Excitation spectra for Andreev billiards of box and disk geometries

We study Andreev billiards of box and disk geometries by matching the wave functions at the interface of the normal and the superconducting region using the exact solutions of the Bogoliubov-de Gennes equation. The mismatch in the Fermi wave numbers and the effective masses of the normal system and the superconductor, as well as the tunnel barrier at the interface are taken into account. A Weyl formula (for the smooth part of the counting function of the energy levels) is derived. The exact quantum mechanical calculations show equally spaced singularities in the density of states. Based on the Bohr-Sommerfeld quantization rule a semiclassical theory is proposed to understand these singularities. For disk geometries two kinds of states can be distinguished: states either contribute through whispering gallery modes or are Andreev states strongly coupled to the superconductor. Controlled by two relevant material, parameters, three kinds of energy spectra exist in disk geometry. The first is dominated by Andreev reflections, the second, by normal reflections in an annular disk geometry. In the third case the coherence length is much larger than the radius of the superconducting region, and the spectrum is identical to that of a full disk geometry

Childcare Remains Out of Reach for Millions in 2021, Leading to Disproportionate Job Losses for Black, Hispanic, and Low-Income Families

Using data from the U.S. Census Bureau’s Household Pulse Survey, collected in late summer through the fall of 2021, this brief documents recent racial and income disparities in reports of inadequate access to childcare and identifies the employment-related consequences of these shortages. The authors find that, in Fall 2021, about 5 million U.S. households had a child under age 12 who was unable to attend childcare as a result of it being closed, unavailable, unaffordable, or because parents were concerned about their child’s safety in the past month. Black and low-income households were more likely to experience inadequate childcare access. About one in five households with inadequate childcare access suffered a related employment loss as a result of time needed to care for children. The authors suggest that swift policy actions to stem the loss of childcare slots are needed in the immediate term, including bolstering the wages of the childcare workforce. Longer-term actions should focus on stabilizing the sector with creative investments to increase supply in low-income neighborhoods and cap costs for the most disadvantaged families

Observation of conduction electron spin resonance in boron doped diamond

We observe the electron spin resonance of conduction electrons in boron doped (6400 ppm) superconducting diamond (Tc =3.8 K). We clearly identify the benchmarks of conduction electron spin resonance (CESR): the nearly temperature independent ESR signal intensity and its magnitude which is in good agreement with that expected from the density of states through the Pauli spin-susceptibility. The temperature dependent CESR linewidth weakly increases with increasing temperature which can be understood in the framework of the Elliott-Yafet theory of spin-relaxation. An anomalous and yet unexplained relation is observed between the g-factor, CESR linewidth, and the resistivity using the empirical Elliott-Yafet relation.Comment: 10 pages, 11 figures, submitted to Phys. Rev.

Diffusion maps tailored to arbitrary non-degenerate Ito processes

We present two generalizations of the popular diffusion maps algorithm. The first generalization replaces the drift term in diffusion maps, which is the gradient of the sampling density, with the gradient of an arbitrary density of interest which is known up to a normalization constant. The second generalization allows for a diffusion map type approximation of the forward and backward generators of general Ito diffusions with given drift and diffusion coefficients. We use the local kernels introduced by Berry and Sauer, but allow for arbitrary sampling densities. We provide numerical illustrations to demonstrate that this opens up many new applications for diffusion maps as a tool to organize point cloud data, including biased or corrupted samples, dimension reduction for dynamical systems, detection of almost invariant regions in flow fields, and importance sampling

Electron spin resonance signal of Luttinger liquids and single-wall carbon nanotubes

A comprehensive theory of electron spin resonance (ESR) for a Luttinger liquid (LL) state of correlated metals is presented. The ESR measurables such as the signal intensity and the line-width are calculated in the framework of Luttinger liquid theory with broken spin rotational symmetry as a function of magnetic field and temperature. We obtain a significant temperature dependent homogeneous line-broadening which is related to the spin symmetry breaking and the electron-electron interaction. The result crosses over smoothly to the ESR of itinerant electrons in the non-interacting limit. These findings explain the absence of the long-sought ESR signal of itinerant electrons in single-wall carbon nanotubes when considering realistic experimental conditions.Comment: 5 pages, 1 figur

From Metastable to Coherent Sets – time-discretization schemes

Given a time-dependent stochastic process with trajectories x(t) in a space \Omega, there may be sets such that the corresponding trajectories only very rarely cross the boundaries of these sets. We can analyze such a process in terms of metastability or coherence. Metastable sets M are defined in space M \subset \Omega, coherent sets M(t) \subset \Omega are defined in space and time. Hence, if we extend the space \Omega by the time-variable t, coherent sets are metastable sets in \Omega \times [0,\infty). This relation can be exploited, because there already exist spectral algorithms for the identification of metastable sets. In this article we show that these well-established spectral algorithms (like PCCA+) also identify coherent sets of non-autonomous dynamical systems. For the identification of coherent sets, one has to compute a discretization (a matrix T) of the transfer operator of the process using a space-time-discretization scheme. The article gives an overview about different time-discretization schemes and shows their applicability in two different fields of application

Singular Value Decomposition of Operators on Reproducing Kernel Hilbert Spaces

Reproducing kernel Hilbert spaces (RKHSs) play an important role in many statistics and machine learning applications ranging from support vector machines to Gaussian processes and kernel embeddings of distributions. Operators acting on such spaces are, for instance, required to embed conditional probability distributions in order to implement the kernel Bayes rule and build sequential data models. It was recently shown that transfer operators such as the Perron-Frobenius or Koopman operator can also be approximated in a similar fashion using covariance and cross-covariance operators and that eigenfunctions of these operators can be obtained by solving associated matrix eigenvalue problems. The goal of this paper is to provide a solid functional analytic foundation for the eigenvalue decomposition of RKHS operators and to extend the approach to the singular value decomposition. The results are illustrated with simple guiding examples