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Theses

Atomes froids fortement corrélés dans un réseau optique.

Abstract : This thesis is concerned with the theoretical study of strongly correlated quantum states of ultra-cold fermionic atoms trapped in optical lattices. This field has grown considerably in recent years, following the experimental progress made in cooling and controlling atomic gases, which has led to the observation of the first Bose-Einstein condensation (in 1995 [4]). The trapping of these gases in optical lattices has opened a new field of research at the interface between atomic physics and condensed matter physics. The observation of the transition from a superfluid to a Mott insulator for bosonic atoms [46] paved the way for the study of strongly correlated phases and quantum phase transitions in these systems. Very recently, the investigation of the Mott insulator state of fermionic atoms [63] provides additional motivation to conduct such theoretical studies. This thesis can be divided broadly into two types of work: • On the one hand, we have proposed a new type of spectroscopy to measure single-particle correlators and associated physical observables in these strongly correlated states. • On the other hand, we have studied the ground state of the fermionic Hubbard model under different conditions (mass imbalance, population imbalance) by using analytical techniques and numerical simulations. In a collaboration with J. Dalibard and C. Salomon (LKB at the ENS Paris) and I. Carusotto (Trento, Italy), we have proposed and studied a novel spectroscopic method for the measurement and characterization of single particle excitations (in particular, the low energy excitations, namely the quasiparticles) in systems of cold fermionic atoms, with energy and momentum resolution. This type of spectroscopy is an analogue of angular-resolved photoemission in solid state physics (ARPES). We have shown, via simple models, that this method of measurement can characterize quasiparticles not only in the "conventional" phases such as the weakly interacting gas in the lattice or in Fermi liquids, but also in unusual phases such as the normal state of high-temperature superconductivity with a pseudogap (leading to a differentiation between nodes and anti-nodes) observed in condensed mater physics. The first experiment implementing a type of spectroscopy (RF spectroscopy) very closely related to our proposal has been recently realized at Boulder in D. S. Jin's group, just as this thesis was being written up. In the second part of this thesis, we have performed theoretical studies of several phases of strongly correlated fermions in optical lattices in the framework of theoretical models such as the Hubbard model. We have implemented and developed analytical methods (Hartree-Fock mean field theory at weak coupling, mapping on a effective spin model at strong coupling) and numerical methods (the dynamic mean field theory approach). This work has led to two particular types of studies. The first one studies the competition between a superfluid phase and a density wave (or phase separation) for fermions with mass imbalance and attractive interaction. We have shown that the superfluid phase is unstable beyond a certain value of the mass ratio, which depends on the interaction. The second study treats a gas with imbalanced populations (polarized gas) with an attractive interaction in a three dimensional optical lattice. The main result is a phase diagram showing the stability of a uniform superfluid phase with polarization (Sarma phase or breached pair phase) in a certain parameter regime. Via an energetic argument, we concluded that the stability of the polarized superfluid phase is due to the reduction of the polarizability and the critical field of the non-polarized superfluid phase. In the strong coupling regime of the Hubbard model, within the DMFT method, we have shown that the formation of the preformed pair in the normal state reduces the polarizability and favors the stability of the breached pair phase. Although some aspects have been addressed in this thesis, many interesting questions still remain open for future work. In the first part, the framework of the novel spectroscopy method established in chapter 2 can allow for different concrete studies of the nature of strongly correlated states. For example, it should be very interesting to understand the spectra of single particle excitation in non trivial phases such as the Mott insulator, the preformed-pairs or phases with long range order. In the second part, the construction of the improved (BCS-Slater) mean field theory including the Hartree correction allows for a better comparison to modern methods (DMFT and Slave Bosons). For the system with the same population for both species, the region close to the Falicov-Kimball model is not yet well understood in our DMFT analysis because of problems in numerical convergence within the exact-diagonalization method. However, within the mean field theory analysis, we see that a novel uniform phase of charge density wave (doped-CDW) can be stabilized thanks to the high asymmetry of hopping. In order to clarify this question, a study by Slave Boson mean-field theory could be very useful. This method has two advantages: First, it contains the strongly correlated physics (including quantum fluctuations); second, in some simple cases we can extract the analytical behavior of the solution. In addition, a full treatment within MFT for both order parameters, the superfluid and the CDW, should be useful for understanding the nature of the phase transition in this limit. Another perspective of this thesis is the understanding of the nature of the polarized superfluid phase. The mismatch of the Fermi surfaces considered in this thesis is due to the population imbalance. We can always control this mismatch by introducing furthermore a mass imbalance. In the region with high mass imbalance, it is likely that the stability of the polarized uniform superfluid phase may be further enhanced. Within this thesis, the effects of the confining potential have been included via the local density approximation. For a weak and smooth potential, this approximation is expected to be accurate. However, for stronger confinement, it may become more questionable. Indeed, this issue has been recently debated in the literature, in the context of the interpretation of experiments with population imbalance [92, 126]. Dynamical Mean-Field Theory can be implemented in an inhomogeneous framework, beyond LDA [53, 95, 102] and this could be used to assess the validity of the LDA approximation for problems such as those studied in this thesis. This could be relevant in particular to the current debate on the phases of the fermionic systems with population imbalance.
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Tung-Lam Dao. Atomes froids fortement corrélés dans un réseau optique.. Optique [physics.optics]. Ecole Polytechnique X, 2008. Français. ⟨pastel-00004402⟩

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