Statistical met h od s for boson s

Statistical methods for bosons

Lecture 1.19th December 2012

by Bedřich Velický

2

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

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

http://kfes-80.karlov.mff.cuni.cz/kfes/PackageWeb/php/K12M23ColdAtoms.ppt

4

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"

Eric A. Cornell Wolfgang Ketterle

Carl E. Wieman

1/3 of the prize 1/3 of the prize 1/3 of the prize

USA Federal Republic of Germany

USA

University of Colorado, JILA Boulder, CO, USA

Massachusetts Institute of Technology (MIT) Cambridge, MA, USA

University of Colorado, JILA Boulder, CO, USA

b. 1961 b. 1957 b. 1951

I.

Introductory matter on bosons

6

Bosons and Fermions (capsule reminder)

independent quantum postulate

Identical particles are indistinguishable

7



Identical particles are indistinguishable Cannot be labelled or numbered

8




Permuting particles does not lead to a different state

1 2 2 1 1 2

Two particles

( , ) ( , ) ( , )x x x x x x

Cannot be labelled or numbered

9





1 2 2 1 1 2

Two particles

( , ) ( , ) ( , )x x x x x x

10





1 2 2 1 12

12 2

Two particles

( , ) (( , ,) )( , )x x x x x x x x

11





1 2 2 1 1 22

2

2

1

Two particles

( , ) ( , ) ( , ) ( , )

1

x x x x xx x x

fermions bosons

antisymmetric symmetric

1 1

12





1 2 2 1 1 22

2

2

1

Two particles

( , ) ( , ) ( , ) ( , )

1

x x x x xx x x

fermions bosons


half-integer spin integer spin

1 1 comes from

nowhere

"empirical

fact"

13





1 2 2 1 1 22

2

2

1

Two particles

( , ) ( , ) ( , ) ( , )

1

x x x x xx x x

fermions bosons



1 1 comes from

nowhere

"empirical

fact"

Finds justification in the relativistic quantum field

theory

14





1 2 2 1 1 22

2

2

1

Two particles

( , ) ( , ) ( , ) ( , )

1

x x x x xx x x

fermions bosons



electrons photons

1 1 comes from

nowhere

"empirical

fact"


theory

15





1 2 2 1 1 22

2

2

1

Two particles

( , ) ( , ) ( , ) ( , )

1

x x x x xx x x

fermions bosons



electrons photons

everybody knows our present concern

1 1 comes from

nowhere

"empirical

fact"


theory

16


Independent particles (… non-interacting)

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

x x

17




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

x x

18




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

x x

2

1 2 3

1 3, , , , ,

, , , , , -particle state pn

p

pn n n n nn n

Σ

19


1 2

1 2

3

3

1 2 1 2

, , , , , -particle state

0 0 , 0 , 0 , ,0 , 0-particle state

1 0 , 0 , 0 , ,1 , 1-particle ( )

, , ,

( ) ( ') ( ') ( ) /0 , 1 ,1 , ,0 , 2-particle

, ,

p

pp

p

n

px

x x

n n

x

n n

x

n n n

vacuu

Σ

m

1 10 , , 0 , ,0 , 2-particle

2

( ) ( ' )2 x x

1 , 1 , ,1 , 0 , -particl

not a

e ground state

l wed

l o

F N

Representation of occupation numbers (basically, second quantization)…. for fermions Pauli principle fermions keep apart – as sea-gulls

20


1 2

1 2

3

3

1 2 1 2


0 0 , 0 , 0 , ,0 , 0-particle state

1 0 , 0 , 0 , ,1 , 1-particle ( )

, , ,

( ) ( ') ( ') ( ) /0 , 1 ,1 , ,0 , 2-particle

, ,

p

pp

p

n

px

x x

n n

x

n n

x

n n n

vacuu

Σ

m

1 10 , , 0 , ,0 , 2-particle

2

( ) ( ' )2 x x

1 , 1 , ,1 , 0 , -particl

not a

e ground state

l wed

l o

F N

Representation of occupation numbers (basically, second quantization)…. for fermions Pauli principle fermions keep apart – as sea-gulls

in atoms: filling of the shells (Pauli Aufbau Prinzip)

in metals: Fermi sea

21


3

1 2

1 2

3

1 2 1 2


0 0 , 0 , 0 , ,0 , 0-particle state

1 0 , 0 , 0 , ,1 , 1-particle

0 , 1 ,1 , ,0 , 2-

, , , , ,

par

( )

( ) ( ') ( ') (ticle ) /

p

ppn

p px

x x x x

n n n n n n n

vacuum

Σ

1 2

1 1

1 1 1

0 , 2, 0 , ,0 , 2-particle

, 0 ,0 , ,0 , -particle

all on a single orbital

2

( ) ( ')

( ) ( ) ( )

ground state

N

x x

x x x

B N N

Representation of occupation numbers (basically, second quantization)…. for bosons princip identity bosons prefer to keep close – like monkeys

22


3

1 2

1 2

3

1 2 1 2


0 0 , 0 , 0 , ,0 , 0-particle state

1 0 , 0 , 0 , ,1 , 1-particle

0 , 1 ,1 , ,0 , 2-

, , , , ,

par

( )

( ) ( ') ( ') (ticle ) /

p

ppn

p px

x x x x

n n n n n n n

vacuum

Σ

1 2

1 1

1 1 1

0 , 2, 0 , ,0 , 2-particle

, 0 ,0 , ,0 , -particle

all on a single orbital

2

( ) ( ')

( ) ( ) ( )

ground state

N

x x

x x x

B N N

Representation of occupation numbers (basically, second quantization)…. for bosons princip identity bosons prefer to keep close – like monkeys

Bose-Einstein condensate

Who are bosons ?

• elementary particles• quasiparticles• complex massive particles, like atoms … compound bosons

24

Examples of bosonsbosons

elementary particles

quasi particles

photons

phononsmagnons

simple particlesN not conserved

complex particlesN conserved

atomic nuclei

excited atoms

4 7 23 87

alkali metals

He, Li, Na, Rb

atoms

25

Examples of bosons (extension of the table)bosons

elementary particles

quasi particles

compoundquasi particles

atomic nuclei

excitedatoms

ions

molecules

photons

phononsmagnons

excitonsCooper pairs

simple particlesN not conserved

complex particlesN conserved

4 7 23 87

alkali metals

He, Li, Na, Rb

atoms

26

Question: How a complex particle, like an atom, can behave as a single whole, a compound boson ESSENTIAL CONDITIONS

1)All compound particles in the ensemble must be identical; the identity includes

o detailed elementary particle composition

o characteristics like mass, charge or spin

2)The total spin must have an integer value

3)The identity requirement extends also on the values of observables corresponding to internal degrees of freedom

4)which are not allowed to vary during the dynamical processes in question

5)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

27

Example: How a complex particle, like an atom, can behave as a single whole, a compound boson

A

Z

2 2 6 2 6 10 2 6 10 1

closed shells - spin compensation

electron configuration

1 2 2 3 3 3 4 4 4 5s s p s p d s p d s

2 1SJL

12

12, ,

0L

S

J L S

S L SJ L

RUBIDIUM -- THE FIRST ATOMIC CLOUD TO UNDERGO BEC

• single element Z = 37

• single isotope A = 87

• single electron configuration

1

12

8737

32

Rb

[Kr]52

s

I

S

28

Example: How a complex particle, like an atom, can behave as a single whole, a compound boson

A

Z

RUBIDIUM -- THE FIRST ATOMIC CLOUD TO UNDERGO BEC

• single element Z = 37

• single isotope A = 87

• single electron configuration

1

12

8737

32

Rb

[Kr]52

s

I

S

37 electrons

37 protons

50 neutrons

• total spin of the atom decides

total electron spin

total nuclear spin

12S

32I

1, , 2,S I S I

F S I

F

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

29

http://intro.chem.okstate.edu/1314f00/lecture/chapter7/lec111300.html

1133

1133

Rb

21

25

9

6

9

m =0.1

m

COMP

=

E

3.

AR

3

0.244 10 m

10 10 m

5.7 10 10 m

r

in the air at 0 C and 1 a

d n

d n

tm

vs.

vs.

II.

Homogeneous gas of non-interacting bosons

The basic system exhibiting the Bose-Einstein Condensation (BEC)

original case studied by Einstein

31

Cell size (per k vector)

Cell size (per p vector)

In the (r, p)-phase space

Plane wave in classical terms and its quantum transcription

Discretization ("quantization") of wave vectors in the cavity

volume periodic boundary conditions

Plane waves in a cavity

i0e , ( )

de Brog

, 2 /

lie wave, len, ( ), / t g h

tX X k k

p h p

k r

p k

xL yL

zL

x y zV L L L

2,

2,

2

xx

ymx

znx

kL

k mL

k nL

xk

yk

(2 ) /dk V

d dkV h

/dp h V

32

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

Density of states

xk

yk

( ) ( )p p

( ) dd dp C p

( ) ( ( )) / ( ( )) / dd p dp V p h

1 1

1 d( ) ( )

d

d d ( )( ( ) / ) ( ( ) / )

d dd d

d d

V

pp h dC h p h

D

22

( 2 1)!

d /

dCd /

33

Ideal quantum gases at a finite temperature: a reminder

( ) Boltzmann distribution

high temperatures, dilute g as e s

e n mean occupation number of a one-particle state with

energy

34

Ideal quantum gases at a finite temperature




energy

35


1

e 1n

( )

1

e 1n

( )

1

e 1n

fermions bosons

N NFDBE




energy

36


1

e 1n

( )

1

e 1n

( )

1

e 1n

fermions bosons

N NFDBE




energy

chemical potential

fixes particle number N

37


1 , 1 , ,1 , 0 , , 0 ,0 , ,0 , vacF B N

1

e 1n

( )

1

e 1n

( )

1

e 1n

fermions bosons

0T 0T 0T

N NFDBE




energy

38


1 , 1 , ,1 , 0 , , 0 ,0 , ,0 , vacF B N

1

e 1n

( )

1

e 1n

( )

1

e 1n

fermions bosons

0T 0T 0T

N NFDBE




energy

Aufbau principle

39


1 , 1 , ,1 , 0 , , 0 ,0 , ,0 , vacF B N

1

e 1n

( )

1

e 1n

( )

1

e 1n

fermions bosons

0T 0T 0T

N N

freezing out

FDBE




energy

Aufbau principle

40


1 , 1 , ,1 , 0 , , 0 ,0 , ,0 , vacF B N

1

e 1n

( )

1

e 1n

( )

1

e 1n

fermions bosons

0T 0T 0T

N N

freezing out?

FDBE




energy

Aufbau principle

41


1 , 1 , ,1 , 0 , , 0 ,0 , ,0 , vacF B N

1

e 1n

( )

1

e 1n

( )

1

e 1n

fermions bosons

0T 0T 0T

N N

freezing outBEC?

FDBE




energy

Aufbau principle

Bose-Einstein condensation: elementary approach

43

Einstein's manuscript with the derivation of BEC

http://www.lorentz.leidenuniv.nl/history/Einstein_archive/Einstein_1925_manuscript/Pages/Einstein_1925_01.html

44

A gas with a fixed average number of atomsIdeal boson gas (macroscopic system)

atoms: mass m, dispersion law

system as a whole:

volume V, particle number N, density n=N/V, temperature T.

2

( )2

pp

m

45



system as a whole:


Equation for the chemical potential closes the equilibrium problem:

2

( )2

pp

m

( )

1( , ) ( )

e 1jjj j

nN T

N

46



system as a whole:



2

( )2

pp

m

Always < 0. At high temperatures, in the thermodynamic limit, the continuum approximation can be used:

( )0

1d ( ) ( , )

e 1N V T

D N

( )

1( , ) ( )

e 1jjj j

nN T

N

47



system as a whole:



2

( )2

pp

m


( )0

1d ( ) ( , )

e 1N V T

D N

It holds

For each temperature, we get a critical number of atoms the gas can accommodate. Where will go the rest?

( , 0) ( ,0)T T N N

( )

1( , ) ( )

e 1jjj j

nN T

N

48



system as a whole:



2

( )2

pp

m


( )0

1d ( ) ( , )

e 1N V T

D N

It holds

For each temperature, we get a critical number of atoms the gas can accommodate. Where will go the rest?

( , 0) ( ,0)T T N N

This will beshown in a while

( )

1( , ) ( )

e 1jjj j

nN T

N

49



system as a whole:



2

( )2

pp

m


( )0

1d ( ) ( , )

e 1N V T

D N

It holds

For each temperature, we get a critical number of atoms the gas can accommodate. Where will go the rest? To the condensate

( , 0) ( ,0)T T N N

This will beshown in a while

( )

1( , ) ( )

e 1jjj j

nN T

N

50

Gas particle concentration

0

3

23 32 22

3

2

2

1( ,0) d ( )

e 1

2 4 ( ) ( )

2 2 2,612

B

B

T V

mk TV

h

mk TV

h

N D

The integral is doable:

Riemann function

use the general formula

51


0

3

23 32 22

3

2

2

1( ,0) d ( )

e 1

2 4 ( ) ( )

2 2 2,612

B

B

T V

mk TV

h

mk TV

h

N D


/ 2

Riemann function

3

2

2

2( ) 2

m

h

D

Bose-Einstein condensation: critical temperature

53


0

3

23 32 22

3

2

2

1( ,0) d ( )

e 1

2 4 ( ) ( )

2 2 2,612

B

B

T V

mk TV

h

mk TV

h

N D

The integral is doable:3

2

2

2( ) 2

m

h

D

/ 2

Riemann function

( ,0)cT NN

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

54

Critical temperature

0

3

23 32 22

3

2

2

1( ,0) d ( )

e 1

2 4 ( ) ( )

2 2 2,612

B

B

T V

mk TV

h

mk TV

h

N D


/ 2

Riemann function


( ,0)cT NN

2 222 2 3 33

180,52725 1,6061 104 2,612 4c

B B

h N h n nT

mk V uk M M

atomic mass

3

2

2

2( ) 2

m

h

D

55

Critical temperature

A few estimates:

system M n TC

He liquid 4 11028 3.54 K

Na trap 23 11020 2.86 K

Rb trap 87 11019 95 nK


2 222 2 3 33

180,52725 1,6061 104 2,612 4c

B B

h N h n nT

mk V uk M M

56

Digression: simple interpretation of TC

1

3

B c

V h

N mk T

Rearranging the formula for critical temperature

we get

22 3

4 2,612cB

h NT

mk V

mean interatomic distance

thermal de Broglie

wavelength

The quantum breakdown sets on when

the wave clouds of the atoms start overlapping

57

de Broglie wave length for atoms and molekules

2

p

Thermal energies small … NR formulae valid:

At thermal equilibrium

Two useful equations

kin

2

2mE

3kin 2 BE k T

... at

. (mol.) mass

m Mu

92 2 1 12,5 10

3 3B Bmk u k MT MT

23kin 2 eV K /11600 158

TE T v v

M

thermal wave length

58

Ketterle explains BEC to the King of Sweden

Bose-Einstein condensation: condensate

60

33 22

3 3

2 2

( ,0)= for

1

CG C

C

G BEC C

T Tn BT n T T

V T

T Tn n n n n

T T

N

Condensate concentration

GAScondensate

/ CT T

fract ion

61

33 22

3 3

2 2

( ,0)= for

1

CG C

C

G BEC C

T Tn BT n T T

V T

T Tn n n n n

T T

N


GAScondensate

/ CT T

fract ion

62

33 22

3 3

2 2

( ,0)= for

1

CG C

C

G BEC C

T Tn BT n T T

V T

T Tn n n n n

T T

N


GAScondensate

/ CT T

fract ion

63

Where are the condensate atoms?ANSWER: On the lowest one-particle energy level

For understanding, return to the discrete levels.

( )

1( , ) ( )

e 1jjj j

nN T

N

There is a sequence of energies

For very low temperatures,

all atoms are on the lowest level, so that

0 1 2(0) 0

1 0( ) 1

1 0

0

( )0

( )

0

(e )

1 connecting equati

all atoms are in the condensate

chemical potential is zero on the gross energy sca

o n

e

e

l

1

B

n N O

N

k T

N

64

Where are the condensate atoms? ContinuationANSWER: On the lowest one-particle energy level

For temperatures below

all condensate atoms are on the lowest level, so that

0

0

( )

0

1 connecting equatio

all condensate atoms remain on t

e 1

he lowest level

chemical potential keeps zero o

n

n the gross energy scale

B

B

E

E

B

BE

n N

N

k T

N

CT

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

2

2 31 0 /h m V

2

30 1

0 1

( ), ( ) .... much slower growthB Bk T k Tn O V n O V

65

Where are the condensate atoms? SummaryANSWER: On the lowest one-particle energy level

The final balance equation for is

0( ) ( )0

1 1( , ) d ( )

e 1e 1VTN

N D

CT T

LESSON:

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

III.

Physical properties and discussion of BEC

Off-Diagonal Long Range OrderThermodynamics of BEC

67

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

68

Closer look at BEC











69

Closer look at BEC











70

Closer look at BEC











71

Closer look at BEC











72

Closer look at BEC











Off-Diagonal Long Range Order

Beyond the thermodynamic view:Coherence of the condensate in real space

Analysis on the one-particle level

74

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:

= insert unit operator

change the summation ord

double average, q

er

|

uantum and thermal

define the one-particle density matrix

Tr

X n

n X

X

X

X n

X

= n


75




change the

s

do

umma

uble ave

tion ord

rage, quantum and t

er

| define the one-particle density matrix

Tr

hermal


n X

X

X

X n

X n

X

= n

76




double average, quantum and thermal


chang

| define the one-particle density matrix

Tr

e th

e summation or

e

d r

X

X

n X

X

X

n

n

X

= n

77






chang

de

e the summation

fine the one-particle density mat

Tr

order

|

r x

i

X n

X n

X

n X

X

X

= n

78






chang

de

e the summation


rix

order

|

Tr

X n

X n

n X

X

X

X

= n

79

OPDM for homogeneous systems

In coordinate representation

i ( ')

( , ')

1e

n

nV

kk

k r rk

k

r r r k k r'

• 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 0k

80

OPDM for homogeneous systems: ODLRO

CONDENSATE lowest orbital with

1

30 ( ) 0O V

k

0

0

i ( ') i ( ')0

coherent across of a smooth functionthe sample has a finite range

BE G

1 1( ') e e

( ') ( ')

FT

n nV V

k r r k r rk

k k

r r

r r r r

81



1

30 ( ) 0O V

k

0

0

i ( ') i ( ')0


BE G

1 1( ') e e

( ') ( ')

FT

n nV V

k r r k r rk

k k

r r

r r r r

BE G

BE G

( ) ( ) ( )

+n n n

0 0 0

DIAGONAL ELEMENT r = r' for k0 = 0

82



1

30 ( ) 0O V

k

0

0

i ( ') i ( ')0


BE G

1 1( ') e e

( ') ( ')

FT

n nV V

k r r k r rk

k k

r r

r r r r

BE G

BE G

( ) ( ) ( )

+n n

0 0 0

DIAGONAL ELEMENT r = r'

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

BE BE| '|

G| '|

BE

( ')

( ') 0

( ')

n

n

r r

r r

r r

r r

r r

r r


ODLRO

83

From OPDM towards the macroscopic wave functionCONDENSATE lowest orbital with

1

30 ( ) 0O V

k

0

0

0

i ( ') i ( ')0


i ( ')

1 1( ') e e

1 ( ) ( ) e

dyadic

FT

n nV V

nV

k r r k r rk

k k

k r rk

k k

r r

r r'

MACROSCOPIC WAVE FUNCTION

0i( ) an arbitrary ph ee s( ) , aBEn k r+r

• 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

Capsule on thermodynamics

85

Homogeneous one component phase:boundary conditions (environment) and state variables

not in useS P

isolated, conservativ eS V N

isothermal T V N

isobaricS P Nopen S V

grand T V isothermal-isobaricNT P

not in use T P

additive variables, have dens / "extensive"ities / S V N s S V n N V

dual variables, intensities "intensive "T P

87

Homogeneous one component phase:boundary conditions (environment) and state variables

isolated, conservativ eS V N

isothermal T V N

grand T V isothermal-isobaricNT P

additive variables, have dens / "extensive"ities / S V N s S V n N V

dual variables, intensities "intensive "T P

The important four

The one we use presently

88

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

Adiabatic system Real system Isothermal 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)

S B S B • temperature lag

• interface layer

89

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

d d d dU T S P V N

,

( , , ) , etc.S V N

UT T S V N

,

d

, ( , , )

d d d

, etc.

, ( , ,

T

, )

V N

F S T P V N

F U TS

FS

U U T V N S S T V N

90






d d d dU T S P V N

,

( , , ) , etc.S V N

UT T S V N

,

d

, ( , , )

d d d

, etc.

, ( , ,

T

, )

V N

F S T P V N

F U TS

FS

U U T V N S S T V N

New variables: perform the substitution

everywhere; this shows in the Maxwell identities (partial derivatives)

91






d d d dU T S P V N

,

( , , ) , etc.S V N

UT T S V N

,

d

, ( , , )

d d d

, etc.

, ( , ,

T

, )

V N

F S T P V N

F U TS

FS

U U T V N S S T V N

Legendre transformation: subtract the relevant

product of conjugate (dual) variables

New variables: perform the substitution

everywhere; this shows in the Maxwell identities (partial derivatives)

92




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


d d d dU T S P V N

,

( , , ) , etc.S V N

UT T S V N

,

, ( , , ), ( , , ),

d d d d

, etc.T V N

F U TS U U T V N S S T V N

F S T P V N

FS

93

A table

isolated system microcanonical ensemble

internal energy

, ,S V NU d d d dU T S P V N

isothermic system canonical ensemble

free energy

, ,T V NF U TS d d d dF S T P V N

isothermic-isobaric system isothermic-isobaric ensemble

free enthalpy

, ,T P NU TS NPV d d d dS T V P N

isothermic open system grand canonical ensemble

grand potential

, ,T V U TS VN P d d d dS T P V N

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

Similar consideration holds for

PV V

,

( , ) , ( , )T

T V T PV

( , )T P N

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

95

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

, ,T V

ˆ ˆ ˆˆTr sharp Trˆ ˆH U V N N N

B Tr ln maxˆˆS k

ˆ ˆ1 ( )

ˆ ˆ( ) ( , , )

B

statistical s

e

( , , ) Tr e e

( , , ) ln ( , , )

um

grand canonical poten t ial

ˆ H N

H N V

Z

Z V

V k T Z V

96

Grand canonical statistical sum for ideal Bose gas

ˆ ˆ( ) ( , , )

(

(

)

)( )

( )(

( , , ) Tr e e

= e

statistical sum

eigenstate

e e up to here tr

label with

ivial

1TRICK!! e

1 e

H N V

E

n n

n n

n

n

N

Z V

n n N

)

Recall

97


ˆ ˆ( ) ( , , )

( )

(

(

)( )

)(

( , , ) Tr e e

= e

statistical sum

eigenstate

up to here tr

label with

1TRICK!! e

ive ial

e

1 e

H N V

E N

n

n

n

n

n

n

Z V

n n N

)

Recall

98


ˆ ˆ( ) ( , , )

( )

( )( )

( )(

( , , ) Tr e e

= e

statistical sum

eigenstate

up to here tr

label with

e e

1e

ivial

TRIC1 e

K

!!

H N V

E N

n n

n n

n

n

Z V

n n N

)

( )

1 1( , , )

1 e 1 eZ V

z

Recall

activitye

fugacity z

99


ˆ ˆ( ) ( , , )

( )

( )( )

( )(

( , , ) Tr e e

= e

statistical sum

eigenstate

up to here tr

label with

e e

1e

ivial

TRIC1 e

K

!!

H N V

E N

n n

n n

n

n

Z V

n n N

)

( )

1 1( , , )

1 e 1 ezZ V

Recall

activitye

fugacity z

100


ˆ ˆ( ) ( , , )

( )

( )( )

( )(

( , , ) Tr e e

= e

statistical sum

eigenstate

up to here tr

label with

e e

1e

ivial

TRIC1 e

K

!!

H N V

E N

n n

n n

n

n

Z V

n n N

)

( )

1 1( , , )

1 e 1 ezZ V

Recall

activitye

fugacity z

B

( )B

B

grand canonical potenti( , , ) ln ( , , )

= + ln 1 e

+ ln 1

al

e

V k T Z V

k

k T z

T

101


ˆ ˆ( ) ( , , )

( )

( )( )

( )(

( , , ) Tr e e

= e

statistical sum

eigenstate

up to here tr

label with

e e

1e

ivial

TRIC1 e

K

!!

H N V

E N

n n

n n

n

n

Z V

n n N

)

( )

1 1( , , )

1 e 1 ezZ V

Recall

activitye

fugacity z

B

( )B

B

grand canonical potenti( , , ) ln ( , , )

= + ln 1 e

+ ln 1

al

e

V k T Z V

k

k T z

T

valid for - extended "ïnfinite" gas

- parabolic trapsjust the same

Using the statistical sum

102

ˆ( )ˆ ˆ( )

ˆ ˆ ˆ( )( )

B B

( )B

( )

A. (Re)derivi

ˆ ˆTr Trˆ ˆTr

Tr Tr

ln ( , , )

ln 1

ng the Bose-Einstein distrib

e

1

e 1

ution

nH N

nH N

e n e nn n

e eZ

k T k T Z VZ

k T

o.

2B

For

B.

Fe

Pair

rmions

correlations

...........

(particle number fluctu

..........

ation

( ) (1

For classi

)

bunch

cal gas ..

ing

anti-bunching

..........

)

(1 )

.. .

.

k T n n n n n

n n

n

k.

103

Ideal Boson systems at a finite temperature

1 , 1 , ,1 ,, 0 , va0 ,0 , ,0 , c B NF

1

e 1n

( )

1

e 1n

( )

1

e 1n



e n

fermions bosons

0T 0T

N N

?

FDBE

( )

1( , ) ( )

e 1jjj j

nN T

N


104

Thermodynamic functions for an extended Bose gasFor Born-Karman periodic boundary conditions, the lowest level is

Its contribution has to be singled out, like before:

( ) 0 k 0

10

1( , , ) d ( )

1 e 1

zN T V V

z z

D

3

2

2

2( )

3D D S

2

O

m

h

D

0

( , , ) ln(1 ) d ln(1 e ) ( )V z V zT

D

10

( , , ) d ( )e 1

U T V Vz

D

activitye

fugacity z

23PV U*

108

Specific heat of an ideal Bose gas

,

/V

V N

U NC

T

A weak singularity …

what decides is the coexistence of two phases

109

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

,

/V

V N

U NC

T

Dulong-Petit (classical) limit

Dulong-Petit (classical) limit

- singularity

weak singularity

quadratic dispersion law

linear dispersion law

110

Isotherms in the P-V plane

cv v

P

3 / 2BE3/ 2 B

5 / 25 / 2 B

(1)

(1)

NNg k T

V V

P g k T

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

/V N

2 / 33 / 2

5

3

2

5 / 3 5 /

/

/c c

c c

v AT T v A

P B

P BA

T

v

ISOTHERM

111

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

112

Fig. 151 Experimental isotherms of carbon dioxide (CO2}.

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

IV.

Non-interacting bosons in a trap

114

Useful digression: energy units

-1

B B

B

-B

B

1

/

1K 1eV s

1K / /

1eV / /

s / /

J

J

J

energy

k e k h

e k e h

h k h

e/

e

k

h/

-1

05 10

04

23

19 14

-1 31 15 41

1K 1eV s

1K 8.63 10 2.08 10

1eV 1.16 10 2.41 1

1.38 10

0

s

energy

4.80 10 4.14 1

1.60 10

6.63 100

115

Trap potential (physics involved skipped)

Typical profile

coordinate/ microns

?

evaporation cooling

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

116

Trap potential

Parabolic approximation

in general, an anisotropic harmonic oscillator usually with axial symmetry

2 2 2 2 2 2 21 1 1 1

2 2 2 2x y z

x y z

H m x m y m zm

H H H

p

1D

2D

3D

117

Ground state orbital and the trap potential

level number

200 nK

400 nK

0/ xx a

2

20

0 0 0 0

2 22

0 0 0 2 20 00

( , , )

1 1 1 1( ) e , ,

2 2 2

x y z

u

a

m

x y z x y z

u a Em ma Mu aa

22 2

0

1 1( )

2 2

uV u m u

a

• characteristic energy

• characteristic length

118


level number

200 nK

400 nK

0/ xx a

2

20

0 0 0 0

2 22

0 0 0 2 20 00

( , , )

1 1 1 1( ) e , ,

2 2 2

x y z

u

a

m

x y z x y z

u a Em ma Mu aa

22 2

0

1 1( )

2 2

uV u m u

a

0

6

87

1 m

=10 nK

~ 10 at.

Rba

N



119


level number

200 nK

400 nK

0/ xx a

2

20

0 0 0 0

2 22

0 0 0 2 20 00

( , , )

1 1 1 1( ) e , ,

2 2 2

x y z

u

a

m

x y z x y z

u a Em ma Mu aa

22 2

0

1 1( )

2 2

uV u m u

a

0

6

87

1 m

=10 nK

~ 10 at.

Rba

N



120

Filling the trap with particles: IDOS, DOS1D

2D

1

( ) int( / ) /

( ) '( )

E E E

E E

D

const.E x yE E E

212( ) /( )

( ) '( ) /( )

x y

x y

E E

E E E

D

"thermodynamic limit"

only approximate … finite systems

better for small

meaning wide trap potentials

B ck T

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

x

y

x E

121

Filling the trap with particles3D

316

212

( ) /( )

( ) '( ) /( )

x y z

x y z

E E

E E E

D

Estimate for the transition temperature

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

B

1 1

2 2B

1 1

3 3B

2D / ( )

3D / ( )

c x y

c x y z

N k T

T k N

T k N

For 106 particles,2

B 10ck T


122

Filling the trap with particles3D

316

212

( ) /( )

( ) '( ) /( )

x y z

x y z

E E

E E E

D

Estimate for the transition temperature

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

B

1 1

2 2B

1 1

3 3B

2D / ( )

3D / ( )

c x y

c x y z

N k T

T k N

T k N

For 106 particles,2

B important for therm. limit10 ck T


123

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,

Exact expressions for critical temperature etc.

0( )

1( , )

e 1VTN

N

( )0

1d ( )

e 1

D

( )

1( , ) ( )

e 1jjj j

nN T

N

1 1 1

3 3 3B B( (3)) / 0.94 /cT k N k N

3BE 1 ( / ) ,c cN N T T T T

for0 CT T

124

How sharp is the transition

These are experimental data

fitted by the formula

The rounding is apparent,

but not really an essential feature

3BE 1 ( / ) ,c cN N T T T T

125

Seeing the condensate – reminder



chang

de

e the summation


rix

order

|

Tr

X n

X n

n X

X

X

X

= n



126

OPDM in the Trap

• Use the eigenstates of the 3D oscillator

• Use the BE occupation numbers

• Single out the ground state

( )

= ( , , ), 0, 1, 2, 3,

1

= , = e 1

= = + + x y z

x y z w

x y z

x x x x x x

E

n

E E E E

(000)

BEC TERM

( )

1 1= 000 000

e 1 e 1

E

zero point oscillations

absorbed in the

chemical potential

127

OPDM in the Trap

• Use the eigenstates of the 3D oscillator

• Use the BE occupation numbers

• Single out the ground state

( )

= ( , , ), 0, 1, 2, 3,

1

= , = e 1

= = + + x y z

x y z w

x y z

x x x x x x

E

n

E E E E

(000)

BEC TERM

( )

1 1= 000 000

e 1 e 1

E

zero point oscillations

absorbed in the

chemical potentialCoherent component, be it condensate or not. At , it contains

ALL atoms in the cloud/ BT k

Incoherent thermal component, coexisting with the

condensate. At , it freezes out and contains

NO atoms

/ BT k

128

OPDM in the Trap, Particle Density in Space

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

BEC THERM( )

n r r r r r r r

as we would write down

naively at once

op

2

( )

( )

( ) Tr ( )

Tr d ( ) Tr

1

e 1

1( )

e 1

E

E

n

r r r

r r r r r r r

r r r r

r

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

134

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

2 2 2 2 2 2

3THERM

( ( ))

( )

1 )2

(

( , ) e

( ) d ( , )

e

ex y z

B

B

W U

U

x y zm

f

n f

r

r

r p

r p r p

For comparison:

1

30 0

2

0

0

0

0 ,

1

(

/

)x y z

B

R

m

m

a

a

T

a a

k T

a

a

1

3( )x y z

anisotropygiven by analogous

definitions of thetwo lengths

for each direction

22 2

2 2 20 0 0

22 2

BEC 0 0 0

30 0 0

1( )

e 1

1 1e

e 1x y z

x y z

x y z

yx za a a

n x y z

a a a

r

135

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

136

Particle Velocity (Momentum) DistributionThe procedure is similar, do it quickly:

THERM

BEC THERM

(000)

2 2

000(000)

BEC

( )

( )

known

laborious

( )

1 1= 000 000

e 1 e 1

1

1( ) ( )

e 1 e

1

( ) ( )

E

E

f

f f

p p p p p p p

p p p p

p p

p p

22 2

2 2 20 0 0

2

00

2 2

BEC 0 0 0

1( )

e 1

1e ,

e 1

yx z

x y z

x x y y z z

ww

pp pb b b

f p p p

ba

p

137

Thermal Particle Velocity (Momentum) DistributionAgain, we approximate the thermal distribution by its classical limit.

Boltzmann distribution in an external field:

1 2 2 2

3THERM

( ( ))

1 )2

(

( , ) e

( ) d ( , )

e

ex y z

B

B

W U

W

m p p p

f

f f

rr p

r r r p

22 2

2 2 20 0 0

2

00

2 2

BEC 0 0 0

1( )

e 1

1e ,

e 1

yx z

x y z

x x y y z z

ww

pp pb b b

f p p p

ba

p

Two directly observable characteristic lengths

00

00

1

30 0 0 ,

1

(

/

)x y z

B

ba

b b b

T

k T

m

b

B

b

Remarkable:

Thermal and condensatelengths in the same ratio

for positions and momenta

0 0

0 0

/

/

B

B

B b k T bT

R a k T aT

138

Three crossed laser beams: 3D laser cooling

the probe laser beam excites fluorescence.

the velocity distribution inferred from the cloud size

and shape

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

139




and shape




"molasses"


Temperature measurement: turn off

the lasers. Atoms slowly sink in the field

of gravity


fashion

140




and shape




"molasses"




of gravity

141




and shape




"molasses"




of gravity


fashion

142




and shape




"molasses"




of gravity


fashion

143

BEC observed by TOF in the velocity distribution

144

BEC observed by TOF in the velocity distribution

Qualitative features: all Gaussians

wide vs.narrow

isotropic vs. anisotropic

146

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

Next time !!!

The end

149

V.

Interacting atoms

151

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

152

Fortunate properties of the interactions

1. 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.

2. 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.

3. At the very low energies in question, the effective interaction is typically weak and repulsive … which enhances the formation and stabilization of the condensate.

153

Fortunate properties of the interactions

1. 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.

2. 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.

3. 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

154

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.

6

1

r

12 6

TRUE ( ) 4U rr r

-22Ar =1.6 10 J =0.34 nm

155


6

1

r

12 6

TRUE ( ) 4U rr r

-22Ar =1.6 10 J =0.34 nm

minimum

vdW radius 1

62 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.

156


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

6TRUE 6

( ) repulsive part - C

U rr

26

2 66

26

6

6

1/ 4

"local kinetic energy" "local potential energy

2

2

"

1

m

C

m

C

157


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

6TRUE 6

( ) repulsive part - C

U rr

26

2 66

26

6

6

1/ 4

"local kinetic energy" "local potential energy

2

2

"

1

m

C

m

C

rough estimate of the last bound state energy

comparewith

B collision energy of the

condensate a m

to sCk T

158

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")

3

0sin( )kr

r

sc... the atte, ring length s sr a a

2

( ) ( )

4 s

U r g

ag

m

r

0k

159

Experimental data

as

160

Useful digression: energy units

-1

B B B

B

-1B

B

B Ha

H

Ha

a

Ha

Ha Ha Ha

/J

J

1K 1eV s a.u.

1K / / /

1eV / / /

s / J/ /

a.u. /

ene

J

g

/

r

/

y

h

k e k h k

e k e h e

h k h e

k e h

k

e /

h /

/

-1

05 10 06

04 14 02

-1 11 15 16

05 01 1

23

1

25

9

34

1

1K 1eV s a.u.

1K 8.63 10 2.08 10 3.17 10

1eV 1.16 10 2.41 10 3.67 10

s 4.80 10 4.14 10 1.52 10

a.u. 3

1.38

e

.16 10 2.72 10 6

10

1.60 10

6.63 10

5.8.56 10

y

10

e

8

n rg

161

Experimental data

nm

-1.44.1

-1.7-19.5

5.6127.2

1 a.u. = 1 bohr 0.053 nm

NOTES

weak attraction okweak repulsion ok

weak attractionintermediate attractionweak repulsion okstrong resonant repulsion

"well behaved; monotonous increase

seemingly erratic, very interesting physics of scattering resonances behind

for “ordinary” gases VLT cloudsasnm

3.44.7

6.88.78.7

10.4

162

Experimental data

(K)

5180940

------73---

nm

-1.44.1

-1.7-19.5

5.6127.2

1 a.u. = 1 bohr 0.053 nm

NOTES





for “ordinary” gases VLT cloudsasnm

3.44.7

6.88.78.7

10.4

163

Experimental data

nm

3.44.7

6.88.78.7

10.4

nm

-1.44.1

-1.7-19.5

5.6127.2

1 a.u. = 1 bohr 0.053 nm

NOTES





for “ordinary” gases VLT cloudsas

VI.

Mean-field treatment of interacting atoms

165

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

Many-body Hamiltonian and the Hartree approximation

21 1ˆ ( ) ( )2 2a a a b

a a b

H p V Um

r r r

We start from the mean field approximation.

2GP

2

1ˆ ( ) ( )2

( ) d ( ) ( ) ( )

( )

a a H aa

H a b a b b a

H p V Vm

V U n g n

n n

r r

r r r r r r

r r

21( ) ( )

2 Hp V V Em

r r r r

self-consistentsystem

166

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.

220 0 0 0

1( )

2p V gN E

m

r r r r

0( ) ( )N r r

221( )

2p V g

m

r r r r

This is a single self-consistent equation for a single orbital,the simplest HF like theory ever.

167

Hartree approximation at zero temperature


Putting



220 0 0 0

1( )

2p V gN E

m

r r r r

0( ) ( )N r r

221( )

2p V g

m

r r r r

This is a single self-consistent equation for a single orbital,the simplest HF like theory ever.

single self-consistent equation

for a single orbital

168

Gross-Pitaevskii equation at zero temperature


Putting



220 0 0 0

1( )

2p V gN E

m

r r r r

0( ) ( )N r r

221( )

2p V g

m

r r r r



169


220 0 0 0

1( )

2p V gN E

m

r r r r

0( ) ( )N r r

221( )

2p V g

m

r r r r


Putting



• has the form of a simple non-linear Schrödinger equation

• concerns a macroscopic quantity • suitable for numerical solution.



170



Putting





220 0 0 0

1( )

2p V gN E

m

r r r r

0( ) ( )N r r

221( )

2p V g

m

r r r r

The lowest level coincides with the chemical potential



171



Putting





220 0 0 0

1( )

2p V gN E

m

r r r r

0( ) ( )N r r

221( )

2p V g

m

r r r r


How do we

know??



172



Putting





220 0 0 0

1( )

2p V gN E

m

r r r r

0( ) ( )N r r

221( )

2p V g

m

r r r r


How do we

know??

How much

is it??



173

Gross-Pitaevskii equation – "Bohmian" form

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

The Gross-Pitaevskii equation

becomes

0

23 3

( )

[ ] d (

( )

) d

(

(

)

)

N

n N n

n

N

N

r

r

r

r

r

r r

221( )

2p V g

m

r r r r

2 ( )( )

( )( )

2V

ng

mn

n

r

rr

r

Bohm's quantum potential

the effective mean-field potential

174




becomes

0

23 3

( )

[ ] d (

( )

) d

(

(

)

)

N

n N n

n

N

N

r

r

r

r

r

r r

221( )

2p V g

m

r r r r

2 ( )( )

( )( )

2V

ng

mn

n

r

rr

r



175




becomes

0

23 3

( )

[ ] d (

( )

) d

(

(

)

)

N

n N n

n

N

N

r

r

r

r

r

r r

221( )

2p V g

m

r r r r

2 ( )( )

( )( )

2V

ng

mn

n

r

rr

r



http://en.wikipedia.org/wiki/Bohm_interpretation

176

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).

3 2 22 1

2[ ] d (( ) ( ))

2( )( )n nn V g

mn

E

r r rr r

[ ] minn E

[ ]n NN

[ ] [ ] 0n n E N

177

Gross-Pitaevskii equation – variational interpretation


It is required that


that is

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

3 2 22 1

2[ ] d (( ) ( ))

2( )( )n nn V g

mn

E

r r rr r

[ ] minn E

[ ]n NN

[ ] [ ] 0n n E N LAGRANGE MULTIPLIER

178

Gross-Pitaevskii equation – chemical potential


It is required that


that is

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

3 2 22 1

2[ ] d (( ) ( ))

2( )( )n nn V g

mn

E

r r rr r

[ ] minn E

[ ]n NN

[ ] [ ] 0n n E N

[ ]

[ ]

n

n

EN

179

Gross-Pitaevskii equation – chemical potential


It is required that


that is

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

3 2 22 1

2[ ] d (( ) ( ))

2( )( )n nn V g

mn

E

r r rr r

[ ] minn E

[ ]n NN

[ ] [ ] 0n n E N

[ ]

[ ]

n

n

EN

chemical potential by definition

Interacting atoms in a constant potential

184

The simplest case of all: a homogeneous gas

2 ( )( ) ( )

2 ( )

nV g n

g

n

n V

m

rr r

r

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

the GP equation simplifies

( ) const. ( ) const. n n V V r rNV

185


2 ( )

2 ( )

n

m n

r

r

( ) ( )

V g n

g n V

r r




186


2 ( )

2 ( )

n

m n

r

r

( ) ( )

V g n

g n V

r r




The repulsive interaction increases the chemical potential

If , it would be

and the gas would be thermodynamically unstable.

0g 0

n


187


2 ( )

2 ( )

n

m n

r

r

( ) ( )

V g n

g n V

r r





2 21 12 2

internal presstotal energy ure

gn gn

E

E V PV

188


2 ( )

2 ( )

n

m n

r

r

( ) ( )

V g n

g n V

r r





2 21 12 2


gn gn

E

E V PV

212 gE

NV

189


2 ( )

2 ( )

n

m n

r

r

( ) ( )

V g n

g n V

r r





If , it would be


0g 0n

2 21 12 2


gn gn

E

E V PV

212 gE

NV

190


2 ( )

2 ( )

n

m n

r

r

( ) ( )

V g n

g n V

r r





If , it would be


0g 0n

2 21 12 2


gn gn

E

E V PV

impossible!!!

212 gE

NV

VII.

Interacting atoms in a parabolic trap

193

Reminescence: The trap potential and the ground state

level number

200 nK

400 nK

0/ xx a

2

20

0 0 0 0

2 22

0 0 0 2 20 00

( , , )

1 1 1 1( ) e , ,

2 2 2

x y z

u

a

m

x y z x y z

u a Em ma Mu aa

0

6

87

1 m

=10 nK

~ 10 at.

Rba

N

194


level number

200 nK

400 nK

0/ xx a

2

20

0 0 0 0

2 22

0 0 0 2 20 00

( , , )

1 1 1 1( ) e , ,

2 2 2

x y z

u

a

m

x y z x y z

u a Em ma Mu aa

22 2

0

1 1( )

2 2

uV u m u

a

0

6

87

1 m

=10 nK

~ 10 at.

Rba

N



195

Parabolic trap with interactions

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

2

22 21

2 2m r g

m

r r r

196



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

2

22 21

2 2m r g

m

r r r

23 3d ( ) d ( )n N r r r r

197




2

22 21

2 2m r g

m

r r r

23 3d ( ) d ( )n N r r r r

Dimensionless GP equation for the trap 0 energy = energya r r

2222 2 2

0 0 020

230 3

0

22

0

41

22

d ( )

8

1

s

s

am a r a a

mma

Na

a

a Nr

a

r r r

r r r r

r r r

198




2

22 21

2 2m r g

m

r r r

23 3d ( ) d ( )n N r r r r

Dimensionless GP equation for the trap 0 energy = energya r r

a single dimensionless

para e

2

met r

2222 2 2

0 0 020

230 3

0

22

0

41

22

d ( )

8

1

s

s

am a r a a

mma

Na

a

a Nr

a

r r r

r r r r

r r r

199



(dashed - - - - -)


perfectly reproduced by the solution of the GP equation

200



(dashed - - - - -)


perfectly reproduced by the solution of the GP equation

guess:internal pressure due to repulsioncompetes with the trap potential

201

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

2 3INT 0

2KIN 0 0

s sE N a a NagNn

E N Na a

202


Qualitative



Quantitative


2 3INT 0

2KIN 0 0

s sE N a a NagNn

E N Na a

203


Qualitative



Quantitative


2 3INT 0

2KIN 0 0

s sE N a a NagNn

E N Na a

204


Qualitative



Quantitative


2 3INT 0

2KIN 0 0

s sE N a a NagNn

E N Na a

unlike in an extended homogeneous gas !!

205


INT

KI

20

N

3

20 0 4s sNg N a a Nan

N Na a

EE

Qualitative



Quantitative


206

Qualitative



Quantitative



2

20ma

24 sag

m

2 3INT 0

2KIN 0 0 4

s sN a a NaNgn

N Na a

EE

207


Qualitative



Quantitative


2 3INT 0

2KIN 0 0 4

s sN a a NaNgn

N Na a

EE

208


weak individual collisions

1collective effect weak or strong depending on N

can vary in a wide range

Qualitative



Quantitative


2 3INT 0

2KIN 0 0 4

s sN a a NaNgn

N Na a

EE

209


weak individual collisions

collective effect weak or strong depending on N

can vary in a wide range

realistic value

0

1

Qualitative



Quantitative


2 3INT 0

2KIN 0 0 4

s sN a a NaNgn

N Na a

EE

Variational method of solving the GPE for atoms in a parabolic trap

212


level number

200 nK

400 nK

0/ xx a

2

20

0 0 0 0

2 22

0 0 0 2 20 00

( , , )

1 1 1 1( ) e , ,

2 2 2

x y z

u

a

m

x y z x y z

u a Em ma Mu aa

0

6

87

1 m

=10 nK

~ 10 at.

Rba

N

213


level number

200 nK

400 nK

0/ xx a

2

20

0 0 0 0

2 22

0 0 0 2 20 00

( , , )

1 1 1 1( ) e , ,

2 2 2

x y z

u

a

m

x y z x y z

u a Em ma Mu aa

0

6

87

1 m

=10 nK

~ 10 at.

Rba

N

ground state orbital

214


level number

200 nK

400 nK

0/ xx a

2

20

0 0 0

2 22

0 0 0 2 20 00

0( , , )

1 1 1 1( ) e , ,

2 2 2

x y z

u

a

m

x y z x y z

u a Em ma Mu aa

0

6

87

1 m

=10 nK

~ 10 at.

Rba

N

ground state orbital

215

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.

2

2 1/ 43 20

12e ,

r

bA A b

r

Variational estimate of the condensate properties

SCALING ANSATZ

216

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.

2

2 1/ 43 20

12e ,

r

bA A b

r

Variational estimate of the condensate properties

SCALING ANSATZ

variational parameter b

217

Importance of the interaction: scaling approximation

2

2 1/ 43 20

12e ,

r

bA A b

r

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

218


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

0

1

2sN a

a

Variational parameter is the orbital width in units of a0

ADDITIONAL NOTES

2

2 1/ 43 20

12e ,

r

bA A b

r


dimension-less energy per particle

dimension-less orbital size

0N E E 22 3

0

3 1 1

4

bE

a

self-interaction

219



0

1

2sN a

a


ADDITIONAL NOTES

2

2 1/ 43 20

12e ,

r

bA A b

r




0N E E 22 3

0

3 1 1

4

bE

a

self-interaction

3/ 2

0

1

2

1

2 (2 )sN a

a

220



0

1

2sN a

a


ADDITIONAL NOTES

2

2 1/ 43 20

12e ,

r

bA A b

r




0N E E 22 3

0

3 1 1

4

bE

a

self-interaction

5

minimize the en

2

e gy

0 0

r

E

221



0g 0g




0

1

2sN a

a

222



0g 0g




0

1

2sN a

a

5

minimize the en

2

e gy

0 0

r

E

223



0g 0g


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

( )E

1/ 5

min



0

1

2sN a

a

224



0g 0g


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

( )E

1/ 5

min

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

0.27 0 0.27c



0

1

2sN a

a

The end

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