle evaporation. The measured spectrum of the resulting qubit, which we nickname gralmonium, is indistinguishable from that of a standard fluxonium. Remarkably, the lack of a mesoscopic parallel plate capacitor gives rise to an intrinsically large granular aluminium nanojunction charging energy in the range of tens of gigahertz, comparable to its Josephson energy. We measure coherence times in the microsecond range and we observe spontaneous jumps of the value of the Josephson energy on timescales from milliseconds to days, which offers a powerful diagnostics tool for microscopic defects in superconducting materials.
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Affiliation(s)
- D Rieger
- Physikalisches Institut, Karlsruhe Institute of Technology, Karlsruhe, Germany.
| | - S Günzler
- Physikalisches Institut, Karlsruhe Institute of Technology, Karlsruhe, Germany
- Institute for Quantum Materials and Technologies, Karlsruhe Institute of Technology, Eggenstein-Leopoldshafen, Germany
| | - M Spiecker
- Physikalisches Institut, Karlsruhe Institute of Technology, Karlsruhe, Germany
| | - P Paluch
- Physikalisches Institut, Karlsruhe Institute of Technology, Karlsruhe, Germany
- Institute for Quantum Materials and Technologies, Karlsruhe Institute of Technology, Eggenstein-Leopoldshafen, Germany
| | - P Winkel
- Physikalisches Institut, Karlsruhe Institute of Technology, Karlsruhe, Germany
- Institute for Quantum Materials and Technologies, Karlsruhe Institute of Technology, Eggenstein-Leopoldshafen, Germany
| | - L Hahn
- Institute of Microstructure Technology, Karlsruhe Institute of Technology, Eggenstein-Leopoldshafen, Germany
| | - J K Hohmann
- Institute of Microstructure Technology, Karlsruhe Institute of Technology, Eggenstein-Leopoldshafen, Germany
| | - A Bacher
- Institute of Microstructure Technology, Karlsruhe Institute of Technology, Eggenstein-Leopoldshafen, Germany
| | - W Wernsdorfer
- Physikalisches Institut, Karlsruhe Institute of Technology, Karlsruhe, Germany
- Institute for Quantum Materials and Technologies, Karlsruhe Institute of Technology, Eggenstein-Leopoldshafen, Germany
| | - I M Pop
- Physikalisches Institut, Karlsruhe Institute of Technology, Karlsruhe, Germany.
- Institute for Quantum Materials and Technologies, Karlsruhe Institute of Technology, Eggenstein-Leopoldshafen, Germany.
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19
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Arrachea L. Energy dynamics, heat production and heat-work conversion with qubits: toward the development of quantum machines. REPORTS ON PROGRESS IN PHYSICS. PHYSICAL SOCIETY (GREAT BRITAIN) 2023; 86:036501. [PMID: 36603220 DOI: 10.1088/1361-6633/acb06b] [Citation(s) in RCA: 3] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/27/2022] [Accepted: 01/05/2023] [Indexed: 06/17/2023]
Abstract
We present an overview of recent advances in the study of energy dynamics and mechanisms for energy conversion in qubit systems with special focus on realizations in superconducting quantum circuits. We briefly introduce the relevant theoretical framework to analyze heat generation, energy transport and energy conversion in these systems with and without time-dependent driving considering the effect of equilibrium and non-equilibrium environments. We analyze specific problems and mechanisms under current investigation in the context of qubit systems. These include the problem of energy dissipation and possible routes for its control, energy pumping between driving sources and heat pumping between reservoirs, implementation of thermal machines and mechanisms for energy storage. We highlight the underlying fundamental phenomena related to geometrical and topological properties, as well as many-body correlations. We also present an overview of recent experimental activity in this field.
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Affiliation(s)
- Liliana Arrachea
- Escuela de Ciencia y Tecnología and ICIFI, Universidad de San Martín, Av. 25 de Mayo y Francia, 1650 Buenos Aires, Argentina
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20
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Macrì N, Giannelli L, Paladino E, Falci G. Coarse-Grained Effective Hamiltonian via the Magnus Expansion for a Three-Level System. ENTROPY (BASEL, SWITZERLAND) 2023; 25:234. [PMID: 36832601 PMCID: PMC9954943 DOI: 10.3390/e25020234] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 12/15/2022] [Revised: 01/18/2023] [Accepted: 01/25/2023] [Indexed: 06/18/2023]
Abstract
Quantum state processing is one of the main tools of quantum technologies. While real systems are complicated and/or may be driven by non-ideal control, they may nevertheless exhibit simple dynamics approximately confined to a low-energy Hilbert subspace. Adiabatic elimination is the simplest approximation scheme allowing us to derive in certain cases an effective Hamiltonian operating in a low-dimensional Hilbert subspace. However, these approximations may present ambiguities and difficulties, hindering a systematic improvement of their accuracy in larger and larger systems. Here, we use the Magnus expansion as a systematic tool to derive ambiguity-free effective Hamiltonians. We show that the validity of the approximations ultimately leverages only on a proper coarse-graining in time of the exact dynamics. We validate the accuracy of the obtained effective Hamiltonians with suitably tailored fidelities of quantum operations.
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Affiliation(s)
- Nicola Macrì
- Dipartimento di Fisica e Astronomia “Ettore Majorana”, Università di Catania, 95123 Catania, Italy
- Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Catania, 95123 Catania, Italy
| | - Luigi Giannelli
- Dipartimento di Fisica e Astronomia “Ettore Majorana”, Università di Catania, 95123 Catania, Italy
- CNR-IMM, UoS Università, 95123 Catania, Italy
| | - Elisabetta Paladino
- Dipartimento di Fisica e Astronomia “Ettore Majorana”, Università di Catania, 95123 Catania, Italy
- Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Catania, 95123 Catania, Italy
- CNR-IMM, UoS Università, 95123 Catania, Italy
| | - Giuseppe Falci
- Dipartimento di Fisica e Astronomia “Ettore Majorana”, Università di Catania, 95123 Catania, Italy
- Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Catania, 95123 Catania, Italy
- CNR-IMM, UoS Università, 95123 Catania, Italy
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21
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Potts H, Aspegren M, Debbarma R, Lehmann S, Thelander C. Large-bias spectroscopy of Yu-Shiba-Rusinov states in a double quantum dot. NANOTECHNOLOGY 2023; 34:135002. [PMID: 36595334 DOI: 10.1088/1361-6528/aca90e] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/09/2022] [Accepted: 12/05/2022] [Indexed: 06/17/2023]
Abstract
We have performed tunnel transport spectroscopy on a quantum dot (QD) molecule proximitized by a superconducting contact. In such a system, the scattering between QD spins and Bogoliubov quasiparticles leads to the formation of Yu-Shiba-Rusinov (YSR) states within the superconducting gap. In this work, we investigate interactions appearing when one- and two-electron spin states in a double-QD energetically align with the superconducting gap edge. We find that the inter-dot spin-triplet state interacts considerably stronger with the superconductor than the corresponding singlet, pointing to stronger screening. By forming a ring molecule with a significant orbital contribution to the effectiveg-factor, we observe interactions of all four spin-orbital one-electron states with the superconductor under a weak magnetic field.
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Affiliation(s)
- Heidi Potts
- Division of Solid State Physics and NanoLund, Lund University, SE-221 00 Lund, Sweden
| | - Markus Aspegren
- Division of Solid State Physics and NanoLund, Lund University, SE-221 00 Lund, Sweden
| | - Rousan Debbarma
- Division of Solid State Physics and NanoLund, Lund University, SE-221 00 Lund, Sweden
| | - Sebastian Lehmann
- Division of Solid State Physics and NanoLund, Lund University, SE-221 00 Lund, Sweden
| | - Claes Thelander
- Division of Solid State Physics and NanoLund, Lund University, SE-221 00 Lund, Sweden
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22
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Fatemi V, Kurilovich PD, Hays M, Bouman D, Connolly T, Diamond S, Frattini NE, Kurilovich VD, Krogstrup P, Nygård J, Geresdi A, Glazman LI, Devoret MH. Microwave Susceptibility Observation of Interacting Many-Body Andreev States. PHYSICAL REVIEW LETTERS 2022; 129:227701. [PMID: 36493424 DOI: 10.1103/physrevlett.129.227701] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/17/2021] [Accepted: 09/13/2022] [Indexed: 06/17/2023]
Abstract
Electrostatic charging affects the many-body spectrum of Andreev states, yet its influence on their microwave properties has not been elucidated. We developed a circuit quantum electrodynamics probe that, in addition to transition spectroscopy, measures the microwave susceptibility of different states of a semiconductor nanowire weak link with a single dominant (spin-degenerate) Andreev level. We found that the microwave susceptibility does not exhibit a particle-hole symmetry, which we qualitatively explain as an influence of Coulomb interaction. Moreover, our state-selective measurement reveals a large, π-phase shifted contribution to the response common to all many-body states which can be interpreted as arising from a phase-dependent continuum in the superconducting density of states.
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Affiliation(s)
- V Fatemi
- Departments of Physics and Applied Physics, Yale University, New Haven, Connecticut 06520, USA
| | - P D Kurilovich
- Departments of Physics and Applied Physics, Yale University, New Haven, Connecticut 06520, USA
| | - M Hays
- Departments of Physics and Applied Physics, Yale University, New Haven, Connecticut 06520, USA
| | - D Bouman
- QuTech and Delft University of Technology, 2600 GA Delft, Netherlands
- Kavli Institute of Nanoscience, Delft University of Technology, 2600 GA Delft, Netherlands
| | - T Connolly
- Departments of Physics and Applied Physics, Yale University, New Haven, Connecticut 06520, USA
| | - S Diamond
- Departments of Physics and Applied Physics, Yale University, New Haven, Connecticut 06520, USA
| | - N E Frattini
- Departments of Physics and Applied Physics, Yale University, New Haven, Connecticut 06520, USA
| | - V D Kurilovich
- Departments of Physics and Applied Physics, Yale University, New Haven, Connecticut 06520, USA
| | - P Krogstrup
- Niels Bohr Institute, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark
| | - J Nygård
- Center for Quantum Devices, Niels Bohr Institute, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark
| | - A Geresdi
- QuTech and Delft University of Technology, 2600 GA Delft, Netherlands
- Kavli Institute of Nanoscience, Delft University of Technology, 2600 GA Delft, Netherlands
- Quantum Device Physics Laboratory, Department of Microtechnology and Nanoscience, Chalmers University of Technology, SE 41296 Gothenburg, Sweden
| | - L I Glazman
- Departments of Physics and Applied Physics, Yale University, New Haven, Connecticut 06520, USA
| | - M H Devoret
- Departments of Physics and Applied Physics, Yale University, New Haven, Connecticut 06520, USA
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23
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Lee M, López R, Xu HQ, Platero G. Proposal for Detection of the 0^{'} and π^{'} Phases in Quantum-Dot Josephson Junctions. PHYSICAL REVIEW LETTERS 2022; 129:207701. [PMID: 36462010 DOI: 10.1103/physrevlett.129.207701] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/15/2022] [Accepted: 10/03/2022] [Indexed: 06/17/2023]
Abstract
The competition between the Kondo correlation and superconductivity in quantum-dot Josephson junctions (QDJJs) has been known to drive a quantum phase transition between 0 and π junctions. Theoretical studies so far have predicted that under strong Coulomb correlations the 0-π transition should go through intermediate states, 0^{'} and π^{'} phases. By combining a nonperturbative numerical method and the resistively shunted junction model, we investigated the magnetic-field-driven phase transition of the QDJJs in the Kondo regime and found that the low-field magnetotransport exhibits a unique feature which can be used to distinguish the intermediate phases. In particular, the magnetic-field driven π^{'}-π transition is found to lead to the enhancement of the supercurrent which is strongly related to the Kondo effect.
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Affiliation(s)
- Minchul Lee
- Department of Applied Physics and Institute of Natural Science, College of Applied Science, Kyung Hee University, Yongin 17104, Korea
| | - Rosa López
- Institut de Física Interdisciplinària i de Sistemes Complexos IFISC (CSIC-UIB), E-07122 Palma de Mallorca, Spain
| | - H Q Xu
- Beijing Key Laboratory of Quantum Devices, Key Laboratory for the Physics and Chemistry of Nanodevices, and School of Electronics, Peking University, Beijing 100871, China
- Beijing Academy of Quantum Information Sciences, Beijing 100193, China
| | - Gloria Platero
- Instituto de Ciencia de Materiales de Madrid (ICMM-CSIC), 28049 Cantoblanco, Madrid, Spain
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24
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Moehle C, Rout PK, Jainandunsing NA, Kuiri D, Ke CT, Xiao D, Thomas C, Manfra MJ, Nowak MP, Goswami S. Controlling Andreev Bound States with the Magnetic Vector Potential. NANO LETTERS 2022; 22:8601-8607. [PMID: 36279222 PMCID: PMC9650727 DOI: 10.1021/acs.nanolett.2c03130] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 08/07/2022] [Revised: 10/16/2022] [Indexed: 06/16/2023]
Abstract
Tunneling spectroscopy measurements are often used to probe the energy spectrum of Andreev bound states (ABSs) in semiconductor-superconductor hybrids. Recently, this spectroscopy technique has been incorporated into planar Josephson junctions (JJs) formed in two-dimensional electron gases, a potential platform to engineer phase-controlled topological superconductivity. Here, we perform ABS spectroscopy at the two ends of planar JJs and study the effects of the magnetic vector potential on the ABS spectrum. We show that the local superconducting phase difference arising from the vector potential is equal in magnitude and opposite in sign at the two ends, in agreement with a model that assumes localized ABSs near the tunnel barriers. Complemented with microscopic simulations, our experiments demonstrate that the local phase difference can be used to estimate the relative position of localized ABSs separated by a few hundred nanometers.
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Affiliation(s)
- Christian
M. Moehle
- QuTech
and Kavli Institute of Nanoscience, Delft
University of Technology, 2600 GADelft, The Netherlands
| | - Prasanna K. Rout
- QuTech
and Kavli Institute of Nanoscience, Delft
University of Technology, 2600 GADelft, The Netherlands
| | - Nayan A. Jainandunsing
- QuTech
and Kavli Institute of Nanoscience, Delft
University of Technology, 2600 GADelft, The Netherlands
| | - Dibyendu Kuiri
- Academic
Centre for Materials and Nanotechnology, AGH University of Science and Technology, 30-059Krakow, Poland
| | - Chung Ting Ke
- QuTech
and Kavli Institute of Nanoscience, Delft
University of Technology, 2600 GADelft, The Netherlands
| | - Di Xiao
- Department
of Physics and Astronomy, Purdue University, West Lafayette, Indiana47907, United States
| | - Candice Thomas
- Department
of Physics and Astronomy, Purdue University, West Lafayette, Indiana47907, United States
| | - Michael J. Manfra
- Department
of Physics and Astronomy, Purdue University, West Lafayette, Indiana47907, United States
- Elmore
School of Electrical and Computer Engineering, Purdue University, West Lafayette, Indiana47907, United States
- School
of Materials Engineering, Purdue University, West Lafayette, Indiana47907, United States
- Microsoft
Quantum Lab West Lafayette, West
Lafayette, Indiana47907, United States
| | - Michał P. Nowak
- Academic
Centre for Materials and Nanotechnology, AGH University of Science and Technology, 30-059Krakow, Poland
| | - Srijit Goswami
- QuTech
and Kavli Institute of Nanoscience, Delft
University of Technology, 2600 GADelft, The Netherlands
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25
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Fong KC. Graphene amplifier reaches the quantum limit. NATURE NANOTECHNOLOGY 2022; 17:1128-1129. [PMID: 36319754 DOI: 10.1038/s41565-022-01239-5] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/16/2023]
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26
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Chirolli L, Yao NY, Moore JE. SWAP Gate between a Majorana Qubit and a Parity-Protected Superconducting Qubit. PHYSICAL REVIEW LETTERS 2022; 129:177701. [PMID: 36332252 DOI: 10.1103/physrevlett.129.177701] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/07/2022] [Accepted: 09/15/2022] [Indexed: 06/16/2023]
Abstract
High fidelity quantum information processing requires a combination of fast gates and long-lived quantum memories. In this Letter, we propose a hybrid architecture, where a parity-protected superconducting qubit is directly coupled to a Majorana qubit, which plays the role of a quantum memory. The superconducting qubit is based upon a π-periodic Josephson junction realized with gate-tunable semiconducting wires, where the tunneling of individual Cooper pairs is suppressed. One of the wires additionally contains four Majorana zero modes that define a qubit. We demonstrate that this enables the implementation of a SWAP gate, allowing for the transduction of quantum information between the topological and conventional qubit. This architecture combines fast gates, which can be realized with the superconducting qubit, with a topologically protected Majorana memory.
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Affiliation(s)
- Luca Chirolli
- Department of Physics, University of California Berkeley, Berkeley, California 94720, USA
- Istituto Nanoscienze-CNR, I-56127 Pisa, Italy
| | - Norman Y Yao
- Department of Physics, University of California Berkeley, Berkeley, California 94720, USA
- Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
| | - Joel E Moore
- Department of Physics, University of California Berkeley, Berkeley, California 94720, USA
- Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
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27
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Matute-Cañadas FJ, Metzger C, Park S, Tosi L, Krogstrup P, Nygård J, Goffman MF, Urbina C, Pothier H, Yeyati AL. Signatures of Interactions in the Andreev Spectrum of Nanowire Josephson Junctions. PHYSICAL REVIEW LETTERS 2022; 128:197702. [PMID: 35622049 DOI: 10.1103/physrevlett.128.197702] [Citation(s) in RCA: 6] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/17/2021] [Accepted: 04/01/2022] [Indexed: 06/15/2023]
Abstract
We performed microwave spectroscopy of an InAs nanowire between superconducting contacts implementing a finite-length, multichannel Josephson weak link. Certain features in the spectra, such as the splitting by spin-orbit interactions of the transition lines among Andreev states, have been already understood in terms of noninteracting models. However, we identify here additional transitions, which evidence the presence of Coulomb interactions. By combining experimental measurements and model calculations, we reach a qualitative understanding of these very rich Andreev spectra.
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Affiliation(s)
- F J Matute-Cañadas
- Departamento de Física Teórica de la Materia Condensada, Condensed Matter Physics Center (IFIMAC) and Instituto Nicolás Cabrera, Universidad Autónoma de Madrid, 28049 Madrid, Spain
| | - C Metzger
- Quantronics group, Service de Physique de l'État Condensé (CNRS, UMR 3680), IRAMIS, CEA-Saclay, Université Paris-Saclay, 91191 Gif-sur-Yvette, France
| | - Sunghun Park
- Departamento de Física Teórica de la Materia Condensada, Condensed Matter Physics Center (IFIMAC) and Instituto Nicolás Cabrera, Universidad Autónoma de Madrid, 28049 Madrid, Spain
| | - L Tosi
- Centro Atómico Bariloche and Instituto Balseiro, CNEA, CONICET, 8400 San Carlos de Bariloche, Río Negro, Argentina
| | - P Krogstrup
- Center for Quantum Devices, Niels Bohr Institute, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark
| | - J Nygård
- Center for Quantum Devices, Niels Bohr Institute, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark
| | - M F Goffman
- Quantronics group, Service de Physique de l'État Condensé (CNRS, UMR 3680), IRAMIS, CEA-Saclay, Université Paris-Saclay, 91191 Gif-sur-Yvette, France
| | - C Urbina
- Quantronics group, Service de Physique de l'État Condensé (CNRS, UMR 3680), IRAMIS, CEA-Saclay, Université Paris-Saclay, 91191 Gif-sur-Yvette, France
| | - H Pothier
- Quantronics group, Service de Physique de l'État Condensé (CNRS, UMR 3680), IRAMIS, CEA-Saclay, Université Paris-Saclay, 91191 Gif-sur-Yvette, France
| | - A Levy Yeyati
- Departamento de Física Teórica de la Materia Condensada, Condensed Matter Physics Center (IFIMAC) and Instituto Nicolás Cabrera, Universidad Autónoma de Madrid, 28049 Madrid, Spain
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28
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Alfieri A, Anantharaman SB, Zhang H, Jariwala D. Nanomaterials for Quantum Information Science and Engineering. ADVANCED MATERIALS (DEERFIELD BEACH, FLA.) 2022:e2109621. [PMID: 35139247 DOI: 10.1002/adma.202109621] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/25/2021] [Revised: 02/04/2022] [Indexed: 06/14/2023]
Abstract
Quantum information science and engineering (QISE)-which entails the use of quantum mechanical states for information processing, communications, and sensing-and the area of nanoscience and nanotechnology have dominated condensed matter physics and materials science research in the 21st century. Solid-state devices for QISE have, to this point, predominantly been designed with bulk materials as their constituents. This review considers how nanomaterials (i.e., materials with intrinsic quantum confinement) may offer inherent advantages over conventional materials for QISE. The materials challenges for specific types of qubits, along with how emerging nanomaterials may overcome these challenges, are identified. Challenges for and progress toward nanomaterials-based quantum devices are condidered. The overall aim of the review is to help close the gap between the nanotechnology and quantum information communities and inspire research that will lead to next-generation quantum devices for scalable and practical quantum applications.
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Affiliation(s)
- Adam Alfieri
- Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, PA, 19104, USA
| | - Surendra B Anantharaman
- Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, PA, 19104, USA
| | - Huiqin Zhang
- Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, PA, 19104, USA
| | - Deep Jariwala
- Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, PA, 19104, USA
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29
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Wendin G, Shumeiko V. Coherent manipulation of a spin qubit. Science 2021; 373:390-391. [PMID: 34437104 DOI: 10.1126/science.abk0929] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/02/2022]
Affiliation(s)
- Göran Wendin
- Chalmers University of Technology, 41296 Gothenburg, Sweden.
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