Perfectly conducting boundaries, and their Dirichlet counterparts for quantum scalar fields, predict nonintegrable energy densities. A more realistic model with a finite ultraviolet cutoff yields two inconsistent values for the force on a curved or edged boundary (the "pressure anomaly"). A still more realistic, but still easily calculable, model replaces the hard wall by a power-law potential; because it involves no a posteriori modification of the formulas calculated from the theory, this model should be anomaly-free. Here we first set up the formalism and notation for the quantization of a scalar field in the background of a planar soft wall, and we approximate the reduced Green function in perturbative and WKB limits (the latter being appropriate when either the mode frequency or the depth into the wall is sufficiently large). Then we display numerical calculations of energy density and pressure for the region outside the wall, which show that the pressure anomaly does not occur there. Calculations inside the wall are postponed to later papers, which must tackle regularization and renormalization of divergences induced by the potential in the bulk region.

We study theoretically the Josephson current-phase relationship in a chaotic quantum dot coupled to superconductors by ballistic contacts. In this regime, strong proximity effect induces superconductivity in the quantum dot that leads to a significant modification in the electron density of states and formation of multiple sub-gaps. The magnitude of the resulting supercurrent depends on the phase difference of the superconducting order parameter in the leads and shows strongly anharmonic skewed behavior. We find that when the Thouless energy on the dot exceeds the superconducting energy gap, the second harmonic of the supercurrent becomes comparable in magnitude to the first harmonic. To address these effects on the technical level, we use the nonlinear σ-model Keldysh formalism in the framework of the circuit theory to compute dependence of the density of states, Josephson energy, and current on the superconducting phases in the leads. We analyze how these quantities change as a function of the Thouless energy and the superconducting gap. Finally, we briefly discuss sub-gap tail states, mesoscopic supercurrent fluctuations, weak localization correction, and also touch on anharmonicity of gatemon qubits with quantum dot Josephson junctions. arXiv:1804.02782v2 [cond-mat.supr-con] 30 May 2018

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