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arXiv:2512.17765 · Physical review. B./Physical review. B

Proximity effect in gap-asymmetric superconducting bilayers and regularization of transition rates

G. Marchegiani, G. Catelani

cond-mat.supr-concond-mat.mes-hallquant-phSubmitted Dec 19, 2025• Announced Dec 22, 2025• 0 citations• Revised Sep 3, 2026 (v2)
v1 · Dec 19, 2025v2 · Sep 3, 2026
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Abstract

The standard mean-field treatment of low-temperature superconductors leads to a square-root divergent density of states at the gap value. This feature can lead to unphysical logarithmic divergences in various quantities, such as currents and qubit transition rates. We revisit their possible regularization based on the proximity effect between two superconducting films with different gaps. We derive analytical approximations for the density of states in each superconducting film. We find that the smearing of the density of states grows with the gap asymmetry. As a concrete example, we discuss the regularization of transition rates in qubits with frequency close to resonance with the gap asymmetry between the two films, and the consequent smoothening of the jump discontinuity in the qubit frequency shift. Our results could also aid the development of superconducting detectors and related devices based on superconducting bilayers.

Journal reference
Phys. Rev. B 114, 144504 (2026)
DOI
10.1103/k2d4-tpc4
Comments
21 pages, 8 figures

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In many superconducting devices, including qubits, quasiparticle excitations are detrimental. A normal metal (N)(N)(N) in contact with a superconductor (S)(S)(S) can trap these excitations; therefore, such a trap can potentially improve the device's performances. The two materials influence each other, a phenomenon known as proximity effect which has drawn attention since the 1960's. Here, we study whether this mutual influence places a limitation on the possible performance improvement in superconducting qubits. We first revisit the proximity effect in uniform NSNSNS bilayers; we show that the density of states is of the Dynes type above the minigap. We then extend our results to describe a nonuniform system in the vicinity of a trap edge. Using these results together with a phenomenological model for the suppression of the quasiparticle density due to the trap, we find in a transmon qubit an optimum trap-junction distance at which the qubit relaxation rate is minimized. This optimum distance, of the order of 4 to 20 coherence lengths, originates from the competition between proximity effect and quasiparticle density suppression. We conclude that the harmful influence of the proximity effect can be avoided so long as the trap is farther away from the junction than this optimum.
Superconducting qubits hold promise for quantum computing, but their operation is challenged by various sources of noise, including excitations known as quasiparticles. Qubits with gap asymmetry larger than their transition energy are less susceptible to quasiparticle decoherence as the quasiparticles are mostly trapped in the low-gap side of the junction. Because of this trapping, the gap asymmetry can contribute to maintaining the quasiparticles out of equilibrium. Here we address the temperature dependence of the quasiparticle densities in the two sides of the junction. We show that four qualitatively different regimes are possible with increasing temperature: (i) nonequilibrium, (ii) local quasiequilibrium, (iii) global quasiequilibrium, and (iv) full equilibrium. We identify shortcomings in assuming global quasiequilibrium when interpreting experimental data, highlighting how measurements in the presence of magnetic field can aid the accurate determination of the junction parameters, and hence the identification of the nonequilibrium regimes. Quasiparticle poisoning challenges the operation of superconducting qubits, but errors can be suppressed by engineering gap asymmetry between the electrodes of the qubit’s Josephson junction. The authors present a model of the qubit-quasiparticle system valid beyond the low-temperature limit and identify different nonequilibrium regimes, which can affect the interpretation of experimental data.
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We evaluate the quasiparticle contribution to the frequency shift and relaxation rates of a transmon with the Josephson junctions connecting superconductors that have unequal energy gaps. The gap difference substantially affects the transmon characteristics. We investigate their dependence on the density and effective temperature of the quasiparticles, and on the nominal (unperturbed by the quasiparticles) transmon frequency. At temperatures low compared to the qubit frequency, the gap difference can induce an anomalous positive frequency shift, resulting in a non-monotonic temperature dependence of the transmon frequency. The qubit relaxation rate exhibits a resonance when the qubit frequency matches the gap difference; the shape of the resonance is strongly temperature-dependent. We propose to use these effects to access the details of the quasiparticle energy distribution.