IN two previous notes1, Prof. Max Born and I have shown that one can obtain a theory of superconductivity by taking account of the fact that the interaction of the electrons with the ionic lattice is appreciable only near the boundaries of Brillouin zones, and particularly strong near the corners of these. This leads to the criterion that the metal should be superconductive if a set of corners of a Brillouin zone is lying very near the Fermi surface, considered as a sphere, which limits the region in the momentum space completely filled with electrons.

On the theory of superconductivity

This review paper presents an overview of the theoretical and experimental progress on the study of matter-wave dark solitons in atomic Bose–Einstein condensates. Upon introducing the general framework, we discuss the statics and dynamics of single and multiple matter-wave dark solitons in the quasi one-dimensional setting, in higher dimensional settings, as well as in the dimensionality crossover regime. Special attention is paid to the connection between theoretical results, obtained by various analytical approaches, and relevant experimental observations.

https://iopscience.iop.org/article/10.1088/1751-8113/43/21/213001/pdf

Dark solitons in atomic Bose–Einstein condensates: from theory to experiments

This review paper presents an overview of the theoretical and experimental progress on the study of matter-wave dark solitons in atomic Bose-Einstein condensates. Upon introducing the general framework, we discuss the statics and dynamics of single and multiple matter-wave dark solitons in the quasi one-dimensional setting, in higher-dimensional settings, as well as in the dimensionality crossover regime. Special attention is paid to the connection between theoretical results, obtained by various analytical approaches, and relevant experimental observations.

/pdf/dark-solitons-in-atomic-bose-einstein-condensates-from-5a8fcdp426.pdf

Dark solitons in atomic Bose-Einstein condensates: from theory to experiments

The aim of this review is to introduce the reader to some of the physical notions and the mathematical methods that are relevant to the study of nonlinear waves in Bose–Einstein condensates (BECs). Upon introducing the general framework, we discuss the prototypical models that are relevant to this setting for different dimensions and different potentials confining the atoms. We analyse some of the model properties and explore their typical wave solutions (plane wave solutions, bright, dark, gap solitons as well as vortices). We then offer a collection of mathematical methods that can be used to understand the existence, stability and dynamics of nonlinear waves in such BECs, either directly or starting from different types of limits (e.g. the linear or the nonlinear limit or the discrete limit of the corresponding equation). Finally, we consider some special topics involving more recent developments, and experimental setups in which there is still considerable need for developing mathematical as well as computational tools.

/pdf/nonlinear-waves-in-bose-einstein-condensates-physical-o6ouu967m4.pdf

Nonlinear waves in Bose-Einstein condensates: physical relevance and mathematical techniques

We study the low-intensity light pulse propagation through an asymmetric double quantum well via Fano-type interference based on intersubband transitions. The propagation of the pulse across the quantum well is studied analytically and numerically with the coupled Maxwell-Schr\"odinger equations. We show the generation of ultraslow bright and dark optical solitons in this system. Whether the solitons are dark and bright can be controlled by the ratio of dipole moments of the intersubband transitions. Such investigation of ultraslow optical solitons in the present work may lead to important applications such as high-fidelity optical delay lines and optical buffers in semiconductor quantum wells structure.

/pdf/ultraslow-bright-and-dark-solitons-in-semiconductor-quantum-4qavp06n4z.pdf

Ultraslow bright and dark solitons in semiconductor quantum wells

We develop a systematic analytical approach to consider the dynamics of linear and nonlinear excitations in trapped quasi-one-dimensional Bose-Einstein condensates with repulsive atom-atom interactions. We show that, for a condensate strongly confined in two transverse directions, the ground state of the system involves the high-order eigenmodes of the transverse confining potential in the transverse directions and effective high-order Thomas-Fermi wave functions in the axial direction. The linear excitations of the system have a Bogoliubov-type spectrum with the excitation frequency varying slowly along the axial direction. We find that, in a weak nonlinear approximation, the amplitude of a nonlinear excitation is governed by a variable coefficient Korteweg\char21{}de Vries equation with additional terms contributed from the transverse structure and the inhomogeneity in the axial direction of the condensate, which results in varying amplitude, width, and velocity for dark solitons. Because of the inhomogeneity the dark solitons undergo deformation and emit radiations when traveling along the axial direction. We finally demonstrate that a dark soliton will disintegrate into several ones plus a residual wave train when passing over a steplike potential.

Dynamics of dark solitons in quasi-one-dimensional Bose-Einstein condensates

The Meissner effect is studied by using an approach based on Newton and Maxwell's equations. The objective is to assess the relevance of London's equation and shed light on the connection between the Meissner and skin effects. The properties of a superconducting cylinder, cooled in a magnetic field, are accounted for within the same framework. The radial Hall effect is predicted. The energy, associated with the Meissner effect, is calculated and compared with the binding energy of the superconducting phase with respect to the normal one.

https://hal.archives-ouvertes.fr/hal-01866579/document

Comparative Study of the Meissner and Skin Effects in Superconductors

A second-harmonic generation (SHG) is predicted for the Bogoliubov excitations in a two-component Bose-Einstein condensate. It is shown that, because the linear dispersion curve of the excitations displays two branches, the phase-matching condition for the SHG can be fulfilled if the wave vectors and frequencies of fundamental and second-harmonic waves are selected suitably from different branches. The nonlinearly coupled envelope equations for the SHG are derived by using a method of multiple scales. The explicit solutions of these envelope equations are provided and the conversion efficiency of the SHG is also discussed.

Second-harmonic generation of Bogoliubov excitations in a two-component Bose-Einstein condensate

On the existence of moving breathers in one-dimensional anharmonic lattices

A classical model is presented for persistent currents in superconductors. Their existence is argued to be warranted because their decay would violate the second law of thermodynamics. This conclusion is achieved by analyzing comparatively Ohm's law and the Joule effect in normal metals and supercon-ducting materials. Whereas Ohm's law applies in identical terms in both cases, the Joule effect is shown to cause the temperature of a superconducting sample to decrease. An experiment is proposed to check the validity of this work in superconductors of both types I and II.

https://hal.archives-ouvertes.fr/hal-02085943/document

Jacob Szeftel

Papers

Dynamics of dark solitons in quasi-one-dimensional Bose-Einstein condensates

Comparative Study of the Meissner and Skin Effects in Superconductors

Second-harmonic generation of Bogoliubov excitations in a two-component Bose-Einstein condensate

On the existence of moving breathers in one-dimensional anharmonic lattices

An observable prerequisite for the existence of persistent currents