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Statistical met h od s for boson s. Lecture 1. 19 th December 2012 by Be dřich Velický. Short version of the lecture plan. Lecture 1 Lecture 2.

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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
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

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

nobelist s
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"

bosons and fermions capsule reminder
Bosons and Fermions (capsule reminder)

independent quantum postulate

Identical particles are indistinguishable

bosons and fermions capsule reminder1
Bosons and Fermions (capsule reminder)

independent quantum postulate

Identical particles are indistinguishable

Cannot be labelled or numbered

bosons and fermions capsule reminder2
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 reminder3
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 reminder4
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 reminder5
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 reminder6
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 reminder7
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 reminder8
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 reminder9
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 reminder10
Bosons and Fermions (capsule reminder)

Independent particles (… non-interacting)

basis of single-particle states (  complete set of quantum numbers)

bosons and fermions capsule reminder11
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 reminder12
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 reminder13
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 reminder14
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 reminder15
Bosons and Fermions (capsule reminder)

Representation of occupation numbers (basically, second quantization)

…. for bosons princip identity

bosonsprefer to keep close – like monkeys

bosons and fermions capsule reminder16
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

Who are bosons ?

elementary particles

quasiparticles

complex massive particles, like atoms

… compound bosons

examples of 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
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
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
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 boson1
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
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

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 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
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

slide33

mean occupation number of a one-particle state with energy

Ideal quantum gases at a finite temperature: a reminder

slide34

mean occupation number of a one-particle state with energy

Ideal quantum gases at a finite temperature

CLASSICAL LIMIT

34

ideal quantum gases at a finite temperature1
Ideal quantum gases at a finite temperature

mean occupation number of a one-particle state with energy 

fermions

bosons

N

N

FD

BE

36

a gas with a fixed average number of atoms
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 atoms1
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 atoms2
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 atoms3
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 atoms4
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 atoms5
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
Gas particle concentration

use the

general formula

The integral is doable:

Riemann function

gas particle concentration1
Gas particle concentration

The integral is doable:

Riemann function

gas particle concentration2
Gas particle concentration

The integral is doable:

Riemann function

CRITICAL TEMPERATURE

the lowest temperature at which all atoms are still accomodated in the gas:

critical temperature
Critical temperature

The integral is doable:

Riemann function

CRITICAL TEMPERATURE

the lowest temperature at which all atoms are still accomodated in the gas:

atomic mass

critical temperature1
Critical temperature

CRITICAL TEMPERATURE

the lowest temperature at which all atoms are still accomodated in the gas:

A few estimates:

digression simple interpretation of t c
Digression: simple interpretation of TC

Rearranging the formula for critical temperature

we get

thermal de Broglie wavelength

mean interatomic

distance

The quantum breakdown sets on when

the wave clouds of the atoms start overlapping

de broglie wave length for atom s a nd molekul es
de Broglie wave length for atoms and molekules

Thermal energies small … NR formulae valid:

At thermal equilibrium

Two useful equations

thermal wave length

condensate concentration
Condensate concentration

fract ion

GAS

condensate

condensate concentration1
Condensate concentration

fract ion

GAS

condensate

condensate concentration2
Condensate concentration

fract ion

GAS

condensate

where are the condensate atoms
Where are the condensate atoms?

ANSWER: On the lowest one-particle energy level

For understanding, return to the discrete levels.

There is a sequence of energies

For very low temperatures,

all atoms are on the lowest level, so that

where are the condensate atoms continuation
Where are the condensate atoms? Continuation

ANSWER: On the lowest one-particle energy level

For temperaturesbelow

all condensate atoms are on the lowest level, so that

question … what happens with the occupancy of the next level now? Estimate:

where are the condensate atoms summary
Where are the condensate atoms? Summary

ANSWER: On the lowest one-particle energy level

The final balance equation for is

LESSON:

be slow with making the thermodynamic limit (or any other limits)

iii physical properties and discussion of bec

III.Physical properties and discussion of BEC

Off-Diagonal Long Range Order

Thermodynamics of BEC

closer look at bec
Closer look at BEC
  • Thermodynamically, this is a real phase transition, although unsual
  • Pure quantum effect
  • There are no real forces acting between the bosons, but there IS a real correlation in their motion caused by their identity (symmetrical wave functions)
  • BEC has been so difficult to observe, because other (classical G/L or G/S) phase transitions set on much earlier
  • BEC is a "condensation in the momentum space", unlike the usual liquefaction of classical gases, which gives rise to droplets in the coordinate space.
  • This is somewhat doubtful, especially now, that the best observed BEC takes place in traps, where the atoms are significantly localized
  • What is valid on the "momentum condensation": BEC gives rise to quantum coherence between very distant places, just like the usual plane wave
  • BEC is a macroscopic quantum phenomenon in two respects:
  •  it leads to a correlation between a macroscopic fraction of atoms
  •  the resulting coherence pervades the whole macroscopic sample
closer look at bec1
Closer look at BEC
  • Thermodynamically, this is a real phase transition, although unsual
  • Pure quantum effect
  • There are no real forces acting between the bosons, but there IS a real correlation in their motion caused by their identity (symmetrical wave functions)
  • BEC has been so difficult to observe, because other (classical G/L or G/S) phase transitions set on much earlier
  • BEC is a "condensation in the momentum space", unlike the usual liquefaction of classical gases, which gives rise to droplets in the coordinate space.
  • This is somewhat doubtful, especially now, that the best observed BEC takes place in traps, where the atoms are significantly localized
  • What is valid on the "momentum condensation": BEC gives rise to quantum coherence between very distant places, just like the usual plane wave
  • BEC is a macroscopic quantum phenomenon in two respects:
  •  it leads to a correlation between a macroscopic fraction of atoms
  •  the resulting coherence pervades the whole macroscopic sample
closer look at bec2
Closer look at BEC
  • Thermodynamically, this is a real phase transition, although unsual
  • Pure quantum effect
  • There are no real forces acting between the bosons, but there IS a real correlation in their motion caused by their identity (symmetrical wave functions)
  • BEC has been so difficult to observe, because other (classical G/L or G/S) phase transitions set on much earlier
  • BEC is a "condensation in the momentum space", unlike the usual liquefaction of classical gases, which gives rise to droplets in the coordinate space.
  • This is somewhat doubtful, especially now, that the best observed BEC takes place in traps, where the atoms are significantly localized
  • What is valid on the "momentum condensation": BEC gives rise to quantum coherence between very distant places, just like the usual plane wave
  • BEC is a macroscopic quantum phenomenon in two respects:
  •  it leads to a correlation between a macroscopic fraction of atoms
  •  the resulting coherence pervades the whole macroscopic sample
closer look at bec3
Closer look at BEC
  • Thermodynamically, this is a real phase transition, although unsual
  • Pure quantum effect
  • There are no real forces acting between the bosons, but there IS a real correlation in their motion caused by their identity (symmetrical wave functions)
  • BEC has been so difficult to observe, because other (classical G/L or G/S) phase transitions set on much earlier
  • BEC is a "condensation in the momentum space", unlike the usual liquefaction of classical gases, which gives rise to droplets in the coordinate space.
  • This is somewhat doubtful, especially now, that the best observed BEC takes place in traps, where the atoms are significantly localized
  • What is valid on the "momentum condensation": BEC gives rise to quantum coherence between very distant places, just like the usual plane wave
  • BEC is a macroscopic quantum phenomenon in two respects:
  •  it leads to a correlation between a macroscopic fraction of atoms
  •  the resulting coherence pervades the whole macroscopic sample
closer look at bec4
Closer look at BEC
  • Thermodynamically, this is a real phase transition, although unsual
  • Pure quantum effect
  • There are no real forces acting between the bosons, but there IS a real correlation in their motion caused by their identity (symmetrical wave functions)
  • BEC has been so difficult to observe, because other (classical G/L or G/S) phase transitions set on much earlier
  • BEC is a "condensation in the momentum space", unlike the usual liquefaction of classical gases, which gives rise to droplets in the coordinate space.
  • This is somewhat doubtful, especially now, that the best observed BEC takes place in traps, where the atoms are significantly localized
  • What is valid on the "momentum condensation": BEC gives rise to quantum coherence between very distant places, just like the usual plane wave
  • BEC is a macroscopic quantum phenomenon in two respects:
  •  it leads to a correlation between a macroscopic fraction of atoms
  •  the resulting coherence pervades the whole macroscopic sample
closer look at bec5
Closer look at BEC
  • Thermodynamically, this is a real phase transition, although unsual
  • Pure quantum effect
  • There are no real forces acting between the bosons, but there IS a real correlation in their motion caused bytheir identity (symmetrical wave functions)
  • BEC has been so difficult to observe, because other (classical G/L or G/S) phase transitions set on much earlier
  • BEC is a "condensation in the momentum space", unlike the usual liquefaction of classical gases, which gives rise to droplets in the coordinate space.
  • This is somewhat doubtful, especially now, that the best observed BEC takes place in traps, where the atoms are significantly localized
  • What is valid on the "momentum condensation": BEC gives rise to quantum coherence between very distant places, just like the usual plane wave
  • BEC is a macroscopic quantum phenomenon in two respects:
  •  it leads to a correlation between a macroscopic fraction of atoms
  •  the resulting coherence pervades the whole macroscopic sample
off diagonal long range order

Off-Diagonal Long Range Order

Beyond the thermodynamic view:

Coherence of the condensate in real space

Analysis on the one-particle level

coherence in bec odlro
Coherence in BEC: ODLRO

Off-Diagonal Long Range Order

Without field-theoretical means, the coherence of the condensate may be studied using the one-particle density matrix.

Definition of OPDM for non-interacting particles: Take an additive observable, like local density, or current density. Its average value for the whole assembly of atoms in a given equilibrium state:

coherence in bec odlro1
Coherence in BEC: ODLRO

Without field-theoretical means, the coherence of the condensate may be studied using the one-particle density matrix.

Definition of OPDM for non-interacting particles: Take an additive observable, like local density, or current density. Its average value for the whole assembly of atoms in a given equilibrium state:

coherence in bec odlro2
Coherence in BEC: ODLRO

Without field-theoretical means, the coherence of the condensate may be studied using the one-particle density matrix.

Definition of OPDM for non-interacting particles: Take an additive observable, like local density, or current density. Its average value for the whole assembly of atoms in a given equilibrium state:

coherence in bec odlro3
Coherence in BEC: ODLRO

Without field-theoretical means, the coherence of the condensate may be studied using the one-particle density matrix.

Definition of OPDM for non-interacting particles: Take an additive observable, like local density, or current density. Its average value for the whole assembly of atoms in a given equilibrium state:

coherence in bec odlro4
Coherence in BEC: ODLRO

Without field-theoretical means, the coherence of the condensate may be studied using the one-particle density matrix.

Definition of OPDM for non-interacting particles: Take an additive observable, like local density, or current density. Its average value for the whole assembly of atoms in a given equilibrium state:

opdm for homogeneous systems
OPDM for homogeneous systems

In coordinate representation

  • depends only on the relative position (transl. invariance)
  • Fourier transform of the occupation numbers
  • isotropic … provided thermodynamic limit is allowed
  • in systems without condensate, the momentum distribution is smooth and the density matrix has a finite range.

CONDENSATE lowest orbital with

opdm for homogeneous systems odlro2

CONDENSATE lowest orbital with

OPDM for homogeneous systems: ODLRO

DIAGONAL ELEMENT r = r'

DISTANT OFF-DIAGONAL ELEMENT |r - r' |

Off-Diagonal Long Range Order

ODLRO

from opdm towards the macroscopic wave function
From OPDM towards the macroscopic wave function

CONDENSATE lowest orbital with

MACROSCOPIC WAVE FUNCTION

  • expresses ODLRO in the density matrix
  • measures the condensate density
  • appears like a pure state in the density matrix, but macroscopic
  • expresses the notion that the condensate atoms are in the same state
  • is the order parameter for the BEC transition
homogeneous one component phase boundary conditions environment and state variables
Homogeneous one component phase:boundary conditions (environment) and state variables
homogeneous one component phase boundary conditions environment and state variables1
Homogeneous one component phase:boundary conditions (environment) and state variables

The important four

The one we use presently

87

digression which environment to choose

S B

S B

Digression: which environment to choose?

THE ENVIRONMENT IN THE THEORY SHOULD CORRESPOND TO THE EXPERIMENTAL CONDITIONS

… a truism difficult to satisfy

  •  For large systems, this is not so sensitive for two reasons
    • System serves as a thermal bath or particle reservoir all by itself
    • Relative fluctuations (distinguishing mark) are negligible
  • AdiabaticsystemRealsystemIsothermal system
  • SB heat exchange – the slowest medium fast the fastest
  • process
  •  Atoms in a trap: ideal model … isolated. In fact: unceasing energy exchange (laser cooling). A small number of atoms may be kept (one to, say, 40). With 107, they form a bath already. Besides, they are cooled by evaporation and they form an open (albeit non-equilibrium) system.
  •  Sometime, N =const. crucial (persistent currents in non-SC mesoscopic rings)
  • temperature lag
  • interface layer
thermodynamic potentials and all that
Thermodynamic potentials and all that

Basic thermodynamic identity (for equilibria)

For an isolated system,

For an isothermic system, the independent variables are T, V, N.

The change of variables is achieved by means of the Legendre transformation. Define Free Energy

thermodynamic potentials and all that1
Thermodynamic potentials and all that

Basic thermodynamic identity (for equilibria)

For an isolated system,

For an isothermic system, the independent variables are T, V, N.

The change of variables is achieved by means of the Legendre transformation. Define Free Energy

New variables: perform the substitution everywhere; this shows in the Maxwell identities (partial derivatives)

thermodynamic potentials and all that2
Thermodynamic potentials and all that

Basic thermodynamic identity (for equilibria)

For an isolated system,

For an isothermic system, the independent variables are T, V, N.

The change of variables is achieved by means of the Legendre transformation. Define Free Energy

Legendre transformation: subtract the relevant product of conjugate (dual) variables

New variables: perform the substitution everywhere; this shows in the Maxwell identities (partial derivatives)

thermodynamic potentials and all that3
Thermodynamic potentials and all that

Basic thermodynamic identity (for equilibria)

For an isolated system,

For an isothermic system, the independent variables are T, S, V.

The change of variables is achieved by means of the Legendre transformation. Define Free Energy

a table
A table

How comes ? is additive, is the only additive independent variable. Thus,

Similar consideration holds for

grand canonical thermodynamic functions of an ideal bose gas

Grand canonical thermodynamic functions of an ideal Bose gas

 General procedure for independent particles

 Elementary treatment of thermodynamic functions of an ideal gas

Specific heat

Equation of the state

quantum statistics with g rand canonical ensemble
Quantum statistics with grand canonical ensemble

Grand canonical ensemble admits both energy and particle number exchange between the system and its environment.

The statistical operator (many body density matrix) acts in the Fock space

External variables are . They are specified by the conditions

Grand canonical statistical operator has the Gibbs' form

grand canonical statistical sum for ideal bose gas5
Grand canonical statistical sum for ideal Bose gas

Recall

valid for

- extended "ïnfinite" gas

- parabolic traps

just the same

ideal boson systems at a finite temperature
Ideal Boson systems at a finite temperature

fermions

bosons

N

N

FD

BE

Equation for the chemical potential closes the equilibrium problem:

?

thermodynamic functions for an extended bose gas
Thermodynamic functions for an extended Bose gas

For Born-Karman periodic boundary conditions, the lowest level is

Its contribution has to be singled out, like before:

*

specific heat of an ideal bose gas
Specific heat of an ideal Bose gas

A weak singularity …

what decides is the coexistence of two phases

specific heat comparison liquid 4 he vs ideal bose gas
Specific heat: comparison liquid 4He vs. ideal Bose gas

Dulong-Petit (classical) limit 

- singularity

linear dispersion law

Dulong-Petit (classical) limit 

weak singularity

quadratic dispersion law

isotherms in the p v plane
Isotherms in the P-V plane

For a fixed temperature, the specific volume can be arbitrarily small. By contrast, the pressure is volume independent …typical for condensation

ISOTHERM

compare with condensation of a real gas
Compare with condensation of a real gas

CO2

Basic similarity:

increasing pressure with compression

critical line

beyond is a plateau

Differences:

at high pressures

at high compressions

compare with condensation of a real gas1
Compare with condensation of a real gas

CO2

Basic similarity:

increasing pressure with compression

critical line

beyond is a plateau

Differences:

at high pressures

at high compressions

 no critical point

Conclusion:

BEC in a gas is a phase transition

of the first order

Fig. 151 Experimental isotherms of

carbon dioxide (CO2}.

trap potential physics involved skipped
Trap potential (physics involved skipped)

evaporation cooling

Typical profile

?

coordinate/ microns 

This is just one direction

Presently, the traps are mostly 3D

The trap is clearly from the real world, the atomic cloud is visible almost by a naked eye

115

trap potential
Trap potential

Parabolic approximation

in general, an anisotropic harmonic oscillator usually with axial symmetry

1D

2D

3D

116

ground state orbital and the trap potential
Ground state orbital and the trap potential

400 nK

level number

200 nK

  • characteristic energy
  • characteristic length

117

ground state orbital and the trap potential1
Ground state orbital and the trap potential

400 nK

level number

200 nK

  • characteristic energy
  • characteristic length

118

ground state orbital and the trap potential2
Ground state orbital and the trap potential

400 nK

level number

200 nK

  • characteristic energy
  • characteristic length

119

filling the trap with particles idos dos
Filling the trap with particles: IDOS, DOS

1D

For the finite trap, unlike in the extended gas, is not divided by volume !!

2D

"thermodynamic limit"

only approximate … finite systems

better for small

meaning wide trap potentials

120

filling the trap with particles
Filling the trap with particles

3D

Estimate for the transition temperature

particle number comparable with the number of states in the thermal shell

  • characteristic energy

For 106 particles,

121

filling the trap with particles1
Filling the trap with particles

3D

Estimate for the transition temperature

particle number comparable with the number of states in the thermal shell

  • characteristic energy

For 106 particles,

122

exact expressions for critical temperature etc
Exact expressions for critical temperature etc.

The general expressions are the same like for the homogeneous gas.

Working with discrete levels, we have

and this can be used for numerics without exceptions.

In the approximate thermodynamic limit, the old equation holds, only the volume V does not enter as a factor:

In 3D,

123

how sharp is the transition
How sharp is the transition

These are experimental data

fitted by the formula

The rounding is apparent,

but not really an essential feature

124

seeing the condensate reminder
Seeing the condensate – reminder

Without field-theoretical means, the coherence of the condensate may be studied using the one-particle density matrix.

Definition of OPDM for non-interacting particles: Take an additive observable, like local density, or current density. Its average value for the whole assembly of atoms in a given equilibrium state:

125

opdm in the trap
OPDM in the Trap
  • Use the eigenstates of the 3D oscillator
  • Use the BE occupation numbers
  • Single out the ground state

zero point oscillations

absorbed in the

chemical potential

126

opdm in the trap1
OPDM in the Trap

Coherent component, be it condensate or not. At , it contains ALL atoms in the cloud

Incoherent thermal component, coexisting with the condensate. At , it freezes out and contains NO atoms

  • Use the eigenstates of the 3D oscillator
  • Use the BE occupation numbers
  • Single out the ground state

zero point oscillations

absorbed in the

chemical potential

127

opdm in the trap particle density in space
OPDM in the Trap, Particle Density in Space

The spatial distribution of atoms in the trap is inhomogeneous. Proceed by definition:

as we would write down

naively at once

Split into the two parts, the coherent and the incoherent phase

128

particle density in space boltzmann limit
Particle Density in Space: Boltzmann Limit

We approximate the thermal distribution by its classical limit.

Boltzmann distribution in an external field:

Two directly observable characteristic lengths

For comparison:

anisotropy

given by analogous

definitions of the

two lengths

for each direction

134

real space image of an atomic cloud
Real space Image of an Atomic Cloud
  • the cloud is macroscopic
  • basically, we see the thermal distribution
  • a cigar shape: prolate rotational ellipsoid
  • diffuse contours: Maxwell – Boltzmann
  • distribution in a parabolic potential

135

particle velocity momentum distribution
Particle Velocity (Momentum) Distribution

The procedure is similar, do it quickly:

136

thermal particle velocity momentum distribution
Thermal Particle Velocity (Momentum) Distribution

Again, we approximate the thermal distribution by its classical limit.

Boltzmann distribution in an external field:

Two directly observable characteristic lengths

Remarkable:

Thermal and condensate

lengths in the same ratio

for positions and momenta

137

three crossed laser beams 3d laser cooling
Three crossed laser beams: 3D laser cooling

20 000 photons needed to stop from the room temperature

braking force propor-tional to velocity: viscous medium, "molasses"

For an intense laser a matter of milliseconds

temperature measure-ment: turn off the lasers. Atoms slowly sink in the field of gravity

simultaneously, they spread in a ballistic fashion

the probe laser beam excites fluorescence.

the velocity distribution inferred from the cloud size and shape

138

three crossed laser beams 3d laser cooling1
Three crossed laser beams: 3D laser cooling

20 000 photons needed to stop from the room temperature

braking force propor-tional to velocity: viscous medium, "molasses"

For an intense laser a matter of milliseconds

Temperature measurement: turn off the lasers. Atoms slowly sink in the field of gravity

simultaneously, they spread in a ballistic fashion

the probe laser beam excites fluorescence.

the velocity distribution inferred from the cloud size and shape

139

three crossed laser beams 3d laser cooling2
Three crossed laser beams: 3D laser cooling

20 000 photons needed to stop from the room temperature

braking force propor-tional to velocity: viscous medium, "molasses"

For an intense laser a matter of milliseconds

Temperature measurement: turn off the lasers. Atoms slowly sink in the field of gravity

the probe laser beam excites fluorescence.

the velocity distribution inferred from the cloud size and shape

140

three crossed laser beams 3d laser cooling3
Three crossed laser beams: 3D laser cooling

20 000 photons needed to stop from the room temperature

braking force propor-tional to velocity: viscous medium, "molasses"

For an intense laser a matter of milliseconds

Temperature measurement: turn off the lasers. Atoms slowly sink in the field of gravity

simultaneously, they spread in a ballistic fashion

the probe laser beam excites fluorescence.

the velocity distribution inferred from the cloud size and shape

141

three crossed laser beams 3d laser cooling4
Three crossed laser beams: 3D laser cooling

20 000 photons needed to stop from the room temperature

braking force propor-tional to velocity: viscous medium, "molasses"

For an intense laser a matter of milliseconds

Temperature measurement: turn off the lasers. Atoms slowly sink in the field of gravity

simultaneously, they spread in a ballistic fashion

the probe laser beam excites fluorescence.

the velocity distribution inferred from the cloud size and shape

142

bec observed by tof in the velocity distribution1
BEC observed by TOF in the velocity distribution

Qualitative features:  all Gaussians

wide vs.narrow

 isotropic vs. anisotropic

144

importance of the interaction synopsis
Importance of the interaction – synopsis

Without interaction, the condensate would occupy the ground state of the oscillator

(dashed - - - - -)

In fact, there is a significant broadening of the condensate of 80 000 sodium atoms in the experiment by Hau et al. (1998),

The reason … the interactions experiment perfectly reproduced by the solution of the Gross – Pitaevski equation

146

are the interactions important
Are the interactions important?
  • In the dilute gaseous atomic clouds in the traps, the interactions are incomparably weaker than in liquid helium.
  • That permits to develop a perturbative treatment and to study in a controlled manner many particle phenomena difficult to attack in HeII.
  • Several roles of the interactions
  • the atomic collisions take care of thermalization
  • the mean field component of the interactions determines most of the deviations from the non-interacting case
  • beyond the mean field, the interactions change the quasi-particles and result into superfluidity even in these dilute systems

151

fortunate properties of the interactions
Fortunate properties of the interactions
  • Strange thing: the cloud lives for seconds, or even minutes at temperatures, at which the atoms should form a crystalline cluster. Why?
  • For binding of two atoms, a third one is necessary to carry away the released binding energy and momentum. Such ternary collisions are very unlikely in the rare cloud, however.
  • The interactions are elastic and spin independent: they do not spoil the separation of the hyperfine atomic species and preserve thus the identity of the atoms.
  • At the very low energies in question, the effective interaction is typically weak and repulsive … which enhances the formation and stabilization of the condensate.

152

fortunate properties of the interactions1
Fortunate properties of the interactions
  • Strange thing: the cloud lives for seconds, or even minutes at temperatures, at which the atoms should form a crystalline cluster. Why?
  • For binding of two atoms, a third one is necessary to carry away the released binding energy and momentum. Such ternary collisions are very unlikely in the rare cloud, however.
  • The interactions are elastic and spin independent: they do not spoil the separation of the hyperfine atomic species and preserve thus the identity of the atoms.
  • At the very low energies in question, the effective interaction is typically weak and repulsive … which enhances the formation and stabilization of the condensate.

Quasi-Equilibrium

153

interatomic interactions
Interatomic interactions
  • For neutral atoms, the pairwise interaction has two parts
  • van der Waals force
  • strong repulsion at shorter distances due to the Pauli principle for electrons
  • Popular model is the 6-12 potential:
  • Example:
  •  corresponds to ~12 K!!
  • Many bound states, too.

154

interatomic interactions1
Interatomic interactions

minimum

vdW radius

  • For neutral atoms, the pairwise interaction has two parts
  • van der Waals force
  • strong repulsion at shorter distances due to the Pauli principle for electrons
  • Popular model is the 6-12 potential:
  • Example:
  •  corresponds to ~12 K!!
  • Many bound states, too.

155

interatomic interactions2
Interatomic interactions

The repulsive part of the potential – not well known

The attractive part of the potential can be measured with precision

Even this permits to define a characteristic length

156

interatomic interactions3
Interatomic interactions

The repulsive part of the potential – not well known

The attractive part of the potential can be measured with precision

Even this permits to define a characteristic length

rough estimate of the last bound state energy

compare

with

157

scattering length pseudopotential
Scattering length, pseudopotential

Beyond the potential radius, say , the scattered wave propagates in free space

For small energies, the scattering is purely isotropic , the s-wave scattering. The outside wave is

For very small energies, , the radial part becomes just

This may be extrapolated also into the interaction sphere

(we are not interested in the short range details)

Equivalent potential ("Fermi pseudopotential")

158

experimental data1
Experimental data

for “ordinary” gases

VLT clouds

NOTES

weak attraction ok

weak repulsion ok

weak attraction

intermediate attraction

weak repulsion ok

strong resonant repulsion

nm

3.4

4.7

6.8

8.7

8.7

10.4

nm

-1.4

4.1

-1.7

-19.5

5.6

127.2

as

"well behaved; monotonous increase

seemingly erratic, very interesting physics of scattering resonances behind

161

experimental data2
Experimental data

for “ordinary” gases

VLT clouds

NOTES

weak attraction ok

weak repulsion ok

weak attraction

intermediate attraction

weak repulsion ok

strong resonant repulsion

 (K)

5180

940

---

---

73

---

nm

3.4

4.7

6.8

8.7

8.7

10.4

nm

-1.4

4.1

-1.7

-19.5

5.6

127.2

as

"well behaved; monotonous increase

seemingly erratic, very interesting physics of scattering resonances behind

162

experimental data3
Experimental data

for “ordinary” gases

VLT clouds

NOTES

weak attraction ok

weak repulsion ok

weak attraction

intermediate attraction

weak repulsion ok

strong resonant repulsion

nm

3.4

4.7

6.8

8.7

8.7

10.4

nm

-1.4

4.1

-1.7

-19.5

5.6

127.2

as

"well behaved; monotonous increase

seemingly erratic, very interesting physics of scattering resonances behind

163

many body hamiltonian and the hartree approximation
Many-body Hamiltonian and the Hartree approximation

We start from the mean field approximation.

This is an educated way, similar to (almost identical with) the HARTREE APPROXIMATION we know for many electron systems.

Most of the interactions is indeed absorbed into the mean field and what remains are explicit quantum correlation corrections

self-consistent

system

165

hartree approximation at zero temperature
Hartree approximation at zero temperature

Consider a condensate. Then all occupied orbitals are the same and

Putting

we obtain a closed equation for the order parameter:

This is the celebrated Gross-Pitaevskii equation.

This is a single self-consistent equation for a single orbital,

the simplest HF like theory ever.

166

hartree approximation at zero temperature1
Hartree approximation at zero temperature

single self-consistent equation for a single orbital

Consider a condensate. Then all occupied orbitals are the same and

Putting

we obtain a closed equation for the order parameter:

This is the celebrated Gross-Pitaevskii equation.

This is a single self-consistent equation for a single orbital,

the simplest HF like theory ever.

167

gross pitaevskii equation at zero temperature
Gross-Pitaevskii equation at zero temperature

single self-consistent equation for a single orbital

Consider a condensate. Then all occupied orbitals are the same and

Putting

we obtain a closed equation for the order parameter:

This is the celebrated Gross-Pitaevskii equation.

168

gross pitaevskii equation at zero temperature1
Gross-Pitaevskii equation at zero temperature

single self-consistent equation for a single orbital

  • Consider a condensate. Then all occupied orbitals are the same and
  • Putting
  • we obtain a closed equation for the order parameter:
  • This is the celebrated Gross-Pitaevskii equation.
  • has the form of a simple non-linear Schrödinger equation
  • concerns a macroscopic quantity 
  • suitable for numerical solution.

169

gross pitaevskii equation at zero temperature2
Gross-Pitaevskii equation at zero temperature

single self-consistent equation for a single orbital

  • Consider a condensate. Then all occupied orbitals are the same and
  • Putting
  • we obtain a closed equation for the order parameter:
  • This is the celebrated Gross-Pitaevskii equation.
  • has the form of a simple non-linear Schrödinger equation
  • concerns a macroscopic quantity 
  • suitable for numerical solution.

The lowest level coincides with the chemical potential

170

gross pitaevskii equation at zero temperature3
Gross-Pitaevskii equation at zero temperature

single self-consistent equation for a single orbital

  • Consider a condensate. Then all occupied orbitals are the same and
  • Putting
  • we obtain a closed equation for the order parameter:
  • This is the celebrated Gross-Pitaevskii equation.
  • has the form of a simple non-linear Schrödinger equation
  • concerns a macroscopic quantity 
  • suitable for numerical solution.

The lowest level coincides with the chemical potential

How do we know??

171

gross pitaevskii equation at zero temperature4
Gross-Pitaevskii equation at zero temperature

single self-consistent equation for a single orbital

  • Consider a condensate. Then all occupied orbitals are the same and
  • Putting
  • we obtain a closed equation for the order parameter:
  • This is the celebrated Gross-Pitaevskii equation.
  • has the form of a simple non-linear Schrödinger equation
  • concerns a macroscopic quantity 
  • suitable for numerical solution.

The lowest level coincides with the chemical potential

How do we know??

How much is it??

172

gross pitaevskii equation bohmian form
Gross-Pitaevskii equation – "Bohmian" form

For a static condensate, the order parameter has ZERO PHASE. Then

The Gross-Pitaevskii equation

becomes

Bohm's quantum potential

the effective mean-field potential

173

gross pitaevskii equation bohmian form1
Gross-Pitaevskii equation – "Bohmian" form

For a static condensate, the order parameter has ZERO PHASE. Then

The Gross-Pitaevskii equation

becomes

Bohm's quantum potential

the effective mean-field potential

174

gross pitaevskii equation bohmian form2
Gross-Pitaevskii equation – "Bohmian" form

For a static condensate, the order parameter has ZERO PHASE. Then

The Gross-Pitaevskii equation

becomes

Bohm's quantum potential

the effective mean-field potential

175

gross pitaevskii equation variational interpretation
Gross-Pitaevskii equation – variational interpretation

This equation results from a variational treatment of the Energy Functional

It is required that

with the auxiliary condition

that is

which is the GP equation written for the particle density (previous slide).

176

gross pitaevskii equation variational interpretation1
Gross-Pitaevskii equation – variational interpretation

This equation results from a variational treatment of the Energy Functional

It is required that

with the auxiliary condition

that is

which is the GP equation written for the particle density (previous slide).

LAGRANGE MULTIPLIER

177

gross pitaevskii equation chemical potential
Gross-Pitaevskii equation – chemical potential

This equation results from a variational treatment of the Energy Functional

It is required that

with the auxiliary condition

that is

which is the GP equation written for the particle density (previous slide). From there

178

gross pitaevskii equation chemical potential1
Gross-Pitaevskii equation – chemical potential

This equation results from a variational treatment of the Energy Functional

It is required that

with the auxiliary condition

that is

which is the GP equation written for the particle density (previous slide). From there

chemical potential by definition

179

the simplest case of all a homogeneous gas
The simplest case of all: a homogeneous gas

In an extended homogeneous system (… Born-Kármán boundary condition),

the GP equation simplifies

184

the simplest case of all a homogeneous gas1
The simplest case of all: a homogeneous gas

In an extended homogeneous system (… Born-Kármán boundary condition),

the GP equation simplifies

185

the simplest case of all a homogeneous gas2
The simplest case of all: a homogeneous gas

In an extended homogeneous system (… Born-Kármán boundary condition),

the GP equation simplifies

The repulsive interaction increases the chemical potential

The repulsive interaction increases the chemical potential

If , it would be

and the gas would be thermodynamically unstable.

186

the simplest case of all a homogeneous gas3
The simplest case of all: a homogeneous gas

In an extended homogeneous system (… Born-Kármán boundary condition),

the GP equation simplifies

The repulsive interaction increases the chemical potential

187

the simplest case of all a homogeneous gas4
The simplest case of all: a homogeneous gas

In an extended homogeneous system (… Born-Kármán boundary condition),

the GP equation simplifies

The repulsive interaction increases the chemical potential

188

the simplest case of all a homogeneous gas5
The simplest case of all: a homogeneous gas

In an extended homogeneous system (… Born-Kármán boundary condition),

the GP equation simplifies

The repulsive interaction increases the chemical potential

If , it would be

and the gas would be thermodynamically unstable.

189

the simplest case of all a homogeneous gas6
The simplest case of all: a homogeneous gas

In an extended homogeneous system (… Born-Kármán boundary condition),

the GP equation simplifies

The repulsive interaction increases the chemical potential

If , it would be

and the gas would be thermodynamically unstable.

impossible!!!

190

reminescence the trap potential and the ground state1
Reminescence: The trap potential and the ground state

400 nK

level number

200 nK

  • characteristic energy
  • characteristic length

194

parabolic trap with interactions
Parabolic trap with interactions

GP equation for a spherical trap ( … the simplest possible case)

195

parabolic trap with interactions1
Parabolic trap with interactions

GP equation for a spherical trap ( … the simplest possible case)

Where is the particle number N? ( … a little reminder)

196

parabolic trap with interactions2
Parabolic trap with interactions

GP equation for a spherical trap ( … the simplest possible case)

Where is the particle number N? ( … a little reminder)

Dimensionless GP equation for the trap

197

parabolic trap with interactions3
Parabolic trap with interactions

GP equation for a spherical trap ( … the simplest possible case)

Where is the particle number N? ( … a little reminder)

Dimensionless GP equation for the trap

198

importance of the interaction synopsis1
Importance of the interaction – synopsis

Without interaction, the condensate would occupy the ground state of the oscillator

(dashed - - - - -)

In fact, there is a significant broadening of the condensate of 80 000 sodium atoms in the experiment by Hau et al. (1998),

perfectly reproduced by the solution of the GP equation

199

importance of the interaction synopsis2
Importance of the interaction – synopsis

Without interaction, the condensate would occupy the ground state of the oscillator

(dashed - - - - -)

In fact, there is a significant broadening of the condensate of 80 000 sodium atoms in the experiment by Hau et al. (1998),

perfectly reproduced by the solution of the GP equation

guess:

internal pressure due to repulsion

competes with the trap potential

200

importance of the interaction
Importance of the interaction

Qualitative

for g>0, repulsion, both inner "quantum pressure" and the interaction broaden the condensate.

for g<0, attraction, "quantum pressure" and the interaction together allow the condensate to form. It shrinks and becomes metastable. Onset of instability with respect to three particle recombination processes

Quantitative

The decisive parameter for the "importance" of interactions is

201

importance of the interaction1
Importance of the interaction

Qualitative

for g>0, repulsion, both inner "quantum pressure" and the interaction broaden the condensate.

for g<0, attraction, "quantum pressure" and the interaction together allow the condensate to form. It shrinks and becomes metastable. Onset of instability with respect to three particle recombination processes

Quantitative

The decisive parameter for the "importance" of interactions is

202

importance of the interaction2
Importance of the interaction

Qualitative

for g>0, repulsion, both inner "quantum pressure" and the interaction broaden the condensate.

for g<0, attraction, "quantum pressure" and the interaction together allow the condensate to form. It shrinks and becomes metastable. Onset of instability with respect to three particle recombination processes

Quantitative

The decisive parameter for the "importance" of interactions is

203

importance of the interaction3
Importance of the interaction

Qualitative

for g>0, repulsion, both inner "quantum pressure" and the interaction broaden the condensate.

for g<0, attraction, "quantum pressure" and the interaction together allow the condensate to form. It shrinks and becomes metastable. Onset of instability with respect to three particle recombination processes

Quantitative

The decisive parameter for the "importance" of interactions is

unlike in an extended homogeneous gas !!

204

importance of the interaction4
Importance of the interaction

Qualitative

for g>0, repulsion, both inner "quantum pressure" and the interaction broaden the condensate.

for g<0, attraction, "quantum pressure" and the interaction together allow the condensate to form. It shrinks and becomes metastable. Onset of instability with respect to three particle recombination processes

Quantitative

The decisive parameter for the "importance" of interactions is

205

importance of the interaction5
Importance of the interaction

Qualitative

for g>0, repulsion, both inner "quantum pressure" and the interaction broaden the condensate.

for g<0, attraction, "quantum pressure" and the interaction together allow the condensate to form. It shrinks and becomes metastable. Onset of instability with respect to three particle recombination processes

Quantitative

The decisive parameter for the "importance" of interactions is

206

importance of the interaction6
Importance of the interaction

Qualitative

for g>0, repulsion, both inner "quantum pressure" and the interaction broaden the condensate.

for g<0, attraction, "quantum pressure" and the interaction together allow the condensate to form. It shrinks and becomes metastable. Onset of instability with respect to three particle recombination processes

Quantitative

The decisive parameter for the "importance" of interactions is

207

importance of the interaction7
Importance of the interaction

Qualitative

for g>0, repulsion, both inner "quantum pressure" and the interaction broaden the condensate.

for g<0, attraction, "quantum pressure" and the interaction together allow the condensate to form. It shrinks and becomes metastable. Onset of instability with respect to three particle recombination processes

Quantitative

The decisive parameter for the "importance" of interactions is

can vary in a wide range

collective effect weak or strong depending on N

weak individual collisions

208

importance of the interaction8
Importance of the interaction

Qualitative

for g>0, repulsion, both inner "quantum pressure" and the interaction broaden the condensate.

for g<0, attraction, "quantum pressure" and the interaction together allow the condensate to form. It shrinks and becomes metastable. Onset of instability with respect to three particle recombination processes

Quantitative

The decisive parameter for the "importance" of interactions is

can vary in a wide range

0

realistic value

collective effect weak or strong depending on N

weak individual collisions

209

reminescence the trap potential and the ground state3
Reminescence: The trap potential and the ground state

400 nK

level number

200 nK

ground state orbital

213

reminescence the trap potential and the ground state4
Reminescence: The trap potential and the ground state

400 nK

level number

200 nK

ground state orbital

214

variational estimate of the condensate properties
Variational estimate of the condensate properties

SCALING ANSATZ

The condensate orbital will be taken in the form

It is just like the ground state orbital for the isotropic oscillator, but with a rescaled size. This is reminescent of the well-known scaling for the ground state of the helium atom.

215

variational estimate of the condensate properties1
Variational estimate of the condensate properties

SCALING ANSATZ

The condensate orbital will be taken in the form

It is just like the ground state orbital for the isotropic oscillator, but with a rescaled size. This is reminescent of the well-known scaling for the ground state of the helium atom.

variational parameter

b

216

importance of the interaction scaling approximation
Importance of the interaction: scaling approximation

Variational ansatz: the GP orbital is a scaled ground state for g = 0

217

importance of the interaction scaling approximation1
Importance of the interaction: scaling approximation

Variational ansatz: the GP orbital is a scaled ground state for g = 0

dimension-less energy per particle

dimension-less orbital size

self-interaction

Variational estimate of the total energy of the condensate as a function of the parameter

ADDITIONAL NOTES

Variational parameter is the orbital width in units of a0

218

importance of the interaction scaling approximation2
Importance of the interaction: scaling approximation

Variational ansatz: the GP orbital is a scaled ground state for g = 0

dimension-less energy per particle

dimension-less orbital size

self-interaction

Variational estimate of the total energy of the condensate as a function of the parameter

ADDITIONAL NOTES

Variational parameter is the orbital width in units of a0

219

importance of the interaction scaling approximation3
Importance of the interaction: scaling approximation

Variational ansatz: the GP orbital is a scaled ground state for g = 0

dimension-less energy per particle

dimension-less orbital size

self-interaction

Variational estimate of the total energy of the condensate as a function of the parameter

ADDITIONAL NOTES

Variational parameter is the orbital width in units of a0

220

importance of the interaction scaling approximation4
Importance of the interaction: scaling approximation

Variational estimate of the total energy of the condensate as a function of the parameter

Variational estimate of the total energy of the condensate as a function of the parameter

Variational parameter is the orbital width in units of a0

Variational parameter is the orbital width in units of a0

221

importance of the interaction scaling approximation5
Importance of the interaction: scaling approximation

Variational estimate of the total energy of the condensate as a function of the parameter

Variational estimate of the total energy of the condensate as a function of the parameter

Variational parameter is the orbital width in units of a0

Variational parameter is the orbital width in units of a0

222

importance of the interaction scaling approximation6
Importance of the interaction: scaling approximation

Variational estimate of the total energy of the condensate as a function of the parameter

Variational estimate of the total energy of the condensate as a function of the parameter

Variational parameter is the orbital width in units of a0

Variational parameter is the orbital width in units of a0

The minimum of the curve gives the condensate size for a given . With increasing , the condensate stretches with an asymptotic power law

223

importance of the interaction scaling approximation7
Importance of the interaction: scaling approximation

Variational estimate of the total energy of the condensate as a function of the parameter

Variational estimate of the total energy of the condensate as a function of the parameter

Variational parameter is the orbital width in units of a0

Variational parameter is the orbital width in units of a0

For , the condensate is metastable, below , it becomes unstable and shrinks to a 'zero' volume. Quantum pressure no more manages to overcome the attractive atom-atom interaction

The minimum of the curve gives the condensate size for a given . With increasing , the condensate stretches with an asymptotic power law

224