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Theory of the photonic Joule effect in superconducting circuits

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Pith's one-line read A voltage-biased junction can overheat its own photonic bath into a non-equilibrium state that makes the I-V curve bistable.

desk verdict The photonic Joule effect is a robust qualitative claim; the quantitative thermal steady state and bistability boundaries still need tighter support. read the letter →

arxiv 2411.19912 v2 pith:WPTKQBVH submitted 2024-11-29 cond-mat.mes-hall cond-mat.supr-conquant-ph

Samuel Cailleaux , Quentin Ficheux , Nicolas Roch , Denis M. Basko This is my paper · ORCID
classification cond-mat.mes-hallcond-mat.supr-conquant-ph
keywords photonicJouleeffectJosephsonjunctionchainP(E)theorynon-equilibriumsteadystatebistabilityhigh-impedanceenvironmentcircuitquantumelectrodynamics
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper argues that a small voltage-biased Josephson junction can overheat the microwave photons in a long Josephson-junction chain that acts as its electromagnetic environment. Such a chain is normally treated as a passive thermal bath, but its photons decay mainly by escaping through the boundary, so energy deposited by Cooper-pair tunneling can accumulate inside. The authors generalize the standard P(E)-theory tunneling-rate calculation into a self-consistent kinetic equation for the mode occupations and show that the steady state can have strongly elevated, mode-dependent temperatures. This 'photonic Joule effect' qualitatively changes the junction's I-V curve and can make it bistable, meaning the usual fixed-bath assumption can miss the dominant physics in realistic superconducting circuits.

What carries the argument

The central object is a self-consistent kinetic equation for the average photon occupations $\bar n(\omega)$ of the chain modes, Eq. (6), which replaces the fixed thermal occupations of standard P(E) theory with occupations fixed by balance between Cooper-pair tunneling rates $W_\pm(\omega)$ and boundary decay $\kappa(\omega)$. The tunneling rates are built from the P(E) function, the probability that a tunneling Cooper pair exchanges energy $E$ with the environment; that function depends on the occupations through the phase-correlation function $J(t)$, closing the loop. Solving the closed system gives the stationary occupations, the effective mode temperatures $T_m$, and the dc current. A complementary classical treatment, in which mode amplitudes follow damped driven equations and the junction phase is classical, reproduces the same I-V curves and shows that the hot state is chaotic with Gaussian amplitude statistics.

What would settle it

Measure the microwave spectrum emitted into the external $50\,\Omega$ line while biasing the junction below the plasma voltage, $2eV<\hbar\omega_p$: a cold bath would emit only up to $2eV$, while the photonic-Joule prediction is photon flux across the whole band $0<\omega<\omega_p$. A complementary check is to sweep the junction's Josephson energy with a magnetic field through a SQUID; the hot branch makes the I-V curve reach higher voltages as $E_J$ grows, whereas a passive bath would only rescale the current as $E_J^2$ without changing the voltage dependence.

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Extended reading notes

Core claim

The central claim is that the environment's photonic modes do not remain in the cryostat-temperature thermal state: inelastic Cooper-pair tunneling feeds energy into the chain faster than boundary damping removes it, so the modes settle into a non-equilibrium steady state with strongly elevated occupations. Each mode is approximately thermal, but different modes have different temperatures. A self-consistent P(E) calculation, a simplified single-temperature energy-balance version, and a classical simulation of the mode amplitudes all agree on the resulting I-V curve over a wide voltage range. Well above the plasma frequency the curve has two stable branches, a cold low-current branch and a hot high-current branch, indicating hysteresis. The authors conclude that the standard assumption of a passive, unchanged bath is violated for photonic environments and propose experimental signatures in the emitted microwave spectrum and in the dependence of the I-V curve on the junction's Josephson energy.

Load-bearing premise

The quantitative predictions rely on the assumption that mode phases randomize and that a single Cooper-pair tunneling event changes any one mode's photon number by at most one; the paper notes this can fail for the lowest modes, where the coupling is of order one, so the exact shape and existence of the predicted I-V branches could change under a more complete treatment.

Editorial extensions

If this is right

  • In circuits where a small junction is coupled to a high-impedance photonic environment, assuming a passive thermal bath can be wrong, and measured I-V curves may already carry heating signatures.
  • Tuning the junction's Josephson energy, for example with a magnetic field through a SQUID, should change the voltage range over which current flows; for an equilibrium bath only the current scale would change, not the curve's shape.
  • When the bias obeys $2eV<\hbar\omega_p$, an overheated bath should emit photons across the entire band $0<\omega<\omega_p$, while a cold bath emits only up to $2eV$.
  • The system can be bistable with a hysteretic I-V curve, analogous to self-heating bistability in electronic conductors.
  • The bath's low-frequency modes, whose occupancy each tunneling event can change by more than one photon, are the places where the quantitative prediction is least secure.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the photonic Joule effect is confirmed, emission spectra already recorded from high-impedance environments could be reinterpreted: broadband photon flux below the plasma frequency may be heating rather than an intrinsic junction response.
  • A direct numerical treatment of the first few modes without the small-coupling assumption would test whether the predicted bistability survives; this is the natural next calculation.
  • The comparison with a dispersionless transmission line suggests a platform-selection rule: chains with curved dispersion favor chaotic thermalization, while more equidistant spectra favor coherent Josephson-laser behavior, so engineering the dispersion could choose which regime appears.
  • A quantitative map of the predicted mode-temperature profile from emitted-photon spectroscopy would be a sharper test than the I-V curve alone, since different modes carry different effective temperatures.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 3 minor

Summary. The paper considers a voltage-biased small Josephson junction coupled to a long chain of large Josephson junctions that acts as a photonic bath. It argues that the standard P(E)-theory assumption of a passive thermal bath can fail: inelastic Cooper-pair tunneling can strongly heat the chain modes, driving the system to a non-equilibrium steady state with mode-dependent temperatures and a qualitatively modified I-V characteristic, possibly including a bistable region. The claim is supported by three complementary calculations: a single-temperature energy-balance model, a self-consistent kinetic equation for the mode occupations derived by Fermi's Golden Rule, and numerical solution of the classical equations of motion. The three approaches agree over a wide voltage range for the Josephson-junction chain. The paper also gives an estimate that quasiparticle heating does not destroy superconductivity, and it discusses a dispersionless transmission line where the quantitative agreement is worse and a coherent laser-like tendency appears; this limitation is explicitly acknowledged.

Significance. If the qualitative claim holds, the result matters for circuit QED and engineered environments: it shows that a high-impedance photonic bath can be driven far from the passive thermal state assumed in standard P(E) theory. The paper is self-contained: the central equations follow from the model Hamiltonian (1) with no fitted parameters, and the appendices give detailed derivations of the mode spectrum, the kinetic equation, the quasiparticle-stability estimate, and the classical dynamics. The three-method agreement and the explicit falsifiable predictions (SQUID-tunable I-V shape, photon emission spectrum) are strengths. The authors are transparent about the regimes in which their approximations are uncontrolled, particularly the low-voltage and low-mode sector, which is a credit to the paper.

minor comments (3)
  1. [Fig. 11 caption] The caption states δ1 = πvg/l = 2π × 0.5 MHz; with vg = 5 × 10^6 m/s and l = 5 mm, the correct value is π × 10^9 s^-1 = 2π × 0.5 GHz, so the unit should be GHz, not MHz.
  2. [Appendix C, assumption (ii)] The statement that the contribution of the modes with Λ_m ~ 1 to the observables will be checked a posteriori is not backed by a quantitative check; the agreement in Fig. 4 is suggestive, but a quantitative statement (e.g., the fraction of the total phase variance or of the dc current carried by modes with Λ_m > 0.5) would make the argument more transparent.
  3. [Conclusions] The phrase 'the state of each mode is thermal, although thermalization is incomplete' is ambiguous; it would be clearer to state that each mode has a Bose-Einstein distribution with a mode-dependent temperature, so the joint state is not a global thermal state.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central derivation is self-contained from the model Hamiltonian.

full rationale

The paper's central claim is derived by standard open-quantum-system methods: the Hamiltonian (1) specifies the driven small junction coupled to chain photons and to the external circuit; Eq. (3) fixes the dimensionless couplings from the known impedance without fitting to the target I-V curves; the single-temperature estimate (4) and the kinetic equation (6) are solved self-consistently for the unknown mode occupations nbar(omega) and the resulting dc current, rather than fitting those outputs. Each of the three approaches (single-temperature P(E), Fock-space kinetic equation, and classical equations of motion) is independently formulated, and the discrepancies between them are exposed rather than hidden. The acknowledged limitations—Lambda_m of order unity for the lowest modes, the classical mean-field replacement, and the missing coherent component in the P(E) ansatz—are approximations with stated validity ranges, not quantities inserted as the predicted result. Self-citations (Refs. [5] and [37]) are used only for a detection scheme and for the quasiparticle-heating stability estimate, and the latter is rederived and extended in Appendix B with explicit parameter values; they do not supply the central mode occupations or I-V curves. Hence no step in the derivation reduces by construction to its own input.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central claim depends on a chain of assumptions: harmonic long-lived photon modes, weak perturbative tunneling, and an incoherent (diagonal) environment density matrix. The most fragile are the incoherence and the single-photon-per-tunneling-event assumptions, which are validated only by classical simulations and only for the non-equidistant JJ-chain spectrum. The only seriously hand-picked parameter is Qint; the rest are typical experimental values.

free parameters (3)
  • Internal quality factor Qint = 5e4 (assumed)
    Controls the internal (non-radiative) photon decay through κ_nr = ω_m/Qint in Eq. (A6). The overheating magnitude and the bistability region depend on it; it is not measured in the paper.
  • Small-junction Josephson energy E_J = 0.2 hbar ω_p
    Hand-chosen to satisfy the perturbative condition E_J << hbar ω_p while being large enough to produce heating. The current scales as E_J^2; the existence of heating requires E_J^2 ≳ N hbar^2 κ ω_p. It is tunable via a SQUID in the proposed experiment.
  • Chain parameters (Z0, ωp, C/Cg, N, Zext, CJ) = Z0=5 kΩ, ωp=2π×20 GHz, C/Cg=1600, N=5000, Zext=50 Ω, CJ=2 fF
    Chosen as typical values for high-impedance Josephson chains; they determine the mode spectrum, coupling constants, and boundary damping. They are not fitted to any target data, and the qualitative conclusions are argued to be generic for long chains.
assumptions (6)
  • domain assumption The large junctions of the chain are in the harmonic regime (E'_J >> e^2/C), so the chain Hamiltonian is quadratic and supports linear photonic modes.
    Stated in the Model section; used to write Eq. (1) and the dispersion (A1).
  • domain assumption The photonic modes are long-lived: at subgap frequencies, quasiparticles are scarce, photons do not couple directly to phonons, and the dominant loss is escape through the boundary, with a small phenomenological internal loss.
    Central to the heating effect. The estimate in Appendix B argues quasiparticle heating is weak, but the argument depends on the electron-phonon dynamics and assumes parameters such as τ0 ~ 100 ns.
  • ad hoc to paper The environment's density matrix is diagonal in the Fock basis of the chain modes (phases randomize); coherences are irrelevant.
    This is assumption (i) in Appendix C, needed for the kinetic equation (6). It is supported a posteriori by the chaotic classical dynamics, but no quantum proof is given.
  • ad hoc to paper Single Cooper-pair tunneling events change the occupation of each mode by at most one quantum; inter-mode correlations are neglected (Λ_m << 1).
    Assumptions (ii)-(iii) in Appendix C. Acknowledged to break down for the lowest modes where Λ_1 ~ 1; the paper uses the classical calculation to check that this does not invalidate the results.
  • standard math Fermi's Golden Rule / perturbation theory in E_J applies to the inelastic Cooper-pair tunneling ('P(E) theory').
    Standard for weak tunneling (E_J small compared to the relevant photon energy scales); used in Eqs. (5) and (C5).
  • domain assumption For the kinetic equation, the finite-chain impedance can be replaced by the infinite-chain smooth Ztot(ω) when P(E) is smooth, i.e., for T >~ 0.1 hbar ω_p.
    Stated after Eq. (6) and in Appendix C; the low-voltage regime where this fails is explicitly excluded from the quantitative analysis.

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Pith. "Pith review of Theory of the photonic Joule effect in superconducting circuits." pith.science (2026). https://pith.science/paper/WPTKQBVH

@misc{pith2026241119912,
  author       = {Pith},
  title        = {Pith review of: Theory of the photonic Joule effect in superconducting circuits},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WPTKQBVH}},
  note         = {Machine review of arXiv:2411.19912}
}
abstract

When a small system is coupled to a bath, it is generally assumed that the state of the bath remains unaffected by the system due to the bath's large number of degrees of freedom. Here we show theoretically that this assumption can be easily violated for photonic baths typically used in experiments involving superconducting circuits. We analyze the dynamics of a voltage-biased Josephson junction coupled to a photonic bath, represented as a long Josephson junction chain. Our findings show that the system can reach a non-equilibrium steady state where the photonic degrees of freedom become significantly overheated, leading to a qualitative change in the current-voltage $I-V$ curve. This phenomenon is analogous to the Joule effect observed in electrical conductors, where flowing current can substantially heat up electrons. Recognizing this effect is crucial for the many applications of high-impedance environments in quantum technologies.

Figures

Figures reproduced from arXiv: 2411.19912 by the authors.

Figure 1
Figure 1. FIG. 1. (a) The open-system paradigm: a small subsystem is [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Mode frequencies [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Steady-state dc current (solid line, left axis) and [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Re [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Probability distributions of [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Trajectories of the slow envelopes [PITH_FULL_IMAGE:figures/full_fig_p013_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Evolution of [PITH_FULL_IMAGE:figures/full_fig_p014_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11 [PITH_FULL_IMAGE:figures/full_fig_p014_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. Probability distributions of [PITH_FULL_IMAGE:figures/full_fig_p014_12.png]

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Reviewed August 12, 2026 · model on record in the stance chip above.