DOCSLIB.ORG

„Fermi-Bose Mixtures of 40K and 87Rb Atoms: Does a Bose Einstein Condensate Float in a Fermi Sea?" Klaus Sengstock

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
„Fermi-Bose Mixtures of 40K and 87Rb Atoms: Does a Bose Einstein Condensate Float in a Fermi Sea? Krynica, June 2005 Quantum Optics VI „Fermi-Bose mixtures of 40K and 87Rb atoms: Does a Bose Einstein condensate float in a Fermi sea?" Klaus Sengstock Mixtures of ultracold Bose- and Fermi-gases Bright Fermi-Bose solitons Dynamics of the system: e.g.: mean field driven collapse Universität Hamburg Institut für Laserphysik ColdCold QuantumQuantum GasGas GroupGroup HamburgHamburg Fermi-Bose-Mixture Spinor-BEC BEC ‘in Space‘ Atom-Guiding in PBF ColdCold QuantumQuantum GasGas GroupGroup HamburgHamburg Fermi-Bose-Mixture Spinor-BEC Poster by Silke Ospelkaus Poster by Jochen Kronjäger on Tuesday on Monday Bose-Einstein Condensation Bose-Einstein distribution 1 f (ε) = e(ε −µ )/ kT −1 critical temperature for BEC 1 S. N. Bose A. Einstein 3 kTc ≈ 0.94 hω N T>Tc T<Tc N0/N 1-(T/T )3 1 c Tc T Bose-Einstein Condensation Bose-Einstein distribution 1 f (ε) = High-temperature effect !!! e(ε −µ )/ kT −1 critical temperature for BEC 1 3 kTc ≈ 0.94 hω N T>Tc T<Tc N0/N 1-(T/T )3 1 c Tc T Fermions in a Harmonic Trap Fermi-Dirac distribution 1 f (ε) = ( ) / e ε −µ kT + 1 Fermi temperature 1 3 E. Fermi P.A.M. Dirac kTF ≈1,81 hω N T>T T=0 F f(ε) T=0 T~TF 1 εF T>TF εF ε Fermions in a Harmonic Trap Fermi-Dirac distribution 1 Quantum statistical effects also for f (ε) = ( ) / ε −µ kT + 1 T~TF, but more difficult to see... e Fermi temperature 1 3 kTF ≈1,81 hω N T>T T<T F F f(ε) T=0 T~TF 1 T>TF εF ε Fermionic Quantum Gases difficulty to reach low temperatures for Fermi gases: no s-wave scattering of identical fermions! Æ no thermalization in evaporative cooling a) Æ use different spin components (D. Jin et al. 98) b) Æ use e.g. a BEC to cool a Fermi sea (and look to the details...) thermal Bosons condensate fraction Fermions e.g.: Momentum Distributions of Fermions and Bosons P(p) P(p) T>>Tc,TF 0 p -pF 0 pF p P(p) P(p) T<Tc,TF p p 0 -pF 0 pF P(p) P(p) T<<Tc,TF p p 0 -pF 0 pF e.g.: Momentum Distributions of Fermions and Bosons P(p) P(p) T>>Tc,TF 0 p -pF 0 pF p P(p) P(p) T<Tc,TF p p 0 -pF 0 pF e.g.: Superfluidity in Quantum Gases: a) Bosons • drag free motion MIT C. Raman et al., PRL. 83, 2502-2505 (1999). • scissors modes Oxford O.M. Maragò et al., PRL 84, 2056 (2000) • vortices, vortex lattice JILA, ENS, MIT Image from: P. Engels and E. A. Cornell Superfluidity in Quantum Gases: b) Fermions Cooper pairs - BCS superfluidity T60 exponentially difficult to reach r k π − 2kF a TBCS ≈ 0.28TF e r − k (valid for kF|a|<<1) -4 e.g.: kFa=-0.2 -> TBCS ~ 10 TF (very very small) (very) low-temperature effect Superfluidity in Quantum Gases: b) Fermions ways out of it: manipulate TBCS using a Feshbach resonance BEC of molecules BEC/BCS crossover •Duke •ENS •Innsbruck •JILA •MIT •Rice use additional particles to mediate interactions - Bosons • ? ... ÆÆFermi-Bose Mixtures • boson mediated superfluidity L. Viverit, Phys. Rev. A 66, 023605 (2002) F. Matera, Phys. Rev. A 68, 043624 (2003) T. Swislocki, T. Karpiuk, M. Brewsczyk, Poster 1, Monday ... • boson mediated superfluidity in a lattice F. Illuminati and A. Albus, Phys. Rev. Lett. 93, 090406 (2004) ... Æ interplay between tunneling and various on-site-interactions Fermi-Bose Mixtures there is even more: • special interest: mixtures in optical lattices Æ new phases, composite particles, ... • composite fermions II M. Lewenstein et al., FD Phys. Rev. Lett. 92, 050401 (2004) 2 IISF M. Cramer et al., 1 IDM U IIFL Phys. Rev. Lett. 93, 190405 (2004) bf IFL Ubb 0 IIDM IIFL IDM -1 IISF IIFL . IIDM . -201 µb/Ubb Fermi-Bose Mixtures effective interactions: Bose-Bose int. Bose-Fermi int. (B) 2 N ∂ϕ 2 F 2 bosons h 2 (B) (B) (B) (B) (B) (F ) (B) ih = − ∇ ϕ + Vtrap ϕ + gB N B ϕ ϕ + gBF ∑ ϕi ϕ , ∂t 2m i =1 (F ) 2 ∂ϕ 2 fermions j h 2 (F ) (F ) (F ) (B) (F ) ih = − ∇ ϕ j + Vtrap ϕ j + gBF N B ϕ ϕ j ∂t 2m new degrees of freedom due to additional interactions e.g.: 40K - 87Rb mixture: gB > 0 (aBB ~ 100 a0) gBF < 0 (aBF ~ -280 a0) tunable by Feshbach resonances! S. Inouye et al., PRL 93, 183201 (2004) see also: G. Modugno et al., Science 297, 2240 (2002) Fermi-Bose Mixtures ÆÆ detailed understanding of interactions and also of loss processes is necessary Bose-FermiBose-Fermi interactioninteraction physicsphysics --ssystemystembboundaryoundary conditionsconditions --ccoupledoupled excitationsexcitations (e.g.(e.g.eexp.xp. in inJinJin g group,roup, JILA JILA and andInguscioInguscio group, group, LENS) LENS) --BBose-Fermiose-Fermi interactionsinteractions --interspeciesinterspecies correlationscorrelations --nnovelovel phasesphases --hheteronucleareteronuclear moleculesmolecules 6Li/7Li at Duke U., ENS Paris, Innsbruck U., Rice U. 6Li/23Na at MIT 40K/87Rb at LENS Florence, Jila Boulder, Hamburg U., ETH Zürich Hamburg Setup two-species 2D-MOT flux: 87Rb ~ 5 · 109 s-1 40K ~ 5·106 s-1 two-species 3D-MOT Rb ~ 1010 K ~ 3·107 within 10..20 s in addition: dipole trap magnetic trap νax ~ 11 Hz (Rb) νrad ~ 260 Hz (Rb) soon: optical lattice Hamburg Setup Mai 2003 laser systems experimental setup first BEC 7/2004 first degenerate Fermi gas 8/2004 Sympathetic Cooling state of the art 76 5x10 Li at T~0.05TF (temperature): 640 1x10 K at T~0.15TF (for K-Rb cooling) νax=11Hz, νr=330Hz ν =11Hz, ν =267Hz state of the art ax r (particle numbers): f K-atoms o only BEC: >5*106 only Fermions: >1*106 number number of Rb-atoms Attractive Boson-Fermion Interaction aK-Rb ~ -279 a0 Æeffective potential for fermions: + = BEC experimental signatures: Fermion cloud without BEC Fermion cloud with BEC Mean Field Instability of the System BEC BEC attraction of fermions Fermi-Sea BEC density increase collapsecollapse runaway Collapse Experiments 7Li collapse Sackett et al., PRL 82, 876 (1999) J.M. Gerton et al., Nature 8, 692 (2000) 85Rb "Bosenova" Donley et al., Nature 412, 295 (2001) Images from: http://spot.colorado.edu/~cwieman/Bosenova.html 40K / 87Rb Fermi-Bose collapse G. Modugno et al., Science 297, 2240 (2002) Fermi-Bose Mixtures in the Large Particle Limit: Local Collapse Dynamics Fermi-Bose Mixtures in the Large Particle Limit: Collapse but...: is it just losses?? Æ locally high density: enhanced two- and three-body losses?? Lifetime Regimes τ = 197ms τ = 21ms time/frequency scales: -> collapse-time - ν (K) = 394 Hz 3-body-loss r due to trap dynamics - νax(K) = 17 Hz - thermalization 10..50 ms - collapse: ~ 20 ms - loss processes 100..200 ms loss and collapse dynamics can be distinguished! 3-Body Losses measurement of the 3-body KRb decay rate model for 3-body inelastic N K 1 K K Rb Rb 3 2 decay in thermal mixture: d r nB r,t nF r,t N K N K 3 2 T T d rnB r,t nF r,t integration over time: ln N T ln N 0 K dt K K K Rb Rb 0 N K t T ln N T ln N 0 K K Result: 0 cm 6 K ( 3.5 +/- 0.2) 10 28 -0.5 KRbRb s -1 Measurement does not depend on K atom number calibration -1.5 For 87Rb |2,2> decay, we reproduce the value from Söding et al. -2 [Appl. Phys. B69, 257 (1999)] -2.5 0 20 40 60 80 100 120 140 160 180 3 2 T d rnB r,t nF r,t 38 6 0 dt 10 m s N K t Fermi-Bose Mixtures in the Large Particle Limit: NBoson Stability Diagram stable mixture non stable mixture aKRb=-281 a0 (S. Inouye et al., PRL 93, 183201 (2004) NFermion Does a Bose Einstein condensate float in a Fermi sea? ... it depends ... Solitons in Matter Waves g>0 g<0 dark solitons filled solitons bright solitons quantum pressure K.S. Strecker et al., Nature 417, 150 (2002) interactions B. P. Anderson et al., PRL 86, 2926 (2001) gap solitons "negative mass" L. Khaykovich et al., Science 296, 1290 (2002) 4 NSoliton< 10 S. Burger et al., PRL 83, 5198 (1999) quasi-1D regime collapse for Eint>Eradial J. Denschlag et al., Science 287, 97 (2000) B. Eiermann et al. PRL 92, 230401(2004) 1D: Bright Mixed ‘‘Solitons‘‘ cr Bose-Bose repulsion versus Fermi-Bose attraction gBnB = gBF nF after switching behaviour in off the trap: cr the trap: g BF < g BF our data theory cr g BF > g BF theory by T. Karpiuk, M. Brewczyk, M. Gaida, K. Rzazewski dynamics: constant envelope 9 simulation from M. Brewczyk et al. T. Karpiuk, M. Brewczyk, S. Ospelkaus-Schwarzer, K. Bongs, M. Gajda, and K. Rzążewski, PRL 93, 100401 (2004) Collision simulation shows complex dynamics: - repulsive - shape oscillations - particle exchange Simulation from M. Brewczyk et al. fermionic character due to the Pauli-principle ? Bose-Fermi Mixtures with Attractive Interactions Physics in the High Density Limit effective interaction ("density") bright collapse mixed soliton attractive boson-induced BCS ? repulsive trap aspect ratio InfluenceInfluenceooffllossoss processesprocesses?? HamburgHamburg TeamTeam K. Se Kai Bongs - Atom optics V. M. Baev - Fibre lasers Spinor BEC: Jochen Kronjäger Stefan Salewski Christoph Becker Ortwin Hellmig Thomas Garl Arnold Stark Martin Brinkmann Sergej Wexler Fermi-Bose mixtures K-Rb: Oliver Back Silke Ospelkaus-Schwarzer Gerald Rapior Christian Ospelkaus Philipp Ernst Oliver Wille Q.
Load more

Recommended publications
  • Competition Between Paramagnetism and Diamagnetism in Charged
    Competition between paramagnetism and diamagnetism in charged Fermi gases Xiaoling Jian, Jihong Qin, and Qiang Gu∗ Department of Physics, University of Science and Technology Beijing, Beijing 100083, China (Dated: November 10, 2018) The charged Fermi gas with a small Lande-factor g is expected to be diamagnetic, while that with a larger g could be paramagnetic. We calculate the critical value of the g-factor which separates the dia- and para-magnetic regions. In the weak-field limit, gc has the same value both at high and low temperatures, gc = 1/√12. Nevertheless, gc increases with the temperature reducing in finite magnetic fields. We also compare the gc value of Fermi gases with those of Boltzmann and Bose gases, supposing the particle has three Zeeman levels σ = 1, 0, and find that gc of Bose and Fermi gases is larger and smaller than that of Boltzmann gases,± respectively. PACS numbers: 05.30.Fk, 51.60.+a, 75.10.Lp, 75.20.-g I. INTRODUCTION mK and 10 µK, respectively. The ions can be regarded as charged Bose or Fermi gases. Once the quantum gas Magnetism of electron gases has been considerably has both the spin and charge degrees of freedom, there studied in condensed matter physics. In magnetic field, arises the competition between paramagnetism and dia- the magnetization of a free electron gas consists of two magnetism, as in electrons. Different from electrons, the independent parts. The spin magnetic moment of elec- g-factor for different magnetic ions is diverse, ranging trons results in the paramagnetic part (the Pauli para- from 0 to 2.
    [Show full text]
  • Solid State Physics 2 Lecture 5: Electron Liquid
    Physics 7450: Solid State Physics 2 Lecture 5: Electron liquid Leo Radzihovsky (Dated: 10 March, 2015) Abstract In these lectures, we will study itinerate electron liquid, namely metals. We will begin by re- viewing properties of noninteracting electron gas, developing its Greens functions, analyzing its thermodynamics, Pauli paramagnetism and Landau diamagnetism. We will recall how its thermo- dynamics is qualitatively distinct from that of a Boltzmann and Bose gases. As emphasized by Sommerfeld (1928), these qualitative di↵erence are due to the Pauli principle of electons’ fermionic statistics. We will then include e↵ects of Coulomb interaction, treating it in Hartree and Hartree- Fock approximation, computing the ground state energy and screening. We will then study itinerate Stoner ferromagnetism as well as various response functions, such as compressibility and conduc- tivity, and screening (Thomas-Fermi, Debye). We will then discuss Landau Fermi-liquid theory, which will allow us understand why despite strong electron-electron interactions, nevertheless much of the phenomenology of a Fermi gas extends to a Fermi liquid. We will conclude with discussion of electrons on the lattice, treated within the Hubbard and t-J models and will study transition to a Mott insulator and magnetism 1 I. INTRODUCTION A. Outline electron gas ground state and excitations • thermodynamics • Pauli paramagnetism • Landau diamagnetism • Hartree-Fock theory of interactions: ground state energy • Stoner ferromagnetic instability • response functions • Landau Fermi-liquid theory • electrons on the lattice: Hubbard and t-J models • Mott insulators and magnetism • B. Background In these lectures, we will study itinerate electron liquid, namely metals. In principle a fully quantum mechanical, strongly Coulomb-interacting description is required.
    [Show full text]
  • Lecture 24. Degenerate Fermi Gas (Ch
    Lecture 24. Degenerate Fermi Gas (Ch. 7) We will consider the gas of fermions in the degenerate regime, where the density n exceeds by far the quantum density nQ, or, in terms of energies, where the Fermi energy exceeds by far the temperature. We have seen that for such a gas μ is positive, and we’ll confine our attention to the limit in which μ is close to its T=0 value, the Fermi energy EF. ~ kBT μ/EF 1 1 kBT/EF occupancy T=0 (with respect to E ) F The most important degenerate Fermi gas is 1 the electron gas in metals and in white dwarf nε()(),, T= f ε T = stars. Another case is the neutron star, whose ε⎛ − μ⎞ exp⎜ ⎟ +1 density is so high that the neutron gas is ⎝kB T⎠ degenerate. Degenerate Fermi Gas in Metals empty states ε We consider the mobile electrons in the conduction EF conduction band which can participate in the charge transport. The band energy is measured from the bottom of the conduction 0 band. When the metal atoms are brought together, valence their outer electrons break away and can move freely band through the solid. In good metals with the concentration ~ 1 electron/ion, the density of electrons in the electron states electron states conduction band n ~ 1 electron per (0.2 nm)3 ~ 1029 in an isolated in metal electrons/m3 . atom The electrons are prevented from escaping from the metal by the net Coulomb attraction to the positive ions; the energy required for an electron to escape (the work function) is typically a few eV.
    [Show full text]
  • Chapter 13 Ideal Fermi
    Chapter 13 Ideal Fermi gas The properties of an ideal Fermi gas are strongly determined by the Pauli principle. We shall consider the limit: k T µ,βµ 1, B � � which defines the degenerate Fermi gas. In this limit, the quantum mechanical nature of the system becomes especially important, and the system has little to do with the classical ideal gas. Since this chapter is devoted to fermions, we shall omit in the following the subscript ( ) that we used for the fermionic statistical quantities in the previous chapter. − 13.1 Equation of state Consider a gas ofN non-interacting fermions, e.g., electrons, whose one-particle wave- functionsϕ r(�r) are plane-waves. In this case, a complete set of quantum numbersr is given, for instance, by the three cartesian components of the wave vector �k and thez spin projectionm s of an electron: r (k , k , k , m ). ≡ x y z s Spin-independent Hamiltonians. We will consider only spin independent Hamiltonian operator of the type ˆ 3 H= �k ck† ck + d r V(r)c r†cr , �k � where thefirst and the second terms are respectively the kinetic and th potential energy. The summation over the statesr (whenever it has to be performed) can then be reduced to the summation over states with different wavevectork(p=¯hk): ... (2s + 1) ..., ⇒ r � �k where the summation over the spin quantum numberm s = s, s+1, . , s has been taken into account by the prefactor (2s + 1). − − 159 160 CHAPTER 13. IDEAL FERMI GAS Wavefunctions in a box. We as- sume that the electrons are in a vol- ume defined by a cube with sidesL x, Ly,L z and volumeV=L xLyLz.
    [Show full text]
  • Electron-Electron Interactions(Pdf)
    Contents 2 Electron-electron interactions 1 2.1 Mean field theory (Hartree-Fock) ................ 3 2.1.1 Validity of Hartree-Fock theory .................. 6 2.1.2 Problem with Hartree-Fock theory ................ 9 2.2 Screening ..................................... 10 2.2.1 Elementary treatment ......................... 10 2.2.2 Kubo formula ............................... 15 2.2.3 Correlation functions .......................... 18 2.2.4 Dielectric constant ............................ 19 2.2.5 Lindhard function ............................ 21 2.2.6 Thomas-Fermi theory ......................... 24 2.2.7 Friedel oscillations ............................ 25 2.2.8 Plasmons ................................... 27 2.3 Fermi liquid theory ............................ 30 2.3.1 Particles and holes ............................ 31 2.3.2 Energy of quasiparticles. ....................... 36 2.3.3 Residual quasiparticle interactions ................ 38 2.3.4 Local energy of a quasiparticle ................... 42 2.3.5 Thermodynamic properties ..................... 44 2.3.6 Quasiparticle relaxation time and transport properties. 46 2.3.7 Effective mass m∗ of quasiparticles ................ 50 0 Reading: 1. Ch. 17, Ashcroft & Mermin 2. Chs. 5& 6, Kittel 3. For a more detailed discussion of Fermi liquid theory, see G. Baym and C. Pethick, Landau Fermi-Liquid Theory : Concepts and Ap- plications, Wiley 1991 2 Electron-electron interactions The electronic structure theory of metals, developed in the 1930’s by Bloch, Bethe, Wilson and others, assumes that electron-electron interac- tions can be neglected, and that solid-state physics consists of computing and filling the electronic bands based on knowldege of crystal symmetry and atomic valence. To a remarkably large extent, this works. In simple compounds, whether a system is an insulator or a metal can be deter- mined reliably by determining the band filling in a noninteracting cal- culation.
    [Show full text]
  • Schedule for Rest of Quarter • Review: 3D Free Electron Gas • Heat Capacity
    Lecture 16 Housekeeping: schedule for rest of quarter Review: 3D free electron gas Heat capacity of free electron gas Semi classical treatment of electric conductivity Review: 3D fee electron gas This model ignores the lattice and only treats the electrons in a metal. It assumes that electrons do not interact with each other except for pauli exclusion. 풌∙풓 Wavelike-electrons defined by their momentum eigenstate k: 휓풌(풓) = 푒 Treat electrons like quantum particle in a box: confined to a 3D box of dimensions L in every direction Periodic boundary conditions on wavefunction: 휓(푥 + 퐿, 푦, 푧) = 휓(푥, 푦, 푧) 휓(푥, 푦 + 퐿, 푧) = 휓(푥, 푦, 푧) 휓(푥, 푦, 푧 + 퐿) = 휓(푥, 푦, 푧) Permissible values of k of the form: 2휋 4휋 푘 = 0, ± , ± , … 푥 퐿 퐿 Free electron schrodinger’s eqn in 3D + boundary conditions gives energy eigenvalues: ℏ2 휕2 휕2 휕2 − ( + + ) 휓 (푟) = 휖 휓 (푟) 2푚 휕푥2 휕푦2 휕푧2 푘 푘 푘 ℏ2 ℏ2푘2 (푘2 + 푘2 + 푘2) = 휖 = 2푚 푥 푦 푧 푘 2푚 There are N electrons and we put them into available eigenstates following these rules: o Every state is defined by a unique quantized value of (푘푥, 푘푦, 푘푧) o Every state can hold one spin up and one spin down electron o Assume that single-electron solution above applies to collection of electrons (i.e. electrons do not interact and change the eigenstates) o Fill low energy states first. In 3D, this corresponds to filling up a sphere in k space, one 2 2 2 2 ‘shell’ at a time.
    [Show full text]
  • Chapter 6 Free Electron Fermi
    Chapter 6 Free Electron Fermi Gas Free electron model: • The valence electrons of the constituent atoms become conduction electrons and move about freely through the volume of the metal. • The simplest metals are the alkali metals– lithium, sodium, potassium, Na, cesium, and rubidium. • The classical theory had several conspicuous successes, notably the derivation of the form of Ohm’s law and the relation between the electrical and thermal conductivity. • The classical theory fails to explain the heat capacity and the magnetic susceptibility of the conduction electrons. M = B • Why the electrons in a metal can move so freely without much deflections? (a) A conduction electron is not deflected by ion cores arranged on a periodic lattice, because matter waves propagate freely in a periodic structure. (b) A conduction electron is scattered only infrequently by other conduction electrons. Pauli exclusion principle. Free Electron Fermi Gas: a gas of free electrons subject to the Pauli Principle ELECTRON GAS MODEL IN METALS Valence electrons form the electron gas eZa -e(Za-Z) -eZ Figure 1.1 (a) Schematic picture of an isolated atom (not to scale). (b) In a metal the nucleus and ion core retain their configuration in the free atom, but the valence electrons leave the atom to form the electron gas. 3.66A 0.98A Na : simple metal In a sea of conduction of electrons Core ~ occupy about 15% in total volume of crystal Classical Theory (Drude Model) Drude Model, 1900AD, after Thompson’s discovery of electrons in 1897 Based on the concept of kinetic theory of neutral dilute ideal gas Apply to the dense electrons in metals by the free electron gas picture Classical Statistical Mechanics: Boltzmann Maxwell Distribution The number of electrons per unit volume with velocity in the range du about u 3/2 2 fB(u) = n (m/ 2pkBT) exp (-mu /2kBT) Success: Failure: (1) The Ohm’s Law , (1) Heat capacity Cv~ 3/2 NKB the electrical conductivity The observed heat capacity is only 0.01, J = E , = n e2 / m, too small.
    [Show full text]
  • Fermi and Bose Gases
    Fermi and Bose gases Anders Malthe-Sørenssen 25. november 2013 1 1 Thermodynamic identities and Legendre transforms 2 Fermi-Dirac and Bose-Einstein distribution When we discussed the ideal gas we assumed that quantum effects were not important. This was implied when we introduced the term 1=N! for the partition function for the ideal gas, because this term only was valid in the limit when the number of states are many compared to the number of particles. Now, we will address the general case. We start from an individual particle. For a single particle, we address its orbitals: Its possible quantum states, each of which corresponds to a solution of the Schrodinger equation for a single particle. For many particles we will address how the “fill up" the various orbitals { where the underlying assump- tion is that we can treat a many-particle system as a system with the same energy levels as for a single particular system, that is as a system with no interactions between the states of the various particles. However, the number of particles that can occupy the same state depends on the type of particle: For Fermions (1/2 spin) Only 0 or 1 fermion can be in a particular state. For Bosons (integer spin) Any number of bosons can be in a particular state. We will now apply a Gibbs ensemble approach to study this system. This means that we will consider a system where µ, V; T is constant. We can use the Gibbs sum formalism to address the behavior of this system.
    [Show full text]
  • One-Dimensional Multicomponent Fermions with Δ-Function Interaction in Strong- and Weak-Coupling Limits: Κ-Component Fermi Gas
    PHYSICAL REVIEW A 85, 033633 (2012) One-dimensional multicomponent fermions with δ-function interaction in strong- and weak-coupling limits: κ-component Fermi gas Xi-Wen Guan,1 Zhong-Qi Ma,2 and Brendan Wilson1 1Department of Theoretical Physics, Research School of Physics and Engineering, Australian National University, Canberra ACT 0200, Australia 2Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China (Received 23 January 2012; published 26 March 2012) We derive the first few terms of the asymptotic expansion of the Fredholm equations for one-dimensional κ- component fermions with repulsive and attractive δ-function interaction in strong- and weak-coupling regimes. We thus obtain a highly accurate result for the ground-state energy of a multicomponent Fermi gas with polarization for these regimes. This result provides a unified description of the ground-state properties of the Fermi gas with higher spin symmetries. However, in contrast to the two-component Fermi gas, there does not exist a mapping that can unify the two sets of Fredholm equations as the interacting strength vanishes. Moreover, we find that the local pair-correlation functions shed light on the quantum statistic effects of the κ-component interacting fermions. For the balanced spin case with repulsive interaction, the ground-state energy obtained confirms Yang and You’s result [Chin. Phys. Lett. 28, 020503 (2011)] that the energy per particle as κ →∞is the same as for spinless Bosons. DOI: 10.1103/PhysRevA.85.033633 PACS number(s): 03.75.Ss, 03.75.Hh, 02.30.Ik, 34.10.+x I. INTRODUCTION Although the model was solved long ago by Sutherland [4], solutions of the Fredholm equations for the model are far from The recently established experimental control over the being thoroughly investigated except for the two-component effective spin-spin interaction between atoms is opening up Fermi gas [16].
    [Show full text]
  • Ideal Quantum Gases from Statistical Physics Using Mathematica © James J
    Ideal Quantum Gases from Statistical Physics using Mathematica © James J. Kelly, 1996-2002 The indistinguishability of identical particles has profound effects at low temperatures and/or high density where quantum mechanical wave packets overlap appreciably. The occupation representation is used to study the statistical mechanics and thermodynamics of ideal quantum gases satisfying Fermi- Dirac or Bose-Einstein statistics. This notebook concentrates on formal and conceptual developments, liberally quoting more technical results obtained in the auxiliary notebooks occupy.nb, which presents numerical methods for relating the chemical potential to density, and fermi.nb and bose.nb, which study the mathematical properties of special functions defined for Ferm-Dirac and Bose-Einstein systems. The auxiliary notebooks also contain a large number of exercises. Indistinguishability In classical mechanics identical particles remain distinguishable because it is possible, at least in principle, to label them according to their trajectories. Once the initial position and momentum is determined for each particle with the infinite precision available to classical mechanics, the swarm of classical phase points moves along trajectories which also can in principle be determined with absolute certainty. Hence, the identity of the particle at any classical phase point is connected by deterministic relationships to any initial condition and need not ever be confused with the label attached to another trajectory. Therefore, in classical mechanics even identical particles are distinguishable, in principle, even though we must admit that it is virtually impossible in practice to integrate the equations of motion for a many-body system with sufficient accuracy. Conversely, in quantum mechanics identical particles are absolutely indistinguishable from one another.
    [Show full text]
  • 8.044 Lecture Notes Chapter 9: Quantum Ideal Gases
    8.044 Lecture Notes Chapter 9: Quantum Ideal Gases Lecturer: McGreevy 9.1 Range of validity of classical ideal gas . 9-2 9.2 Quantum systems with many indistinguishable particles . 9-4 9.3 Statistical mechanics of N non-interacting (indistinguishable) bosons or fermions 9-9 9.4 Classical ideal gas limit of quantum ideal gas . 9-13 2 9.5 Ultra-high temperature (kBT mc and kBT µ) limit . 9-15 9.6 Fermions at low temperatures . 9-18 9.7 Bosons at low temperatures . 9-33 Reading: Baierlein, Chapters 8 and 9. 9-1 9.1 Range of validity of classical ideal gas For a classical ideal gas, we derived the partition function N 3=2 Z1 V 2πmkBT Z = ;Z1 = 3 = V 2 ; N! λth h h where the length scale λth ≡ is determined by the particle mass and the temperature. 2πmkB T When does this break down? 1. If `idealness' fails, i.e. if interactions become important. This is interesting and impor- tant but out of bounds for 8.044. We'll still assume non-interacting particles. 2. If `classicalness' fails. Even with no interactions, at low enough temperatures, or high enough densities, quantum effects modify Z. An argument that classicalness must fail comes by thinking harder about λth, the \thermal de Broglie wavelength". Why is it called that? Recall from 8.04 that the de Broglie wavelength h for a particle with momentum p is λdB = p . For a classical ideal gas, we know the RMS momentum from equipartition ~p2 3 p h i = k T =) p ≡ ph~p2i = 3mk T: 2m 2 B RMS B We could have defined ~ h h λth ≡ = p ; pRMS 3mkBT there's no significance to the numerical prefactor of p1 instead of p1 .
    [Show full text]
  • Density of States
    Density of States Franz Utermohlen February 13, 2018 Contents 1 Preface 1 2 Summation definition 2 2.1 Examples of Eqs. 2 and 3 for a fermionic system . 2 2.2 Example: Ideal Fermi gas . 3 3 Derivative definition 4 3.1 Example: Ideal Fermi gas . 4 4 Useful expressions 6 1 Preface In these notes, the expressions Vd denotes the volume of a d-dimensional sphere of radius R and Ωd−1 denotes the solid angle a d-dimensional sphere (or equivalently, the surface area of a d-dimensional sphere of unit radius).1 These expressions are given by d=2 d=2 π d 2π Vd = d R ; Ωd−1 = d : (1) Γ(1 + 2 ) Γ( 2 ) For example, for d = 1; 2; and 3, these are d = 1 : V1 = 2R; Ω0 = 2 ; 2 d = 2 : V2 = πR ; Ω1 = 2π ; 4 d = 3 : V = πR3 ; Ω = 4π : 3 3 2 1The d − 1 subscript refers to the fact that the fact that the surface area of a d-dimensional sphere is (d − 1)-dimensional. It's also useful to use d − 1 as the subscript, since the surface area of a d-dimensional d−1 sphere of radius R is given by Sd−1 = Ωd−1R . 1 2 Summation definition Intensive quantities A can generally be expressed in the form 1 X A = a( ) ; (2) V i d i where Vd is the d-dimensional volume of the system, the sum is over all possible single- particle states i, and i is the energy of the single-particle state i.
    [Show full text]