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Strong Correlations in Ultracold Fermi Gases

W. R. Schneider

Published Jan 1, 2011• 2 citations
OpenAlex

Abstract

Ultracold atomic gases provide an ideal system with which to study fundamental many-body physics.Exhibiting universal interactions in clean and controllable environments, long-used simple models as well as more exotic models can now be realized.The interplay between theory and experiment is therefore very active, and, in this thesis, I will detail several works, both exact analytic results and numerical calculations, which have impacts on current experiments.I begin with an introduction to the field including a brief discussion of experiments, the microscopic model of two species of interacting fermions, the BCS-BEC crossover and an overview of the various phases of atomic Fermi gases.I then describe the various results of my theoretical investigations, which are divided into five chapters.First, I describe radio frequency (RF) spectroscopy experiments and how they probe the single-particle spectral function.This leads to my results on an exact feature of the spectral lineshape, a universal large-momentum structure which exists for all states of interacting Fermi systems and has been verified in recent angle-resolved RF experiments.Second, I focus on gases which have a normal Fermi liquid ground state and show that their lineshape exhibits a characteristic jump discontinuity.I illustrate this Fermi surface singularity and the previously mentioned universal large momentum tail with explicit calculations.Third, I turn to the low energy structure of the single-particle spectral function in the superfluid state.I argue that sharp low energy quasiparticle excitations exist across the BCS-BEC crossover using a general argument that includes the interaction of fermions with the low-energy collective mode.This is illustrated with an explicit calculation within an approximation scheme.Fourth, I address the trap-induced inhomogeneity and use a Bogoliubov-deGennes analysis to test if a simple local density approximation (LDA) can provide an adequate description of a spin imbalanced Fermi system.Finally, I discuss dynamics of atomic gases motivated by a recent experiment in which two spin-polarized clouds are allowed to collide with each other.Despite the underlying attractive interactions, the clouds are seen to bounce off of each other.Using a hydrodynamic description, I argue that the short and intermediate time dynamics reflects a metastable state in which the effective interaction is repulsive.It is my hope that the work presented in this thesis can be used to gain key insights which inspire future endeavors.6.2 Time dependence of the center-of-mass positions zσ of the two spin clouds at unitarity for an initial separation z↑ (t = 0) = R z , z↓ (t = 0) = -R z using the upper branch energy functional and damping γ = 1.Red (upper) and blue (lower) curves denote the different spin species. . . . . . . . . . . . . . . .71 6.3 The upper branch energy functional obtained from lowest order constraint variational method.(a), energy as a function of the polarization P for various values of k F a s as indicated in the figure.(b), energy as a function of the interaction parameter k F a s for various values of polarization P as indicated in the figure.(c), phase diagram of the interacting two-component Fermi gases.The two vertical red dashed lines indicates the separation between three phases.For 1/(k F a s ) > 1.124, the system is in the Fermi liquid (FL) phase.For 1.124 > 1/(k F a s ) > 0.91, the system is in the partially polarized phase (PP) and for 1/(k F a s ) < 0.91, the system is fully polarized. . . . . .74 6.4 (Color online) Time dependence of the center-of-mass positions zσ of the two spin clouds for k F (0)a s = 2. . . . . . . . . . . . . . . . . . . . . . . . . . .75 6.5 (Color online) Time dependence of the center-of-mass positions zσ of the two spin clouds for k F (0)a s = 0.5. . . . . . . . . . . . . . . . . . . . . . . . . .

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Ultracold atomic gases provide an ideal system with which to study fundamental many-body physics.Exhibiting universal interactions in clean and controllable environments, long-used simple models as well as more exotic models can now be realized.The interplay between theory and experiment is therefore very active, and, in this thesis, I will detail several works, both exact analytic results and numerical calculations, which have impacts on current experiments.I begin with an introduction to the field including a brief discussion of experiments, the microscopic model of two species of interacting fermions, the BCS-BEC crossover and an overview of the various phases of atomic Fermi gases.I then describe the various results of my theoretical investigations, which are divided into five chapters.First, I describe radio frequency (RF) spectroscopy experiments and how they probe the single-particle spectral function.This leads to my results on an exact feature of the spectral lineshape, a universal large-momentum structure which exists for all states of interacting Fermi systems and has been verified in recent angle-resolved RF experiments.Second, I focus on gases which have a normal Fermi liquid ground state and show that their lineshape exhibits a characteristic jump discontinuity.I illustrate this Fermi surface singularity and the previously mentioned universal large momentum tail with explicit calculations.Third, I turn to the low energy structure of the single-particle spectral function in the superfluid state.I argue that sharp low energy quasiparticle excitations exist across the BCS-BEC crossover using a general argument that includes the interaction of fermions with the low-energy collective mode.This is illustrated with an explicit calculation within an approximation scheme.Fourth, I address the trap-induced inhomogeneity and use a Bogoliubov-deGennes analysis to test if a simple local density approximation (LDA) can provide an adequate description of a spin imbalanced Fermi system.Finally, I discuss dynamics of atomic gases motivated by a recent experiment in which two spin-polarized clouds are allowed to collide with each other.Despite the underlying attractive interactions, the clouds are seen to bounce off of each other.Using a hydrodynamic description, I argue that the short and intermediate time dynamics reflects a metastable state in which the effective interaction is repulsive.It is my hope that the work presented in this thesis can be used to gain key insights which inspire future endeavors.6.2 Time dependence of the center-of-mass positions zσ of the two spin clouds at unitarity for an initial separation z↑ (t = 0) = R z , z↓ (t = 0) = -R z using the upper branch energy functional and damping γ = 1.Red (upper) and blue (lower) curves denote the different spin species. . . . . . . . . . . . . . . .71 6.3 The upper branch energy functional obtained from lowest order constraint variational method.(a), energy as a function of the polarization P for various values of k F a s as indicated in the figure.(b), energy as a function of the interaction parameter k F a s for various values of polarization P as indicated in the figure.(c), phase diagram of the interacting two-component Fermi gases.The two vertical red dashed lines indicates the separation between three phases.For 1/(k F a s ) > 1.124, the system is in the Fermi liquid (FL) phase.For 1.124 > 1/(k F a s ) > 0.91, the system is in the partially polarized phase (PP) and for 1/(k F a s ) < 0.91, the system is fully polarized. . . . . .74 6.4 (Color online) Time dependence of the center-of-mass positions zσ of the two spin clouds for k F (0)a s = 2. . . . . . . . . . . . . . . . . . . . . . . . . . .75 6.5 (Color online) Time dependence of the center-of-mass positions zσ of the two spin clouds for k F (0)a s = 0.5. . . . . . . . . . . . . . . . . . . . . . . . . .
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
In this Thesis we will theoretically address some issues concerning the Physics of ultracold Fermi gases, all of them with experimental relevance. The field of ultracold gases, and more recently of ultracold Fermi gases, is gathering a lot of experimental and theoretical interest for two main reasons: the great control on the relevant parameters of the problem and the relative simplicity of the minimal theory which is able to correctly describe the system. Building upon this solid background, ultracold gases provided an ideal laboratory for testing more refined theories and also for addressing fundamental issues of quantum mechanics as well as for simulating more complex physical systems such as those encountered in condensed matter. Even if the effective hamiltonian of ultracold gases can be simple, due to the diluteness of the system and the low temperature, which imply low energy physics, the solution of the quantum mechanic equations governing the state of these systems is not always simple. For the static properties usually mean-field solutions exist or perturbative expansions can be produced in some regimes. However Quantum Monte Carlo (QMC) techniques provide more accurate results especially in the strongly interacting regimes. For confined systems it is possible to use QMC only for a few particles, so that, for large number of particles, a fruitful combined use of Density Functional Theory (DFT) and QMC is necessary. The study of the dynamics of ultracold gases has received little attention with QMC techniques, due to the intrinsic computational difficulty of the many-body problem, so that general hydrodynamic equations are often used for studying the propagation of smoothly varying perturbations. In this Thesis we use QMC techniques for studying the problem of ferromagnetism in repulsive or effectively repulsive ultracold gases without a lattice and the problem of the Bardeen-Cooper-Schrieffer to Bose-Einstein-Condensation (BCS-BEC) crossover in two dimensions. We use DFT in the Local Density Approximation (LDA) for calculating the density profiles of ultracold Fermi gases in harmonic magneto-optical traps, starting from QMC equations of state. We study the propagation of first and second sound in ultracold Fermi gases in cylindrical geometry, using the hydrodynamic equations of superfluids.
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.
The ability to study ultracold atomic Fermi gases holds the promise of significant advances in testing fundamental theories of many-body quantum physics. Of particular interest are strongly interacting Fermi gases in the BCS to BEC crossover that exhibit a transition to a superfluid state at temperatures near 0.2TF , where TF is the Fermi temperature. This transition, as a fraction of TF, is extremely high compared to any known superfluid or superconductor. These gases are also in a universal regime where the physics is independent of the details of the atomic interactions and is therefore relevant to fields as diverse as condensed matter, nuclear physics and astrophysics. In this thesis, I present an experimental probe of atomic gases that uses momentum-resolved RF spectroscopy to realize an analog of angle-resolved photoemission spectroscopy (ARPES) in materials. This measurement reveals the energy and momentum of single-particle states in the strongly interacting Fermi gas. In condensed matter, ARPES has proved to be one of the most powerful experimental techniques for studying the electronic structure of strongly correlated electron materials. The ability to perform analogous measurements in ultracold Fermi gases constitutes a significant advance in our ability to directly connect ultracold atomic gases to strongly correlated electron systems. Taking advantage of this new measurement technique, I investigate a long-standing problem in the field of strongly interacting fermions, namely whether a pseudogap state consisting of incoherent fermion pairs exists at temperatures above the critical temperature for superfluidity. The photoemission data I present provide strong evidence for this state and have implications for fundamental theories of strongly interacting Fermi gases and strongly correlated electron materials. I also discuss the experimental confirmation of recently predicted universal relations for strongly interacting Fermi gases, as well as some of the first experiments involving atomic Fermi gases with p-wave pairing.
In this thesis, we study the coherence and the dynamics of ultracold atomic quantum gases, with a particular regard on Fermi gases. For what concerns the coherence, we study a Fermi gas confined on a ring threaded by an artificial gauge field. First, we show that the persistent current flowing in the ring can be used as probe for the different regimes of the BCS-BEC crossover. We propose a readout protocol for our results, based on the co-expansion of the gas trapped on the ring and a reference gas placed in the center of the ring itself. Next, we show that using the same expansion protocol we can extract information about the phase of the quantum gas. At intermediate times, the expanding cloud provides information about the first order coherence of the Fermi gas. In few-body systems, we can also access the number of particles in the system. At long times, we extract information about the many-body coherence coherence of the gas. Subsequently we focus on the dynamics of a strongly interacting two-component Fermi gas trapped in a harmonic potential. We consider a spin-separated state and then we follow the dynamics until long times. At short times, the dynamics is superdiffusive with dynamical exponents of the Kardar-Parisi-Zhang universality class. At intermediate times, the dynamics is dominated by damped spin oscillations unveleing a spin-drag effect. At long times, we observe thermalization. Ultimately, we study a multicomponent Bose gas in the strongly interacting regime. We first provide the exact solution for the wavefunction and we discuss its symmetry. Afterwards, we study the response of the gas to an artificial gauge field, with particular regard to the period of the persistent current as a function of the gauge flux.
This thesis studies radio-frequency spectroscopy of superfluid alkali Fermi gases. Radio-frequency and laser fields provide a coherent and well understood tool for manipulating atomic gases. However, new phenomena are expected and have already been seen experimentally when the alkali atoms become strongly interacting. Here we use a simple perturbative treatment of the atom-rf-field coupling for studying the spectroscopy of a strongly interacting gas. At ultracold temperatures fermionic atoms become paired in analogy to electrons in superconductors. The resulting binding energy, or the pairing gap, can be observed as a shift in the radio-frequency spectrum. With strong interactions, the fermions begin pairing already above the critical superfluid transition temperature. The radio-frequency spectroscopy grants a way to study these fluctuation effects quantitatively. The inhomogeneous trapping potential of the atom cloud also adds its own flavour, resulting in distinct mesoscopic effects in the radio-frequency spectra. Our theoretical models are in very good agreement with the experimental results. First order perturbation theory provides qualitatively correct lineshapes, and our generalised perturbative approach gives quantitatively correct magnitudes for the transfer rates. The inhomogeneity plays an important part also in more exotic systems in which the simple BCS-type pairing of atoms is prevented by polarising the gas. Superfluidity and phase separation in such spin-imbalanced gases has been experimentally observed. We have suggested the use of radio-frequency spectroscopy for observing exotic polarised superfluid states, present at the edges of the atom cloud.