We advocate the use of the T -matrix of the pair potential to study the properties of ultracold Fermi gases in the mean-field approximation. Our approach does not require renormalization procedures even in the limit of contact interaction, and it provides a rigorous definition of the range of the potential. We also rewrite the Bogoliubov-de Gennes equation for the pairing function as a function of the T-matrix, and use it to investigate finite-range effects on the main thermodynamic observables in a gas of 6Li atoms at unitarity, calculating the pair potential with ab initio quantum chemical methods.
We advocate the use of the T -matrix of the pair potential to study the properties of ultracold Fermi gases in the mean-field approximation. Our approach does not require renormalization procedures even in the limit of contact interaction, and it provides a rigorous definition of the range of the potential. We also rewrite the Bogoliubov-de Gennes equation for the pairing function as a function of the T-matrix, and use it to investigate finite-range effects on the main thermodynamic observables in a gas of 6Li atoms at unitarity, calculating the pair potential with ab initio quantum chemical methods.
We develop a theoretical method going beyond the contact-interaction approximation frequently used in mean-field theories of many-fermion systems, based on the low-energy T matrix of the pair potential to rigorously define the effective radius of the interaction. One of the main consequences of our approach is the possibility to investigate finite-density effects, which are outside the range of validity of approximations based on $\ensuremath{\delta}$-like potentials. We apply our method to the calculation of density-dependent properties of an ultracold gas of 6Li atoms at unitarity, whose two-body interaction potential is calculated using ab initio quantum chemistry methods. We find that density effects will be significant in ultracold gases with densities 1 order of magnitude higher than those attained in current experiments.
We calculate Bardeen-Cooper-Schrieffer (BCS) state of a unitary Fermi gas of atoms interacting with the finite-ranged Jost-Kohn potential which has been recently shown to account for the resonant interactions [2019 {\rm J. Phys. B: At. Mol. Opt. Phys.} {\bf 52}, 165004]. Using exact scattering solution of the potential, we derive two-body T-matrix element which is employed to construct the BCS Hamiltonian in momentum space. We present results on the energy- and range-dependence of the pairing gap and superfluid density and the range-dependence of the chemical potential for a wide variation of the scattering length including the unitarity regime. In the zero range limit our calculated gap at the Fermi energy is found to be nearly equal to that calculated with contact potential. The mean gap averaged over the full width at half maximum of the gap function in the zero range and unitarity limits is found to be 0.42EF which is quite close to the recent result of the quantum Monte Carlo simulation [2018 {\rm Phys. Rev.A} {\bf 97}, 013601]. The chemical potential in the zero range limit also agrees well with that for the contact potential.
Abstract We calculate Bardeen–Cooper–Schrieffer (BCS) state of a unitary Fermi gas of atoms interacting with the finite-ranged Jost-Kohn potential which has been recently shown to account for the resonant interactions (2019 J. Phys. B: At. Mol. Opt. Phys. 52 165004). Using exact scattering solution of the potential, we derive two-body T-matrix element which is employed to construct the BCS Hamiltonian in momentum space. We present results on the energy- and range-dependence of the pairing gap and superfluid density and the range-dependence of the chemical potential for a wide variation of the scattering length including the unitary regime. In the zero range limit our calculated gap at the Fermi energy is found to be nearly equal to that calculated in mean-field theory with contact potential. The mean gap averaged over the full width at half maximum of the gap function in the zero range and unitary limits is found to be 0.42E F which is quite close to the recent result of the quantum Monte-Carlo simulation (2018 Phys. Rev. A 97 013601). The chemical potential in the zero range limit also agrees well with that for the contact potential.
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].
We develop an analytical approach for the description of an atomic Bose gas at unitarity. By focusing in first instance on the evaluation of the single-particle density matrix, we derive several universal properties of the unitary Bose gas, such as the chemical potential, the contact, the speed of sound, the condensate density and the effective interatomic interaction. The theory is also generalized to describe Bose gases with a finite scattering length and then reduces to the Bogoliubov theory in the weak-coupling limit.
Abstract This is the first of a series of chapters devoted to interacting Fermi gases, with special focus here on the effects of superfluidity. It starts with a brief discussion of the ideal Fermi gas and then focuses on the properties of dilute interacting Fermi gases. Topics include the weakly repulsive Fermi gas; the gas of composite bosons; the BCS limit of a weakly interacting Fermi gas; the strongly interacting, but still dilute, unitary Fermi gas where the scattering length is much larger than the average interatomic distance; the BCS–BEC crossover; and the Bogoliubov–de Gennes approach. A discussion of the main physical quantities predicted by mean-field theory and by quantum Monte Carlo simulations, including, in particular, the equation of state, the momentum distribution, and the condensation of pairs, is also presented.