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arxiv: 2402.02118 · v1 · submitted 2024-02-03 · ❄️ cond-mat.mes-hall · quant-ph

Generalized transmon Hamiltonian for Andreev spin qubits

Luka Pave\v{s}i\'c , Rok \v{Z}itko This is my paper

Pith reviewed 2026-05-24 03:52 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall quant-ph
keywords Andreev spin qubitstransmon HamiltonianJosephson junctionquantum dotcharging energyRichardson modelphase fluctuationsexact diagonalization
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The pith

Flat-band Richardson approximation yields an exact solvable generalized transmon Hamiltonian that treats quantum-dot, Josephson, and charging physics on equal footing.

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

The paper develops a method for an interacting quantum dot placed inside a Josephson junction that also carries finite charging energy. It starts from the Richardson model of superconductivity and imposes a flat-band approximation that shrinks the Hilbert space until exact diagonalization becomes feasible. The resulting Hamiltonian simultaneously includes the dot's spin and charge degrees of freedom, the superconducting phase and its quantum fluctuations, and the Coulomb repulsion. Because all these elements are kept at the same level of description, the model directly supplies the low-energy spectrum, time-dependent dynamics, and microwave transition matrix elements needed for Andreev spin qubits embedded in transmon circuits.

Core claim

By applying the flat-band approximation to the Richardson model, the full Hilbert space of the quantum-dot–Josephson-junction system with charging energy reduces to a tractable size that still contains every state required for low-energy physics; exact diagonalization then produces a generalized transmon Hamiltonian whose eigenstates encode the coupled fluctuations of the superconducting phase and the quantum-dot spin.

What carries the argument

Flat-band approximation of the Richardson model, which truncates the superconducting quasiparticle spectrum to allow exact diagonalization while preserving all low-energy states and the phase–dot coupling.

If this is right

  • The Hamiltonian can be used for Andreev spin qubits in transmon circuits across the full range of charging energies and tunnel couplings.
  • Time-dependent driving protocols can be simulated without additional approximations.
  • Transition matrix elements for charge-, current-, and spin-flip microwave processes become directly computable.
  • Quantum phase fluctuations and their back-action on the dot spin are automatically included.

Where Pith is reading between the lines

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

  • The same diagonalization procedure could be reused to extract effective qubit-qubit interactions when multiple dots share a common transmon.
  • Adding a small number of non-flat bands would provide a controlled way to test the approximation's accuracy at higher energies.
  • The method supplies a microscopic starting point for deriving reduced master equations that include both charge and spin noise channels.

Load-bearing premise

The flat-band truncation keeps every state that matters for the low-energy spectrum and dynamics.

What would settle it

Direct comparison of the calculated low-energy spectrum or microwave transition rates against measurements performed on an actual Andreev spin qubit inside a transmon circuit.

Figures

Figures reproduced from arXiv: 2402.02118 by Luka Pave\v{s}i\'c, Rok \v{Z}itko.

Figure 1
Figure 1. Figure 1: Model sketch. The QD is a single energy level embedded [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Block diagonalizing the Hamiltonian at Ec = 0 with the Fourier transform. In the analogy with the tight-binding chain, the t2 block denotes coupling within each chain site, while t1 is hopping between the sites. thus related via Fourier transform. When performed on the m-states, we obtain a basis in the ϕ-space: |ϕ, fL, fR⟩ = 1 √ mtot + 1 X m e iϕ(m+mtot)/2 |m, fL, fR⟩, (14) with mtot the total number of C… view at source ↗
Figure 3
Figure 3. Figure 3: Spectra of the SC-QD-SC junction for Ec = 0, tp = 0. (a, b) ϕ dependence of the lowest singlet (red) and the lowest doublet (blue) states for (a) the resonant case of ϵ = 0 and (b) for the generic case of ϵ = 0.1∆. Here U = 0. (c) v dependence of the two lowest singlets and the lowest doublet for ϕ = 0 (dashed) and ϕ = π (solid), for U = 0. (d) U dependence of the spectra at ϕ = 0 close to the particle-hol… view at source ↗
Figure 4
Figure 4. Figure 4: Effective Josephson energy E eff J , defined as the differ￾ence in the energy of the lowest state at ϕ = 0 and ϕ = π. (a) and (b) U = 0 results. Full lines: singlet (red) and doublet (blue); black dashed lines: expected perturbative corrections. (c) E eff J for the singlet ground state for different U with ϵ = −0.8(U/2), sightly shifted away from the particle-hole symmetric point. (d) E eff J for the doubl… view at source ↗
Figure 5
Figure 5. Figure 5: Effect of the tunneling amplitude of Cooper pairs across the reference junction, [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Effect of the charge repulsion Ec. (a) Evolution of the excitation spectrum with increasing Ec in the doublet (blue) and the singlet (red) sector. (b) Absolute values of the amplitudes in the ϕ-basis for the ground state and first two excitations in the doublet sector in the intermediate regime, Ec/(E eff J /2) = 10−3 . (c) Absolute values of amplitudes in the ϕ-basis (αϕ) and m-basis (αm) for the doublet … view at source ↗
Figure 7
Figure 7. Figure 7: Spin-orbit splitting of the doublet states at zero mag [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: SOC-induced level repulsion. (a) Spectra of the doublet manifold with increasing Zeeman splitting of the QD level with (dashed) no spin-orbit coupling and with (solid) tsc = 0.2∆, v↑↓ = 0.2∆. In both cases, the ground state at EZ = 0 is set to zero energy. (b) Amplitudes in the ϕ-space of the |0, ↑⟩ and |1, ↓⟩ eigenstates through the avoided crossing. Other parameters: U/∆ = 3, ϵ = − U 2 , v/∆ = 0.5, tp/∆ … view at source ↗
Figure 9
Figure 9. Figure 9: Coherent evolution in response to spin-rotating pulses. [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Matrix elements for transitions: overview of the [PITH_FULL_IMAGE:figures/full_fig_p013_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Magnetic field dependence: parallel field [PITH_FULL_IMAGE:figures/full_fig_p013_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Magnetic field dependence: perpendicular field [PITH_FULL_IMAGE:figures/full_fig_p014_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Asymptotic low-field behavior of current matrix ele [PITH_FULL_IMAGE:figures/full_fig_p015_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Quantum dot level [PITH_FULL_IMAGE:figures/full_fig_p015_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Quantum dot electron-electron repulsion U dependence. Parameters: ϵ/∆ = −1.5, v/∆ = 0.5, tsc = 0.2∆, v↑↓ = 0.2∆, tp/∆ = 0.1, Ec/∆ = 0.02, ϕext = π/2, Ez/∆ = 0.02 [PITH_FULL_IMAGE:figures/full_fig_p016_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Charging energy Ec dependence. Parameters: U/∆ = 3, ϵ = −U/2, v/∆ = 0.5, tsc = 0.2∆, v↑↓ = 0.2∆, tp/∆ = 0.1, ϕext = π/2, Ez/∆ = 0.02. N = 101 and n0 = 50 [PITH_FULL_IMAGE:figures/full_fig_p016_16.png] view at source ↗
Figure 18
Figure 18. Figure 18: Perpendicular magnetic field dependence. Parameters as [PITH_FULL_IMAGE:figures/full_fig_p017_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Ground state expectation value of charge and Josephson [PITH_FULL_IMAGE:figures/full_fig_p021_19.png] view at source ↗
read the original abstract

We solve the problem of an interacting quantum dot embedded in a Josephson junction between two superconductors with finite charging energy described by the transmon (Cooper pair box) Hamiltonian. The approach is based on the flat-band approximation of the Richardson model, which reduces the Hilbert space to the point where exact diagonalisation is possible while retaining all states that are necessary to describe the low energy phenomena. The presented method accounts for the physics of the quantum dot, the Josephson effect and the Coulomb repulsion (charging energy) at the same level. In particular, it captures the quantum fluctuations of the superconducting phase as well as the coupling between the superconducting phase and the quantum dot (spin) degrees of freedom. The method can be directly applied for modelling Andreev spin qubits embedded in transmon circuits in all parameter regimes, for describing time-dependent processes, and for the calculation of transition matrix elements for microwave-driven transmon, spin-flip and mixed transitions that involve coupling to charge or current degree of freedom.

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

2 major / 2 minor

Summary. The paper develops a generalized transmon Hamiltonian for an interacting quantum dot embedded in a Josephson junction with finite charging energy. It employs the flat-band approximation of the Richardson model to reduce the Hilbert space sufficiently for exact diagonalization while retaining states needed for low-energy physics, including superconducting phase fluctuations and their coupling to the quantum-dot spin. The resulting framework is claimed to treat the quantum dot, Josephson effect, and Coulomb repulsion on equal footing and is positioned for direct application to Andreev spin qubits in transmon circuits, including time-dependent dynamics and microwave transition matrix elements.

Significance. If the central approximation is validated, the work supplies a unified, numerically tractable description of hybrid superconducting systems that simultaneously incorporates charging energy, phase fluctuations, and spin-phase coupling. This addresses a practical modeling gap for Andreev spin qubits and enables calculation of mixed transitions involving charge or current degrees of freedom.

major comments (2)
  1. [§2] §2 (flat-band Richardson approximation): the assertion that the approximation retains every state required for low-energy phase-spin coupling is load-bearing for the central claim, yet the manuscript provides no explicit numerical benchmark against the dispersive (finite-bandwidth) Richardson model once finite E_C is restored. Without such a check, it remains unclear whether band dispersion introduces additional low-energy matrix elements that would alter the effective Hamiltonian.
  2. [§3] §3 (exact diagonalization and effective Hamiltonian): the derivation of the generalized transmon Hamiltonian from the reduced Hilbert space must demonstrate that the retained basis is closed under the charging-energy term; otherwise the claimed quantitative reliability across all parameter regimes is not guaranteed.
minor comments (2)
  1. Notation for the phase operator and its conjugate charge should be introduced with an explicit commutation relation at first use to avoid ambiguity when the effective Hamiltonian is written.
  2. Figure captions for the energy spectra should state the precise parameter values (E_C, E_J, dot level position) used, rather than referring only to 'typical' regimes.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and insightful comments on our manuscript. We address each of the major comments below and outline the revisions we plan to make.

read point-by-point responses

  1. Referee: [§2] §2 (flat-band Richardson approximation): the assertion that the approximation retains every state required for low-energy phase-spin coupling is load-bearing for the central claim, yet the manuscript provides no explicit numerical benchmark against the dispersive (finite-bandwidth) Richardson model once finite E_C is restored. Without such a check, it remains unclear whether band dispersion introduces additional low-energy matrix elements that would alter the effective Hamiltonian.

    Authors: The flat-band approximation is central to enabling exact diagonalization while preserving the low-energy physics of the Andreev spin qubit coupled to the transmon. We justify its use in the manuscript by noting that it retains all states necessary for the low-energy phenomena as per the Richardson model properties. However, we acknowledge the value of an explicit benchmark. In the revised manuscript, we will add a section or appendix providing a numerical comparison for small systems between the flat-band results and a truncated finite-bandwidth Richardson model with finite E_C, to verify that dispersion effects do not introduce significant alterations to the low-energy effective Hamiltonian. revision: yes

  2. Referee: [§3] §3 (exact diagonalization and effective Hamiltonian): the derivation of the generalized transmon Hamiltonian from the reduced Hilbert space must demonstrate that the retained basis is closed under the charging-energy term; otherwise the claimed quantitative reliability across all parameter regimes is not guaranteed.

    Authors: The reduced basis is obtained by diagonalizing the Richardson Hamiltonian in the flat-band limit, which spans the relevant charge and spin configurations for the quantum dot. The charging energy is the transmon charging term, which depends on the total excess charge. Since the basis states are eigenstates with well-defined parity or charge properties in the superconducting context, the charging term preserves the subspace. To address the concern, we will include in the revised manuscript an explicit proof or demonstration that the charging-energy operator maps the retained basis to itself, ensuring the effective Hamiltonian is well-defined within this space. revision: yes

Circularity Check

0 steps flagged

No circularity; method is an explicit approximation choice without self-referential reduction

full rationale

The paper states its central construction as the flat-band approximation of the Richardson model, which is presented as an input assumption that reduces the Hilbert space for exact diagonalization while retaining low-energy states. No quoted step shows a derived quantity defined in terms of itself, a fitted parameter renamed as a prediction, or a load-bearing claim justified only by self-citation. The abstract and description treat the approximation as a modeling choice whose validity is external to the derivation chain itself, making the overall approach self-contained against the stated model rather than circular.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of the flat-band approximation of the Richardson model for retaining all relevant low-energy states and on the standard transmon (Cooper pair box) Hamiltonian description of charging energy.

axioms (1)
  • domain assumption Flat-band approximation of the Richardson model reduces the Hilbert space while retaining all states necessary for low-energy phenomena
    Invoked to enable exact diagonalization of the interacting quantum-dot-plus-Josephson-junction system.

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Reference graph

Works this paper leans on

79 extracted references · 79 canonical work pages

  1. [1]

    In this simplest case the phase difference ϕ is a con- served scalar quantity (good quantum number) that labels the eigenstates of the system

    Subgap spectrum of the SC-QD-SC junction We begin by reproducing the basic properties of a JJ with an embedded QD, starting with tp = 0, i.e., without the refer- 7 ence junction, and Ec = 0, i.e., without any charging energy terms. In this simplest case the phase difference ϕ is a con- served scalar quantity (good quantum number) that labels the eigenstat...

    work page

  2. [2]

    perpendicular field

    Effective Josephson energy In analogy with a standard JJ, we define the effective Josephson energy as the half-width of the energy band ob- tained by varying ϕ, Eeff J = 1 2 max ϕ E(ϕ) − min ϕ E(ϕ) , (17) where E(ϕ) is the energy of the lowest state in the singlet or doublet subspace (the result is state-dependent). The extreme values are taken at ϕ = 0 a...

    work page

  3. [3]

    The model can be applied to other states as well

    the calculation of QD spin flipping induced by a pulse, 3) the investigation of transition matrix elements. The model can be applied to other states as well. While in this work we have focused on the doublet subspace, as appro- priate for Andreev spin qubits, it is equally possible to study the singlet subspace which is relevant for Andreev level qubits b...

    work page

  4. [4]

    Singlet states The singlet manifold has dimension 14, consisting of one state with no unpaired particles: • |mL, 0, mR, 0⟩, six states with two unpaired particles: • |mL, 0, mR, 2⟩, • |mL, 2, mR, 0⟩, • d† ↓d† ↑|mL, 0, mR, 0⟩, • d† ↑|mL, ↓, mR, 0⟩ − d† ↓|mL, ↑, mR, 0⟩ / √ 2, • d† ↑|mL, 0, mR, ↓⟩ − d† ↓|mL, 0, mR, ↑⟩ / √ 2, • (|mL, ↑, mR, ↓⟩ − |mL, ↓, mR, ↑...

    work page

  5. [5]

    Doublet states The doublet manifold also has dimension 14. It contains three states with one unpaired particle: • d† ↑|mL, 0, mR, 0⟩, • |mL, ↑, mR, 0⟩, • |mL, 0, mR, ↑⟩, eight states with three unpaired particles: • d† ↓d† ↑|mL, ↑, mR, 0⟩, • d† ↓d† ↑|mL, 0, mR, ↑⟩, • d† ↑|mL, 2, mR, 0⟩, • d† ↑|mL, 0, mR, 2⟩, • |mL, ↑, mR, 2⟩, • |mL, 2, mR, ↑⟩, • d† ↓|mL, ...

    work page

  6. [6]

    The off-diagonal part is generated by first calculating ana- lytical expressions for all matrix elements ⟨d′, m′ L, f ′ L, m′ R, f ′ R|Hhyb|d, mL, fL, mR, fR⟩

    Hamiltonian generation The diagonal part of the Hamiltonian is Hdiag =ϵnQD + U nQD↑nQD↓ + g 2 X βσ f † βσ fβσ + X β Eβ c X σ f † βσ fβσ + 2mβ − n0 !2 (A1) and can be read out directly for each basis state. The off-diagonal part is generated by first calculating ana- lytical expressions for all matrix elements ⟨d′, m′ L, f ′ L, m′ R, f ′ R|Hhyb|d, mL, fL, ...

    work page

  7. [7]

    Kjaergaard, M

    M. Kjaergaard, M. E. Schwartz, J. Braum ¨uller, P. Krantz, J. I.- J. Wang, S. Gustavsson, and W. D. Oliver, Annual Review of Condensed Matter Physics 11, 369 (2020)

    work page 2020

  8. [8]

    M. H. Devoret, A. Wallraff, and J. M. Martinis, Superconduct- ing qubits: A short review (2004)

    work page 2004

  9. [9]

    Devoret, Quantum Fluctuations: Volume 63 , edited by S

    M. Devoret, Quantum Fluctuations: Volume 63 , edited by S. Reynaud, E. Giacobino, and F. David, Les Houches (North- Holland, Oxford, England, 1997)

    work page 1997

  10. [11]

    Wendin, Reports on Progress in Physics 80, 106001 (2017)

    G. Wendin, Reports on Progress in Physics 80, 106001 (2017)

    work page 2017

  11. [12]

    M. Hays, V . Fatemi, D. Bouman, J. Cerrillo, S. Diamond, K. Serniak, T. Connolly, P. Krogstrup, J. Nyg ˚ard, A. L. Yey- ati, A. Geresdi, and M. H. Devoret, Science 373, 430 (2021)

    work page 2021

  12. [13]

    Pita-Vidal, A

    M. Pita-Vidal, A. Bargerbos, R. ˇZitko, L. J. Splitthoff, L. Gr ¨unhaupt, J. J. Wesdorp, Y . Liu, L. P. Kouwenhoven, R. Aguado, B. van Heck, A. Kou, and C. K. Andersen, Nature Physics 10.1038/s41567-023-02071-x (2023)

    work page doi:10.1038/s41567-023-02071-x 2023

  13. [14]

    Pita-Vidal, J

    M. Pita-Vidal, J. J. Wesdorp, L. J. Splitthoff, A. Bargerbos, Y . Liu, L. P. Kouwenhoven, and C. K. Andersen, Strong tun- able coupling between two distant superconducting spin qubits (2023), arXiv:2307.15654 [quant-ph]

    work page arXiv 2023

  14. [15]

    N. M. Chtchelkatchev and Y . V . Nazarov, Phys. Rev. Lett. 90, 226806 (2003)

    work page 2003

  15. [17]

    K. G. Wilson, Reviews of Modern Physics 47, 773 (1975)

    work page 1975

  16. [18]

    Bulla, T

    R. Bulla, T. A. Costi, and T. Pruschke, Reviews of Modern Physics 80, 395 (2008)

    work page 2008

  17. [19]

    Tinkham, Introduction to superconductivity, 2nd ed

    M. Tinkham, Introduction to superconductivity, 2nd ed. (Dover Publications, 2004)

    work page 2004

  18. [20]

    Gobert, U

    D. Gobert, U. Schollw ¨ock, and J. von Delft, The European Physical Journal B - Condensed Matter and Complex Systems 38, 501 (2004)

    work page 2004

  19. [21]

    Glazman and G

    L. Glazman and G. Catelani, SciPost Physics Lecture Notes 10.21468/scipostphyslectnotes.31 (2021)

    work page doi:10.21468/scipostphyslectnotes.31 2021

  20. [22]

    M. T. Tuominen, J. M. Hergenrother, T. S. Tighe, and M. Tin- kham, Physical Review Letters 69, 1997 (1992)

    work page 1997

  21. [23]

    Lafarge, P

    P. Lafarge, P. Joyez, D. Esteve, C. Urbina, and M. H. Devoret, Physical Review Letters 70, 994 (1993)

    work page 1993

  22. [24]

    Bouchiat, D

    V . Bouchiat, D. Vion, P. Joyez, D. Esteve, and M. H. Devoret, Physica Scripta T76, 165 (1998)

    work page 1998

  23. [25]

    Nakamura, Y

    Y . Nakamura, Y . A. Pashkin, and J. S. Tsai, Nature 398, 786 (1999)

    work page 1999

  24. [26]

    Cottet, Implementation of a quantum bit in a superconduct- ing circuit , Ph.D

    A. Cottet, Implementation of a quantum bit in a superconduct- ing circuit , Ph.D. thesis, Ecole normale superieure de Paris (2002)

    work page 2002

  25. [27]

    T. W. Larsen, K. D. Petersson, F. Kuemmeth, T. S. Jespersen, P. Krogstrup, J. Nyg ˚ard, and C. M. Marcus, Physical Review Letters 115 (2015)

    work page 2015

  26. [28]

    de Lange, B

    G. de Lange, B. van Heck, A. Bruno, D. J. van Woerkom, A. Geresdi, S. R. Plissard, E. P. A. M. Bakkers, A. R. Akhmerov, and L. DiCarlo, Physical Review Letters 115 (2015)

    work page 2015

  27. [29]

    Zheng, L

    H. Zheng, L. Y . Cheung, N. Sangwan, A. Kononov, R. Haller, J. Ridderbos, C. Ciaccia, J. H. Ungerer, A. Li, E. P. A. M. Bakkers, A. Baumgartner, and C. Sch ¨onenberger, Coherent control of a few-channel hole type gatemon qubit (2023), arXiv:2312.06411 [cond-mat.mes-hall]

    work page arXiv 2023

  28. [30]

    Clarke, A

    J. Clarke, A. N. Cleland, M. H. Devoret, D. Esteve, and J. M. Martinis, Science 239, 992–997 (1988)

    work page 1988

  29. [31]

    Yu, Acta Phys

    L. Yu, Acta Phys. Sin. 21, 75 (1965)

    work page 1965

  30. [32]

    Shiba and T

    H. Shiba and T. Soda, Progress of Theoretical Physics 41, 25 (1969)

    work page 1969

  31. [33]

    A. I. Rusinov, JETP Lett. 9, 85 (1969), zh. Eksp. Teor. Fiz. Pisma Red. 9, 146 (1968)

    work page 1969

  32. [34]

    A. C. Hewson, The Kondo Problem to Heavy Fermions (Cam- bridge University Press, 1993)

    work page 1993

  33. [35]

    A. V . Rozhkov and D. P. Arovas, Physical Review Letters 82, 2788 (1999)

    work page 1999

  34. [36]

    A. A. Clerk and V . Ambegaokar, Physical Review B 61, 9109 (2000)

    work page 2000

  35. [37]

    Vecino, A

    E. Vecino, A. Mart ´ın-Rodero, and A. L. Yeyati, Physical Re- view B 68, 10.1103/physrevb.68.035105 (2003)

    work page doi:10.1103/physrevb.68.035105 2003

  36. [38]

    M.-S. Choi, M. Lee, K. Kang, and W. Belzig, Physical Review B 70, 10.1103/physrevb.70.020502 (2004)

    work page doi:10.1103/physrevb.70.020502 2004

  37. [39]

    Oguri, Y

    A. Oguri, Y . Tanaka, and A. C. Hewson, Journal of the Physical Society of Japan 73, 2494 (2004)

    work page 2004

  38. [40]

    J. A. van Dam, Y . V . Nazarov, E. P. A. M. Bakkers, S. D. Franceschi, and L. P. Kouwenhoven, Nature442, 667 (2006)

    work page 2006

  39. [41]

    ˇZitko, Josephson potentials for single impurity Anderson im- purity in a junction between two superconductors (2022)

    R. ˇZitko, Josephson potentials for single impurity Anderson im- purity in a junction between two superconductors (2022)

    work page 2022

  40. [42]

    Bargerbos, M

    A. Bargerbos, M. Pita-Vidal, R. ˇZitko, J. ´Avila, L. J. Split- thoff, L. Gr ¨unhaupt, J. J. Wesdorp, C. K. Andersen, Y . Liu, L. P. Kouwenhoven, R. Aguado, A. Kou, and B. van Heck, PRX Quantum 3, 10.1103/prxquantum.3.030311 (2022)

    work page doi:10.1103/prxquantum.3.030311 2022

  41. [43]

    M. R. Sahu, F. J. Matute-Ca ˜nadas, M. Benito, P. Krogstrup, J. Nyg˚ard, M. F. Goffman, C. Urbina, A. L. Yeyati, and H. Poth- ier, Ground state phase diagram and ”parity flipping” mi- crowave transitions in a gate-tunable josephson junction (2023), arXiv:2312.12914 [cond-mat.mes-hall]

    work page arXiv 2023

  42. [44]

    Bulaevskii, V

    L. Bulaevskii, V . Kuzii, and A. Sobyanin, Solid State Commu- nications 25, 1053 (1978)

    work page 1978

  43. [45]

    D. Vion, A. Aassime, A. Cottet, P. Joyez, H. Pothier, C. Urbina, D. Esteve, and M. H. Devoret, Physica Scripta T102, 162 (2002)

    work page 2002

  44. [46]

    D. Vion, A. Aassime, A. Cottet, P. Joyez, H. Pothier, C. Urbina, D. Esteve, and M. Devoret, Fortschritte der Physik 51, 462 (2003)

    work page 2003

  45. [47]

    M. L. Della Rocca, M. Chauvin, B. Huard, H. Pothier, D. Es- teve, and C. Urbina, Physical Review Letters 99, 127005 (2007)

    work page 2007

  46. [48]

    L. V . Ginzburg, I. E. Batov, V . V . Bol’ginov, S. V . Egorov, V . I. Chichkov, A. E. Shchegolev, N. V . Klenov, I. I. Soloviev, S. V . Bakurskiy, and M. Y . Kupriyanov, JETP Letters 107, 48–54 (2018)

    work page 2018

  47. [49]

    Bargerbos, M

    A. Bargerbos, M. Pita-Vidal, R. ˇZitko, L. J. Splitthoff, L. Gr ¨unhaupt, J. J. Wesdorp, Y . Liu, L. P. Kouwenhoven, R. Aguado, C. K. Andersen, A. Kou, and B. van Heck, Phys. Rev. Lett. 131, 097001 (2023)

    work page 2023

  48. [50]

    Pave ˇsi´c, M

    L. Pave ˇsi´c, M. Pita-Vidal, A. Bargerbos, and R. ˇZitko, SciPost Physics 15, 2542 (2023)

    work page 2023

  49. [51]

    R. W. Richardson and N. Sherman, Nuclear Physics 52, 221 (1964)

    work page 1964

  50. [52]

    Dukelsky and G

    J. Dukelsky and G. Sierra, Physical Review B61, 12302 (2000)

    work page 2000

  51. [53]

    Pave ˇsi´c, D

    L. Pave ˇsi´c, D. Bauernfeind, and R. ˇZitko, Physical Review B 104, 10.1103/physrevb.104.l241409 (2021)

    work page doi:10.1103/physrevb.104.l241409 2021

  52. [54]

    Rom ´an, G

    J. Rom ´an, G. Sierra, and J. Dukelsky, Nuclear Physics B 634, 483–510 (2002). 23

    work page 2002

  53. [55]

    Dukelsky, S

    J. Dukelsky, S. Pittel, and G. Sierra, Reviews of Modern Physics 76, 643 (2004)

    work page 2004

  54. [56]

    von Delft and D

    J. von Delft and D. Ralph, Physics Reports 345, 61 (2001)

    work page 2001

  55. [58]

    Manca, S

    L. Pave ˇsi´c and R. ˇZitko, Physical Review B105, 10.1103/phys- revb.105.075129 (2022)

    work page doi:10.1103/phys- 2022

  56. [59]

    For detailed derivations and expressions at general filling see Ref. 51

    work page

  57. [60]

    S. T. Belyaev, JETP Letters 12, 968 (1961)

    work page 1961

  58. [61]

    V . A. Khodel’ and V . R. Shaginyan, JETP Letters 51, 488 (1990)

    work page 1990

  59. [62]

    G. E. V olovik, JETP Letters107, 516 (2018)

    work page 2018

  60. [63]

    F. J. Matute-Ca ˜nadas, L. Tosi, and A. L. Yeyati, Quantum cir- cuits with multiterminal josephson-andreev junctions (2023), arXiv:2312.17305 [cond-mat.mes-hall]

    work page arXiv 2023

  61. [64]

    Wolfram Research, Inc., Mathematica, Version 13.3 (2023), Champaign, IL, 2023

    work page 2023

  62. [65]

    Pave ˇsiˇc, Flat band two channel, https://github.com/ PavesicL/flat_band_two_channel (2023)

    L. Pave ˇsiˇc, Flat band two channel, https://github.com/ PavesicL/flat_band_two_channel (2023)

    work page 2023

  63. [66]

    M. H. Devoret, Journal of Superconductivity and Novel Mag- netism 34, 1633–1642 (2021)

    work page 2021

  64. [67]

    Affleck, J.-S

    I. Affleck, J.-S. Caux, and A. M. Zagoskin, Physical Review B 62, 1433 (2000)

    work page 2000

  65. [68]

    F. S. Bergeret, A. L. Yeyati, and A. Mart ´ın-Rodero, Physical Review B 76, 10.1103/physrevb.76.174510 (2007)

    work page doi:10.1103/physrevb.76.174510 2007

  66. [69]

    Grove-Rasmussen, G

    K. Grove-Rasmussen, G. Steffensen, A. Jellinggaard, M. H. Madsen, R. ˇZitko, J. Paaske, and J. Nyg ˚ard, Nature Commu- nications 9, 10.1038/s41467-018-04683-x (2018)

    work page doi:10.1038/s41467-018-04683-x 2018

  67. [70]

    ˇZonda, P

    M. ˇZonda, P. Zalom, T. Novotn ´y, G. Loukeris, J. B ¨atge, and V . Pokorn´y, Generalized atomic limit of a double quantum dot coupled to superconducting leads (2022)

    work page 2022

  68. [71]

    Hermansen, A

    C. Hermansen, A. L. Yeyati, and J. Paaske, Physical Review B 105, 10.1103/physrevb.105.054503 (2022)

    work page doi:10.1103/physrevb.105.054503 2022

  69. [72]

    Schmid, J

    H. Schmid, J. F. Steiner, K. J. Franke, and F. von Oppen, Phys- ical Review B 105, 10.1103/physrevb.105.235406 (2022)

    work page doi:10.1103/physrevb.105.235406 2022

  70. [73]

    Pave ˇsiˇc, R

    L. Pave ˇsiˇc, R. Aguado, and R. ˇZitko, Strong-coupling theory of quantum dot josephson junctions: role of the residual quasipar- ticle (2023)

    work page 2023

  71. [74]

    A. F. Andreev, Zh. Eksperim. i Teor. Fiz. 46, (1964)

    work page 1964

  72. [75]

    C. P. Koch, U. Boscain, T. Calarco, G. Dirr, S. Filipp, S. J. Glaser, R. Kosloff, S. Montangero, T. Schulte-Herbr ¨uggen, D. Sugny, and F. K. Wilhelm, EPJ Quantum Technology 9, 10.1140/epjqt/s40507-022-00138-x (2022)

    work page doi:10.1140/epjqt/s40507-022-00138-x 2022

  73. [76]

    Rossignolo, T

    M. Rossignolo, T. Reisser, A. Marshall, P. Rembold, A. Pagano, P. J. Vetter, R. S. Said, M. M. M ¨uller, F. Motzoi, T. Calarco, F. Jelezko, and S. Montangero, Computer Physics Communica- tions 291, 108782 (2023)

    work page 2023

  74. [77]

    V . N. Golovach, M. Borhani, and D. Loss, Physical Review B 74, https://doi.org/10.1103/PhysRevB.74.165319 (2006)

    work page doi:10.1103/physrevb.74.165319 2006

  75. [78]

    K. C. Nowack, F. H. L. Koppens, Y . V . Nazarov, and L. M. K. Vandersypen, Science 318, 1430 (2007)

    work page 2007

  76. [79]

    Pioro-Ladri `ere, T

    M. Pioro-Ladri `ere, T. Obata, Y . Tokura, Y .-S. Shin, T. Kubo, K. Yoshida, T. Taniyama, and S. Tarucha, Nature Physics 4, 776 (2008)

    work page 2008

  77. [80]

    Nadj-Perge, S

    S. Nadj-Perge, S. M. Frolov, E. P. A. M. Bakkers, and L. P. Kouwenhoven, Nature 468, 1084 (2010)

    work page 2010

  78. [81]

    J. W. G. van den Berg, S. Nadj-Perge, V . S. Prib- iag, S. R. Plissard, E. P. A. M. Bakkers, S. M. Frolov, and L. P. Kouwenhoven, Physical Review Letters 110, https://doi.org/10.1103/PhysRevLett.110.066806 (2013)

    work page doi:10.1103/physrevlett.110.066806 2013

  79. [82]

    B. Li, T. Calarco, and F. Motzoi, PRX Quantum 3, 10.1103/prxquantum.3.030313 (2022)

    work page doi:10.1103/prxquantum.3.030313 2022