We use mean-field theory to investigate the dynamics of superfluid fermions. This thesis includes our two works. The first one is to study Josephson oscillations and self-trapping of superfluid fermions in a double-well potential with time-dependent Bogoliubov-de Gennes equations. We investigate the behaviour of a two-component Fermi superfluid. We numerically solve the time-dependent Bogoliubov-de Gennes equations and characterize the regimes of Josephson oscillations and self-trapping for different potential barriers and initial conditions. In the weak link limit the results agree with a two-mode model where the relative population and the phase difference between the two wells obey coupled nonlinear Josephson equations. A more complex dynamics is predicted for large amplitude oscillations and large tunneling. The second one is to calculate the dynamic structure factor of unitary fermions. We have studied the dynamic structure factor of unitary fermions both at zero and finite temperature using the Bogoliubov-de Gennes theory and also Superfluid Local Density Approximation. We have derived the expression of the linear response function and the dynamic structure factor in the random phase approximation. At zero temperature, the SLDA+RPA formalism indeed provides a better accuracy at low momentum transfer and also its static structure factor is closer to quantum Monte Carlo value than that in BdG+RPA; however SLDA seems to give worse results for the molecular excitations at large momentum transfer. We have discussed the role of temperature and the comparison between SLDA and BdG, as well as with experimental data. The analysis is still at a preliminary level, but it suggests that mean-field theories can indeed be used to extract quantitative information about the order parameter and the excitations of the system by two-photon Bragg scattering experiments.
We use mean-field theory to investigate the dynamics of superfluid fermions. This thesis includes our two works. The first one is to study Josephson oscillations and self-trapping of superfluid fermions in a double-well potential with time-dependent Bogoliubov-de Gennes equations. We investigate the behaviour of a two-component Fermi superfluid. We numerically solve the time-dependent Bogoliubov-de Gennes equations and characterize the regimes of Josephson oscillations and self-trapping for different potential barriers and initial conditions. In the weak link limit the results agree with a two-mode model where the relative population and the phase difference between the two wells obey coupled nonlinear Josephson equations. A more complex dynamics is predicted for large amplitude oscillations and large tunneling. The second one is to calculate the dynamic structure factor of unitary fermions. We have studied the dynamic structure factor of unitary fermions both at zero and finite temperature using the Bogoliubov-de Gennes theory and also Superfluid Local Density Approximation. We have derived the expression of the linear response function and the dynamic structure factor in the random phase approximation. At zero temperature, the SLDA+RPA formalism indeed provides a better accuracy at low momentum transfer and also its static structure factor is closer to quantum Monte Carlo value than that in BdG+RPA; however SLDA seems to give worse results for the molecular excitations at large momentum transfer. We have discussed the role of temperature and the comparison between SLDA and BdG, as well as with experimental data. The analysis is still at a preliminary level, but it suggests that mean-field theories can indeed be used to extract quantitative information about the order parameter and the excitations of the system by two-photon Bragg scattering experiments.
We theoretically investigate the dynamic structure factor of a strongly interacting Fermi gas at the crossover from Bardeen–Cooper–Schrieffer superfluids to Bose–Einstein condensates, by developing an improved random phase approximation within the framework of a density functional theory (DFT)—the so-called superfluid local density approximation. Compared with the previous random-phase-approximation studies based on the standard Bogoliubov–de Gennes equations, the use of the DFT greatly improves the accuracy of the equation of state at the crossover, and leads to a better description of both collective Bogoliubov-Anderson-Goldstone phonon mode and single-particle fermionic excitations at small transferred momentum. Near unitarity, where the s -wave scattering length diverges, we show that the single-particle excitations start to significantly contribute to the spectrum of dynamic structure factor once the frequency is above a threshold of the energy gap at . The sharp rise in the spectrum at this threshold can be utilized to measure the pairing gap Δ. Together with the sound velocity determined from the phonon branch, the dynamic structure factor provides us some key information of the crossover Fermi superfluid. Our predictions could be examined in experiments with 6 Li or 40 K atoms using Bragg spectroscopy.
This thesis summarizes our study on the properties of ultracold Fermions near a Feshbach resonance. The Feshbach resonance allows for a continuous tuning of the interaction strength between the atoms. With this tunability, many properties predicted for the BCS-BEC crossover region in solids can be shown to exist in this ultracold Fermi gas. With the highly controllable parameters and the clean environment in the experiments, the study of the Bardeen-Cooper-Schrieffer (BCS) to Bose-Einstein Condensation (BEC) crossover in ultracold Fermions provides valuable insights to the strongly interacting crossover region, which has never been achieved before in solids. We first examine the properties of 6 Li and 40 K gases. We find that in the unitary region the properties of the system do not depend on the atom species: outside the unitary region, the properties of the system do depend on the scattering properties of the different atom species. We study the dynamics of the ultracold Fermions in the crossover region in response to the variations in the external magnetic field. We find that the response depends on the time scale and the amplitude of the variations; which is in qualitative agreement with the experimental observations. We derive a set of dynamic mean-field equations for a Fermi gas in an arbitrary potential trap near a wide Feshbach resonance. The equations are more general than the local density approximation (LDA), and less demanding computationally as the Bogoliubov-de-Gennes (BdG) equation. We then analyze the solubility of the equations, as well as solve a simplified version of the equations for illustration and for self-consistency check on the assumptions made during the derivation. We systematically study the possible phases for a trapped Fermi gas with population imbalance near a Feshbach resonance. Our results agree qualitatively well with the recent experimental results, and predict the existence of two exotic non-BCS superfluid phases for the strongly interacting gas with population unbalance: the Breached Pair (BP) phase and the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase. Finally, to guide the experimental efforts in the search of new quantum phases, we propose a detection scheme for the possible exotic phases.
This Thesis analyzes the role of a mean-field theory and of the fluctuations in two different fermionic systems: the BCS-BEC crossover and high-Tc superconducting cuprates. In the first part of the Thesis we introduce the mean-field theory and the Gaussian-level fluctuations for the BCS-BEC crossover, i.e. the continuous evolution observed in ultracold neutral fermionic systems, from a weakly-coupled regime where fermions form Cooper pairs, to a strongly-coupled regime where the fermions at first form bosonic molecular dimers, subsequently undergoing Bose-Einstein condensation for low enough temperatures. We then analyze some problems related to the crossover: the condensate fraction for a spin-unbalanced Fermi gas, the Beliaev decay for collective excitations and the case of a two-dimensional Fermi gas, for which we calculate the first sound velocity and the Berezinskii-Kosterlitz-Thouless critical temperature, in very good agreement with recent experimental data. Finally we analyze a regularization procedure for the equation of state in the deep-BEC limit, allowing one to analytically derive the correct ratio between the fermionic and bosonic scattering lengths. In the second part of the Thesis we introduce a gauge approach to superconductivity in cuprates in which the hole is decomposed as the product of a spinful neutral particle, the spinon, and a spinless charged particle, the holon. The statistics of each particle is modified by binding it to a gauge field providing a statistical flux, this approach being analogous to Chern-Simons bosonization. We thus obtain a model characterized by three characteristic temperatures corresponding, respectively, to the appearance of a finite density of incoherent holon pairs, of a finite density of incoherent spinon pairs and, finally, to the phase coherence leading to superconductivity. Specifically in the present Thesis within this model we study the superfluid density, demonstrating that the spinon and holon contributions sum according to a Ioffe-Larking-like rule, analogous to that they found for resistivity, showing excellent agreement with experimental data in the moderate underdoping up to optimal doping region. We also demonstrated that the formalism reproduces the quasi-universality observed in experimental data at different dopings and allows for an analytical derivation of a relation that approximates the Uemura relation for moderate dopings.
By considering the contribution of the higher-order term representing the lowest approximation of beyond mean field corrections, we investigate a superfluid Fermi gas confined in a double-well potential in Bose–Einstein Condensation (BEC) side of the Bardeen–Cooper–Schrieffer (BCS) to BEC crossover. Two limited cases of deep BEC regime and BEC regime of BCS–BEC crossover, corresponding to the two-body scattering length a sc is small enough and large enough, respectively. We derive a simple two-mode model that could depict the dynamics effectively. With making thorough analysis on the two-mode model and its corresponding classical Hamiltonian, we find that the Josephson oscillation or self-trapping phenomenon could emerge at certain parameters. We find three kinds of the phase states: Josephson oscillation (JO), oscillating-phase-type self-trapping (OPTST) and running-phase-type self-trapping (RPTST). The dependence of these three phase states on the dimensionless interaction parameter y = 1/(kFa sc ) and the initial system energy are given in this paper.
In this paper, we study the interaction-modulated tunneling dynamics of a Bose-Fermi superfluid mixture, where a Bose-Einstein condensate (BEC) with weak repulsive interaction is confined in a symmetric deep double-well potential and an equally populated two-component Fermi gas in a harmonic potential symmetrically is positioned in the center of the double-well potential. The tunneling between the two wells is modulated by fermions trapped in a harmonic potential. When the temperature is adequately low and the bosonic particle number is adequately large, we can employ the mean-field theory to describe the evolution of the BEC in the double-well potential through the time-dependent Gross-Pitaevskii equation. For the Fermi gas in the harmonic potential trap, we consider the case where the inter-fermion interaction is tuned on the deep Bose-Einstein condensate of the inter-fermion Feshbach resonance, where two fermions of spin-up and spin-down form a two-body bound state. Within the regime, the Fermi gas is well described by a condensate of these fermionic dimers, and hence can be simulated as well by a Gross-Pitaevskii equation of dimers. The inter-species interactions couple the dynamics of the two species, which results in interesting features in the tunneling oscillations. The dynamic equations of the BEC in the double-well potential is described by a two-mode approximation. Coupling it with time-dependent Gross-Pitaevskii equation of the harmonically potential trapped molecular BEC, we numerically investigate the dynamical evolution of the Boson-Fermi hybrid system under different initial conditions. It is found that the interaction among fermions in a harmonic potential leads to strong non-linearity in the oscillations of the bosons in the double-well potential and enriches the tunneling dynamics of the bosons. Especially, it strengthens macroscopic quantum self-trapping. And the macroscopic quantum self-trapping can be expressed in three forms: the phase tends to be negative and monotonically decreases with time, the phase evolves with time, and the phase tends to be positive and increases monotonically with time. This means that it is possible the tunneling dynamics of the BEC in double-well potential is adjustable. Our results can be verified experimentally in a Bose-Fermi superfluid mixture by varying different interaction parameters via Feshbach resonance and confinement-induced resonance.
Abstract Based on random phase approximation, we numerically calculate dynamical structure factors of a balanced two-dimensional (2D) Fermi superfluid, and discuss their energy, momentum and interaction strength dependence in the 2D BEC–BCS crossover. At a small transferred momentum, a stable Higgs mode is observed in the unitary 2D Fermi superfluid gas where the particle–hole symmetry is not satisfied. Stronger interaction strength will make the visibility of the dispersion of Higgs mode harder to be observed. We also discuss the dimension effect and find that the signal of the Higgs mode in two dimension is more obvious than that in 3D case. At a large transferred momentum regime, stronger interaction strength will induce the weight of the molecules excitation increasing, while in verse the atomic one decreasing, which shows the pairing information of Fermi superfluid. The theoretical results qualitatively agree with the corresponding quantum Monte Carlo data.