Héloïse Nonne

PhD thesis

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Development of my PhD

I did the research work for my PhD under the direction of Prof. Philippe Lecheminant at the Laboratoire de Physique Théorique et Modélisation of the University of Cergy-Pontoise, from October 2008 to September 2011.

My PhD research was incorporated within the framework of a scientific collaboration in a team that included my PhD advisor and myself, as well as researchers from different laboratories. For my PhD thesis, my main collaborators were:

My PhD defense committee

The public defense was held on September 21st 2011. The members of the defense committee were:

M. Antoine Georges CPHT, Ecole polytechnique
Collège de France
President
M. Thierry Giamarchi DPMC
Université de Genève
Referee
M. Pierre Pujol LPT, Toulouse
Université Paul Sabatier
Referee
M. Patrick Azaria LPTMC, Jussieu
Université Pierre et Marie Curie
Examiner
M. Thierry Jolicoeur LPTMS
Université Paris-Sud Orsay
Examiner
M. Philippe Lecheminant LPTM
Université de Cergy-Pontoise
PhD advisor

Motivation

Artificial solids

In the past thirty years, a new research field has opened: the artificial solids. Quantum boxes represent a beautiful example of nanotechnology. It is possible to engineer architectures that confine particles in a limited number of spatial dimensions. Quantum well confine particle in one dimension and allow their propagation along the two remaining dimensions, while in quantum wires, confinement takes place in two dimensions (see figure 1); in quantum dots, the three spatial dimensions are confined: they are like artificial atoms that can be manipulated at will to simulate a solid.

Carbon nanotube
Figure 1: An example of a quantum wire: a carbon nanotube (multi-wall) obtained by transmission electron microscopy (Christopher M. Evans, University of Rochester).

These artificial solids allow us to push back the frontiers of condensed matter physics and probe the matter at a very fundamental level. In particular, they provide us with an experimental access to the unidimensional world. Indeed, if unidimensional systems do exist in some (quasi-unidimensional) materials such as Bechgaard salts, there are nevertheless very uncommon in nature and the experimental parameters are barely tunable.

Cold atoms

On the other hand, these last years have seen a new discipline emerge in physics: the cold atoms. The considerable interest of cold atom gases is that, at very low temperature, the atoms populate the lowest energy levels and the quantum statistics (Bose-Einstein statistic for boson and Fermi-Dirac for fermions) then come forward at a macroscopic level. This provides an ideal playground to the study of quantum systems. Cornell and Wieman's team (see Anderson 1995) and that of Ketterle (see Davis 1995) have realized the first Bose-Einstein condensates en 1995, reaching temperatures of order 170 nanokelvins (nK) (see Anderson 1995). Cornell, Wieman and Ketterle were awarded the Nobel prize in 2001. Quantum degeneracy was achieved for the first time in 1999 for fermionic gases with temperatures of order 300 nK (see deMarco 1999).

In a cold atom gas, on can create an artificial solid with the help of laser beams that form an optical lattice. Figure 2 show such a lattice of potential wells that trap the atoms: in the cold atoms field, this optical lattice is equivalent to what the crystal is in conventional condensed matter systems: metals, insulators, semiconductor, ...

Optical lattic in a cold atom gas
Figure 2: An optical lattice a) with two pairs of perpendicular counter-propagating laser beams interacting with the cold atom gas, one builds a lattice of tubes where one spatial dimension remains free; b) three pairs of perpendicular beams create a lattice where all spatial dimensions are constrained. At each site (in gray), there is a potential well that confines the atoms: this is an example of an artificial solid. [From Bloch 2005]

Cold atoms has a considerable asset compared to material physics and to the aforementioned artificial solids: the experimental conditions are exceptional since the system parameters are highly tunable. Adjusting the intensities and wavelength of the laser beams, or tuning their respective directions, one creates lattice in one, two or three dimensions, simple or complex (see Petsas 1994). Furthermore, the strong interaction regime is today within our reach and the interaction can even chosen to be repulsive or attractive (see for instance Bloch 2008). Hence, from the high tunability of the system, and from the point of view of the lattice geometry or the interactions, cold atoms in optical lattice provide us with an ideal playground to study the strong interaction regime and explore the exotic behaviors of condensed matter physics.

One can thus consider cold atoms as genuine quantum simulators that offer unrivaled opportunities in material physics where the system parameters are usually very constrained in essence. In 2002, Anglin and Ketterle summed up the situation in the following way (Anglin 2002) : "Our field is now at a historic turning point, in which we are moving from studying physics in order to learn about atom cooling to studying cold atoms in order to learn about physics."

Purpose of my thesis

The goal of my PhD thesis was to study cold atom gases confined in one dimension, and where the interaction between atoms are important. More precisely, I studied fermionic atoms with a large number of different individual states, that is, with an access to many possible interaction channels - degrees of freedom. Alkaline elements such as lithium 6 or potassium 40 or some earth alkaline atoms like strontium 87 or ytterbium 173 allow to engineer such systems. This complexification with respect to more "classical" condensed matter systems (that have in general far less degrees of freedom) allow to observe the emergence of new exotic physical phenomena.

By my thesis work, I thus endeavored to study new types of Mott insulators that can occur in multi-components (spin and orbital components) fermionic systems.

Methods

In order to study these systems, we addressed the problem with three different and complementary approaches. First of all, a strong coupling approach that is mainly based on symmetry considerations and group theory; this approach is a good starting point that allows to get an intuitive vision of the phenomena at stake and to anticipate some results that other approaches must confirm afterwards.

Then, a low energy approach enabled us to confirm and complement the former; I was able to compute phase diagrams that are a lot more quantitative than those obtained in the strong coupling regime. For this purpose, I have used very powerful techniques of field theory specific to the one dimensional systems: conformal theory, abelian bosonization, non-abelian bosonization, refermionization, renormalization group (see the 2nd chapter of my thesis that is specifically devoted to these methods). A numerical solution (with an adaptative Runge-Kutta algorithm) of the renormalization group differential equations allowed me to draw the phase diagrams of the studied systems.

Phase diagram for the earth-alkaline model
Figure 3: Phase diagram obtained by numerical solution of the renormalization group differential equations for the (1D) model of an interacting earth-alkaline gas (4 components).

Lastly, numerical simulations, made by our collaborators Sylvain Capponi (Laboratoire de Physique Théorique, Université Paul Sabatier, Toulouse) and Guillaume Roux (Laboratoire de Physique Théorique et Modèles Statistiques, Orsay), with methods specific to one dimensional systems (density matrix renormalization group or DMRG, exact diagonalization) allowed to confirm the results of the two former approaches.

Results

The main contribution of my PhD was to highlight the Haldane physics. This exotic behavior has very interesting characteristics. In some cases, it displays a non trivial topologic structure, that is reflected by the presence of Majorana fermion edge states. These fermions are exotic quantum particles, predicted in 1937 by Ettore Majorana, but that had, up to 2012, never been observed. These past few years, Majorana fermions have aroused a lot of interest in the scientific community that puts a lot of efforts to observe them experimentally (see for instance Mourik 2012 and Anindya Das 2012). My work thus open the possibility to see such fermions in cold atom systems, where the imaging techniques make a lot of progress every year. On could thus hope to be able to observe the very rich physics studied in my PhD thesis in the near future.

Haldane (charge) physics in a 1D gas of earth-alkaline atoms
Figure 4: a cartoon of the ground state of the Haldane charge insulating phase, a new exotic phase that we discovered in one dimensional cold atoms gases with four components (see Nonne 2011 or my publications).

Bibliography