DOCSLIB.ORG

BCS to BEC Evolution

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
BCS to BEC Evolution BCS to BEC Evolution S¸.Furkan Ozt¨urk¨ September 30, 2019 Abstract This report presents a literature review and a conceptual understanding of the crossover from the Bardeen{Cooper{Schrieffer (BCS) state of weakly-correlated pairs of fermions to the Bose{Einstein con- densation (BEC) of diatomic molecules in the atomic Fermi gas. In the context of ultracold Fermi gases, a BCS-BEC crossover means that by tuning the interaction strength, one goes from a BCS state to a BEC state without encountering a phase transition. BEC state is a Bose-Einstein condensate of two-atom molecules (bound fermions), while the BCS state is made up of pair of atoms. The difference between the pairs and the molecules is that the molecules are localized in the real space, whereas the BCS pairs are made of two particles with opposite momenta in the momentum space [1,2]. 1 History In this section, I listed the progress of our understanding of superfluidity and superconductivity. Superfluidity is a collective behavior of atoms in which particles can flow coherently without any friction or dissipation of heat and superconductivity is basically a charged superfluidity. The first observation of superfluid behavior was made by Heike Kamerlingh Onnes in 1911, much before the development of quantum mechanics. He cooled a metallic sample of Hg below to 4.2 K, using liquid helium-4 and found that the sample conducted electricity without dissipation which coined the term superconductivity [3]. Afterwards the superfluidity of bosonic liquid helium-4 was observed by Kapitza and Misener in 1938 [4]. In the same year, Fritz London proposed a connection between the superfluidity of helium-4 and Bose{Einstein condensation (BEC) [5]. In BEC, a finite condensate fraction of the total number of bosons occupies the lowest- energy (zero momentum) single-particle state at sufficiently low temperatures. For non-interacting bosons, all of the zero particles are in the ground state at zero temperature, yet interactions in helium-4 are sufficiently strong to reduce the condensate fraction to 10% at zero temperature [2]. In 1946, Bogoliubov found a microscopic explanation for the superfluidity and put forward that weak repulsive interactions do not destroy the BEC state. More importantly he found that the formation of the BEC state does not guarantee superfluidity which depends on the existence of currents that flow without dissipation and requires correlations|interactions among bosons [6]. Also, in the same year, to account for the superconductivity of metal ammonia solutions, Richard Ogg proposed the possibility of pairing of electrons in real space such that they obey the bosonic statistics [7]. After that, in 1954, Max Schafroth developed a theoretical framework for such pairing, but it was not supported by experimental evidence as no preformed pairs were observed and the BEC temperature was much too high for known superconductors [8]. Also, their theory had problems in dealing with real metals, because the pairs are hugely overlapping in real space, hardly describable as point bosons, and the anti-symmetry of the electronic wavefunction was crucial to the problem of superconductivity. However, in 1957, Bardeen, Cooper, and Schrieffer proposed a theory of superconductivity known as the BCS theory and suggested a pairing of electrons in the momentum space. BCS theory faced up to all these challenges and made quantitative predictions for the properties of superconductors [9]. Having got a successfully theory of superconductivity, the differences between the BCS and BEC have begun to be stressed. In 1969, Eagles studied the superconductivity in doped semiconductors like with a very low carrier density, where the attraction between electrons need not be small compared with the Fermi energy. This led to the first mean-field treatment of the BCS-BEC crossover [10]. Following this, in 1980, Leggett gave picture of the BCS-to-BEC evolution at T = 0 by a description in real space of paired fermions with opposite spins for a dilute gas of fermions. Although this is mostly in the BCS limit, Leggett wanted to understand the extent to which some of its properties, such as the total angular momentum of the superfluid might be similar to that of a BEC of diatomic molecules [11]. In 1985, Nozieres and Rink (NSR), extended Leggett's description to finite temperatures with finite ranged attraction near the critical temperature for superfluidity along with the evolution of the critical temperature [12]. The discovery of high-Tc cuprate superconductors in 1986 by Bednorz and M¨uller,created a new paradigm for superconductivity for which the BCS theory failed dramatically [13]. In 1993, motivated by the inapplicability of the BCS theory to cuprate superconductors, Melo, Randeria, and Engelbrecht extended the preliminary results of Leggett and NSR as an attempt to understand cuprates. They used a zero-ranged attraction characterized by the experimentally 1 measurable length scale as, the scattering length [14]. However, still an experimental realization of the BCS- to-BEC crossover had not been observed. Even though some tunability of the fermion density can be achieved through chemical doping there was no control over the interactions until the application of Feshbach resonance in ultra-cold atoms. Using magnetically driven Feshbach resonances the tuning of interactions became possible in ultracold Fermi atoms. Following this, in 2003, three experimental groups simultaneously succeeded in producing BEC of bound fermion molecules in optical traps at ultracold temperatures. Two groups, Rudolf Grimm's at the University of Innsbruck and Wolfgang Ketterle's at MIT, used Li-6 atoms, and a third group, Deborah Jin's at JILA, used K-40 [15, 16, 17]. And two years later, a direct evidence for the superfluidity came by the experimental detection of vortex lattice when the cloud of trapped fermions was first rotated and then allowed to expand [18]. 2 BCS theory and Superconductivity A theory of superconductivity formulated by John Bardeen, Leon Cooper, and Robert Schrieffer, explaining the phenomenon in which a current of electron pairs flows without resistance in certain materials at low tempera- tures. According to theory, superconductivity arises when a single negatively charged electron slightly distorts the lattice of atoms in the superconductor, drawing toward it a small excess of positive charge creating an electron-phonon pair. This excess, afterwards, attracts a second electron (electron-phonon interaction). It is this weak and indirect attraction that binds the electrons together, into a Cooper pair in momentum space as shown in the figure below. The Cooper pairs within the superconductor carry the current without any resistance, but why? Mathe- matically, because the Cooper pair is more stable than a single electron within the lattice, it experiences less resistance. Physically, the Cooper pair is more resistant to vibrations within the lattice as the attraction to its partner will keep it more stable. Hence, Cooper pairs move through the lattice relatively unaffected by thermal vibrations (phonons) below the critical temperature. The existence of a single fermion pair, however, is not sufficient to describe the macroscopic behavior of superconductors. It is necessary to invent a collective and correlated state in which many electron (fermion) pairs acting together to produce a zero-resistance state. For that, BCS theory proposes a many particle wave- function corresponding to largely overlapping fermion pairs with zero center of mass momentum, zero angular momentum (s-wave), and zero total spin (singlet) [2]. Moreover, one of the most fundamental features of the BCS state is the existence of correlations between fermion pairs, which lead to an order parameter 1=kF as, for the superconducting state. This order parameter is found to be directly related to the energy gap Eg in the elementary excitation spectrum. Because all relevant fermions participating in the ground state of an s-wave superconductor are paired, creating a single fermion excitation requires breaking a Cooper pair with a energy cost. Thus the contribution of elementary excitations to the specific heat and other thermodynamic properties shows exponential behavior ∼ exp(−Eg=T ) at low temperature [1]. BCS theory successfully applied to many experimental results and explained the physics behind the conventional superconductors until the discovery of cuprates. 3 From BCS to BEC superfluids Even though the BCS theory became successfully and widely applicable to many phenomena, it is basically a weak attraction theory. A generalization of the BCS theory has been developed to include the strong attraction regime in which fermion pairs become tightly bound diatomic Bose molecules and undergo Bose{Einstein con- densation. BEC state is on the the strong attraction side of the phase space and is formed by the condensation of bound fermions in real space known as molecular bosons. 2 The simplest conceptual picture of the BCS to BEC superfluidity evolution for s-wave (l = 0) pairing can be constructed for low fermion densities and short-ranged interactions that are much smaller than the average separation between fermions [2]. The essence of the evolution at zero temperature is indicated in the ratio of the size of fermion pairs to the average separation between the fermions. In the BCS regime, the attraction is weak, the pairs are much larger than their average separation, and they overlap substantially in the momentum space. In the BEC regime, the attraction is strong, the pairs are much smaller than their average separation, and they overlap very little in the real space. However the clear picture of evolution between these two regimes is understood at zero temperature, in 1980, when Leggett proposed a simple description in real space of paired fermions with opposite spins. Leggett considered a zero-ranged attractive potential (contact interaction) between fermions and showed that when the attraction is weak, a BCS superfluid appeared, and when the attraction is strong, a BEC superfluid emerged.
Load more

Recommended publications
  • Lecture Notes: BCS Theory of Superconductivity
    Lecture Notes: BCS theory of superconductivity Prof. Rafael M. Fernandes Here we will discuss a new ground state of the interacting electron gas: the superconducting state. In this macroscopic quantum state, the electrons form coherent bound states called Cooper pairs, which dramatically change the macroscopic properties of the system, giving rise to perfect conductivity and perfect diamagnetism. We will mostly focus on conventional superconductors, where the Cooper pairs originate from a small attractive electron-electron interaction mediated by phonons. However, in the so- called unconventional superconductors - a topic of intense research in current solid state physics - the pairing can originate even from purely repulsive interactions. 1 Phenomenology Superconductivity was discovered by Kamerlingh-Onnes in 1911, when he was studying the transport properties of Hg (mercury) at low temperatures. He found that below the liquifying temperature of helium, at around 4:2 K, the resistivity of Hg would suddenly drop to zero. Although at the time there was not a well established model for the low-temperature behavior of transport in metals, the result was quite surprising, as the expectations were that the resistivity would either go to zero or diverge at T = 0, but not vanish at a finite temperature. In a metal the resistivity at low temperatures has a constant contribution from impurity scattering, a T 2 contribution from electron-electron scattering, and a T 5 contribution from phonon scattering. Thus, the vanishing of the resistivity at low temperatures is a clear indication of a new ground state. Another key property of the superconductor was discovered in 1933 by Meissner.
    [Show full text]
  • The Superconductor-Metal Quantum Phase Transition in Ultra-Narrow Wires
    The superconductor-metal quantum phase transition in ultra-narrow wires Adissertationpresented by Adrian Giuseppe Del Maestro to The Department of Physics in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the subject of Physics Harvard University Cambridge, Massachusetts May 2008 c 2008 - Adrian Giuseppe Del Maestro ! All rights reserved. Thesis advisor Author Subir Sachdev Adrian Giuseppe Del Maestro The superconductor-metal quantum phase transition in ultra- narrow wires Abstract We present a complete description of a zero temperature phasetransitionbetween superconducting and diffusive metallic states in very thin wires due to a Cooper pair breaking mechanism originating from a number of possible sources. These include impurities localized to the surface of the wire, a magnetic field orientated parallel to the wire or, disorder in an unconventional superconductor. The order parameter describing pairing is strongly overdamped by its coupling toaneffectivelyinfinite bath of unpaired electrons imagined to reside in the transverse conduction channels of the wire. The dissipative critical theory thus contains current reducing fluctuations in the guise of both quantum and thermally activated phase slips. A full cross-over phase diagram is computed via an expansion in the inverse number of complex com- ponents of the superconducting order parameter (equal to oneinthephysicalcase). The fluctuation corrections to the electrical and thermal conductivities are deter- mined, and we find that the zero frequency electrical transport has a non-monotonic temperature dependence when moving from the quantum critical to low tempera- ture metallic phase, which may be consistent with recent experimental results on ultra-narrow MoGe wires. Near criticality, the ratio of the thermal to electrical con- ductivity displays a linear temperature dependence and thustheWiedemann-Franz law is obeyed.
    [Show full text]
  • The Development of the Science of Superconductivity and Superfluidity
    Universal Journal of Physics and Application 1(4): 392-407, 2013 DOI: 10.13189/ujpa.2013.010405 http://www.hrpub.org Superconductivity and Superfluidity-Part I: The development of the science of superconductivity and superfluidity in the 20th century Boris V.Vasiliev ∗Corresponding Author: [email protected] Copyright ⃝c 2013 Horizon Research Publishing All rights reserved. Abstract Currently there is a common belief that the explanation of superconductivity phenomenon lies in understanding the mechanism of the formation of electron pairs. Paired electrons, however, cannot form a super- conducting condensate spontaneously. These paired electrons perform disorderly zero-point oscillations and there are no force of attraction in their ensemble. In order to create a unified ensemble of particles, the pairs must order their zero-point fluctuations so that an attraction between the particles appears. As a result of this ordering of zero-point oscillations in the electron gas, superconductivity arises. This model of condensation of zero-point oscillations creates the possibility of being able to obtain estimates for the critical parameters of elementary super- conductors, which are in satisfactory agreement with the measured data. On the another hand, the phenomenon of superfluidity in He-4 and He-3 can be similarly explained, due to the ordering of zero-point fluctuations. It is therefore established that both related phenomena are based on the same physical mechanism. Keywords superconductivity superfluidity zero-point oscillations 1 Introduction 1.1 Superconductivity and public Superconductivity is a beautiful and unique natural phenomenon that was discovered in the early 20th century. Its unique nature comes from the fact that superconductivity is the result of quantum laws that act on a macroscopic ensemble of particles as a whole.
    [Show full text]
  • Introduction to Superconductivity Theory
    Ecole GDR MICO June, 2010 Introduction to Superconductivity Theory Indranil Paul [email protected] www.neel.cnrs.fr Free Electron System εεε k 2 µµµ Hamiltonian H = ∑∑∑i pi /(2m) - N, i=1,...,N . µµµ is the chemical potential. N is the total particle number. 0 k k Momentum k and spin σσσ are good quantum numbers. F εεε H = 2 ∑∑∑k k nk . The factor 2 is due to spin degeneracy. εεε 2 µµµ k = ( k) /(2m) - , is the single particle spectrum. nk T=0 εεεk/(k BT) nk is the Fermi-Dirac distribution. nk = 1/[e + 1] 1 T ≠≠≠ 0 The ground state is a filled Fermi sea (Pauli exclusion). µµµ 1/2 Fermi wave-vector kF = (2 m ) . 0 kF k Particle- and hole- excitations around kF have vanishingly 2 2 low energies. Excitation spectrum EK = |k – kF |/(2m). Finite density of states at the Fermi level. Ek 0 k kF Fermi Sea hole excitation particle excitation Metals: Nearly Free “Electrons” The electrons in a metal interact with one another with a short range repulsive potential (screened Coulomb). The phenomenological theory for metals was developed by L. Landau in 1956 (Landau Fermi liquid theory). This system of interacting electrons is adiabatically connected to a system of free electrons. There is a one-to-one correspondence between the energy eigenstates and the energy eigenfunctions of the two systems. Thus, for all practical purposes we will think of the electrons in a metal as non-interacting fermions with renormalized parameters, such as m →→→ m*. (remember Thierry’s lecture) CV (i) Specific heat (C V): At finite-T the volume of πππ 2∆∆∆ 2 ∆∆∆ excitations ∼∼∼ 4 kF k, where Ek ∼∼∼ ( kF/m) k ∼∼∼ kBT.
    [Show full text]
  • Electron-Phonon Interaction in Conventional and Unconventional Superconductors
    Electron-Phonon Interaction in Conventional and Unconventional Superconductors Pegor Aynajian Max-Planck-Institut f¨ur Festk¨orperforschung Stuttgart 2009 Electron-Phonon Interaction in Conventional and Unconventional Superconductors Von der Fakult¨at Mathematik und Physik der Universit¨at Stuttgart zur Erlangung der W¨urde eines Doktors der Naturwissenschaften (Dr. rer. nat.) genehmigte Abhandlung vorgelegt von Pegor Aynajian aus Beirut (Libanon) Hauptberichter: Prof. Dr. Bernhard Keimer Mitberichter: Prof. Dr. Harald Giessen Tag der m¨undlichen Pr¨ufung: 12. M¨arz 2009 Max-Planck-Institut f¨ur Festk¨orperforschung Stuttgart 2009 2 Deutsche Zusammenfassung Die Frage, ob ein genaueres Studium der Phononen-Spektren klassischer Supraleiter wie Niob und Blei mittels inelastischer Neutronenstreuung der M¨uhe wert w¨are, w¨urde sicher von den meisten Wissenschaftlern verneint werden. Erstens erk¨art die ber¨uhmte mikroskopische Theorie von Bardeen, Cooper und Schrieffer (1957), bekannt als BCS Theorie, nahezu alle Aspekte der klassischen Supraleitung. Zweitens ist das aktuelle Interesse sehr stark auf die Hochtemperatur-Supraleitung in Kupraten und Schwere- Fermionen Systemen fokussiert. Daher waren die ersten Experimente dieser Arbeit, die sich mit der Bestimmung der Phononen-Lebensdauern in supraleitendem Niob und Blei befaßten, nur als ein kurzer Test der Aufl¨osung eines neuen hochaufl¨osenden Neutronen- spektrometers am Forschungsreaktor FRM II geplant. Dieses neuartige Spektrometer TRISP (triple axis spin echo) erm¨oglicht die Bestimmung von Phononen-Linienbreiten uber¨ große Bereiche des Impulsraumes mit einer Energieaufl¨osung im μeV Bereich, d.h. zwei Gr¨oßenordnungen besser als an klassische Dreiachsen-Spektrometern. Philip Allen hat erstmals dargelegt, daß die Linienbreite eines Phonons proportional zum Elektron-Phonon Kopplungsparameter λ ist.
    [Show full text]
  • Sounds of a Supersolid A
    NEWS & VIEWS RESEARCH hypothesis came from extensive population humans, implying possible mosquito exposure long-distance spread of insecticide-resistant time-series analysis from that earlier study5, to malaria parasites and the potential to spread mosquitoes, worsening an already dire situ- which showed beyond reasonable doubt that infection over great distances. ation, given the current spread of insecticide a mosquito vector species called Anopheles However, the authors failed to detect resistance in mosquito populations. This would coluzzii persists locally in the dry season in parasite infections in their aerially sampled be a matter of great concern because insecticides as-yet-undiscovered places. However, the malaria vectors, a result that they assert is to be are the best means of malaria control currently data were not consistent with this outcome for expected given the small sample size and the low available8. However, long-distance migration other malaria vectors in the study area — the parasite-infection rates typical of populations of could facilitate the desirable spread of mosqui- species Anopheles gambiae and Anopheles ara- malaria vectors. A problem with this argument toes for gene-based methods of malaria-vector biensis — leaving wind-powered long-distance is that the typical infection rates they mention control. One thing is certain, Huestis and col- migration as the only remaining possibility to are based on one specific mosquito body part leagues have permanently transformed our explain the data5. (salivary glands), rather than the unknown but understanding of African malaria vectors and Both modelling6 and genetic studies7 undoubtedly much higher infection rates that what it will take to conquer malaria.
    [Show full text]
  • ANTIMATTER a Review of Its Role in the Universe and Its Applications
    A review of its role in the ANTIMATTER universe and its applications THE DISCOVERY OF NATURE’S SYMMETRIES ntimatter plays an intrinsic role in our Aunderstanding of the subatomic world THE UNIVERSE THROUGH THE LOOKING-GLASS C.D. Anderson, Anderson, Emilio VisualSegrè Archives C.D. The beginning of the 20th century or vice versa, it absorbed or emitted saw a cascade of brilliant insights into quanta of electromagnetic radiation the nature of matter and energy. The of definite energy, giving rise to a first was Max Planck’s realisation that characteristic spectrum of bright or energy (in the form of electromagnetic dark lines at specific wavelengths. radiation i.e. light) had discrete values The Austrian physicist, Erwin – it was quantised. The second was Schrödinger laid down a more precise that energy and mass were equivalent, mathematical formulation of this as described by Einstein’s special behaviour based on wave theory and theory of relativity and his iconic probability – quantum mechanics. The first image of a positron track found in cosmic rays equation, E = mc2, where c is the The Schrödinger wave equation could speed of light in a vacuum; the theory predict the spectrum of the simplest or positron; when an electron also predicted that objects behave atom, hydrogen, which consists of met a positron, they would annihilate somewhat differently when moving a single electron orbiting a positive according to Einstein’s equation, proton. However, the spectrum generating two gamma rays in the featured additional lines that were not process. The concept of antimatter explained. In 1928, the British physicist was born.
    [Show full text]
  • BCS Thermal Vacuum of Fermionic Superfluids and Its Perturbation Theory
    www.nature.com/scientificreports OPEN BCS thermal vacuum of fermionic superfuids and its perturbation theory Received: 14 June 2018 Xu-Yang Hou1, Ziwen Huang1,4, Hao Guo1, Yan He2 & Chih-Chun Chien 3 Accepted: 30 July 2018 The thermal feld theory is applied to fermionic superfuids by doubling the degrees of freedom of the Published: xx xx xxxx BCS theory. We construct the two-mode states and the corresponding Bogoliubov transformation to obtain the BCS thermal vacuum. The expectation values with respect to the BCS thermal vacuum produce the statistical average of the thermodynamic quantities. The BCS thermal vacuum allows a quantum-mechanical perturbation theory with the BCS theory serving as the unperturbed state. We evaluate the leading-order corrections to the order parameter and other physical quantities from the perturbation theory. A direct evaluation of the pairing correlation as a function of temperature shows the pseudogap phenomenon, where the pairing persists when the order parameter vanishes, emerges from the perturbation theory. The correspondence between the thermal vacuum and purifcation of the density matrix allows a unitary transformation, and we found the geometric phase associated with the transformation in the parameter space. Quantum many-body systems can be described by quantum feld theories1–4. Some available frameworks for sys- tems at fnite temperatures include the Matsubara formalism using the imaginary time for equilibrium systems1,5 and the Keldysh formalism of time-contour path integrals3,6 for non-equilibrium systems. Tere are also alterna- tive formalisms. For instance, the thermal feld theory7–9 is built on the concept of thermal vacuum.
    [Show full text]
  • Topological Superconductors, Majorana Fermions and Topological Quantum Computation
    Topological Superconductors, Majorana Fermions and Topological Quantum Computation 0. … from last time: The surface of a topological insulator 1. Bogoliubov de Gennes Theory 2. Majorana bound states, Kitaev model 3. Topological superconductor 4. Periodic Table of topological insulators and superconductors 5. Topological quantum computation 6. Proximity effect devices Unique Properties of Topological Insulator Surface States “Half” an ordinary 2DEG ; ¼ Graphene EF Spin polarized Fermi surface • Charge Current ~ Spin Density • Spin Current ~ Charge Density Berry’s phase • Robust to disorder • Weak Antilocalization • Impossible to localize Exotic States when broken symmetry leads to surface energy gap: • Quantum Hall state, topological magnetoelectric effect • Superconducting state Even more exotic states if surface is gapped without breaking symmetry • Requires intrinsic topological order like non-Abelian FQHE Surface Quantum Hall Effect Orbital QHE : E=0 Landau Level for Dirac fermions. “Fractional” IQHE 2 2 e xy 1 2h B 2 0 e 1 xy n -1 h 2 2 -2 e n=1 chiral edge state xy 2h Anomalous QHE : Induce a surface gap by depositing magnetic material † 2 2 Hi0 ( - v - DM z ) e e - Mass due to Exchange field 2h 2h M↑ M↓ e2 xyDsgn( M ) EF 2h TI Egap = 2|DM| Chiral Edge State at Domain Wall : DM ↔ -DM Topological Magnetoelectric Effect Qi, Hughes, Zhang ’08; Essin, Moore, Vanderbilt ‘09 Consider a solid cylinder of TI with a magnetically gapped surface M 2 e 1 J xy E n E M h 2 J Magnetoelectric Polarizability topological “q term” 2 DL EB E e 1 ME n e2 h 2 q 2 h TR sym.
    [Show full text]
  • Lecture Notes for Quantum Matter
    Lecture Notes for Quantum Matter MMathPhys c Professor Steven H. Simon Oxford University July 24, 2019 Contents 1 What we will study 1 1.1 Bose Superfluids (BECs, Superfluid He, Superconductors) . .1 1.2 Theory of Fermi Liquids . .2 1.3 BCS theory of superconductivity . .2 1.4 Special topics . .2 2 Introduction to Superfluids 3 2.1 Some History and Basics of Superfluid Phenomena . .3 2.2 Landau and the Two Fluid Model . .6 2.2.1 More History and a bit of Physics . .6 2.2.2 Landau's Two Fluid Model . .7 2.2.3 More Physical Effects and Their Two Fluid Pictures . .9 2.2.4 Second Sound . 12 2.2.5 Big Questions Remaining . 13 2.3 Curl Free Constraint: Introducing the Superfluid Order Parameter . 14 2.3.1 Vorticity Quantization . 15 2.4 Landau Criterion for Superflow . 17 2.5 Superfluid Density . 20 2.5.1 The Andronikoshvili Experiment . 20 2.5.2 Landau's Calculation of Superfluid Density . 22 3 Charged Superfluid ≈ Superconductor 25 3.1 London Theory . 25 3.1.1 Meissner-Ochsenfeld Effect . 27 3 3.1.2 Quantum Input and Superfluid Order Parameter . 29 3.1.3 Superconducting Vortices . 30 3.1.4 Type I and Type II superconductors . 32 3.1.5 How big is Hc ............................... 33 4 Microscopic Theory of Bosons 37 4.1 Mathematical Preliminaries . 37 4.1.1 Second quantization . 37 4.1.2 Coherent States . 38 4.1.3 Multiple orbitals . 40 4.2 BECs and the Gross-Pitaevskii Equation . 41 4.2.1 Noninteracting BECs as Coherent States .
    [Show full text]
  • How the Electron-Phonon Coupling Mechanism Work in Metal Superconductor
    How the electron-phonon coupling mechanism work in metal superconductor Qiankai Yao1,2 1College of Science, Henan University of Technology, Zhengzhou450001, China 2School of physics and Engineering, Zhengzhou University, Zhengzhou450001, China Abstract Superconductivity in some metals at low temperature is known to arise from an electron-phonon coupling mechanism. Such the mechanism enables an effective attraction to bind two mobile electrons together, and even form a kind of pairing system(called Cooper pair) to be physically responsible for superconductivity. But, is it possible by an analogy with the electrodynamics to describe the electron-phonon coupling as a resistivity-dependent attraction? Actually so, it will help us to explore a more operational quantum model for the formation of Cooper pair. In particularly, by the calculation of quantum state of Cooper pair, the explored model can provide a more explicit explanation for the fundamental properties of metal superconductor, and answer: 1) How the transition temperature of metal superconductor is determined? 2) Which metals can realize the superconducting transition at low temperature? PACS numbers: 74.20.Fg; 74.20.-z; 74.25.-q; 74.20.De ne is the mobile electron density, η the damping coefficient 1. Introduction that is determined by the collision time τ . In the BCS theory[1], superconductivity is attributed to a In metal environment, mobile electrons are usually phonon-mediated attraction between mobile electrons near modeled to be a kind of classical particles like gas molecules, Fermi surface(called Fermi electrons). The attraction is each of which performs a Brown-like motion and satisfies the sometimes referred to as a residual Coulomb interaction[2] that Langevin equation can glue Cooper pair together to cause superconductivity.
    [Show full text]
  • Theory of Superconductivity
    Theory of Superconductivity Kwon Park KIAS-SNU Physics Winter Camp Camp Winter KIAS-SNU Physics Phoenix Park Jan 20 – 27, 2013 Outline • Why care about superconductivity? • BCS theory as a trial wave function method • BCS theory as a mean-field theory • High-temperature superconductivity and strong correlation • Effective field theory for superconductivity: Ginzburg- Landau theory Family tree of strongly correlated electron systems Quantum magnetism Topological Mott insulator FQHE insulator HTSC Quantum Hall effect Superconductivity Wigner crystal Breakdown of the Landau-Fermi liquid Collective behavior of a staring crowd Disordered State Ordered State Superconductivity as an emergent phenomenon Superconducting phase coherence: Josephson effect • Cooper-pair box: An artificial two-level system composed of many superconducting electron pairs in a “box.” reservoir box - - - - + + + + gate Josephson effect in the Cooper-pair box Nakamura, Pashkin, Tsai, Nature 398, 786 (99) • Phase vs. number uncertainty relationship: When the phase gets coherent, the Cooper-pair number becomes uncertain, which is nothing but the Josephson effect. Devoret and Schoelkopf, Nature 406, 1039 (00) Meissner effect • mv-momentum = p-momentum − e/c × vector potential • current density operator = 2e × Cooper pair density × velocity operator • θ=0 for a coherent Cooper-pair condensate in a singly connected region.region London equation • The magnetic field is expelled from the inside of a superconductor: Meissner effect. London equation where Electromagnetic field, or wave are attenuated inside a superconductor, which means that, in quantum limit, photons become massive while Maxwell’s equations remain gauge-invariant! Anderson-Higgs mechanism • Quantum field theories should be renormalizable in order to produce physically meaningful predictions via systematic elimination of inherent divergences.
    [Show full text]