absnoise

Andreev-bound-state occupation noise in proximity Josephson junctions, computed from the level structure to the detector budget: the exact finite-length Andreev spectrum, the intrinsic occupation-noise limit of Andreev thermometry, and the resulting temperature, frequency-noise and calorimetric energy resolutions of an inductively read junction. The package exists because the occupation of Andreev levels fluctuates even in equilibrium, and that fluctuation, not the readout, is what ultimately limits a proximity Josephson thermal detector.

Status

v0.2.0 (alpha). Implemented and tested:

• exact finite-length ABS solver from the closed-form secular equation cos(2 arccos(E/Delta) - eta(E)) = 1 - tau + tau cos(phi), with the continuum (E > Delta) free energy from the scattering phase
• weak-coupling BCS gap Delta(T) solved from the gap equation itself (tabulated once per process; no data file), because the widely used tanh interpolation misrepresents dDelta/dT at low temperature
• closed-form short-junction ensemble: current-phase relation, critical current, Josephson inductance, occupation-channel responsivity and noise sums, Andreev heat capacity, and the Cauchy-Schwarz temperature-resolution bound var(T) >= 2 kB T^2 tauA / (C_A t)
• level-resolved finite-length sums and the bound-saturation deficit
• four-state (pair-process) master equation with exact spectra, activated exchange times, and the nonequilibrium occupation penalty
• telegraph-noise Monte Carlo and single-sided PSD estimation
• device budgets: phonon thermal-fluctuation noise, resonator fractional-frequency noise spectra, matched-filter calorimetric energy resolution
• six cited graphene junction contact recipes (Jung et al., Table I) and graphene electronic thermodynamics (heat capacity, electron-phonon cooling, Lee et al.)
• matched-level design and nonlinear click dynamics (new in v0.2): the matched condition Delta*(T0) = 2.3994 kB T0 solved from the gap equation (matched_Tc, matched_recipe), the full nonlinear post-deposit response with a stiffness-safe exponential integrator (click_template), a whitened matched-filter single-photon click Monte Carlo with all three noise channels (click_monte_carlo), and the exact self-heating steady state (steady_temperature)

Verified against closed forms in the test suite rather than asserted: the short-junction limit E = Delta sqrt(1 - tau sin^2(phi/2)) to 1e-12; Kulik levels at tau = 1 to 1e-10; the ballistic anchor Ic = 2 e Delta / hbar per orbital mode to 5e-3 (phase-grid limited); exact vanishing of the continuum phase dependence at L = 0; the BCS midpoint u(0.5) = 0.956887 and both asymptotes; exact saturation of the Cauchy-Schwarz bound by uniform-transparency short junctions to 1e-9, and its non-violation by dispersing finite-length levels; the analytic occupation responsivity to 1e-5; the pair-process master equation's exact singles limit, equilibrium-variance invariance, monotone shortening of the correlation time, probability conservation, and spectrum-to-variance sum rule; the telegraph Monte Carlo Lorentzian plateau, knee, and the variance convention var = S(0)/(2t) (statistics-limited tolerances); and the exact Lorentzian knee of the predicted resonator frequency-noise spectrum.

For v0.2 the test suite additionally asserts: the matched-level condition to 1e-9 from the solved gap equation; energy conservation of the click integrator to machine precision through the exact peak-temperature identity; dip-and-recovery of the occupation template in both exchange scenarios; photon-versus-dark ordering of the click Monte Carlo; and the exact closed-form round trip of the self-heating steady state.

Not yet implemented, stated plainly because they matter physically: continuum contributions to the occupation channel are neglected (bound levels dominate for L < xi; the largest L/xi in the recipe set is 0.43; continuum_share quantifies the supercurrent-channel analog), and phonon-bath heating by the substrate is treated only through the steady-state electron temperature, not dynamically.

Install and use

pip install absnoise

For development, clone the repository and pip install -e .[test].

import numpy as np
from absnoise import RECIPES, SensorBudget

budget = SensorBudget(RECIPES[1])          # Ti/Al/Au recipe, Jung et al.
T = 0.3 * budget.recipe.Tc                 # operating temperature (K)
tauA = 1e-6                                # occupation correlation time (s)

achieved, sums = budget.dT_andreev(T, tauA, t=1.0)
print(f"temperature resolution {achieved*1e6:.2f} uK in 1 s")

Sy, Snu = budget.freq_noise_spectrum(T, tauA, np.array([0.0, 1e3, 1e6]))
sigE = budget.energy_resolution(T, tauA)
print(f"matched-filter energy resolution {sigE:.3e} J")

Units are SI throughout; PSDs are single-sided with the variance convention var(t-average) = S(0)/(2t), validated by Monte Carlo in the test suite.

Cited constants

The six junction recipes ship with full provenance: Table I of W. Jung, E. G. Arnault, B. Huang, J. Park, S. Jang, K. Watanabe, T. Taniguchi, D. Englund, K. C. Fong and G.-H. Lee, "Engineering Andreev Bound States for Thermal Sensing in Proximity Josephson Junctions", Phys. Rev. Applied 26, 014078 (2026) (arXiv:2503.06850). Graphene electron-phonon cooling follows the measured coupling of G.-H. Lee et al., Nature 586, 42 (2020) (resonant-supercollision regime, delta = 3, Sigma ~ 2 W m^-2 K^-3). Physical constants are CODATA 2018. The test suite locks every recipe number to the source; a change to any of them must arrive with a new source.

Methodological basis

T. M. Mahim, A. S. M. Mohsin and M. M. Rahman, "Andreev occupation noise sets the sensitivity limit of proximity Josephson thermal detectors"; code for the paper: https://github.com/Tanvir-Mahmud-Mahim/andreev-occupation-noise

This package is the general-purpose engine (v0.2 includes the matched-level design and the nonlinear click Monte Carlo); the paper repository reproduces the specific study: the device grids, the figures, the approximation-resolution analysis, and the archived trial data.

Citing the tool

The repository carries a CITATION.cff file with citation metadata. If this software contributes to a publication, please cite the versioned DOI you used.

License

Apache-2.0