This is an introductory lecture to the theory of degenerate Fermi gases, in the context of present experiments on atomic Fermi gases. In part one, some properties of the ideal Fermi gas are presented, including a discussion of the fluctuations of the number of fermions in a given spatial zone in 1D, 2D and 3D. In part two, two-body aspects of the interaction potential are discussed and several possible models for the interaction are analyzed, including the two-channel model for the Feshbach resonance. In part three, basic predictions of zero temperature BCS theory are presented, including a derivation of superfluid hydrodynamic equations from time dependent BCS theory.
This is an introductory lecture to the theory of degenerate Fermi gases, in the context of present experiments on atomic Fermi gases. In part one, some properties of the ideal Fermi gas are presented, including a discussion of the fluctuations of the number of fermions in a given spatial zone in 1D, 2D and 3D. In part two, two-body aspects of the interaction potential are discussed and several possible models for the interaction are analyzed, including the two-channel model for the Feshbach resonance. In part three, basic predictions of zero temperature BCS theory are presented, including a derivation of superfluid hydrodynamic equations from time dependent BCS theory.
The main topic of these lectures is the physics of quantum degenerate atomic Fermi gases both under uniform confinement and in harmonic traps. To begin with, a general introduction on ideal Fermi gases is proposed. This "academic" limit is relevant in experiments where ultracold fermionic atoms are all polarized in a single spin state: the low atom momenta together with the fermionic nature of the atoms forbid atomic s-wave scattering and the atoms can be considered as free. Interactions are then introduced for the case of a gas with two spin components. Interactions can be tuned via Feshbach resonances and control the microscopic features of the superfluid phase, ranging from the BCS limit (formation of Cooper pairs) to the BEC limit (Bose-Einstein condensation of dimers). Various theoretical approaches for the description of the BCS-BEC crossover and their predictions in comparison with the experimental results are discussed.
We first consider an exactly solvable classical field model to understand the coherence properties and the density fluctuations of a one-dimensional (1D) weakly interacting degenerate Bose gas with repulsive interactions at temperatures larger than the chemical potential. In a second part, using a lattice model for the quantum field, we explain how to carefully generalize the usual Bogoliubov approach to study a degenerate and weakly interacting Bose gas in 1D, 2D or 3D in the regime of weak density fluctuations. In the last part, using the mapping to an ideal Fermi gas in second quantized formalism, we calculate and discuss physically the density fluctuations and the coherence properties of a gas of impenetrable bosons in 1D.
PACS. 03.75.Hh – Static properties of condensates; thermodynamical, statistical and structural properties. PACS. 03.75.Ss – Degenerate Fermi gases. PACS. 05.30.Fk – Fermion systems and electron gas. Abstract. – We present a theory for a superfluid Fermi gas near the BCS-BEC crossover, including pairing fluctuation contributions to the free energy similar to that considered by Nozières and Schmitt-Rink for the normal phase. In the strong coupling limit, our theory is able to recover the Bogoliubov theory of a weakly interacting Bose gas with a molecular scattering length very close to the known exact result. We compare our results with recent Quantum Monte Carlo simulations both for the ground state and at finite temperature. Excellent agreement is found for all interaction strengths where simulation results are available. The recent experimental realization of strongly interacting Fermi gases of 6 Li and 40 K atoms near a Feshbach resonance has opened up the exciting possibility of investigating the crossover from a Bardeen-Cooper-Schrieffer (BCS) superfluid to a Bose-Einstein condensate (BEC) [1–5]. In these systems, the inter-atomic interaction strength can be varied by tuning
The physics of quantum degenerate atomic Fermi gases in uniform as well as in harmonically trapped configurations is reviewed from a theoretical perspective. Emphasis is given to the effect of interactions that play a crucial role, bringing the gas into a superfluid phase at low temperature. In these dilute systems, interactions are characterized by a single parameter, the s-wave scattering length, whose value can be tuned using an external magnetic field near a broad Feshbach resonance. The BCS limit of ordinary Fermi superfluidity, the Bose-Einstein condensation (BEC) of dimers, and the unitary limit of large scattering length are important regimes exhibited by interacting Fermi gases. In particular, the BEC and the unitary regimes are characterized by a high value of the superfluid critical temperature, on the order of the Fermi temperature. Different physical properties are discussed, including the density profiles and the energy of the ground-state configurations, the momentum distribution, the fraction of condensed pairs, collective oscillations and pair-breaking effects, the expansion of the gas, the main thermodynamic properties, the behavior in the presence of optical lattices, and the signatures of superfluidity, such as the existence of quantized vortices, the quenching of the moment of inertia, and the consequences of spin polarization. Various theoretical approaches are considered, ranging from the mean-field description of the BCS-BEC crossover to nonperturbative methods based on quantum Monte Carlo techniques. A major goal of the review is to compare theoretical predictions with available experimental results.
This thesis discusses two important topics in ultracold atomic gases: strong interactions in quantum gases, and quantum Hall physics in neutral atoms.First we give a brief introduction on basic scattering models in atomic physics, and an approach to adjust the interactions between atoms.We also include a list of experimental probes in cold atom physics.After these introductions, in Chapter 3, we report a few interesting problems in strongly interacting quantum gases.We introduce the BCS-BEC crossover model and relevant many-body techniques at the beginning, and discuss the details of several specific systems.We find the Fermi gases across narrow Feshbach resonances are strongly interacting at low temperature even when the magnetic field is several widths away from the resonance.We also discuss an approach to describe the metastable repulsive branch of Bose and Fermi gases across the resonance, and find a stable region of repulsive Bose gas close to unitarity.Some studies in two dimensional Fermi gases with spin imbalance are also included, and they are closely related to a number of recent experiments.In Chapter 4, we discuss quantum Hall physics in the context of neutral atomic gases.After illustrating how the Berry phase experienced by neutral atoms is equivalent to the magnetic field in electrons, we introduce the newly developed synthetic gauge field scheme in which a gauge potential is coupled to the neutral atoms.We give a detail introduction to this Raman coupling scheme developed by NIST group, and derive the theoretical model of the system.Then we make some predictions on the evolution of quantum Hall states when an extra anisotropy is applied from the external trap.Finally, we propose some experiments to verify our predictions.4.9 Phase diagram of the cloud in rotating anisotropic traps: two phases of the BEC and QH are identified. . . . . . . . . . . . . . . . . . . . . . . . . . . .4.10 Distortion and transition in vortex lattices of a rapidly rotating BEC in an anisotropic trap. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.11 Density profiles at cut x = 0 and y = 0 of the Laughlin wavefunctions at different positions in figure 4.9. . . . . . . . . . . . . . . . . . . . . . . . . .4.12 Distribution of zeros in the relative coordinate of the ground state of two body problem in rotating anisotropic traps. . . . . . . . . . . . . . . . . . .4.13 Variation in the distance of splitting zeros as a function of inverse interaction strength, for fixed anisotropy α = 0.96. . . . . . . . . . . . . . . . . . . . . .4.14 Density profiles after TOF expansion of the gas in a rotating isotropic trap.4.15 Density profiles after TOF expansion of anisotropy α = 0.94. . . . . . . . .4.16 Second order correlation in equation 4.64, in a rotating isotropic (left) and anisotropic (right) trap. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.17 A sketch of high-resolution in situ imaging technique in neutral atoms. . . .4.18 In situ images of the quantity |ψ(r/2, -r/2)| for an anisotropy α = 0.96. .x
This doctoral dissertation is concerned with the physics of strongly interacting cold alkali atoms at low temperatures near a Feshbach resonance. In Chapter 1, we establish a connection between superfluid 4He and the BCS theory of superconductivity, and cold alkali atoms. We give the history of cold atoms, describing the significant achievements, pitfalls and challenges. In Chapter 2, we explore the thermodynamics of strongly interacting Bosonic atoms. We explore the stability of atomic Bosonic condensates near a Feshbach resonance. We show that the experimentally attained atomic condensate is a saddle point of the free energy, but the kinetics of its decay is slow. We also show that there is a second, higher density condensate branch which has an Ising-like phase transition to a molecular (paired) condensate when ramped across the Feshbach resonance. We argue that due to the high density, inelastic 3-body processes possibly render this transition unobservable. In Chapter 3, we explore the thermodynamics of Fermionic atoms near a Feshbach resonance. We determine the zero-temperature (T less than/less than T sub(F) ) pair propagator for a spinimbalanced mixture of up and down spin Fermions, and use it to show that such a mixture becomes completely polarized at mu sub(down arrow)= -0.9 mu sub(up arrow) . We also determine the Thouless criterion for superfluidity in a spin-imbalanced Fermi mixture, and construct a phase diagram of such a system at zero temperature. We then compare our results with experiments performed by two different groups. We find that interaction modifications to the minority spin self-energy inferred from our analysis is roughly double those observed in experiments. This discrepancy is consistent with the expected accuracy of the theory. In Chapter 4, we extend our analysis of the preceding chapter to calculate the surface tension of an interface between spin-polarized Fermions in the normal and superfluid phases. We show that, as expected, this surface tension decreases with increasing temperature and vanishes at a tricritical temperature, above which the transition becomes continuous. We also calculate the thickness of the interface; at T = 0, this is a few interparticle spacings, but diverges at the tricritical temperature. To compare with a set of relevant experiments, we also develop a phenomenological model for surface tension, and conclude that experimental surface tensions are an order of magnitude higher than what our microscopic calculation yields. We hypothesize possible mechanisms. In Chapter 5, we calculate the finite temperature phase diagram of a Bose-Fermi mixture produced from a spin-imbalanced two-component Fermi gas deep in the BEC phase. We show that there is a discontinuous transition between the superfluid and normal phase, with an entropy of mixing sufficient to cool the system down. We detail the construction of such a cooling scheme to cool a Fermi system below what is possible evaporatively, and find that the cooling efficiency is comparable to typical evaporative schemes. In Chapter 6, we shift our focus from thermodynamics to dynamics. We calculate shifts in the energy spectrum of a spin-balanced Fermionic superfluid of Cooper pairs due to the presence of energetically close states coupled by a Feshbach resonance. These shifts manifest themselves as clock-shifts in the radio-frequency spectrum of the superfluid. In addition to a broad asymmetric peak coming from the break-up of Cooper pairs, we find (for certain parameter ranges) a sharp, symmetric "bound-bound" spectral line coming from the conversion of Cooper pairs in one channel to pairs or molecules in another channel. Our theory shows remarkable quantitative agreement with experiments performed by an experimental group.