We consider a cold Bose gas in the unitarity limit. The unitarity limit corresponds to the limit where the scattering length goes to infinity. In this thesis we first review basic quantummechanical scattering theory and explain the physics of Feshbach resonances. After that we discuss a many-body theory, which exactly incorporates the two-body physics of a Feshbach resonance. In a mean-field approximation we obtain a divergence of the energy per particle in the unitarity limit. Beyond mean field the spectral function is ill behaved. To get more insight into these problems, we also discuss a model of linearly coupled atoms and phonons. A generalized Bogoliubov transformation is used to obtain expressions for the quantized atomic and phonon field operators. Finally, we use these operators to calculate some correlation functions.
We consider a cold Bose gas in the unitarity limit. The unitarity limit corresponds to the limit where the scattering length goes to infinity. In this thesis we first review basic quantummechanical scattering theory and explain the physics of Feshbach resonances. After that we discuss a many-body theory, which exactly incorporates the two-body physics of a Feshbach resonance. In a mean-field approximation we obtain a divergence of the energy per particle in the unitarity limit. Beyond mean field the spectral function is ill behaved. To get more insight into these problems, we also discuss a model of linearly coupled atoms and phonons. A generalized Bogoliubov transformation is used to obtain expressions for the quantized atomic and phonon field operators. Finally, we use these operators to calculate some correlation functions.
We develop an analytical approach for the description of the crossover of an atomic Bose gas from small to infinitely large scattering length where the two-body interactions satisfy the unitarity bound. We obtain several properties of the Bose gas as a function of interaction strength, namely the chemical potential, the contact, the speed of sound, the condensate density, the effective interatomic interaction and the three-body recombination rate. Also, the energy dependence of the effective interaction at unitarity and the evolution of the Feshbach bound state are discussed. It is shown how the approach can be systematically improved with renormalization-group methods and how it reduces to the Bogoliubov theory in the weak-coupling limit.
We argue that the energy of a strongly interacting gas in the unitary limit is parametrized by a universal constant β. For fermions, this parameter is known, but not for bosons. Finding β for bosons is the goal of this thesis. We present the basics of scattering theory, including the s-wave scattering length, which is the characteristic length scale for the interactions at low energies and which goes to infinity at unitarity. Shape and Feshbach resonances are explained. The conventional theory for Bose gases, Bogoliubov theory, is presented, which takes into account quantum fluctuations. It is used to derive identities for the groundstate energy, the condensate fraction and the contact. These are given as expansions in the scattering length, which therefore only hold true for weak interactions. To improve on this, we present an effective action from an atom-molecule theory, which also allows bound states. By a mean-field approach we derive universal identities for the associated energy scales, and in the unitary limit an analytical expression is found for the chemical potential, from which the value β = −0.54 is found. However, this mean-field result is qualitatively not reliable, since quantum depletion is neglected. A new approach is introduced, starting with a Jastrow Ansatz, from which the energy is expressed in the Jastrow function f and radial distribution function g. A cluster expansion is made for g, where a diagrammatic analogy is found. The Hyper-Netted Chain approximation is introduced to calculate g systematically by summing infinitely many diagrams. In HNC/0, we minimize the energy by a variational method, which is based on a well-constructed Ansatz for g. Numerical results using the HNC/0 approximation are given not only for the ground-state energy, but also for the condensate density and the contact. For large scattering lengths the calculations became increasingly less accurate, but the results from Bogoliubov theory are correctly included for small a. An extrapolation for the ground-state energy is made to find β ' −0.20, which is expected to be an upper bound. Moreover, quantum depletion is found to be higher when calculated in HNC/0 for large scattering lengths. Also, it is found that the contact at this value for unitarity remains finite and acquires the value of C ' 10.3n4/3, which is remarkably close to the value for fermions, C ' 11n4/3. Cover image courtesy of [18].
A Feshbach resonance in the s-wave scattering length occurs if the energy of the two atoms in the incoming open channel is close to the energy of a bound state in a coupled closed channel. Starting from the microscopic Hamiltonian that describes this situation, we derive the effective atom–molecule theory for a Bose gas near a Feshbach resonance. In order to take into account all two-body processes, we have to dress the bare couplings of the atom–molecule model with ladder diagrams. This results in a quantum field theory that exactly reproduces the scattering amplitude of the atoms and the bound-state energy of the molecules. Since these properties are incorporated at the quantum level, the theory can be applied both above and below the critical temperature of the gas. Moreover, making use of the true interatomic potentials ensures that no divergences are encountered at any stage of the calculation. We also present the mean-field theory for the Bose–Einstein condensed phase of the gas.
Summary form only given. For non-resonant atomic interactions the properties of ultra-cold scattering can be accurately represented by the scattering length. When the interactions are resonant, for instance when the atoms are colliding via a Feshbach resonance, the scattering properties become strongly dependent on energy, and the scattering length approximation is no longer valid. We show that there is a limited set of parameters that fully characterizes the scattering in the ultra-cold regime, and that this description can be easily incorporated into a theory of many-body physics. In effect, it can be shown that with a Hamiltonian where the resonant bound-state is explicitly treated, the resulting many-body mean-field equations have built in the microscopic two-body coupled-channels scattering equations. We have used this many-body theory to derive a theory of resonance superfluidity, and applied it to both fermionic /sup 40/K and /sup 6/Li. Since the scattering length a does not appear as an expansion parameter of the theory it can be shown that this field theory does not break down as a goes to infinity. In fact, the theory describes the high-T/sub c/ behavior of the system and predicts a critical temperature which can be as high as half the Fermi temperature.
In the wake of successful experiments in Fermi condensates, experimental attention is broadening to include resonant interactions in degenerate Bose-Fermi mixtures. In this thesis we wish to study the equilibrium properties of the fermionic molecules that can be created in such a mixture. To this end, we first discuss the two body properties of the system, and introduce the model Hamiltonian we use to describe the resonant physics, highlighting its virtues, as well as its limitations. We then proceed by analyzing the mean field solution of this model, by studying both the equilibrium problem, and the non-equilibrium equations of motion, thus developing a powerful language to discuss the system. We then highlight the limitations of the mean-field approach, and develop a numerically tractable generalized version of this theory, which is able to correctly describe the two-body properties of the system in the low-density limit. Finally, we study the properties of the system using this generalized mean-field theory, by first analyzing the two-body scattering matrix in the many-body environment, assessing its complex poles in order to understand the stability properties of the Feshbach molecules in the gas. Secondly we solve the equilibrium equations self-consistently, to study the molecular populations and density distributions at equilibrium, as a function of external bias magnetic field.
In the first part of this thesis, we present a unified kinetic theory that describes the finite-temperature, non-equilibrium dynamics of a Bose-Einstein condensed gas interacting with a thermal cloud in a trap. This theory includes binary interactions to second order in the interaction potential and reduces to a diagonal quantum Boltzmann equation for Bogoliubov quasiparticles. The Hartree-Fock-Bogoliubov interactions include the pairing field and are expressed as many-body T matrices to second order. The interactions thus include the correct renormalized scattering physics. This renormalized theory is automatically gapless. Thus, the excited Bogoliubov modes are naturally orthogonal to the condensate ground state. This kinetic theory is a complete second-order theory that reduces to the Gross-Pitaevskii equation and the quantum Boltzmann equation in the respective limits and thus is capable of describing the system over a wide temperature range. In the second part, we consider a many-body theory of a dilute Fermi gas near a Feshbach resonance. Experiments explore the crossover physics between the Bardeen-Cooper-Schrieffer (BCS) superfluidity of a two-spin Fermi gas, and the Bose-Einstein condensation (BEC) of composite bosons. We consider correlations between a composite boson and a fermion pair and show that such correlations are the minimal ingredients needed in a many-body theory to generate the correct boson-boson scattering length in the Bose-Einstein limit of the crossover. We also use imaginary-time propagation to find zero-temperature ground states in the BCS/BEC crossover. A cumulant expansion allows us to systematically include higher-order interactions between bosons and fermions. In particular, we calculate the Hartree term across the resonance. We further apply the cumulant-expansion method to thermal fermions and composite bosons interacting above the transition temperature in the normal phase. We numerically calculate the full time dependence in ramps across the resonance in this regime and find different two-body and many-body time scales in the system. We calculate molecular conversion efficiencies as a function of temperature and phase-space density, and find good agreement with results from JILA potassium experiments.