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My main activities focus on the theory of quantum phases and phase transitions in condensed matter physics and ultracold gases. Specifically, I am interested on strongly correlated bosonic systems in lattices. These systems are described by the Bose-Hubbard model which I study with exact numerical methods such as quantum Monte Carlo and exact diagonalization. In my booming research field, quantum magnetism in ultracold gases, novel complex phases arise from the competition between spin interactions, repulsive interactions and delocalization of particles due to tunnel effect. These systems involve many species or colors of particles and are actively investigated experimentally, e.g. in the group of W. Ketterle, E. A. Cornell, D.M. Stamper-Kurn and M. Greiner in USA, I. Bloch, and G. Rempe in Germany, J. Dalibardin France, and T. Esslingerin Switzerland.
Quantum Magnetism in Ultracold Gases
Cold atoms allow since few years to simulate artificial quantum magnetism using neutral atoms with a high degree of control. And this is amazing! Not only because one can study condensed matter Hamiltonians, but also because one can create new type of magnetic textures! Cold atoms offer the possibility to study the influence of many phenomena, for instance magnetism and the Bose-Einstein condensation. My research objectives are to understand the competition between quantum coherence, repulsive interactions and magnetic interaction. Such systems are generically described by the following extended Bose-Hubbard model with U(1)xSU(2) symmetry,
Such Hamiltonians involve many species, corresponding to different internal states or zeeman sublevels of effective spins. But these systems are not simple mixtures since the spin term includes spin conversions, or similarly species conversions, which bring novel and interesting features. When the kinetic term dominates, the system is superfluid whereas the systems adopts a Mott insulating phase for large repulsive interactions for integer filling. As shown below, the magnetic interactions bring a rich physics in the superfluid and Mott insulating phases.
Spin-1 Bosons in 2D optical lattice
The case of spin-1 atoms is very interesting: many magnetic orders are observed depending on the sign of the magnetic interactions. Ferromagnetism is present in both Mott and superfluid phases for negative spin interaction amplitude; whereas for positive amplitude the phase diagram is more subtle (see left). The nematic order, present in the superfluid phase and in the Mott phases with odd filling, reduces SU(2) symmetry into Z2 symmetry with zero magnetization. An intuitive point of view is the one of half a percent spin pointing up and the rest pointing down with a random spatial repartition. The singlet phase has also a vanishing magnetization but has no longe range order: two particles form a zero spin on each lattice site and the state is well described by a factorized wave function (Gutzwiller ansatz).
Our quantum Monte Carlo simulations (PRB 88, 104509 (2013)) gave for the first time the exact phase diagrams and we have elucidate the nature of the quantum phase transitions. Interestingly enough, we observe a first order Mott-superfluid phase transition for even filling, and more surprisingly, a magnetic singlet-nematic transition with broken SU(2) symmetry takes place in the Mott phase. We have used another interesting numerical method (exact diagonalization) to obtain more information. This method has shown the existence of an Anderson tower of states in the nematic phase, in perfect agreement with our analytical calculations (PRL 113, 200402 (2014)). The position of the singlet-nematic transition, identified by crossing of energy levels in the spectrum, is in very good agreement with our quantum Monte Carlo simulations.
Spin-1/2 Bosons in 1D and 2D optical lattices
Mean field studies have reported interesting magnetic features in a system of bosonic atoms with two effective spin degree of freedom - here after called spin-1/2 bosons (e.g. spin up and down). Despite these original predictions, there has been no exact studies before us. A major feature of such systems is the possibility to convert a type of specie into the other by considering the local scattering s-wave process. The physics is discribed by an extended Bosonic-Hubbard with an on-site converting term.
First, we have studied this system using mean field theory (Gutzwiller ansatz) and quantum Monte Carlo in 1D (PRA 82, 063602 (2010)). The phase diagram is characterized both based on the mobility of the particles, i.e. Mott insulating or superfluid, and whether or not the system is magnetic, i.e. different populations for the two species. The phase diagram is shown to be population balanced for negative spin-dependent interactions, regardless of whether it is insulating or superfluid. For positive spin dependent interactions (see left phase diagram), the superfluid phase is always polarized, the two populations are imbalanced. On the other hand, the Mott insulating phase with even commensurate filling has balanced populations while the odd commensurate filling Mott phase is polarized. The Mott-superfluid transition is of the Berezinskii-Kosterlitz-Thouless type at integer filling.
Second, we have studied the effect of dimensionality by studying the system considering a 2D square lattice (PRB 84, 064529 (2011)). The polar superfluid is still observed and is now Bose-Einstein condensed. The Mott insulating phases present the same behavior than in 1D, the agreement with the mean field is much better, as expected. A major difference concerns the phase transitions: their nature is first (second) order at even (odd) filling.
Lastly, we have investigated the thermal effects and derived the phase diagrams(EPJB 85, 169 (2012)). One important question is the robustness of the magnetic (polarized) phase versus the temperature. We have shown that the magnetic effects, observed at zero temperature, are very sensitive when increasing the temperature and the first order transition singularly disappears. We have also focused on the universal jump of the Berezinskii Kosterlitz Thouless transitions.
Ultracold atomic and molecular mixtures
The success of cold-atom quantum simulators for quantum many-body systems strongly relies on their ability of accessing the strongly correlated regime, either making use of deep optical lattices, or via the use of Feshbach resonances. Deep lattices precisely mimic strong correlations in condensed matter whereas Feshbach resonances are a unique tool in atomic physics to control directly the scattering length for inter-particle interactions. Thanks to the coherent coupling between an unbound atomic state and a bound molecular state, brought into resonance by changing a magnetic field, the atomic scattering length can be tuned from positive to negative values.
An enormous body of literature (both theoretical and experimental) exists on the rich physics induced by a divergent interaction strength. Yet much less work has been devoted to considering the many-body implications of the microscopic mechanism behind Feshbach resonances: namely, the coherent atom-molecule conversion (conversion term in the above Hamiltonian). The presence of a coherent transmutation between particles of different species is quite unusual when coming from a condensed-matter perspective, and it is particularly important for the physics of bosonic systems: indeed such a coherent coupling amounts to a Josephson coupling between two species and would correspond to a material showing quantum-coherent chemical reactions in its elementary dynamics. The fundamentally asymmetric nature of this coupling leads to rather new physics which has been only sparsely investigated in the past.
Feshbach insulator: an insulating phase stabilized by atomic-molecular conversions
A convenient picture to visualize the atom-molecule coupling demands to add an extra dimension to the system, and to imagine that molecules live in a region of space separate from that of atoms with respect to the extra dimension. For instance, in one dimension, atoms and molecules live on two parallel chains. Then the atom-molecule conversion amounts to a Josephson coupling between two such chains, by which two atoms move to the "molecular" chain transforming into a molecule.
Our study (PRL 114, 195302 (2015)) shows that the coherent coupling between atoms and molecules close to a Feshbach resonance can have dramatic consequences on the Bose-condensation properties of the two species, when considering the system in an optical lattice. Indeed the Josephson coupling between the lattice of atoms and the lattice of molecules (stacked in the fictitious extra dimension) favors the presence of two atoms (or more) on a site. If molecules are " traced out" sufficiently far from resonance, this amounts to an attractive coupling between atoms induced by the (virtual) conversion to a molecule - whence the ability to tune the scattering length. On the other hand, if the resonance is approached more closely, the coherent nature of the Josephson coupling becomes very prominent. Josephson atom-molecule coupling entangles atoms and molecules into a Bell pair in the fictitious dimension; similarly, the Josephson coupling associated with hopping of atoms and molecules would entangle different sites in the real-space dimensions - and such entanglement is at the basis of condensation. Now, these two forms of entanglement are incompatible with each other due to the non-linearity of the atom-molecule coupling: as it takes two atoms to make a molecule, atom-molecule conversion leads to atom number fluctuations with a well defined parity, while atomic hopping does not preserve parity. A sufficiently strong atom-molecule coupling is then found to suppress real-space coherence of atoms and of molecules, and to lead to a novel, gapped insulating phase dominated by entangled and localized trios of two atoms and a molecule on each site. We dub this new state of matter a "Feshbach insulator".
If molecules were eliminated from the model at sufficiently large detuning from the atomic state, one would be left with atoms with a scattering length which vanishes and becomes negative as one approaches the transition - this would inevitably lead to collapse of the bosonic cloud. On the other hand, the full treatment of the coherent atom-molecule coupling surprisingly shows that, for sufficiently narrow Feshbach resonance, lattice bosons are in fact very stable. Not only does collapse not occur, but also a Feshbach insulator with two atoms/one molecule per site protects the bosonic cloud from rapid decay due to three-body recombination, given that triple occupancy of the lattice sites is suppressed.Our study provides a thorough microscopic investigation of phase diagram of nearly resonant bosonic atom-molecule mixtures in optical lattice, focusing on the appearance of the Feshbach insulator (FI), as well as on the quantum phase transitions connecting it to atomic/molecular condensate phases. In particular we perform a joint study based on Gutzwiller mean-field theory and on numerically exact quantum Monte Carlo, allowing to determine quantitatively the phase diagram as well as the nature of the transition.
Artificial microwave Graphene
We consider both an idealized model with artificially symmetric parameters between atoms and molecules, for which the Feshbach insulator is only possible insulating phase appearing in the phase diagram (see phase diagram left above); as well as a model with microscopic parameters estimated ab-initio from the physics of Rubidium-87 in a deep optical lattice close to its well-known Feshbach resonance at 414 G. A fundamental requirement for the stability of the Feshbach insulator is the narrowness of the Feshbach resonance, conveniently featured by the above cited resonance in Rubidium-87 as well as by other experimental resonances, e.g. the one at 853 G for Sodium-23. In particular we find that the quantum phase transition between a Feshbach insulator and a mixed atom/molecule condensate (BECam) has a prominent first-order nature (red dashed line in the phase diagram), which is clearly observable both via time-of-flight measurements of the momentum distribution (with jumps in the coherent fraction); as well as by imaging the density profile of a trapped system, which exhibits such a transition via density jumps when moving from the center of the cloud towards the wings (see also PRA 95, 013606 (2017)).
Our results are very timely, as they directly connect to very recent experimental and theoretical studies on the resonant Bose gas in continuum space. Recent experiments on the subject leave the question open whether a stable phase of resonant bosons in continuum space will ever be observed. Here, on the contrary, we provide strong evidence that resonant bosons on a lattice can indeed be stable, and that their phase diagram is much richer than what previously thought. As we provide quantitative predictions for a realistic microscopic model of a Feshbach resonance in Rubidium-87, our results lend themselves to an immediate experimental test. Our study is of direct interest for a broad community focusing on strongly correlated quantum systems, few-body and many-body physics of degenerate quantum particles, and state-of-the-art experiments with cold-atom quantum simulators.
In PRA 95, 013606 (2017) we focus our attention on a two-dimensional atom-molecule mixture at zero and finite temperature by using quantum Monte Carlo simulations and mean field theory. This study provides a reliable analysis of the quantum phase transitions, completing the characterization of the ground state phase diagram. Using finite size scaling method, we elucidate the universality class of the quantum phase transitions and observe a large variety of quantum phase transitions: the transition from molecular to mixed atomic-molecular condensate (BECm to BECam transition) is found to be of the 3D Ising type due to the restoration of the Ising Z2 symmetry associated with the phase of the atomic field; the molecular condensate to Feshbach insulator transition belongs to the universality class of the 3D XY model; and interestingly enough, the transition from mixed condensate to disordered phase (vacuum or Feshbach insulator) associated with the spontaneous symmetry breaking of both U(1) and Z2 is systematically found of the first-order; otherwise the transitions are second order.
Furthermore, we unveiled the thermal phase diagram by using quantum Monte Carlo simulations. In two dimensions, fluctuations have dramatic effects, drastically altering the picture that one could get from simple mean-field theory: indeed Bose-Einstein condensation at finite temperature is impossible in the proper sense, leaving space to quasi-condensation via a Berezinskii-Kosterlitz-Thouless (BKT) transition. The coupling between atomic and molecular critical behavior, induced by the conversion term, leads to a very rich phase diagram, including non-usual BKT transitions since the topological defects (vortex-antivortex) of both atomic and molecular fields are coupled. This leads to non-usual molecular superfluid to normal Bose liquid BKT transition, involving a renormalized 8T_BKT/pi stiffness jump, instead of the standard 2T_BKT/pi one for the single component case. The transition from mixed superfluid to normal Bose liquid also required a careful treatment since only the quasi-condensation of the atoms is conventional whereas the thermal disintegration of the molecular superfluid satisfies the scaling of the atom-pair. Finally, we observe a classical first-order transition between the mixed superfluid and the normal Bose liquid at low temperature (dashed line), reminiscence of the quantum first-order transition between the Feshbach insulator and the mixed condensate.
Our results set the stage for future experiments on coherently mixtures of atomic and (collisionally stable) molecules, showing the vast richness in the many-body physics which results from the quantum coherence established between two different particle numbers. The phenomena we discuss are all amenable to experimental verification using state-of-the-art setups in cold-atom physics. The finite-temperature phase diagram bears substantial analogies with the one predicted for one dimensional mixtures at zero temperature, where (as already mentioned) an Ising transition for atomic quasi-condensation is observed. A finite-temperature and two-dimensional realization of this physics has obvious advantages for the experimental feasibility, and moreover it bears the potential to unveil the spatial structure of the unconventional 2D topological excitations appearing in the system, namely weakly bound molecular vortex dipoles and (half) atomic vortices. Such excitations can be experimentally imaged via the interference of independently prepared atomic clouds. Finally, the relevance of our study goes beyond the study of many-body physics for cold atoms: as an example, our model of interest is also relevant for liquid crystal physics, and specifically as a realization of the smectic-A--hexatic-B transition.
In PRA 93, 023639 (2016) we focus our attention on the case of two-dimensional atom-molecule mixtures, where the ground state is a simultaneous Bose-Einstein condensate of both atoms and molecules. While a direct study of the full quantum many-body problem is rather challenging, the universal features of the phase diagram can be unveiled by resorting to a much simpler, classical model which assumes nearly homogeneous densities for both species, and which focuses uniquely on the local phases of the atomic and molecular field operators. The atomic and molecular phases are asymmetrically coupled in a double XY model, which has been very little investigated in the past. Here we provide an extensive Monte Carlo study of its critical properties, based on a newly developed generalization of the powerful Wolff cluster algorithm to the case of asymmetric XY interactions.
Our original numerical approach allows us to reconstruct the phase diagram (see right) in the different regimes of strong versus weak imbalance between atomic and molecular states (i.e. Ja/Jm), and of strong versus weak atom molecule coupling (C parameter); as well as to access unprecedented systems sizes. The latter ingredient revealed to be crucial to critically revisit erroneous claims on the physics of the model and of related models. In particular we unveiled a complex interplay between the atom-molecule imbalance and coupling. Far on the molecular side of the resonance, the atom-molecule mixture is found to exhibit two transitions: a high temperature BKT transition with onset of quasi-condensation for the molecules, and a low-temperature, rather unusual Ising transition for the quasi-condensation of atoms. The Ising transition, occuring between two phases with algebraically decaying phase correlations (or quasi-long range order) exhibits a significant subtlety: since long-range Ising order would imply automatically atomic condensation, it is forbidden by the Mermin–Wagner–Hohenberg theorem (which typically applies only to continuous variables). This means that, despite the existence of an Ising transition, the Ising Z2 symmetry remains effectively unbroken below the critical point. This scenario, fully corroborated by our data, disproves the scenario of S. Ejima and al. in PRL 88, 106 (2011), claiming the existence of long-range Ising order in a related quantum model possessing the same transition.
Moving to the atomic side of the phase diagram, the two transitions merge into a unique BKT transition, with conventional quasi-condensation of the atoms and rather unconventional one for molecules, as the latter is fully driven by the coupling of molecules to atom pairs and not to single atoms. The conventional BKT criticality of atoms contradicts an older claim (contained in F. Shahbazi and R. Ghanbari, PRE 74, 021705 (2006)) about parameter-dependent critical behavior realizing a whole new set of universality classes for 2d transitions – a claim which is found to be misled by the insufficient size of the s imulation boxes. Most importantly we show that the quasi-condensate phase is further characterized by an unconventional, strong crossover phenomenon between a low-temperature regime of tightly bound molecular vortex-antivortex pairs, and an intermediate regime in which pairs proliferate but remain (weakly) bound.
Multiple transitions: unconventional Berezinskii-Kosterlitz-Thouless and Ising transitions
In PRB 82, 094308 (2010), we have presented the first microwave realization of graphene in the tight-binding approximation. Using high index of refraction disks inside two metallic plates, where the disks resonnaces are only coupled evanescently, a tight-binding realization became possible. The Dirac point and a linear increase in the density of sates close to it have been found as well as edge and corner states. It was essential for this approach to conceive a system where each disk brings in just one state coupling evanescently with its neighbors. This is a qualitavely new feature for the microwave experiments as a whole, enabling a new class of experiments in the context of molecular orbital theory, going by far the specific example of graphene studied in the present work.Due to the flexibility of the set-up and the closeness of the system additionally graphene like systems with disorder, defects, graphene quantum dots or even Dirac oscillators can investigated. By introducing different types of disks with different resonance frequencies it is also possible to vestigate systems with different atom per unit cell like boron-nitride, corresponding to a Dirac equation for particles with masses.
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