Multiple Quasiparticle Bound States in a Trap Created by a Local Superconducting Gap Variation

TL;DR

The paper tackles how local variations of the superconducting gap trap quasiparticles by analyzing a disk-shaped gap suppression. Using both semiclassical Bohr–Sommerfeld quantization and full Bogoliubov–de Gennes calculations, it shows that in 2D and 3D there exists an infinite number of bound states accumulating near the gap edge for any finite trap size, with high-angular-momentum states forming rings at radii far from the trap. The results recover the delta-impurity limit for small traps and the semiclassical spectrum for large traps, but reveal additional states beyond these limits. This has direct implications for quasiparticle dynamics, recombination, and the interpretation of traps in disordered superconductors, potentially challenging two-level-system pictures of dissipation.

Abstract

At low temperature, the concentration of quasiparticles observed in superconducting circuits far exceeds the predictions of microscopic BCS theory at equilibrium. As a source of dissipation, these excess quasiparticles degrade the performance of various devices. Therefore, understanding their dynamics, especially their recombination into Cooper pairs, is an active topic of current research. In disordered superconductors, spatial fluctuations in the superconducting gap can trap quasiparticles and modify their eigenspectrum. Since this spectrum plays a key role in quasiparticle dynamics, it must be carefully investigated. To this end, we introduce a toy model of a single trap. Specifically, we consider a shallow disk-shaped gap variation in a clean superconductor. Using a semiclassical approximation, we demonstrate the existence of multiple bound states and give the dependence of their number on the size and depth of the gap suppression. Extending our analysis beyond the semiclassical regime, in dimensions larger than one, we observe an infinite number of bound states very close to the gap edge, even for an arbitrarily small trap. These results deepen our understanding of trapped quasiparticles and may have important implications for their recombination in disordered superconductors.

Figures (4)

• Figure 1: In two dimensions, different semiclassical trajectories are possible. They are characterized by their '' impact parameter'' $a=R\sin\theta$ (red), which determines the angular momentum $\ell=k_Fa$ of the corresponding bound state. The path length of the trajectory is then given by $2L_{\rm eff}$ with the effective length $L_{\rm eff}=2R\cos\theta$ (blue).
• Figure 2: Bound state energies for $\ell=10$ and $\gamma^2=10^{-3}$ as a function of trap size $s$ in a two-dimensional trap. (a) Energy of the lowest-lying bound state at small trap sizes. Analytical results obtained with semiclassics (Eq. \ref{['BS-energy']}) and with the full Bogoliubov-de Gennes equation (Eq. \ref{['res-simple']}), as well as numerical results using Eq. \ref{['eq-sol']} are shown. The semiclassical threshold value is indicated by a star. While the numerics show how the bound state detaches from the gap edge around the semiclassical threshold, the numerical accuracy is not sufficient to resolve the bound state energy at much smaller trap sizes. (b) Energy spectrum of the first few bound states showing good agreement between the semiclassical result, Eq. \ref{['sc-2D']}, and the BdG result, Eq. \ref{['eq-sol']}, at larger scales.
• Figure 3: Left-hand side of Eq. \ref{['s-threshold']} for $\ell=1,5,10$ and $\gamma^2=10^{-3}$ as a function of $x_\ell=2\gamma/\pi\sqrt{s^2-\ell^2}$. Thresholds are given by zero crossings. All the curves cross zero very close to the semiclassical thresholds $s_{n,\ell}$ corresponding to $x_\ell=n$ with $n>0$.
• Figure 4: Wave functions for $\ell=10$, $\gamma^2=10^{-3}$, and different trap sizes. We show the probability density $|\psi^{(10)}(r/R)|^2$ in arbitrary units for different trap sizes. As $s$ decreases, the maximum of $|\psi^{(10)}(r/R)|^2$ moves outward. For states that are beyond the semiclassical approximation, $s\ll s_{n\ell}$, it lies far outside the trap (red).