arXiv subjects

Raphaël Leone

Publications and source records attributed to Raphaël Leone.

8 recordsLinked to original sources

arXiv2020

On the interplay between Noether's theorem and the theory of adiabatic invariants

This article focuses on an important quantity that will be called the Rund-Trautman function. It already plays a central role in Noether's theorem since its vanishing characterizes a symmetry and leads to a conservation law. The main aim of the paper is to show how, in the realm of classical mechanics, an 'almost' vanishing Rund-Trautman function accompanying an 'almost' symmetry leads to an 'almost' constant of motion within the adiabatic assumption, that is, to an adiabatic invariant. To this end, the Rund-Trautman function is first introduced and analysed in detail, then it is implemented for the general one-dimensional problem. Finally, its relevance in the adiabatic context is examined through the example of the harmonic oscillator with a slowly varying frequency. Notably, for some frequency profiles, explicit expansions of adiabatic invariants are derived through it and an illustrative numerical test is realized.

physics.class-ph↗

arXiv2019

On the parallel transport in quantum mechanics with an application to three-state systems

The aim of this article is to give a rigorous although simple treatment of the geometric notions around parallel transport in quantum mechanics. I start by defining the teleparallelism (or generalized Pancharatnam connection) between n-dimensional vector subspaces (or n-planes) of the whole Hilbert space. It forms the basis of the concepts of parallel transport and of both cyclic and non-cyclic holonomies in the Grassmann manifold of n-planes. They are introduced in the discrete case (broken lines) before being rendered 'continuous' (smooth curves) and the role of the geodesics is stressed. Then, I discuss the interest of such a construction to geometrize a part of the dynamics when a (quasi-)dynamical invariant is known, especially in the adiabatic limit. Finally, I illustrate the general theory with a three-state toy model allowing for non-Abelian adiabatic transports.

math-ph↗

arXiv2018

On the wonderfulness of Noether's theorems, 100 years later, and Routh reduction

This paper is written in honour of the centenary of Emmy Amalie Noether's famous article entitled Invariante Variationsprobleme. It firstly aims to give an exposition of what we believe to be the most significant and elegant issues regarding her theorems, through the lens of classical mechanics. Despite the limitation to this field, we try to illustrate the key ideas of her work in a rather complete and pedagogical manner which, we hope, presents some original aspects. The notion of symmetry coming naturally with the idea of simplification, the last part is devoted to the interplay between Noether point symmetries and the reduction procedure introduced by Edward John Routh in 1877.

physics.hist-ph↗

arXiv2016

Li(e)nearity

We demonstrate the fact that linearity is a meaningful symmetry in the sense of Lie and Noether. The role played by that `linearity symmetry' in the quadrature of linear ordinary second-order differential equations is reviewed, by the use of canonical coordinates and the identification of a Wronskian-like conserved quantity as Lie invariant. The Jacobi last multiplier associated with two independent linearity symmetries is applied to derive the Caldirola-Kanai Lagrangian from symmetry principles. Then the symmetry is recognized to be also a Noether one. Finally, the study is extended to higher-order linear ordinary differential equations, derivable or not from an action principle.

math-ph↗

arXiv2015

Classical Noether's theory with application to the linearly damped particle

This paper provides a modern presentation of Noether's theory in the realm of classical dynamics, with application to the problem of a particle submitted to both a potential and a linear dissipation. After a review of the close relationships between Noether symmetries and first integrals, we investigate the variational point symmetries of the Lagrangian introduced by Bateman, Caldirola and Kanai. This analysis leads to the determination of all the time-independent potentials allowing such symmetries, in the one-dimensional and the radial cases. Then we develop a symmetry-based transformation of Lagrangians into autonomous others, and apply it to our problem. To be complete, we enlarge the study to Lie point symmetries which we associate logically to Noether ones. Finally, we succinctly address the issue of a `weakened' Noether's theory, in connection with on-flows symmetries and non-local constant of motions, for it has a direct physical interpretation in our specific problem. Since the Lagrangian we use gives rise to simple calculations, we hope that this work will be of didactic interest to graduate students, and give teaching material as well as food for thought for physicists regarding Noether's theory and the recent developments around the idea of symmetry in classical mechanics.

math-ph↗

arXiv2013

Merging diabolical points of a superconducting circuit

We present the first theoretical study of the merging of diabolical points in the context of superconducting circuits. We begin by studying an analytically solvable four-level model which may serve as theoretical pattern for such a phenomenon. Then, we apply it to a circuit named Cooper pairs pump, whose diabolical points are already known.

cond-mat.mes-hall↗

arXiv2007

Topological quantization by controlled paths: application to Cooper pairs pumps

When physical systems are tunable by three classical parameters, level degeneracies may occur at isolated points in parameter space. A topological singularity in the phase of the degenerate eigenvectors exists at these points. When a path encloses such point, the accumulated geometrical phase is sensitive to its presence. Furthermore, surfaces in parameter space enclosing such point can be used to characterize the eigenvector singularities through their Chern indices, which are integers. They can be used to quantize a physical quantity of interest. This quantity changes continuously during an adiabatic evolution along a path in parameter space. Quantization requires to turn this path into a surface with a well defined Chern index. We analyze the conditions necessary to a {\em Topological Quantization by Controlled Paths}. It is applied to Cooper pair pumps. For more general problems, a set of four criteria are proposed to check if topological quantization is possible.

cond-mat.mes-hall↗

arXiv2007

The Cooper Pair Pump as a Quantized Current Source

A new charge quantization in a phase-polarized Cooper Pair Pump (CPP) is proposed, based on the topological properties of its Hamiltonian ground state over a three-dimensional parameter space $\mathbb{P}$. The charge is quantized using a set of path in $\mathbb{P}$ covering the surface of a torus, and is a multiple of the integer Chern index $c_1$ of this surface. This quantization is asymptotic but the pumped charge converges rapidly to the quantized value with the increase in the path frequency. The topological nature of the current makes this CPP implementation an excellent candidate for a metrological current standard.

cond-mat.mes-hall↗