Probing topological properties of Bloch bands with ultracold atoms in a honeycomb optical lattice

More than thirty years ago, Berry delineated the effects of the geometric structure of Hilbert space on the adiabatic evolution of quantum mechanical systems. Since then, the concept of geometrical phases – the Berry phases – associated with adiabatic evolution has been applied to a large variety of physical systems. In condensed matter, analogous topological properties of the energy bands are responsible for a wide range of many-body phenomena, such as the integer quantum Hall effect and the existence of robust conducting edge states in topological insulators. The Berry curvature, the parameter which describes the geometry of the band eigenstates, is of fundamental importance to the characterization of the physical properties of electronic states in condensed matter systems and it determines the topological invariant of a band, the Chern number. Despite the central role of Berry phases and Berry curvature, fully mapping out the geometric structure of an energy band remains a major challenge for experiments. This thesis reports on the experimental realization of a honeycomb optical lattice for ultracold atoms which has properties similar to those of solid state graphene. This lattice structure presents peculiar topological features associated with conical intersections of energy bands, the Dirac cones. The topological properties of a Bloch band can manifest themselves when a particle is adiabatically moved in a closed loop in reciprocal space around one of the conical intersection of bands present in the energy spectrum. By doing so, the particle acquires a geometric phase which is analogous to the phase shift experienced by electrons moving around a solenoid, the well known Aharonov-Bohm effect. This thesis presents an interferometric technique to detect the topological properties of such cones and to probe the distribution of Berry curvature in reciprocal space with high quasimomentum resolution. The first part of this thesis introduces the honeycomb optical lattice and the experimental setup for cooling ultracold atoms. Then, an interferometric protocol for ultracold atoms, analog of the Aharonov-Bohm effect in reciprocal space, is developed to detect the geometric phase associated with the Dirac cones. The interferometric technique combines the motion of atoms in the band with spin-echo interferometry and it allows the topological properties of the lowest band of the optical lattice to be characterized by probing the localization of the Berry curvature of an individual Dirac cone. The detected sharp phase jump to π and the strong reduction in interference contrast at the Dirac cones demonstrate the winding of the band eigenstates and the strong localization of the Berry curvature at the cones. Moreover, the interferometry is used to detect the movement in reciprocal space and the annihilation of the topological features upon lattice imbalance. The observed disappearance of the π Berry phase confirms the transition to a topologically trivial band structure. The last part of this thesis presents preliminary results concerning interferometry in the two lowest bands of the lattice. The Stückelberg interferometry, realized by diabatic motion of atoms in reciprocal space, is used to map the dispersion of the bands and to probe the symmetry of the band eigenstates. Extensions of this work would allow the characterization of topological invariants of multi-band systems, like the Z2 invariant of the quantum spin Hall effect. The results reported in this thesis demonstrate the non-trivial winding of the band eigenstates around the Dirac cones of an optical honeycomb lattice for ultracold atoms. The high degree of control over the lattice parameters makes the system suitable for investigating topological properties of energy bands in a well controlled environment. The two interferometric techniques presented constitute possible approaches for reconstructing the local topological properties of a 2D optical lattice and, thereby, of its topological invariant.