## Abstract

One of the long standing questions in quantum physics is the reason electrical conductivity varies so widely between materials, over 16 orders of magnitude, much more than can be explained by the availability of electrons. The accepted explanation lies in Anderson localisation [1], which relies on the quantum wave nature of the electrons, which carry electrical current.

*Different paths from A to B in a random distribution of scatterers*

For a wave to propagate from A to B in the figure by scattering off a random distribution of scatterers, the different paths have different lengths and therefore different phases. After summing over all the possible paths, these interfere destructively, and propagation is suppressed. On the other hand, the backward and forward going paths going from A to B *and back* have the same path lengths, and will interfere constructively. An insulator is an example of a system that creates such a random potential for electrons. The motion of electrons in an insulator cannot be tracked directly, but this phenomenon has been studied in a range of equivalent systems [2-5]. Recent experiments using ultra-cold atoms show Anderson localisation in a potential generated by laser speckle [6-9]. This exciting development is closely related to the quantum chaos work on the delta-kicked rotor system by both the P.I. [10-13] and the A.I. [14-17].

*Critical remaining questions that we want to answer are:*

- How do interactions influence Anderson localisation?
- How does the nature of the randomness of the potential influence Anderson localisation?
- What happens if the potential depends on time?

Answering these will significantly enhance our understanding of one of the most important properties of materials: their electrical conductivity. We propose to investigate the motion of ultra-cold atoms in a programmable quasi-random potential, which can be arbitrarily shaped. The atoms are so cold, that the wavelength of their quantum-mechanical wave function is much larger than the characteristic distance scale of the potential. We intend to use different potential length scales, as well as different random functions, thereby mapping out the transition from classical to quantum motion. We will confine the atoms to a wave guide to study the role of interactions. Furthermore, the atoms can be given a velocity with respect to the potential, which will allow us to observe a tell-tale signature of Anderson localisation, which is coherent back scattering [18].

We trap and cool rubidium atoms in a dipole trap, formed by the focus of a CO_{2} laser, to the point of Bose-Einstein Condensation (BEC) [19,20]. At this point, all atoms in the trap share the same quantum-mechanical wave function, and are at an equivalent temperature of 50 nanoKelvin above absolute zero. The atoms now are moving very slowly, and consequently have a large “deBroglie” wavelength, larger than the wavelength of light we use for the random potential.

After creating the BEC, the CO_{2} laser trap is turned off and replaced by the quasi-random potential, which is formed by a laser beam that has been shaped by a Spatial Light Modulator (SLM). The SLM consists of a two-dimensional array of 1024×768 microscopic mirrors, that can each be addressed separately, and dynamically. We then obtain an image of the atomic distribution after a propagation time. If Anderson localisation occurs, we should see a drastically reduced spread of the atoms after a time-of-flight, even when the maximum height of the potential is much less than the kinetic energy of the atoms.

The experiments described here build on the existing infrastructure at the University of Auckland, which has routinely made BECs since 2006, and was established by the P.I., and on a strong tradition in New Zealand in Quantum and Atom Optics research. It provides a great opportunity for graduate students to be involved in ground-breaking research.

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