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Oxford University Press eBooks

Superfluids

Tom Lancaster, Stephen J. Blundell

Published Apr 17, 2014• 0 citations
Publisher

Abstract

The first condensed matter physics example is the superfluid. The chapter shows how an approximation due to Bogoliubov allows weakly interacting Bose systems to be treated. The quasiparticles are called bogolons and the ground state of the system can be treated as a coherent state. The Noether current is shown to arise from a gradient in the phase of this coherent state.

DOI
10.1093/acprof:oso/9780199699322.003.0043

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1. Define Bose-Einstein condensation in an interacting boson system in term of macroscopic occupation. 2. Introduce the Gross-Pitaevskii equation and its applicable conditions. 3. Derive the hydrodynamic equations and various physical consequences of the hydrodynamics with or without harmonic trap. 4. Introduce sound wave and emphasize the difference between the wave behavior and the diffusive behavior. 5. Introduce the criterion for superfluidity and determine the superfluid critical velocity. 6. Introduce two-fluid hydrodynamics. 7. Introduce the Bogoliubov theory. 8. Introduce various concepts related to quasi-particle, including ground state as vacuum of quasi-particles, the zero-point energy of quasi-particle, quasi-particle lifetime and vacuum fluctuation. 9. Introduce healing length as the characteristic length scale for low-energy physics of an interacting Bose condensate. 10. Summarize the difference between an interacting and non-interacting Bose condensate. 11. Introduce the Bragg spectroscopy. 12. Show that the Bogoliubov theory fails in one-dimension due to the infrared divergence. 13. Discuss the basic idea of the Bethe-ansatz solution and the Tonks-Girardeau limit. 14. Introduce scale invariant quantum gases. 15. Introduce the Josephson effect and the self-trapping as two different transport phenomena driven by phase coherence. 16. Discuss the conjugate relation between the relative phase fluctuation and the relative particle number fluctuation, and show strong enough phase fluctuation renders the system into a Fock state. 17. Discuss an example of quantum measurement that can project two independent condensates to a coherent superposition state. 18. Discuss interference effects in density and density-density correlations. 19. Discuss the relation between the instability of the Sch\"odinger Cat state and the stability of symmetry breaking.

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•Jan 1, 2011•0 citations
The first condensed matter physics example is the superfluid. The chapter shows how an approximation due to Bogoliubov allows weakly interacting Bose systems to be treated. The quasiparticles are called bogolons and the ground state of the system can be treated as a coherent state. The Noether current is shown to arise from a gradient in the phase of this coherent state.
The effective interaction energy of quasiparticles in the Bose fluid, which is described by ”ladder state diagrams” in the Bogoliubov approximation, was studied for a model potential. It is shown that in the Bogoliubov approximation, the interaction of excitations in liquid He can not be described even qualitatively. It means that a model of the weakly interacting dilute Bose gas fails in describing the interaction between excitation in Bose-condensed liquid, although this model qualitatively incorporates superfluidity in itself.
This chapter introduces the basic physics of Bose--Einstein condensation. We first start with why there is a difference between distinguishable and indistinguishable particles in terms of the number of states that are available in a composite system. Then Bose and Einstein's argument of why one expects a high probability of occupation of the ground state is discussed. This is derived then more rigorously for the grand canonical ensemble, showing that at some critical temperature there should be a macroscopic occupation of the ground state. Next, the low-lying energy spectrum of an interacting Bose--Einstein condensate is derived, leading to the Bogoliubov dispersion. The significance of the Bogoliubov dispersion as the origin of superfluidity is then discussed, in terms of superfluditysuperfluidity. Laudau'sLandau's criterion for superfluidity is derived, by general principles of Galilean transformations of Schrodinger's equation.
The aim of this book is to introduce a graduate student to selected concepts in condensed matter physics for which the language of field theory is ideally suited. The examples considered in this book are those of superfluidity for weakly interacting bosons, collinear magnetism, and superconductivity. Quantum phase transitions are also treated in the context of quantum dissipative junctions and interacting fermions constrained to one-dimensional position space. The style of presentation is sufficiently detailed and comprehensive that it only presumes familiarity with undergraduate physics. Readership: Graduate students and researchers interested in field theory in condensed matter physics.
In 1995 two different research groups observed for the first time the Bose-Einstein condensation (BEC) in ultracold gases. When the confining magnetic trap was turned off the gas was left free to expand, and the velocity of the particles showed a clear peak: most of the particles were occupying the same single particle state, the one of lowest energy. The Bose-Einstein condensation had been predicted in 1925 by Einstein, written by inspiration of a work on the statistic of the photons by Bose (1924). In this work Bose described the behavior of an ensemble of photons, treating them as massless particles, with no number conservation associated. Einstein extended this approach to particles with a mass and with fixed number, creating what is now called the Bose-Einstein distribution. The particles that follow such a description are called ``bosons'', as opposed to the ``fermions'' of the Fermi-Dirac statistics. Einstein predicted that in a gas of bosons - under a critical temperature - a finite fraction of the total number of particles would have been in the ground state, and act as a single entity. This amusing theoretical discovery found its utility a few years later. In the late thirties, new techniques allowed to cool Helium-4 at few Kelvins above the absolute zero. The properties of the resulting liquid were a puzzlement to the scientific community: among others, it could flow without experiencing friction. The liquid was called a ``superfluid''. A first explanation was given by London in 1938, which linked the superfluid behavior to the presence of a BEC among the bosonic Helium particles. The fermions cannot condense by themselves. On the other hand, they can form bound pairs and act as bosons, as it happens in a metal at low temperature. Using this approach, in 1957 Bardeen, Cooper and Schrieffer created a successful model of superconductivity by describing a superconductor as a superfluid in a charged system. During the course of these years we explored the superfluid properties of Bosons and Fermions in different settings. The original contributions of the thesis are described starting from the third chapter, where we speak about the generation and stability of solitons in a periodic optical lattices, both fixed or in motion. In the fourth chapter we study the generation of giant vortices in cold fermions, by using a generalized hydrodynamical approach. In chapter 5 we study the effect of a quasiperiodic lattice and the glassy phase it produces on a gas of bosons. Finally, we study the interaction of normal matter and superfluids with gravitational waves. While this interaction is seen to be extremely small, we believe that the resulting formalism is interesting by itself.
1. Define Bose-Einstein condensation in an interacting boson system in term of macroscopic occupation. 2. Introduce the Gross-Pitaevskii equation and its applicable conditions. 3. Derive the hydrodynamic equations and various physical consequences of the hydrodynamics with or without harmonic trap. 4. Introduce sound wave and emphasize the difference between the wave behavior and the diffusive behavior. 5. Introduce the criterion for superfluidity and determine the superfluid critical velocity. 6. Introduce two-fluid hydrodynamics. 7. Introduce the Bogoliubov theory. 8. Introduce various concepts related to quasi-particle, including ground state as vacuum of quasi-particles, the zero-point energy of quasi-particle, quasi-particle lifetime and vacuum fluctuation. 9. Introduce healing length as the characteristic length scale for low-energy physics of an interacting Bose condensate. 10. Summarize the difference between an interacting and non-interacting Bose condensate. 11. Introduce the Bragg spectroscopy. 12. Show that the Bogoliubov theory fails in one-dimension due to the infrared divergence. 13. Discuss the basic idea of the Bethe-ansatz solution and the Tonks-Girardeau limit. 14. Introduce scale invariant quantum gases. 15. Introduce the Josephson effect and the self-trapping as two different transport phenomena driven by phase coherence. 16. Discuss the conjugate relation between the relative phase fluctuation and the relative particle number fluctuation, and show strong enough phase fluctuation renders the system into a Fock state. 17. Discuss an example of quantum measurement that can project two independent condensates to a coherent superposition state. 18. Discuss interference effects in density and density-density correlations. 19. Discuss the relation between the instability of the Sch\"odinger Cat state and the stability of symmetry breaking.