Statistical met h od s for boson s

Statistical methodsfor bosons Lecture 1. 19th December 2012 by Bedřich Velický

Short version of the lecture plan Lecture 1 Lecture 2 Introductory matter BEC in extended non-interacting systems, ODLRO Atomic clouds in the traps; Confined independent bosons, what is BEC? Atom-atom interactions, Fermi pseudopotential; Gross-Pitaevski equation for extended gas and a trap Infinite systems: Bogolyubov-de Gennes theory, BEC and symmetry breaking, coherent states Dec 19 Jan 9 2

The classes will turn around the Bose-Einstein condensation in cold atomic clouds comparatively novel area of research largely tractable using the mean field approximation to describe the interactions only the basic early work will be covered, the recent progress is beyond the scope

Nobelists The Nobel Prize in Physics 2001 "for the achievement of Bose-Einstein condensation in dilute gases of alkali atoms, and for early fundamental studies of the properties of the condensates"

I.Introductory matter on bosons

Bosons and Fermions (capsule reminder) independent quantum postulate Identical particles are indistinguishable

Bosons and Fermions (capsule reminder) independent quantum postulate Identical particles are indistinguishable Cannot be labelled or numbered

Bosons and Fermions (capsule reminder) independent quantum postulate Identical particles are indistinguishable Cannot be labelled or numbered Permuting particles does not lead to a different state

Bosons and Fermions (capsule reminder) independent quantum postulate Identical particles are indistinguishable Permuting particles does not lead to a different state

Bosons and Fermions (capsule reminder) independent quantum postulate Identical particles are indistinguishable Permuting particles does not lead to a different state comes from nowhere "empirical fact"

Bosons and Fermions (capsule reminder) independent quantum postulate Identical particles are indistinguishable Finds justification in the relativistic quantum field theory Permuting particles does not lead to a different state comes from nowhere "empirical fact"

Bosons and Fermions (capsule reminder) Independent particles (… non-interacting) basis of single-particle states (  complete set of quantum numbers)

Bosons and Fermions (capsule reminder) Independent particles (… non-interacting) basis of single-particle states (  complete set of quantum numbers) FOCK SPACE Hilbert space of many particle states basis states … symmetrized products of single-particle states for bosons … antisymmetrized products of single-particle states for fermions specified by the set of occupation numbers 0, 1, 2, 3, … for bosons 0, 1 … for fermions

Bosons and Fermions (capsule reminder) Representation of occupation numbers (basically, second quantization) …. for fermions Pauli principle fermionskeep apart– as sea-gulls

Bosons and Fermions (capsule reminder) Representation of occupation numbers (basically, second quantization) …. for bosons princip identity bosonsprefer to keep close – like monkeys

Who are bosons ? elementary particles quasiparticles complex massive particles, like atoms … compound bosons

Examples of bosons bosons complex particles Nconserved simple particles Nnot conserved elementary particles atomic nuclei atoms photons quasi particles phonons magnons excited atoms

Examples of bosons (extension of the table) bosons complex particles Nconserved simple particles Nnot conserved elementary particles atomic nuclei atoms photons quasi particles phonons magnons excited atoms compound quasi particles ions excitons Cooper pairs molecules

Question: How a complex particle, like an atom, can behave as a single whole, a compound boson • ESSENTIAL CONDITIONS • All compound particles in the ensemble must be identical; the identity includes • detailed elementary particle composition • characteristics like mass, charge or spin • The total spin must have an integer value • The identity requirement extends also on the values of observables corresponding to internal degrees of freedom • which are not allowed to vary during the dynamical processes in question • The system of the compound bosons must be dilute enough to make the exchange effects between the component particles unimportant and absorbed in an effective weak short range interaction between the bosons as a whole

Example: How a complex particle, like an atom, can behave as a single whole, a compound boson RUBIDIUM -- THE FIRST ATOMIC CLOUD TO UNDERGO BEC • single element Z = 37 • single isotope A = 87 • single electron configuration

Example: How a complex particle, like an atom, can behave as a single whole, a compound boson RUBIDIUM -- THE FIRST ATOMIC CLOUD TO UNDERGO BEC • single element Z = 37 • single isotope A = 87 • single electron configuration • 37 electrons • 37 protons • 50 neutrons • total spin of the atom decides total electron spin total nuclear spin Two distinguishable species coexist; can be separated by joint effect of the hyperfine interaction and of the Zeeman splitting in a magnetic field

Atomic radius vs. interatomic distance in the cloud http://intro.chem.okstate.edu/1314f00/lecture/chapter7/lec111300.html

II.Homogeneous gas of non-interacting bosons The basic system exhibiting the Bose-Einstein Condensation (BEC) original case studied by Einstein

Plane waves in a cavity Plane wave in classical terms and its quantum transcription Discretization ("quantization") of wave vectors in the cavity volume periodic boundary conditions Cell size (per k vector) Cell size (per p vector) In the (r, p)-phase space

Density of states IDOS Integrated Density Of States: How many states have energy less than  Invert the dispersion law Find the volume of the d-sphere in the p-space Divide by the volume of the cell DOS Density Of States: How many states are around  per unit energy per unit volume

mean occupation number of a one-particle state with energy  Ideal quantum gases at a finite temperature: a reminder

mean occupation number of a one-particle state with energy  Ideal quantum gases at a finite temperature CLASSICAL LIMIT 34

mean occupation number of a one-particle state with energy  Ideal quantum gases at a finite temperature fermions bosons N N FD BE

Ideal quantum gases at a finite temperature mean occupation number of a one-particle state with energy  fermions bosons N N FD BE 36

mean occupation number of a one-particle state with energy  Ideal quantum gases at a finite temperature fermions bosons N N FD BE

mean occupation number of a one-particle state with energy  Ideal quantum gases at a finite temperature fermions bosons N N FD BE Aufbau principle

mean occupation number of a one-particle state with energy  Ideal quantum gases at a finite temperature fermions bosons N N FD BE freezing out Aufbau principle

mean occupation number of a one-particle state with energy  Ideal quantum gases at a finite temperature fermions bosons N N FD BE ? freezing out Aufbau principle

mean occupation number of a one-particle state with energy  Ideal quantum gases at a finite temperature fermions bosons N N FD BE BEC? freezing out Aufbau principle

Bose-Einstein condensation: elementary approach

Einstein'smanuscript with the derivation of BEC

A gas with a fixed average number of atoms Ideal boson gas (macroscopic system) atoms: mass m, dispersion law system as a whole: volume V, particle number N, density n=N/V, temperature T.

A gas with a fixed average number of atoms Ideal boson gas (macroscopic system) atoms: mass m, dispersion law system as a whole: volume V, particle number N, density n=N/V, temperature T. Equation for the chemical potential closes the equilibrium problem:

A gas with a fixed average number of atoms Ideal boson gas (macroscopic system) atoms: mass m, dispersion law system as a whole: volume V, particle number N, density n=N/V, temperature T. Equation for the chemical potential closes the equilibrium problem: Always < 0. At high temperatures, in the thermodynamic limit, the continuum approximation can be used:

A gas with a fixed average number of atoms Ideal boson gas (macroscopic system) atoms: mass m, dispersion law system as a whole: volume V, particle number N, density n=N/V, temperature T. Equation for the chemical potential closes the equilibrium problem: Always < 0. At high temperatures, in the thermodynamic limit, the continuum approximation can be used: It holds For each temperature, we get a critical number of atoms the gas can accommodate. Where will go the rest?

A gas with a fixed average number of atoms Ideal boson gas (macroscopic system) atoms: mass m, dispersion law system as a whole: volume V, particle number N, density n=N/V, temperature T. Equation for the chemical potential closes the equilibrium problem: Always < 0. At high temperatures, in the thermodynamic limit, the continuum approximation can be used: This will be shown in a while It holds For each temperature, we get a critical number of atoms the gas can accommodate. Where will go the rest?

A gas with a fixed average number of atoms Ideal boson gas (macroscopic system) atoms: mass m, dispersion law system as a whole: volume V, particle number N, density n=N/V, temperature T. Equation for the chemical potential closes the equilibrium problem: Always < 0. At high temperatures, in the thermodynamic limit, the continuum approximation can be used: This will be shown in a while It holds For each temperature, we get a critical number of atoms the gas can accommodate. Where will go the rest? To the condensate

Gas particle concentration use the general formula The integral is doable: Riemann function

Statistical met h od s for boson s