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Degenerate Fermi gases

Patrizia Vignolo

in Ultracold Gases and Quantum Information: Lecture Notes of the Les Houches Summer School in Singapore: Volume 91, July 2009

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 ... More

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.Less

Fermi gas

E. R. DOBBS

in Helium Three

The properties of liquid 3He change as temperature is lowered from those typical of a classical, monoatomic liquid above 1 K to those of a Fermi gas with interactions, or Fermi liquid, and finally ... More

The properties of liquid 3He change as temperature is lowered from those typical of a classical, monoatomic liquid above 1 K to those of a Fermi gas with interactions, or Fermi liquid, and finally below a few millikelvin to a Fermi superfluid. This chapter reviews the statistical mechanism of a Fermi gas. Its predictions are compared with early measurements of the equilibrium, thermal, magnetic, and acoustical properties of liquid 3He.Less

Microscopic Physics

VOLOVIK GRIGORY E.

in The Universe in a Helium Droplet

This chapter deals with the Fermi systems, where the low-energy effective theory involves both bosonic and fermionic fields. Above the phase transition to the superconducting or superfluid state, the ... More

This chapter deals with the Fermi systems, where the low-energy effective theory involves both bosonic and fermionic fields. Above the phase transition to the superconducting or superfluid state, the overwhelming majority of systems consisting of fermionic particles (electrons in metals, neutrons in neutron stars, 3He atoms in 3He liquid, etc.) form a so-called Fermi liquid. Below transition new types of fermionic vacua emerge. This chapter discusses the Bardeen–Cooper–Schrieffer (BCS) theory for spin-triplet superfluids, which provide examples of different universality classes of fermionic vacua: fully gapped vacua, vacua with stable and marginal point nodes — Fermi points, and vacua with nodal lines — Fermi lines. It also discusses emergent ‘relativistic’ quasiparticles, fundamental constants and hierarchy of Planck energy scales in fermionic systems, problem of vacuum energy and cosmological term in bi-metric gravity, and mass generation for Standard Model fermions.Less

Perfect Gas

Jochen Rau

in Statistical Physics and Thermodynamics: An Introduction to Key Concepts

The perfect gas is perhaps the most prominent application of statistical mechanics and for this reason merits a chapter of its own. This chapter briefly reviews the quantum theory of many identical ... More

The perfect gas is perhaps the most prominent application of statistical mechanics and for this reason merits a chapter of its own. This chapter briefly reviews the quantum theory of many identical particles, in particular the distinction between bosons and fermions, and then develops the general theory of the perfect quantum gas. It considers a number of limits and special cases: the classical limit; the Fermi gas at low temperature; the Bose gas at low temperature which undergoes Bose–Einstein condensation; as well as black-body radiation. For the latter we derive the Stefan–Boltzmann law, the Planck distribution, and Wien’s displacement law. This chapter also discusses the effects of a possible internal dynamics of the constituent molecules on the thermodynamic properties of a gas. Finally, it extends the theory of the perfect gas to dilute solutions.Less

Quantum gases and condensates

Stephen J. Blundell and Katherine M. Blundell

in Concepts in Thermal Physics

Exchange symmetry affects the occupation of allowed states in quantum gases. If the density of the gas is very low, we can ignore this and forget about exchange symmetry; this is what we do for gases ... More

Exchange symmetry affects the occupation of allowed states in quantum gases. If the density of the gas is very low, we can ignore this and forget about exchange symmetry; this is what we do for gases at room temperature. But if the density is high, the effects of exchange symmetry become very important and it really starts to matter whether the particles you are considering are fermions or bosons. This chapter considers quantum gases in detail. It considers non-interacting fermion and boson gases, and discusses Bose-Einstein condensation.Less

Interacting Fermi Gases and the BCS–BEC Crossover

Lev Pitaevskii and Sandro Stringari

in Bose-Einstein Condensation and Superfluidity

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 ... More

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.Less

Theory of spin-polarized solutions

E. R. DOBBS

in Helium Three

Very dilute solutions of 3He in superfluid 4He behave as an almost ideal Fermi gas at millikelvin temperatures, which can be strongly polarized in high magnetic fields. This has enabled studies of ... More

Very dilute solutions of 3He in superfluid 4He behave as an almost ideal Fermi gas at millikelvin temperatures, which can be strongly polarized in high magnetic fields. This has enabled studies of their equilibrium, transport, and dynamical spin properties to be made over a wide range of temperatures, concentrations, and polarizations covering systems that are highly degenerate to those that are non-degenerate. This chapter discusses polarized Fermi gas, model theories, equilibrium theories, transport properties, and spin dynamics.Less

Bose–Einstein condensation

Massimo Inguscio and Leonardo Fallani

in Atomic Physics: Precise Measurements and Ultracold Matter

Besides the enormous opportunities offered by spectroscopy and atom interferometry, laser cooling has opened up the possibility of experimentally realizing and manipulating quantum degenerate gases: ... More

Besides the enormous opportunities offered by spectroscopy and atom interferometry, laser cooling has opened up the possibility of experimentally realizing and manipulating quantum degenerate gases: Bose–Einstein condensates and degenerate Fermionic systems. This chapter starts with an illustration of key experimental techniques, such as magnetic trapping, evaporative cooling, and Feshbach resonances. It then turns to the physics of atomic Bose–Einstein condensates, discussing their superfluid behaviour, their coherence properties, and their application to precise measurements. It also describes the experimental investigation of ultracold Fermi gases and fermionic superfluidity, as well as ongoing research into ultracold polar molecules.Less

Introduction

E. R. DOBBS

in Helium Three

This introductory chapter begins with a discussion of quantum solids and liquids. It then discusses Fermi gas and liquid, superfluid, and magnetic solid.

Quantum Statistics

Daniel V. Schroeder

in An Introduction to Thermal Physics

This chapter begins by extending the Boltzmann distribution to the case of a system that exchanges particles with its environment. This idea finds direct applications ranging from hemoglobin ... More

This chapter begins by extending the Boltzmann distribution to the case of a system that exchanges particles with its environment. This idea finds direct applications ranging from hemoglobin adsorption to ionization of atoms in stars. But the main applications are to dense “gases” in which the quantum behavior of identical particles comes into play. Examples include conduction electrons in metals and semiconductors; white dwarf and neutron stars; photon gases and thermal radiation from incandescent objects; neutrino and electron-positron production in the early universe; quasiparticles associated with vibrations and magnetic excitations in solids; and Bose-Einstein condensation of ultracold clouds of atoms.Less

Elementary concepts of nuclear physics

Helmut Hofmann

in The Physics of Warm Nuclei: with Analogies to Mesoscopic Systems

This chapter describes the basic concepts of nuclear models beginning with the bare force between two nucleons and ending with the complex configurations of the many-body system. For example those ... More

This chapter describes the basic concepts of nuclear models beginning with the bare force between two nucleons and ending with the complex configurations of the many-body system. For example those encountered in the optical model or in Niels Bohr's pictures of the compound nucleus and the liquid drop model. The role of sub-nuclear degrees of freedom is briefly considered. Elementary many-body properties related to independent particle motion are analyzed for the Fermi gas, and two-body correlations implied by the Pauli principle are introduced. Basic properties of nuclear reactions are explained and the relevance of the nucleonic mean free path is examined, for instance with respect to transport properties of collective motion. The Bethe-Weizsäcker-formula for nuclear masses is shown and the deformed liquid drop model is presented.Less

Fermi–Dirac Statistics

Robert H. Swendsen

in An Introduction to Statistical Mechanics and Thermodynamics

This chapter examines the consequences of Fermi-Dirac statistics. The Fermi function is described, and the Fermi energy is found as the low-temperature limit of the chemical potential. The Sommerfeld ... More

This chapter examines the consequences of Fermi-Dirac statistics. The Fermi function is described, and the Fermi energy is found as the low-temperature limit of the chemical potential. The Sommerfeld expansion is used to find the low-temperature behaviour of a Fermi gas. Among the consequences of these results is an explanation of why the electronic degrees of freedom do not affect the law of Dulong and Petit.Less

Electronic Systems

Hans-Peter Eckle

in Models of Quantum Matter: A First Course on Integrability and the Bethe Ansatz

The Bethe ansatz can be generalized to problems where particles have internal degrees of freedom. The generalized method can be viewed as two Bethe ansätze executed one after the other: nested Bethe ... More

The Bethe ansatz can be generalized to problems where particles have internal degrees of freedom. The generalized method can be viewed as two Bethe ansätze executed one after the other: nested Bethe ansatz. Electronic systems are the most relevant examples for condensed matter physics. Prominent electronic many-particle systems in one dimension solvable by a nested Bethe ansatz are the one-dimensional δ‎-Fermi gas, the one-dimensional Hubbard model, and the Kondo model. The major difference to the Bethe ansatz for one component systems is a second, spin, eigenvalue problem, which has the same form in all cases and is solvable by a second Bethe ansatz, e.g. an algebraic Bethe ansatz. A quantum dot tuned to Kondo resonance and coupled to an isolated metallic ring presents an application of the coupled sets of Bethe ansatz equations of the nested Bethe ansatz.Less

Quantum Gases in Cigar and One-dimensional Regimes

Lev Pitaevskii and Sandro Stringari

in Bose-Einstein Condensation and Superfluidity

This chapter discusses the behaviour of one-dimensional quantum gases. It discusses both the so-called cigar configurations, where the system is locally three-dimensional despite its ... More

This chapter discusses the behaviour of one-dimensional quantum gases. It discusses both the so-called cigar configurations, where the system is locally three-dimensional despite its one-dimensional-like geometrical shape, and the deep one-dimensional regime where the motion is frozen in the radial direction. The cigar configuration is well suited to investigating the novel features exhibited by quantized vortices and solitons. In the deep one-dimensional regime Bose gases exhibits a very different behaviour with respect to usual three-dimensional BECs as a consequence of the quantum and thermal fluctuations of the phase. The Lieb–Liniger transition between the one-dimensional mean-field and the Tonks–Girardeau regimes is discussed, and important features concerning the frequency of the collective oscillations and the superfluid behaviour are pointed out. Finally, the chapter discusses some key features exhibited by one-dimensional Fermi gases.Less

Quantum Gases in Pancake and Two-dimensional Regimes

Lev Pitaevskii and Sandro Stringari

in Bose-Einstein Condensation and Superfluidity

This chapter addresses the question of quantum gases in two dimensions. It investigates both the pancake and the deep two-dimensional regimes. In the former case the gas is locally three-dimensional ... More

This chapter addresses the question of quantum gases in two dimensions. It investigates both the pancake and the deep two-dimensional regimes. In the former case the gas is locally three-dimensional along the axial direction. In contrast, in the deep two-dimensional regime the axial motion is frozen. The chapter investigates the new physical phenomena exhibited by these two-dimensional quantum gases exploring the behaviour of the equation of state and of the collective oscillations. It discusses the behaviour of a fast-rotating Bose gas in two-dimensions, and the structure of vortex lines during the transition between the two-dimensional Thomas–Fermi and the lowest Landau level regimes. In the deep two-dimensional regime quantum gases exhibit novel and important phenomena at finite temperatures, due to the absence of long-range order and the emergence of the Berezinskii–Kosterlitz–Thouless transition. The peculiar behaviour of the BCS–BEC transition in deeply two-dimensional Fermi gases is also discussed.Less

Fermi Gas in the Harmonic Trap

Lev Pitaevskii and Sandro Stringari

in Bose-Einstein Condensation and Superfluidity

This chapter deals with Fermi gases confined in harmonic traps. After offering a reminder of the properties of the harmonically trapped ideal Fermi gas, the chapter focuses on the effect of ... More

This chapter deals with Fermi gases confined in harmonic traps. After offering a reminder of the properties of the harmonically trapped ideal Fermi gas, the chapter focuses on the effect of interactions, employing the local density approximation to calculate the density profiles and the momentum along the BCS–BEC crossover and comparing theoretical predictions with experiments.Less

Competing instabilities in quench experiments with ultracold Fermi gases near a Feshbach resonance

David Pekker and Eugene Demler

in Many-Body Physics with Ultracold Gases: Lecture Notes of the Les Houches Summer School: Volume 94, July 2010

Measuring the tunability of effective two-body interactions near Feshbach resonances is a powerful, experimental tool in systems of ultracold atoms. It has been used to explore a variety of ... More

Measuring the tunability of effective two-body interactions near Feshbach resonances is a powerful, experimental tool in systems of ultracold atoms. It has been used to explore a variety of intriguing phenomena in recent experiments. This chapter applies the method of collective mode instabilities to the concrete example of quenches of the non-interacting Fermi gas to the strongly interacting regime. Section 9.3 begins by discussing the relation between linear response and collective modes. It shows how to compute the pairing and the ferromagnetic responses using the equation of motion formalism. Section 9.4 demonstrates how to describe a Feshbach resonance using a pseudo-potential model. Section 9.5 applies the pseudo-potential model to compute the many-body T-matrix and thus obtain the pairing collective mode. Section 9.6 discusses how to incorporate the many-body T-matrix into the ferromagnetic susceptibility. Section 9.7 summarizes the results for the pairing versus Stoner competition in the context of the MIT experiments. Section 9.8 provides concluding remarks.Less