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An Ultracold Graphene Analog
by Christian MINIATURA, Benoît GREMAUD, Kean Loon LEE,
Rui HAN, Berthold-Georg ENGLERT

We're only at the beginning of the graphene story and its cold atoms offsprings

he realization, in laboratory experiments, of Bose-Einstein condensates (BEC) or degenerate Fermi gases (DFG) over the past 15 years has been a major achievement in the field of atomic physics. Loading ultra-cold degenerate gases, be it fermions, bosons, fermion-boson or fermion-fermion mixtures, into so-called optical lattices has opened fascinating new perspectives because the high degree of control and precision achieved in these experiments has allowed systematic studies of physical phenomena previously observed mostly in condensed-matter systems. The great advantage of cold atoms systems is that the relevant parameters can be accurately controlled (configuration and strength of the optical potential, atom-atom interactions using so-called Feshbach resonances, etc) and one can easily get rid of spurious inelastic effects that generally destroy the quantum coherence.

The Age of Cold Atoms

This success story began around the eighties with a quest for lower and lower temperatures in atomic gases and six physicists won the Nobel prize along the road for their decisive contributions in the field. The dream at the end of the seventies was to reach the degeneracy regime, the regime where the atoms would unveil their bosonic or fermionic character and where quantum statistical effects could no longer be ignored but would drive the physics under study. A major tool for achieving low temperatures was laser cooling where one takes advantage of the energy and momentum exchanges between light and atoms to cool down the atoms. Within ten years, physicists were able to produce atomic clouds with temperatures in the micro-Kelvin range, the lowest temperatures ever achieved in labs. But the samples obtained were still not revealing their quantum nature. Indeed, for this to happen, a certain degeneracy criterion has to be met which indicates that the quantum waves associated with the individual atoms do overlap so that atoms become indistinguishable and quantum statistics sets in. This is achieved only if both the temperature is low enough and the atomic number density is high enough, something that the laser cooling techniques then were unable to beat. In short, the laser cooling techniques were hitting a wall. The breakthrough came with the combination of laser cooling with evaporative cooling, leading to the first ever observed gaseous Bose-Einstein condensates, in 1995. At the beginning of the 21st century, the first degenerate Fermi gases were born, completing the mission. Now there are more than a hundred Bose-Einstein condensates around the world and they have become routine tools in labs. The Centre for Quantum Technologies (CQT) at NUS has already produced its own one. On the other hand, degenerate Fermi gases are less numerous but the field is developing fast (and CQT has recently joined the race), blurring the frontiers between the condensed-matter and the atomic physics communities in their search for a deeper understanding of many-body physics.

Dirac physics in a pencil

In 2004, researchers in Manchester isolated one-atom thick sheets of carbon atoms, the atoms being organized in a planar honeycomb structure. Such a material is referred to as graphene and is of uttermost importance in condensed-matter physics since, by wrapping it, one gets carbon nanotubes or fullerenes. Since then, an intense activity has flourished in the field and nurtured the dreams of a full carbon-based electronics. For theorists, such a system is also of great interest because it provides a physical realization of two-dimensional field theories with quantum anomalies. Indeed, the continuum limit of the effective theory describing the electronic transport in graphene is that of two-dimensional massless Dirac fermions. The reported and predicted phenomena include the Klein paradox (the perfect transmission of relativistic particles through high and wide potential barriers), the anomalous quantum Hall effect induced by Berry phases and its corresponding modified Landau levels and the experimental observation of a surprizing minimal conductivity.

From a crystallographic point of view, the graphene is a triangular Bravais lattice with a diamond-shaped unit tile consisting of two sites (see Figure 1). By repeatedly tiling the plane with this unit tile, one gets the honeycomb structure. The electronic propagation in such a perfect crystal is described by Bloch waves (a generalization to lattices of the ordinary plane waves in free space). The corresponding Bloch wave vector k spans the so-called Brillouin zone and the way the energy of these Bloch waves depends on k is encoded in the band structure. The very unique feature of the graphene band structure is that the two lowest-energy bands, known as the valence and the conduction bands, touch at two isolated points located at the corners of the Brillouin zone. In the immediate vicinity of these degeneracy points, known as the Dirac points, the band structure is a cone. From the point of view of electronic transport, such a situation is referred to as a semi-metal or to a zero-gap semi-conductor. In natural graphene samples, there is exactly one electron per site, and thus, at zero temperature, all levels in the valence band are filled (a situation known as half-filling). As a result, the energy of the last occupied level precisely slices the band structure at the Dirac points. The low-energy excitations of this system are then described by the massless two-dimensional Weyl-Dirac equation and their energy dispersion relation = vF k is that of relativistic massless fermions with particle-hole symmetry. In graphene these massless fermions propagate with a velocity vF which is the equivalent of the speed of light in Special Relativity. It is about one 300th of the actual speed of light.

Optical lattices

Atoms, and more generally all dielectric objects, interacting with a monochromatic laser field get polarized and acquire an induced electrical dipole moment since their electronic clouds get distorted by the electromagnetic forces exerted by the laser field. As a consequence, the atoms experience radiative forces due to photon absorption and emission cycles. When the light frequency is tuned far away from any atomic internal resonance, then the atom-field interaction is dominated by stimulated emission processes where the induced atomic electric dipole absorbs a photon from one Fourier mode of the field and radiates it back into the same or another one of these Fourier modes. In each such stimulated cycle, there is a momentum transfer to the atom. Consequently, the atom experiences an average force in the course of time. This dipole force exerted by the field onto the atom is conservative and derives from the polarization energy of the electronic cloud. The corresponding dipole potential, also known as the light-shift potential, is simply proportional to the light intensity at the position of the atom. Hence, by conveniently tailoring the space and time dependence of the laser field, one can produce a great variety of dipole potentials and thus manipulate the atomic motion. In fact, these dipolar forces, albeit small, allow us to manipulate small dielectric objects and they have become precious tools under the name of optical tweezers. For example, in biophysics, they are used to study the work force of molecular motors copying DNA molecules onto RNA molecules.

Optical lattices are periodic intensity patterns of light obtained through the interference of several monochromatic laser beams. By loading ultracold atoms into such artificial crystals of light one obtains periodic arrays of atoms. Indeed, for a convenient choice of the laser frequency, and at sufficiently low temperatures, atoms are preferentially trapped at the field-intensity minima. Such arrays of ultracold atoms trapped in optical lattices have been used in a wide variety of experiments. They have proven to be a unique tool to mimic, test, and go beyond phenomena observed until now in the condensed-matter realm. They also have a promising potential for the implementation of quantum simulators and for quantum information processing purposes, a subject at the heart of CQT interests.

One has to appreciate that very few parameters need to be known and controlled to describe the dynamics of atoms loaded in optical lattices at very low temperatures. There are basically two processes at hand. First the atoms can quantum mechanically hop (or tunnel) from one potential well to another and in turn possibly change their potential energy if the wells do not have the same depth. Second they can interact with each other through collisions inside the same potential well. These collisions can possibly flip the spin of the atoms. This situation is encapsulated in the so-called Hubbard model and the physics it describes is very rich and complex. One does not even know an analytical solution to the Hubbard model in general and one of the main goal of the many-body physicists is to identify and understand the different phases of the atomic system when the various parameters are varied.

New perspectives from the cold atoms side

From a crystallographic point of view, the graphene situation can be simply reproduced with cold atoms by superposing, for example, 3 coplanar linearly-polarized lasers beams propagating along directions with consecutive angles of 120 degrees (see Figures 2 and 3). In such a situation, one precisely realizes an optical potential where the minima are organized in a honeycomb structure and, at low enough temperatures, the atoms will accumulate in the potential wells where the physics is in turn well described by the Hubbard model. By superposing a red-detuned high-power standing-wave perpendicular to the honeycomb structure, several layers will be produced. Finally, by controlling the tunneling rates between the layers, a two-dimensional confinement will be achieved mimicking a graphene sheet. It is interesting to note that, by playing with the directions of the interfering laser beams, one also opens the possibility to create many coupled or uncoupled layers, a situation of great interest in the condensed-matter community.

But why study graphene-like physics with cold atoms as the natural system already exists and is under scrutiny? In fact, the advantage of cold atoms compared with true graphene samples is that the honeycomb optical potential offers an incomparable stage for many different physical situations that could be hardly studied with graphene flakes. We give here a few instances. For example, playing with the directions and intensities of the optical lattice beams, one can imbalance the tunneling rates between the different honeycomb sites. This makes the Dirac points move inside the Brillouin zone until they merge and disappear, leading to a topological metal-insulator transition. This situation could certainly be obtained in real graphene samples by stretching the layer, but this is not easy to achieve. Furthermore, by applying an external magnetic field, on can control and tune the on-site atom-atom interactions over a wide range, using so-called Feshbach resonances. Up to now, interactions have played little role in all phenomena observed with graphene samples. However their impact on the many-body phenomena is crucial. Playing with the sign and strength of the interactions, one can target far more complex and rich physical situations. Indeed one can induce quantum phase transitions, that is a qualitative change of the many-body ground state and properties of the system at zero temperature just by changing the interactions or the lattice filling. For example, if the interactions are repulsive and the lattice half-filled, one can produce a situation where only one site of two are occupied, - a situation reminiscent of a checkerboard - or to an anti-ferromagnet if we associate an empty site with a down-spin and an occupied site with an up-spin! In fact, very generally, these optical lattice systems have a close correspondence with quantum magnetism. For attractive interactions one induces Cooper pairing and the celebrated Bardeen-Cooper-Schrieffer (BCS) superfluidity sets in, leading to a BEC-BCS crossover as the interaction strength is increased and the loose Cooper pairs become tightly bound. One could also load mixtures of different types, Bose-Bose, Fermi-Fermi or Fermi-Bose. At finite temperatures, the atoms fall under the spell of the Mermin-Wagner theorem and one gets the celebrated Kosterlitz-Thouless transition with the appearance of vortex pairs of opposite circulation.

The list of examples is not over. Needless to say, the graphene story and its cold atoms offsprings is just at its beginning!

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