Thesis of Farid Madani

Defense of thesis Farid Madani - laboratory PhLAM

Abstract :

The Anderson model studies the transport of an electron in a crystal in the presence of disorder. In dimensions 1 and 2, this model exhibits an exponential localization of the electron's wave function in position space, known as Anderson localization. In dimensions D > 2, a quantum phase transition appears depending on the strength of the disorder, between a regime where all the states of the system are localized (insulating behavior) and a regime where they are delocalized (metallic behavior). This is called the Anderson transition, also known as the metal-insulator transition. This phase transition has been studied at D = 3, both theoretically and experimentally, using different systems, and its critical exponents have been measured.

In this thesis, we use the Atomic Kicked Rotor model, which belongs to the universality class of the Anderson model, to study the metal-insulator transition in dimension 4, using an experiment of ultracold potassium-41 atoms. Since this dimension is not physically achievable, we use a method to generate synthetic dimensions through modulations of the amplitude of a pulsed optical potential to increase the system's effective dimensionality. This allows us to observe the quantum phase transition and measure the critical exponents - of the localization length on the localized side, and of the diffusion constant on the delocalized side. We also measure the two-parameter scaling function, which characterizes the behavior around the critical point. The values obtained for the critical exponents confirm Wegner's relation (which links the two exponents), are in very good agreement with numerical simulations of the Anderson model, and differ from the predictions of mean-field theory (here the self-consistent theory). This work is the first experimental demonstration that D = 4 is not the upper critical dimension of the Anderson transition.

Keywords : Anderson localization,Ultracold Atoms,Quantum phase transition,Dynamical localization,Quantum Chaos,Anderson Transition