Anderson Localization - the complete halt of wave transport through multiple scattering and phase coherence - was discovered in 1958 and has been actively studied since then with several types of waves (light, µ-waves, ultrasound, acoustic waves, etc.). In three-dimensional systems, Anderson Localization manifests as a phase transition around a critical energy - the Mobility Edge - that separates the localized (insulator) states from the diffusive (metallic) states.Ultracold atoms are promising candidates for such experiments, and various groups have reported evidence of the 3D Anderson Transition, in particular, our group in 2012. Nevertheless, all these preliminary measurements were limited by the unwanted broadening of the energy distribution when the atoms were transferred into the disorder. Hence, no direct observation of the transition could be achieved.The quest for the direct observation of the 3D Anderson Transition has motivated us to develop a novel spectroscopic scheme using state-dependent disorder. The concept is to transfer, via RF coupling, a small fraction of a Bose-Einstein Condensate from a disorder-insensitive state to a disorder-sensitive state. By doing so, we selectively populate targeted energy levels in the disorder with a narrow energy distribution. This technique has paved the way for successfully achieving the direct observation of the 3D Anderson Transition in a 3D laser speckle. Moreover, it yields a precise measurement of the Mobility Edge, which is in excellent agreement with the numerical predictions.The heart of our achievement lies in the spectroscopic scheme. When transferring only a few atoms, we expect the coupling of a discrete state to a continuum, leading to a Fourier-limited energy distribution. This key assumption has so far been verified only indirectly, but never tested numerically. In the last part of my PhD, I have set up the framework for the numerical description of the RF transfer scheme. While the study remains preliminary, it already demonstrates that the dynamics can be more subtle than expected, and the model deserves further attention in the future.
We study the effect of Anderson localization on the expansion of a Bose-Einstein condensate, released from a harmonic trap, in a 3D random potential. We use scaling arguments and the self-consistent theory of localization to show that the long-time behavior of the condensate density is controlled by a single parameter equal to the ratio of the mobility edge and the chemical potential of the condensate. We find that the two critical exponents of the localization transition determine the evolution of the condensate density in time and space.
Anderson localization is fundamentally controlled by dimensionality, yet the nature of the Anderson transition in continuously tunable noninteger dimensions remains largely unexplored. Here, we introduce a family of three-dimensional fractal lattices with continuously tunable spectral dimension
ds∈[2,3], providing a controlled platform for studying localization physics beyond integer dimensions and across the lower critical dimension
ds=2. Using large-scale finite-size scaling analysis, we systematically investigate the Anderson transition and identify mobility edges throughout the fractal family. The critical disorder strength evolves continuously from
0 to
16.6 as the spectral dimension increases from
2 to
3. We show that the spectral dimension predominantly governs the universality class of the transition, while the precise critical point is additionally influenced by microscopic geometric details of the underlying fractal lattice. The critical exponent exhibits an approximate inverse dependence on
ds, providing quantitative insight into scaling theory in noninteger dimensions. Our results establish tunable fractal lattices as a versatile framework for exploring localization and quantum critical phenomena beyond conventional integer-dimensional systems.