Band theory immediately followed the development of quantum mechanics (1925-1930). It classifies the crystals in insulators or conductors (metals) depending on their energy spectrum - organized in allowed energy bands, separated by band gaps - and its filling by electrons. If the Fermi level, which characterizes the filling, falls in an energy band, it is a metal; if it falls into a gap (a forbidden band between two allowed bands), it is an insulator. A key lesson of the recent discovery of graphene (2004) and topological insulators is that the energy spectrum is not sufficient information to fully describe the electronic properties of a crystal. The band theory needs to be refined. Indeed the full band structure comprises not only energy bands but also wave functions. Essential information is contained in the wave functions, which is not reflected in the energy spectrum. An example of an overall characteristic of the wave functions is an integer called the Chern number. It allows one to further classify band insulators in trivial (i.e. with zero Chern number) or topological insulators (i.e. with non-zero Chern number). In the latter case, there is inevitably conducting states, only at the edges of the sample.

In collaboration with theorists at LPTMC Paris and the Institut d’Optique in Palaiseau, we recently proposed to measure these topological or geometric properties with cold atoms in an optical lattice realizing an "artificial graphene-like crystal". The idea is to use a Stückelberg interferometer. It is an atomic analogue of the famous optical Mach-Zehnder interferometer. It consists of two beam splitters and two arms or paths: the first beam splitter splits the matter wave (i.e. the atom) in two, which then moves along the two paths before being recombined at the second beam splitter. In the case of a Stückelberg interferometer, the two paths are two energy bands (see figure a) and beam splitters are avoided crossings between bands where the gap is minimal: an atom can jump across a gap by tunneling (known as Landau-Zener effect). In a Stückelberg interferometer, an atom interferes with itself by being able to go through two paths from the valence band to the conduction band of the artificial crystal. During its path in both bands, it acquires a phase difference (called geometric) which depends in particular on topological characteristics of the bands (such as the Chern number). The interferometer is used to measure this geometric phase.

Figure : (a) Dispersion relation (energy E as a function of the momentum p_{x}) featuring two avoided crossings (D and D’) when β=0 or π/2. An atom (in red) moves through the Stückelberg interferometer. (b) Dispersion relation when β=π/4. The gap is closed in D’. (c) Probability of detecting the atom in the conduction band as a function of the distance d between the two avoided crossings and the parameter β. The bright fringes are out of phase on both sides of the value β=π/4, revealing a geometric phase of π.

A toy model (see figure) illustrates these concepts in a simple case. This is a one-dimensional band structure which depends continuously on an external parameter β that allows one to go from a trivial to a topological insulating phase via a (semi-)metallic phase at the transition (see figure c). The dispersion relation (energy versus momentum) typically has two bands separated by a gap (see figure a) except when β=π/4 (see figure b) which marks the phase transition. The gap is minimal in two points D and D ’. The two avoided crossings (D and D ’) are used as beam splitters for an atom (in red) initially in the valence band. The atom can then take one of two paths (blue or green) to go from the valence to the conduction band through D and D’, giving rise to Stückelberg interferences in the probability of detecting the atom in the conduction band. In figure c, the interferences are plotted as a function of the distance d between D and D ’and the parameter β. The bright fringes at β=0 and β=π/2 are out of phase although the energy spectrum is the same (see figure a). This is a signature of the geometric phase, which is π in the present case. At β=π/4, interferences are absent because the atom can only take the green path to go from the valence to the conduction band and the fringe contrast vanishes.

**Contact:**

Jean-Noël Fuchs

**Reference:**

Mass and chirality inversion of a Dirac cone pair in Stückelberg interferometry

Lih-King Lim, Jean-Noël Fuchs et Gilles Montambaux*Phys. Rev. Lett.* **112**, 155302 (2014).