To calculate static response properties of a many-body system, local density approximation (LDA) can be safely applied. But, to obtain dynamical response functions, the applicability of LDA is limited bacause dynamics of the system needs to be considered as well. To examine this in the context of cold atoms, we consider a system of non-interacting spin4 fermions confined by a harmonic trapping potential. We have calculated a very important response function, the spectral intensity distribution function (SIDF), both exactly and using LDA at zero temperature and compared with each other for different dimensions, trap frequencies and momenta. The behaviour of the SIDF at a particular momentum can be explained by noting the behaviour of the density of states (DoS) of the free system (without trap) in that particular dimension. The agreement between exact and LDA SIDFs becomes better with increase in dimensions and number of particles.
To calculate static response properties of a many-body system, local density approximation (LDA) can be safely applied. But, to obtain dynamical response functions, the applicability of LDA is limited bacause dynamics of the system needs to be considered as well. To examine this in the context of cold atoms, we consider a system of non-interacting spin4 fermions confined by a harmonic trapping potential. We have calculated a very important response function, the spectral intensity distribution function (SIDF), both exactly and using LDA at zero temperature and compared with each other for different dimensions, trap frequencies and momenta. The behaviour of the SIDF at a particular momentum can be explained by noting the behaviour of the density of states (DoS) of the free system (without trap) in that particular dimension. The agreement between exact and LDA SIDFs becomes better with increase in dimensions and number of particles.
With recent advances in the field of ultra cold atoms one can, by trapping atoms at low temperatures by laser beams, simulate systems which can be adequately described by single-band lattice Hamiltonians. Also, due to the high parameter tunability of the experimental setups, lattice disorder can be introduced in a con- trolled fashion in these systems. This thesis considers disordered/ordered interacting fermion lattice systems in equilibrium in one and two dimensions, subject to trap- ping parabolic potentials, where different levels of description, ranging from exact where possible (in 1D, Density Matrix Renormalization Group, DMRG), to approxi- mate (based on several local-density approximations within the framework of Lattice Density Functional Theory) are used. The exchange-correlation potentials consid- ered come from Many-Body Approximations obtained using Green’s functions, as well as from an exact local-density approximation (LDA) based on the Bethe-Ansatz (BALDA) in 1D. Both one- and two-dimensional systems were studied in equilib- rium, essentially looking at the ground state density profiles. Furthermore, in 2D, a pseudo-dynamics representing the trap-opening in the complete adiabatic limit was also studied. For one-dimensional systems, it is found that, in general, BALDA yields very good results compared to DMRG, except for the low density limit, but DMRG can describe features that none of the LDA:s considered can reproduce. It is also found that the strength of the external potential affects the impact that the exchange-correlation potential has on the system. Many of these features translate to the 2D case. However, a new aspect emerges in two dimensions, related to the competition of disorder and interaction. Here an important outcome is that, on opening the trap at an ideally adiabatic rate, different MBA:s (and thus different LDA:s) provide different minimal vs maximal expansion radii of the particle cloud, as a result of the interplay of disorder and interaction, and the underlying square lattice structure. For the 2D results, exact benchmarks were not available, and our findings may thus need further validation, by e.g. considering several disorder configurations or, ideally, by performing full Green’s function calculations. These considerations are summarised in our conclusions and outlook remarks, where possible directions for future investigation are highlighted.
One-particle properties of noninteracting fermions in a one-dimensional harmonic trap and at zero temperature are studied. Exact expressions and asymptotic results for a large fermion number N are given for the particle density distribution n0(z,N). For large N and near the classical boundary at the Fermi energy, the density displays increasing fluctuations. A simple scaling of these tails of the density distribution with respect to N is established. The Fourier transform of the density distribution is calculated exactly. It displays a small but characteristic hump near 2kF, with kF being a properly defined Fermi wave number. This is due to Friedel oscillations, which are identified and discussed. These quantum effects are missing in the semiclassical approximation. Momentum distributions are also evaluated and discussed. As an example of a time-dependent one-particle problem, we calculate exactly the evolution of the particle density when the trap is suddenly switched off, and find a simple scaling behavior, in agreement with recent general results.
Dynamics of strongly interacting Fermi gases, consisting of a 50-50 mixture of two different fermionic species, is investigated. For the equation of state we consider a Padé [2/2] approximations, which gives the weak-coupling perturbative formula (up to 4th order) in the low density regime, the unitary-limit Monte Carlo result in the high density regime, and reproduces the 4-fermion prediction for dimer-dimer scattering length in the BEC region. We use a time-dependent LDA to derive various properties of the Fermi gas under a harmonic confinement and compare them with the data of very recent experiments of 6Li atoms across a Feshbach resonance.
We evaluate the zero temperature properties and Bose-Einstein condensation (BEC) of hard sphere bosons in a harmonic trap using Quantum Monte Carlo methods. The chief purpose of this work is to go beyond the dilute limit, to test the limits of the Gross-Pitaevskii equation and related mean field approximations and to explore the role of interactions in determining the zero temperature properties of the trapped Bose gas. The density is increased by adjusting both the number of trapped bosons, N , and the s-wave scattering length, a, to study systems from the highly dilute region corresponding to early experiments in trapped BEC [2, 3] up to liquid He densities and beyond. Rather than relying on the perturbative Bogoliubov approximation (which assumes a large condensate fraction) to describe the condensate, we use the one-body density matrix (OBDM) formulation of BEC so that the properties of the condensate for systems with arbitrarily large interactions may be studied. In this formulation of BEC, condensate properties are obtained by diagonalizing the OBDM and obtaining the corresponding single particle “natural orbitals” and their occupation numbers for the system. The condensate wave function and condensate fraction are then obtained from the single particle orbital(s) with macroscopic occupation (N0 >> 1). Within this framework, we calculate the effects of interactions and increased density on the ground state energy, the density profile, the momentum distribution, the condensate fraction and condensate “wave-function” and several other properties.
We calculated and measured the density distribution and cloud size of a trapped two-dimensional (2D) 6 Li Fermi gas near a Feshbach resonance at low temperatures. Density distributions and cloud sizes were calculated for a wide range of interaction parameters using a local density approximation (LDA) and a zero-temperature equation of state obtained from quantum Monte Carlo simulations reported by Bertaina and Giorgini (2011 Phys. Rev. Lett. 106 110403). We found that LDA predictions agree well with experimental measurements across a Feshbach resonance. Theoretical results for Tan's contact parameter in a trapped gas are reported here along with predictions for the static structure factor at large momentum which could be measured in future Bragg spectroscopy experiments on 2D Fermi gases.
Following the idea of the density-functional approach, we develop a generalized Bogoliubov theory of an interacting Bose gas confined in a one-dimensional harmonic trap by using a local chemical potential, calculated with the Lieb-Liniger exact solution, as the exchange energy. At zero temperature, we use the theory to describe collective modes of a finite-particle system in all interaction regimes from the ideal gas limit to the mean-field Thomas-Fermi regime and to the strongly interacting Tonks-Girardeau regime. At finite temperature, we investigate the temperature dependence of collective modes in the weak-coupling regime by means of a Hartree-Fock-Bogoliubov theory with Popov approximation. By emphasizing the effects of finite particle number and nonzero temperature on collective-mode frequencies, we compare our results with a recent experimental measurement [E. Haller et al., Science 325, 1224 (2009)] and some previous theoretical predictions. We show that the experimental data are still not fully explained within current theoretical framework.