Quantum dynamics for a fractal spectrum - Application to simple processes in electrodynamics and quantum optics

Ariane SORET^{1}

Supervised by Eric AKKERMAN^{2}

1ENS Cachan, France ; ^{2} Technion - Israel Institute of Technology, Haifa, Israël 2/25/2015

Introduction

During the past decades, there has been a growing interest for Schroedinger operators H = −\frac{ℏ^{2}}{2m}Δ+V (r) with a fractal spectrum. In fact, while the dynamics and the transport properties of a quantum system for a continuous or discrete spectrum are well understood, little is known about singular continuous spectra (like fractals), which are ubiquitous (quasi-crystals, quasi-periodic potentials, some glasses).

Some recent experiments, pursued at the Laboratoire de Photonique et Nanostructure (CNRS, Marcoussis, France) in partnership with my group at the Technion, on cavity polaritons placed in a quasi-periodic potential with an associated Cantor like spectrum, have evidenced the log-periodic behaviour of some dynamical quantities of the system. Although expected theoretically in the case of a fractal spectrum, such oscillations had never been observed experimentally before.

Objectives

Our goal is to understand, for various quantum systems, how the fractal feature of the spectrum affects the dynamics, to determine which are the relevant physical quantities in order observe the signature of the fractal spectrum, and to derive an analytical expression for those quantities. This study should then allow to pursue more experiments. My initial project contains three main topics, but so far I have focused on two of them : 1 - Analytical and numerical study of a wave packet for a triadic Cantor spectrum ; this study was initiated by the experimental results obtained with the cavity polaritons, and was the subject of my master’s research internship pursued between April and July 2014. 2 - Spontaneous emission and superradiance for atoms coupled to a fractal spectrum.

Figure 1 – Time-averaged return probability of a wave packet in a Fibonacci potential.

Figure 2 – Time-averaged return probability for a triadic Cantor spectrum.

Dynamics of a wave packet for a triadic Cantor spectrum

Consider a quantum system, described by a Hamiltonian H = −\frac{d^{2}}{dx^{2}} + V (x) on L^{2}(ℝ), with associated triadic Cantor set spectrum K. The density of states is degenerate, but can be described by an appropriate measure dμ on K such that, for any function f defined on K, one has (scaling property) :

∫_{K} f(x)dμ(x) = \frac{1}{2} ∫_{K} f(\frac{x}{3})dμ(x) + \frac{1}{2} ∫_{K} f(\frac{x + 2}{3} )dμ(x)

Return probability

As suggested by a numerical study for the Fibonacci potential, the time-averaged return probability, C(t) = \frac{1}{t} ∫_{0}^{t} |⟨ψ(0)|ψ(t′)⟩|^{2}dt′, displays the characteristic logperiodic feature. The derivation of the analytical expression of C(t) was part of my master’s internship project, but I continued with a numerical analysis since.