The progress in ultra-cold atoms experiments allowing to tune fermionic systems through a Feshbach resonance where itinerant fermionic atoms may form tightly bound pairs ("bosonic" molecules) [1,2] has led to a renewed theoretical interest in boson-fermion models. We study a 1D boson fermion resonance model describing itinerant spin-1/2 fermions and itinerant scalar bosons coupled through a local interaction of strength g, which describes the binding of a pair of opposite spin fermions to form a scalar boson and the reverse process. The model also includes a detuning term characterized by a detuning parameter ν. It is found that the model has an exact solution by Bethe Ansatz.
The progress in ultra-cold atoms experiments allowing to tune fermionic systems through a Feshbach resonance where itinerant fermionic atoms may form tightly bound pairs ("bosonic" molecules) [1,2] has led to a renewed theoretical interest in boson-fermion models. We study a 1D boson fermion resonance model describing itinerant spin-1/2 fermions and itinerant scalar bosons coupled through a local interaction of strength g, which describes the binding of a pair of opposite spin fermions to form a scalar boson and the reverse process. The model also includes a detuning term characterized by a detuning parameter ν. It is found that the model has an exact solution by Bethe Ansatz.
Abstract We study a model of two species of 1D linearly dispersing fermions interacting via an s-wave Feshbach resonance at zero temperature. While this model is known to be integrable, it possesses novel features that have not previously been investigated. Here, we present an exact solution based on the coordinate Bethe Ansatz. In the limit of infinite resonance strength, which we term the strongly interacting limit, the two species of fermions behave as free Fermi gases. In the limit of infinitely weak resonance, or the weakly interacting limit, the gases can be in different phases depending on the detuning, the relative velocities of the particles, and the particle densities. When the molecule moves faster or slower than both species of atoms, the atomic velocities get renormalized and the atoms may even become non-chiral. On the other hand, when the molecular velocity is between that of the atoms, the system may behave like a weakly interacting Lieb–Liniger gas.
We develop an effective field theory to describe the superfluid pairing in strongly interacting fermions with arbitrary short-range attractions by extending Kaplan's idea of coupling fermions to a fictitious boson state [presented in Nucl. Phys. B 494, 471 (1997)]. This boson field is assigned with an unconventional kinetic term to recover the exact scattering phase shift obtained either from scattering data or model calculations. The theory works even if the explicit form of the interaction potential has not been constructed from scattering data. The contact boson-fermion coupling allows us to go beyond mean-field to include Gaussian pair fluctuations, yielding reliable predictions on equations of state. As an application, we use our theory to address the nonuniversal ground-state energy of strongly paired fermions due to the nontrivial momentum dependence of the phase shift characterized, for example, by effective range. We find a good agreement between our predictions and recent quantum Monte Carlo simulations on the effective-range dependence in both three and two spatial dimensions. We propose that in cold-atom experiments, the nonuniversal dependence in thermodynamics can be probed using dark-state optical control of Feshbach resonances.
The exact solutions of a one-dimensional mixture of spinor bosons and spinor fermions with δ-function interactions are studied. Some new sets of Bethe ansatz equations are obtained by using the graded nest quantum inverse scattering method. Many interesting features appear in the system. For example, the wave function has the SU(2|2) supersymmetry. It is also found that the ground state of the system is partial polarized, where the fermions form a spin singlet state and the bosons are totally polarized. From the solution of Bethe ansatz equations, it is shown that all the momentum, spin and isospin rapidities at the ground state are real if the interactions between the particles are repulsive; while the fermions form two-particle bounded states and the bosons form one large bound state, which means the bosons condensed at the zero momentum point, if the interactions are attractive. The charge, spin and isospin excitations are discussed in detail. The thermodynamic Bethe ansatz equations are also derived and their solutions at some special cases are obtained analytically.
Strongly interacting one-dimensional (1D) Bose--Fermi mixtures form a tunable XXZ spin chain. Within the spin-chain model developed here, all properties of these systems can be calculated from states representing the ordering of the bosons and fermions within the atom chain and from the ground-state wave function of spinless noninteracting fermions. We validate the model by means of an exact diagonalization of the full few-body Hamiltonian in the strongly interacting regime. Using the model, we explore the phase diagram of the atom chain as a function of the boson-boson (BB) and boson-fermion (BF) interaction strengths and calculate the densities, momentum distributions, and trap-level occupancies for up to 17 particles. In particular, we find antiferromagnetic (AFM) and ferromagnetic (FM) order and a demixing of the bosons and fermions in certain interaction regimes. We find, however, no demixing for equally strong BB and BF interactions, in agreement with earlier calculations that combined the Bethe ansatz with a local-density approximation.
The research described in this thesis is part of an international effort with the aim of studying ultracold quantum degenerate gas samples like Bose-Einstein condensates and Fermi degenerate systems, consisting of atoms confined in magnetic or optical traps. The behavior of such samples is governed by the inter-atomic interactions and the resulting properties of the atom-atom scattering. At the relevant temperatures of 1nK to 10µK a key property is the atom-atom scattering length a. In chapter 2 a theoretical method is presented which enables one to describe the interaction and scattering of (ultra)cold atoms to unprecedented precision. It is also unparalleled in comprehensiveness: it allows the prediction of a large and varied set of experimental data for all isotopes of the same element. The method relies on the extraction from experiments of a few (phase) parameters which completely summarize the behavior of the atoms in the (ultra)cold regime. In chapter 3 the method is applied to the "workhorses" of cold-atom physics: the atomic species 85Rb and 87Rb. We extract the foregoing parameters to a very high precision from several recent high precision experiments, allowing us to predict e.g. the 87Rb spinor condensate to be ferromagnetic: a prediction for which the scattering length has to be calculated with a precision better than 1%. We also predict Feshbach resonances at experimentally accessible magnetic field strengths; resonances searched for and found by the experimental group of Rempe. In close collaboration with his group we "fine-tune" the interaction parameters found previously, by making use of only one of the observed resonances. We then obtain agreement with 42 out of the observed 43 resonance field strengths and are able to identify bound states inducing the Feshbach resonances at these locations. Chapter 4 describes the results of this research. With a thorough understanding of the rubidium interactions, we then switch to lithium which has a fermionic (6Li) and a bosonic (7Li) isotope. Both are being used in cold-atom experiments. In chapter 5 we evaluate the interaction parameters for lithium allowing us to predict magnetic field strengths for which a sample of fermionic 6Li atoms can be regarded as strongly interacting. Furthermore, a three-level method for measuring mean-field shifts, based on radio-frequency techniques, is introduced. For weak interactions we find proportionality of resonance shifts to interaction strengths. In the strongly interacting regime, however, these shifts become very small reflecting the quantum unitarity limit and many-body effects. Most interesting is the fact that in this regime the shifts are small both for large positive a and for large negative a, likely reflecting the universality of the interaction energy. In chapters 6 and 7 the interactions between lithium atoms are reinvestigated, making use of newly available experimental data and with the updated interaction parameters special attention is paid to locating field strengths at which magnetically tunable Feshbach resonances occur in the scattering of lithium atoms. In chapter 6 scattering events in a gas of (fermionic) 6Li atoms are studied. In chapter 7 we show that a consistent description of the 6Li+6Li system can be given. We discuss theoretical uncertainties for the position of the wide 6Li Feshbach resonance and present an analytic scattering model for this resonance, based on the inclusion of a field-dependent virtual open-channel state. We predict new Feshbach resonances for the 6Li-7Li system, and their importance for different types of crossover superfluidity models is discussed. Molecules created by magnetically sweeping over these resonances will have a fermionic character. One magnetic field strength is predicted at which two different fermionic molecules can be created simultaneously.
The use of Feshbach resonances in the field of ultracold gases has received a lot of attention recently. In first instance it was realized that these resonances can be used to alter the interaction strength in an atomic gas, by means of tuning an external magnetic field. Furthermore, by sweeping the magnetic field from one side of the resonance to the other, a degenerate gas of bosonic molecules can be created, a significant accomplishment. By making the interactions between fermionic atoms strongly attractive, a fermionic superfluid was recently made, a feat that has drawn a lot of attention in the physics community and from the media. We present work that describes these many-particle systems near a Feshbach resonance in a comprehensive way by using the concept of dressed molecules. These dressed molecules are essentially two-particle states that are a superposition of a tightly bound bare molecule, and two unbound bare atoms. This description reduces to the correct paired wave-function in the two-body limit. Moreover it gives a physically correct description of all the phenomena mentioned above: the Bose-condensation of a gas near a Feshbach resonance, that contains molecules, is more correctly interpreted as a Bose-condensation of dressed molecules. The fermionic superfluid contains a condensate of bosonic pairs, normally called Cooper pairs, that can be understood as the dressed molecules. For the latter case, the dressed-molecule picture also gives an elegant characterization of the crossover between a Bose condensate of pure molecules on one side of the resonance, to a fermionic Bardeen-Cooper-Schrieffer superfluid on the other side. Then, the dressed molecules change continuously from being mostly a bare molecule, to being mostly a loose pair of two atoms. It is clear that the nature of the dressed molecules is much determined by the probability for it to be in the bare-molecular state. This probability we denote with a single parameter Z. We present various ways of calculating it, and study how it affects the macroscopic physics.