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Hermes terms in Lindblad linear response recover band curvature and protect dispersive readout against decoherence when populations obey detailed balance.

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2026-07-11 10:46 UTC pith:QQBWMRTO

load-bearing objection Clean N-level Lindblad response with closed-form Hermes/Sisyphus terms; the curvature-recovery identity is solid under the stated Markovian assumptions and useful for Kitaev-junction readout.

arxiv 2607.04968 v1 pith:QQBWMRTO submitted 2026-07-06 cond-mat.mes-hall

Lindblad theory of linear response susceptibility and dispersive readout in minimal Kitaev junctions

Tobias Kuhn , Raffael L. Klees , Ram\'on Aguado , M\'onica Benito This is my paper
classification cond-mat.mes-hall
keywords Lindblad linear responseHermes susceptibilitySisyphus susceptibilitydispersive readoutminimal Kitaev chainKitaev-Josephson junctionquantum capacitanceparity qubit
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Hybrid superconductor-semiconductor quantum dots are being built into devices that host Majorana-based parity qubits, but their multi-level structure and coupling to the environment make standard Kubo response incomplete. This paper supplies a closed-form Lindblad linear-response theory for any number of levels, any probe parameter, any observable and any reference state. The theory isolates two new dynamical pieces that come from fluctuations of the bath rates (Sisyphus) and of the jump operators themselves (Hermes). For the natural choice of observable equal to the derivative of the Hamiltonian, the static, Hamiltonian and detailed-balance Hermes contributions together recover the population-weighted band curvature exactly, independent of the decoherence rates. That identity shows why ordinary curvature-based quantum-capacitance or Josephson-inductance formulas remain accurate in the dispersive regime whenever the system is thermalized, and it quantifies the corrections that appear when populations depart from detailed balance. The same formulas are applied to minimal Kitaev chains and Kitaev-Josephson junctions, clarifying which gate and flux spectroscopic signals can distinguish global and local parity and which cannot.

Core claim

When the observable equals the derivative of the Hamiltonian, the sum of the static susceptibility, the Hamiltonian susceptibility and the detailed-balance part of the Hermes susceptibility equals the population-weighted second derivative of the energy levels at zero frequency, irrespective of the decoherence rates. Hermes terms therefore cancel the decoherence that would otherwise suppress the Hamiltonian piece, protecting curvature-based dispersive readout for thermalized populations.

What carries the argument

The Lindblad-based susceptibility split into four pieces (static, Hamiltonian, Sisyphus, Hermes) together with the closed-form expressions for each piece in the frequency domain; the key identity is χ_st + χ_H(0) + χ_L,DB(0) = ∑_m p_m E_m''.

Load-bearing premise

The whole derivation assumes a weakly coupled, memoryless bath that produces a time-local Lindblad generator with rates fixed by an Ohmic bosonic spectrum; strong coupling or non-Markovian memory would invalidate the closed forms and the claimed Hermes protection.

What would settle it

Measure the real part of the dispersive gate or flux susceptibility of a thermalized minimal Kitaev chain or Kitaev-Josephson junction and check whether it equals the independently measured band curvature (quantum capacitance or inverse Josephson inductance) within the predicted O(ω/ε) corrections; a clear mismatch at low temperature would falsify the Hermes-compensation claim.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 4 minor

Summary. The manuscript develops a closed-form Lindblad linear-response theory for an N-level open quantum system, decomposing the finite-frequency susceptibility into static, Hamiltonian, Sisyphus (rate-fluctuation), and Hermes (jump-operator fluctuation) channels. For the observable A = H', the combination of static, Hamiltonian, and detailed-balance Hermes terms recovers the population-weighted band curvature exactly at zero frequency, independent of decoherence rates (Eqs. 19 and C6). The framework is applied to a minimal Kitaev chain and a Kitaev–Josephson junction, comparing common-gate and flux susceptibilities with quantum capacitance and Josephson inductance and quantifying corrections from non-thermal populations.

Significance. If the formal results hold, the work supplies a practical multi-level tool for interpreting dispersive parity and qubit readout in hybrid superconductor–semiconductor devices. The curvature-recovery identity (Hermes compensation of decoherence under detailed balance) is a clean, parameter-free algebraic consequence of the Lindblad generator plus Hellmann–Feynman, and the explicit DB/BT split of both Sisyphus and Hermes channels goes beyond earlier two-level treatments. Analytic two-level limits recover known results; the KJJ application yields concrete, falsifiable predictions for gate versus flux spectroscopy of global and local parity. The complete Appendices A–C make the derivations reproducible.

minor comments (4)
  1. Footnote 1 and Sec. III A correctly flag the Born–Markov/secular and quasiparticle-poisoning assumptions; a single sentence in the conclusions restating that the Hermes compensation is guaranteed only inside this regime would help experimental readers.
  2. Figs. 2–3 and 6–7 would benefit from explicit units or normalization conventions for |χ| (e.g., relative to |C_Q|_max) so that absolute magnitudes can be compared across panels.
  3. The expansion coefficients c_{k,lmn} for N>2 (Eq. 14 / B16) are left numerical; a short remark that they are obtained from the N×N dissipation matrix (A16) would aid re-implementation.
  4. Occasional typographical slips (e.g., “swiched”, “induction” for inductance) and the informal first sentence of the acknowledgements should be cleaned before final production.

Circularity Check

0 steps flagged

No significant circularity: curvature-recovery identity is an algebraic consequence of the Lindblad generator plus Hellmann–Feynman, not an input restated as output.

full rationale

The paper re-derives the finite-frequency Lindblad susceptibilities (static, Hamiltonian, Sisyphus, Hermes) from the master equation under Born–Markov/secular assumptions (Appendices A–B) and obtains closed N-level expressions. The central claim χ_st + χ_H(0) + χ_L,DB(0) = ∑_m p_m E_m'' (Eqs. 19, C6) follows by direct algebra once G_m = 0 (detailed balance) and A = H' are inserted; decoherence rates cancel between Hamiltonian and detailed-balance Hermes pieces. Self-citations to the authors’ prior two-level work [38] and related notes [54] serve only as consistency checks (the N = 2 formulas reduce correctly) and are not used to import uniqueness or to force the multi-level result. No parameters are fitted to data and later called predictions; the Kitaev-chain applications are pure evaluations of the derived formulas. The Markovian and no-poisoning assumptions are scope limitations, not circular steps. Hence the derivation chain is self-contained.

Axiom & Free-Parameter Ledger

3 free parameters · 4 axioms · 2 invented entities

The central claims rest on standard open-system approximations plus a few modeling choices specific to the hybrid devices. No free parameters are fitted to experimental data; all numerical values are illustrative. The only invented conceptual entities are the named Sisyphus and Hermes channels, which are simply the rate- and jump-operator pieces of the already-known Lindblad generator.

free parameters (3)
  • system-bath coupling strengths |s_mn|^2
    Set by hand to 1/2 or 1/10 for plots; they scale overall rates but do not alter the structural claims about Hermes compensation.
  • temperature k_B T (e.g. 4Δ/5)
    Chosen for illustration of thermal versus non-thermal regimes; not fitted to any data set.
  • probe frequencies ω_d, ω_r
    Selected relative to the gap ϵ to illustrate dispersive and resonant limits.
axioms (4)
  • domain assumption Born-Markov and secular approximations yielding a time-local Lindblad generator with rates fixed by bath correlation functions
    Stated in Sec. II and Appendix A.1; required for the closed-form susceptibilities.
  • domain assumption Ohmic spectral density J(ω) ∝ ω so that pure-dephasing rates remain finite
    Used throughout the rate formulas (Eq. 35 and Appendix A.1).
  • domain assumption Quasiparticle poisoning times ≫ readout integration times, so local parity is conserved during the measurement
    Invoked in Sec. III.B.1 with experimental citations; allows restriction to parity-conserving jump operators.
  • domain assumption Reference state ρ is diagonal in the instantaneous energy basis and constant on the measurement timescale
    Required for the time-local Kubo formula (Eq. 4).
invented entities (2)
  • Sisyphus susceptibility χ_Γ no independent evidence
    purpose: Captures first-order response arising from modulation of the Lindblad rates Γ_mn
    Named and isolated in Sec. II.B; reduces to known two-level results and has no independent experimental handle beyond the overall admittance.
  • Hermes susceptibility χ_L (and its DB/BT split) no independent evidence
    purpose: Captures first-order response arising from modulation of the jump operators themselves; shown to cancel decoherence in the dispersive limit
    Named in Sec. II.C; the compensation identity is a derived consequence, not an independent postulate.

pith-pipeline@v1.1.0-grok45 · 38889 in / 2755 out tokens · 24427 ms · 2026-07-11T10:46:05.007181+00:00 · methodology

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The field of hybrid superconductor-semiconductor quantum dots is advancing toward the development of functional devices that leverage the advantages of both types of materials. However, the inherent complexity of these devices demands a comprehensive theoretical framework for a complete understanding of their responses to external probes, readout and the dissipation arising from environmental coupling. We present a Lindblad-based linear response formalism that captures the multi-level nature of these devices, their probe-readout flexibility, and the non-unitary effects of finite-frequency response, including the so-called Sisyphus and Hermes dynamical susceptibilities. These arise from fluctuations in the rates and jump operators, and are hence absent in standard Kubo linear response treatments. We exemplify the framework using quantum dot-based Kitaev chain setups which are promising candidates for topologically protected Majorana-based parity qubits. Our results shed light onto the validity of the standard curvature-based approximation for fermionic parity and qubit readout, show that Hermes terms compensate decoherence in dispersive readout and implement important corrections beyond thermalized states.

Figures

Figures reproduced from arXiv: 2607.04968 by M\'onica Benito, Raffael L. Klees, Ram\'on Aguado, Tobias Kuhn.

Figure 1
Figure 1. Figure 1: FIG. 1. We introduce a parameter fluctuation [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Common gate spectroscopy on a minimal Kitaev chain. [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Susceptibility of the minimal Kitaev chain for ther [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Curvature of the lowest two KJJ energy bands for [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Common gate susceptibility for global even parity [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Flux susceptibility for ground ( [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p025_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Gate susceptibility contributions at dispersive frequency [PITH_FULL_IMAGE:figures/full_fig_p027_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Real part of the gate susceptibility shown in Fig. [PITH_FULL_IMAGE:figures/full_fig_p027_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Flux susceptibility contributions at dispersive frequency [PITH_FULL_IMAGE:figures/full_fig_p028_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Dispersive susceptibility measurements driving only the left chain common gates. [PITH_FULL_IMAGE:figures/full_fig_p029_12.png] view at source ↗

discussion (0)

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Reference graph

Works this paper leans on

97 extracted references · 7 linked inside Pith

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    Spectrum and qubit definiton The Hamiltonian of the Kitaev-Josephson junction (KJJ) is HKJJ =H L +H R +H J ,(32) whereH L andH R describe left and right MKCs, respec- tively, as introduced in Eq. (22). The coupling Hamilto- nian HJ =−t J eiϕ/2d† L,2dR,1 +h.c.(33) depends on the coupling strengtht J between the near- est quantum dots and the relative phase...

  2. [2]

    Curvature of the qubit states First, we investigate the quantum capacitance, a mea- sure of gate curvature; see Fig. 5a. At the flux sweet spotϕ=π, the quantum capacitance of the global even ground state reads Ceven Q,↓ =− X ± ∆2 ∆2 + µ± tJ 2 2 3/2 ≈ − 16∆2 ϵ3 + 96t2 J∆2 ∆2 −4µ 2 ϵ7 ,(37) (see all eigenenergies in Appendix E 2) corresponding to the sum of...

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    Therefore, the modulated parameterfis the chemical potential,f=µ, and the read-out observable is the common gate currentA=H ′ KJJ

    Common gate response As a natural continuation of minimal Kitaev chains, we simulate common gate spectroscopy on the global even parity KJJ. Therefore, the modulated parameterfis the chemical potential,f=µ, and the read-out observable is the common gate currentA=H ′ KJJ . SinceA(µ) =A, χst = 0. In Fig. 6, we compare dispersive measurements at ω=t/5 in the...

  4. [4]

    Flux response Introducing flux modulationsf→ϕand reading out the supercurrent along the junctionA=I S ∝H ′ KJ J will yield the flux susceptibility across the KJJ shown in Fig. 7. In contrast to gate spectroscopy, the static con- tributionχ st (Eq. (5)) leads to a frequency-independent offset to the real part of the susceptibility sinceA(ϕ) is flux-depende...

  5. [5]

    Gate spectroscopy allows for the distinction of global odd from global even parity

    Comparison of qubit readout strategies We have now introduced common gate and flux sus- ceptibility measurements, which allow us to distinguish between different qubit occupations. Gate spectroscopy allows for the distinction of global odd from global even parity. Within global even par- ity, we can discern local even and local odd parity as well. Spectro...

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    The Lindblad equation of a system coupled to a bosonic bath In order to arrive at an effective Lindblad description of the time evolution of the reduced density matrixρ S of an open quantum system, we use the Caldeira-Leggett model [69] and start with a general Hamiltonian of the form H=H S +H B +V. We assume that the system hasNenergy levels and that the...

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    Linear response In this section, we provide an instructive derivation of the Lindblad-based Kubo formula as it was used in [41– 44, 47]. In general, the expectation value⟨A⟩(t) = tr{Aρ(t)}of an operatorAis calculated from the solutionρ(t) of the Lindblad equation∂ tρ(t) =Lρ(t) that was derived in the previous section. In the following, we assume that we p...

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    (A9), we need to apply the superoperatore tL to the operatorL ′ρ

    Diagonalization of the Lindblad superoperator In order to efficiently calculate the total susceptibilityχ(t) in Eq. (A9), we need to apply the superoperatore tL to the operatorL ′ρ. For this, we diagonalize the nonhermitian Lindblad superoperatorLto find its eigenvaluesλ∈C, such thatLr=λrandL †l=λ ∗lwith right and left eigenoperatorsrandl, respectively. T...

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    Hamiltonian contributionχ H(t) In this case, Eq. (A10a) yields the matrix elements ⟨ϕm|L′ H ρ|ϕn⟩=−i⟨ϕ m|[H ′, ρ]|ϕn⟩=i(p m −p n)⟨ϕ m|H ′|ϕn⟩.(B2) Note that⟨ϕ m|L′ H ρ|ϕn⟩= 0 form=n, meaning that the operatorL ′ H ρdoes not contain contributions fromL mm. We can therefore make direct use of the eigenvaluesλ mn =−Γ mn T2 −i(E m −E n) with the decoherence r...

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    Hermes contributionχ L(t) With the matrix elements of the derivative of the jump operators, ⟨ϕm|L′ ab|ϕn⟩=δ nb ⟨ϕm|ϕ′ a⟩ −δ ma ⟨ϕb|ϕ′ n⟩,(B4) a lengthy but straightforward simplification of Eq. (A10c) yields the matrix elements ⟨ϕm|L′ Lρ|ϕn⟩= NX a,b=1 Γab ⟨ϕm| L′ abρL† ab +L abρ(L† ab)′ − 1 2 n (L† ab)′Lab +L † abL′ ab, ρ o |ϕn⟩ = NX a=1 Γnapa − 1 2(pm +p...

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    Sisyphus contributionχ Γ(t) In this case, Eq. (A10b) yields the matrix elements ⟨ϕm|L′ Γρ|ϕn⟩= NX a,b=1 Γ′ ab ⟨ϕm| LabρL† ab − 1 2 n L† abLab, ρ o |ϕn⟩=δ mn NX a,b=1 (a̸=b) Γ′ abpb(δam −δ bm),(B11) such that L′ Γρ= NX m,n=1 (m̸=n) Γ′ mnpn(Lmm −L nn).(B12) Note that dephasing rates Γ mm do not enter the Sisyphus contribution. SinceL ′ Γρis a superposition ...

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    Band curvature from low-frequency spectroscopy of isolated quantum systems For isolated systems, Γ mn = 0, both Sisyphus and Hermes contributions are zero, and the total susceptibility reduces toχ(ω) =χ st +χ H(ω). In the dispersive frequency regime,ω≪ |E m −E n|, we get χH(ω) = NX m,n=1 (m̸=n) (pm −p n) ⟨ϕm|H ′|ϕn⟩ ⟨ϕn|A|ϕm⟩ Em −E n +O(ω).(C1) In cases w...

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    In general, due to their similarities, the Hamiltonian and Hermes contributions in Eqs

    Band curvature from low-frequency spectroscopy of open quantum systems In more general setups in which interactions with the environment are not negligible, the two additional contributions to the total susceptibility, namely Sisyphus and Hermes contributions, can become relevant. In general, due to their similarities, the Hamiltonian and Hermes contribut...

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    (B3) and (B9)

    Band curvature from spectroscopy of open quantum systems with strong dephasing Hermes and Hamiltonian contributions are very similar when comparing the results in Eqs. (B3) and (B9). In the limit in which the environment only causes strong dephasing, Γ mm ≫Γ mn (m̸=n), the decoherence rate in Eq. (A14) becomes Γ mn T2 ≈(Γ mm + Γnn)/2, and, hence, Λ mn ≈ −...

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    Admittance for gate perturbations The simplest way to characterize quantum dot devices is through the gate current response. A small applied time- dependent gate voltageδV(t) induces a time-dependent shift of the onsite energy,δµ(t) =αe δV(t). The coupling strength is determined by the elementary chargeeand the so-called lever armα=C G/(CG +C S), whereC G...

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    Under the perturbationϕ→ϕ+δϕ(t), the Hamiltonian becomesH→ H+ Φ 0 Is δϕ(t), where Is = 1 Φ0 ∂H ∂ϕ (D5) 21 is the Josephson current operator and Φ0 =ℏ/(2e) is the flux quantum

    Admittance for flux perturbations In superconducting systems, the magnetic flux through a superconducting loop controls the superconducting phase difference across a Josephson junction. Under the perturbationϕ→ϕ+δϕ(t), the Hamiltonian becomesH→ H+ Φ 0 Is δϕ(t), where Is = 1 Φ0 ∂H ∂ϕ (D5) 21 is the Josephson current operator and Φ0 =ℏ/(2e) is the flux quan...

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    Minimal chain in parity sectors The effective Hamiltonian for a minimal Kitaev chain is separated into global parity sectors H=   0 ∆ 0 0 ∆−µ 1 −µ 2 0 0 0 0−µ 1 −t 0 0−t−µ 2   (E1) in the many-body basis{|00⟩,|11⟩,|10⟩,|01⟩}, where ∆ denotes CAR andtdenotes ECT. In the even particle number sector, the energies scale with the average onsite energyµ 1...

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    Linear response operators To compute the linear response, we need the matrix forms of the Hamiltonian derivatives, namely∂ f HKJ J for the gate and flux perturbationsf=µandf=ϕ, respectively. For common gate spectroscopy, the dipole operator is first calculated in second quantization as ∂µHKJJ =− X i=L,R j=1,2 d† i,jdi,j .(E9) 23 In the basis of decoupled ...

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    A projective treatment for the even parity low energy subspace has already been introduced in Ref

    Low-energy regime When considering the weak-coupling limitt J ≪ |t+∆|/2, all three energy regimes can be separated from each other. A projective treatment for the even parity low energy subspace has already been introduced in Ref. [19]. It revealed mirror symmetric flux energy bands around the MKC energyE=ϵ even − +ϵ odd − . However, the full spectrum doe...

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