Resonant Bose-Fermi Mixtures : a T-matrix and Quantum Monte Carlo Study

Abstract

In this thesis we develop the theoretical study of a novel physical system, in the context of ultracold gases. We investigate the physical behavior of a resonant Bose-Fermi mix- ture, namely, an ultracold gas made of both bosons and fermions, with a strong attractive interaction between these two components. We study homogeneous density and mass imbalanced mixtures from weak- to strong- coupling limit, comparing the results obtained with two different theoretical approaches, a many-body diagrammatic approach (the T- matrix approximation) and Quantum Monte Carlo method. By using many-body diagrammatic methods we first obtain the finite-temperature phase diagram and the thermodynamic properties of the system. We observe the presence of a quantum phase transition from the condensed (superfluid) to the normal (molecular) phase. Developing the zero-temperature limit of the same Green’s function formalism we study the effect of density and mass imbalances for the Bose-Fermi mixture. For a van- ishingly small boson density we derive the asymptotic expressions for the critical coupling in the limits of small and large mass ratios. These expressions are relevant also for the polaron-molecule transition in a strongly-imbalanced Fermi mixture, thus bridging two quite distinct physical systems. The analysis of the momentum distribution functions at sufficiently large density imbalances shows an interesting effect in the bosonic momentum distribution, due to the simultaneous presence of composite fermions and unpaired bosons. By using the corresponding retarded propagators we calculate the spectral weight func- tions and the dispersions of the bosons and of the fermions. We obtain a wide collection of single-particle intensity plots, for several values of the density and mass imbalances, which could be compared in the near future with experimental data, obtained e.g. with radio-frequency spectroscopy. We use for the first time the Quantum Monte Carlo method with Fixed-Node approxima- tion to investigate resonant Bose-Fermi mixture, from weak to the strong boson-fermion attraction. Two different nodal surfaces are used as trial wave functions in the two regimes. We obtain the equation of state of a density imbalanced mixture and we observe the presence of the quantum phase transition through the crossing of the energies, cal- culated with their respective trial wave functions. By fitting the Quantum Monte Carlo data, we can write the expressions for the energy in the superfluid and in the normal phase. A phase diagram in the coupling and boson-fermion concentration variables is then derived and the occurrence of phase separation is discussed. We compare Quantum Monte Carlo results to T-matrix calculations, finding an interesting agreement between the two results for the bosonic momentum distribution.