Statistical methods for bosons Lecture 2. 9 th January, 2012

Statistical methods for bosons

Lecture 2.9th January, 2012

2

Short version of the lecture plan: New version

Lecture 1

Lecture 2

Dec 19

Jan 9

Reminder ofLecture I.

Offering many new details and alternative angles of view

4

Ideal quantum gases at a finite temperature

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

( ) Boltzmann distribution

high temperatures, dilute g as e s

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

energy

Aufbau principle

5

Trap potential

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

6

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

0

6

87

1 m

=10 nK

~ 10 at.

Rba

N

• characteristic energy

• characteristic length

7

BEC observed by TOF in the velocity distribution

Qualitative features: all Gaussians

wide vs.narrow

isotropic vs. anisotropic

BUT

The non-interacting model is at most qualitative

Interactions need to be accounted for

9

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

Today: BEC for interacting

bosonsMean-field theory for zero temperature condensatesInteratomic interactionsEquilibrium: Gross-Pitaevskii equationDynamics of condensates: TDGPE

Inter-atomic interaction

Interaction between neutral atoms is weak, basically van der Waals. Even

that would be too strong for us, but at very low collision energies, the efective interaction potential is much weaker.

12

Are the interactions important?

Weak interactions In the dilute gaseous atomic clouds in the traps, the interactions are incomparably weaker than in liquid helium.

Perturbative treatment 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

• thermalization the atomic collisions take care of thermalization

• mean field 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

13

Fortunate properties of the interactions

The atomic interactions in the dilute gas favor formation of long lived quasi-equilibrium clouds with condensate

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 almost elastic and spin independent: they only weakly 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.

I.

Interacting atoms

15

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

16


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

17


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

18

Experimental data

as

19

Experimental data

05

1 a.u. length = 1 bohr 0.053 nm

1 a.u. energy = 1 hartree 3.16 10 K

for “ordinary” gases

nm

3.44.7

6.88.78.7

10.4

(K)

5180940

------73---

20

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")

63

0sin( )kr

r

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

2

( ) ( )

4 s

U r g

ag

m

r

0k

21

Experimental data

(K)

5180940

------73---

nm

-1.44.1

-1.7-19.5

5.6127.2

"well behaved”; decrease increase monotonous

seemingly erratic, very interesting physics of Feshbach scattering resonances behind

for “ordinary” gases VLT cloudsasnm

3.44.7

6.88.78.7

10.4

II.

The many-body Hamiltonian for interacting atoms

Many-body Hamiltonian

21 1ˆ ( ) ( )2 2a a a b

a a b

H p V Um

r r r


21 1ˆ ( ) ( )2 2a a a b

a a b

H p V Um

r r r

2

( ) ( )

4, ... the scattering leng ths

s

U r g

ag a

m

r

At low energies (micro-kelvin range), true interaction potential replaced by an effective potential, Fermi pseudopotential


21 1ˆ ( ) ( )2 2a a a b

a a b

H p V Um

r r r

2

( ) ( )


s

U r g

ag a

m

r


Experimental datafor “ordinary” gases VLT clouds

as


21 1ˆ ( ) ( )2 2a a a b

a a b

H p V Um

r r r

2

( ) ( )


s

U r g

ag a

m

r



as


21 1ˆ ( ) ( )2 2a a a b

a a b

H p V Um

r r r

2

( ) ( )


s

U r g

ag a

m

r



as NOTES

weak attraction okweak repulsion ok

weak attractionintermediate attractionweak repulsion okstrong resonant repulsion

Many-body Hamiltonian: summary

21 1ˆ ( ) ( )2 2a a a b

a a b

H p V Um

r r r

2

( ) ( )


s

U r g

ag a

m

r



as NOTES

weak attraction okweak repulsion ok

weak attractionintermediate attractionweak repulsion okstrong resonant repulsion

III.Mean-field treatment of interacting atoms

In the mean-field approximation, the interacting particles are replaced by independent particles moving in an

effective potential they create themselves

2

2GP H const.

1 1ˆ ( ) ( )2 2

1ˆ ( ) ( ) 2

a a a ba a b

a a aa

H p V Um

H p V Vm

r r r

r r

30

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

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

2

2GP H const.

1 1ˆ ( ) ( )2 2

1ˆ ( ) ( ) 2

a a a ba a b

a a aa

H p V Um

H p V Vm

r r r

r r

31

HARTREE APPROXIMATION ~ many electron systems

mean field approximation Most of the interactions absorbed into the mean field explicit quantum correlation corrections remain


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


2

2GP H const.

1 1ˆ ( ) ( )2 2

1ˆ ( ) ( ) 2

a a a ba a b

a a aa

H p V Um

H p V Vm

r r r

r r

32




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


?

2

2GP H const.

1 1ˆ ( ) ( )2 2

1ˆ ( ) ( ) 2

a a a ba a b

a a aa

H p V Um

H p V Vm

r r r

r r

33




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

2

2GP H const.

1 1ˆ ( ) ( )2 2

1ˆ ( ) ( ) 2

a a a ba a b

a a aa

H p V Um

H p V Vm

r r r

r r

34




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


35

On the way to the mean-field Hamiltonian

ADDITIONAL NOTES

36

On the way to the mean-field HamiltonianADDITIONAL NOTES

First, the following exact transformations are performed

TRICK!!

2

3 3

3 3

3 3

1 1ˆ ( ) ( )2 2

( ) d ( ) ( ) d ( )

1 1( ) d d ' ' '

2 2

1d

ˆ

d ' '

ˆ ˆ

'

ˆ

ˆ

ˆ

)

2

(

'

a a a ba a a b

a aa a

a b a ba b a b

a ba b

H p V Um

V V V

U U

U

U

U

V

W V

n

r r

r r r

r r r r r r r

r r r r r r r r r r

r r r r r r r r

r

ˆ( )n r' eliminates SI (self-interaction)

particle density operator

3 3 3ˆ ˆ ˆ(1ˆ d ( ) d d ' ' '2

) ) )ˆ ( (n nU nH VW r r rr 'r r rr r r r

ˆ( )n r

37


Second, a specific many-body state is chosen, which defines the mean field:

Then, the operator of the (quantum) density fluctuation is defined:

The Hamiltonian, still exactly, becomes

3 3

3 3

3 3

ˆ d ( ) d ' ( )

1d d ' ' ( ) ( )

21

d d ' ' '

ˆ( )

ˆ2

ˆ ˆ( ) ( ) ( )

ˆH V U n

U n

W n

n nU n

n

r r r' r r r'

r r r r

r

r r' r

r r'

r r r r r r

ˆ ˆ( ) ( ) ( )n n n r r r

ˆ ˆ

ˆ ˆ ˆ ˆ ˆ ˆ

n n n

n n n n n n n n n n

r r r

r r' r r' r r' r r' r r'

38


In the last step, the third line containing exchange, correlation and the self-interaction correction is neglected. The mean-field Hamiltonian of the main lecture results:

3 3

3 3

3 3

ˆ d ( ) d ' ( )

1d d ' ' ( ) ( )

2

1d d ' ' '

ˆ( )

ˆ2

ˆ ˆ( ) ( ) ( )

ˆH V U n

U n

W n

n nU n

n

r r r' r r r'

r r r r

r

r r' r

r r'

r r r r r r

REMARKS

• Second line … an additive constant compensation for double-counting of the Hartree interaction energy

• In the original (variational) Hartree approximation, the self-interaction is not left out, leading to non-orthogonal Hartree orbitals

• The same can be done for a time dependent Hamiltonian

substitute back

and integrate

(ˆ( )) aa

n r rr HV r

39


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


only occupied orbitals enter the cycle

HARTREE APPROXIMATION

particle-particle pair correlations neglected the only relevant quantity . . . single-particle density

2GP

1ˆ ( ) ( ) const.2 a a H a

a

H p V Vm

r r

40

Hartree approximation for bosons 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.

41


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

Hartree approximation for bosons at zero temperature

42

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



43


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.



44



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



45



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??



46



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??



47

Gross-Pitaevskii equation – "Bohmian" form

For a static condensate, the order parameter has CONSTANT PHASE. Then the Gross-Pitaevskii equation

becomes

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

220

23 3

0i

( ) | ( ) | ( )

[ ] d ( ) d ( )

( )( )) ( en

n N

n N n N

N

r r r

r r

r

r

r

r

r

N

48

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).

[ ] minn E

[ ]n NN

[ ] [ ] 0n n E N

23 2 3 3 2

QP POT INT

12[ ] d ( ) d ( ) ( ) d ( )

2 +

quantum pressure external potential

+

interaction

( )n n V n g nm

E

E E E

r r r r r r r

49



It is required that


that is


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

50



It is required that


that is


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

51

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

52



It is required that


that is


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

in thermodynamic limit

53



It is required that


that is


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

!! !EN

IV.Interacting atoms in a homogeneous gas

55

Gross-Pitaevskii equation – homogeneous gas

The GP equation simplifies

For periodic boundary conditions in a box with

0

0

2

2

23 2

... GP equ

1( )

( ) ( )

1d

ation

2( )

V

NN n

V

g

g gn

nN m

r

r r

r r r

r

rEN

( )V n r 21 1

2 2g n g n

2

2

2g

m

r r r

x y zV L L L

56

The simplest case of all: a homogeneous gas

2 21 12 2

internal presstotal energy ure

gn gn

E

E V PV

212 g

NE

V

57


2 21 12 2


gn gn

E

E V PV

212 g

NE

V

58


If , it would be

and the gas would be thermodynamically unstable.

0g 0n

2 21 12 2


gn gn

E

E V PV

212 g

NE

V

59


If , it would be

and the gas would be thermodynamically unstable.

0g 0n

impossible!!!

2 21 12 2


gn gn

E

E V PV

212 g

NE

V

V.Condensate in a container with hard walls

61

A container with hard walls

A semi-infinite system … exact solution of the GP equation

n ( )n r

2 ( )

(( )

2 )V

n

nm

r

rr

( )g n r

62

n ( )n r


A semi-infinite system … exact analytical solution of the GP equation

Particles evenly distributedDensity highDensity gradient zeroInternal pressure due to interactions prevails

2 ( )

(( )

2 )V

n

nm

r

rr

( )g n r

63



n ( )n r

Particles pushed away from the wallDensity lowDensity gradient largeQuantum pressure prevails

2 ( )

(( )

2 )V

n

nm

r

rr

( )g n r


64



n ( )n r


2 ( )

(( )

2 )V

n

nm

r

rr

( )g n r

12

22

2

healing length

41

2

8 s

san

a n

m m


65



n ( )n r


2 ( )

(( )

2 )V

n

nm

r

rr

( )g n r

12

22

2

healing length

41

2

8 s

san

a n

m m

2( ) tanh /n n x r


66


no interactions

Interactions gradually flatten out the density in the bulk

Near the walls there is a depletion region about wide

VI.Interacting atoms in a parabolic trap

68

Parabolic trap with interactions

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

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

69




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

70




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

71

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

72


Qualitative


for g<0, attraction, "quantum pressure" acts against the interaction so as to form the condensate. Still, it shrinks and becomes metastable. Onset of three particle recombination processes

Quantitative


2 3INT 0

2KIN 0 0

s sE N a a NagNn

E N Na a

Qualitative



Quantitative


73


2 3INT 0

2KIN 0 0

s sE N a a NagNn

E N Na a

unlike in an extended homogeneous gas !!

74


INT

KI

20

N

3

20 0 4s sNg N a a Nan

N Na a

EE

Qualitative



Quantitative


Qualitative



Quantitative


75


2

20ma

24 sag

m

2 3INT 0

2KIN 0 0 4

s sN a a NaNgn

N Na a

EE

76


2 3INT 0

2KIN 0 0 4

s sN a a NaNgn

N Na a

EE

Qualitative



Quantitative


77


weak individual collisions

1collective effect weak or strong depending on N

can vary in a wide range

2 3INT 0

2KIN 0 0 4

s sN a a NaNgn

N Na a

EE

Qualitative



Quantitative


78


weak individual collisions

collective effect weak or strong depending on N

can vary in a wide range

realistic value

0

1

2 3INT 0

2KIN 0 0 4

s sN a a NaNgn

N Na a

EE

Qualitative



Quantitative


VIII.Variational approach to the condensate ground state

Idea:Use the variational estimate

of the condensate properties instead of solving the GP equation

85

General introduction to the variational method

VARIATIONAL PRINCIPLE OF QUANTUM MECHANICS

The ground state and energy are uniquely defined by

In words, is a normalized symmetrical wave function of N particles. The minimum condition in the variational form is

HARTREE VARIATIONAL ANSATZ FOR THE CONDENSATE WAVE F.

For our many-particle Hamiltonian,

the true ground state is approximated by the condensate of non-interacting particles (Hartree Ansatz, here identical with the symmetrized Hartree-Fock)

Sfor allˆ ˆ' ' ' , ' ' 1NE H H H'

equivalent with theˆˆ ˆ0 SR H E1 H E

21 1ˆ ( ) ( ), ( ) ( )2 2a a a b

a a b

H p V U U gm

r r r r r

1 2 0 1 0 2 0 0, , , , ,p N p N r r r r r r r r

86

10 0 0

23 3 3 3

0 0 0 0

0 0 0 0 0 0

]

2 4d 2 d 1 d

2

,

2

N

V N gm

BY PARTS

r r

r r r r

E

Here, is a normalized real spinless orbital. It is a functional variable to be found from the variational condition

Explicit calculation yields the total Hartree energy of the condensate

Variation of energy with the use of a Lagrange multiplier:

This results into the GP equation derived here in the variational way:

0 0 0 0 0 0 0with ˆ] ] ] 0 ] ] 1 1 H E

0

2

2 2 43 3 30 0 0 0

1] d d 1 d

2 2N N V N N g

m r r r r r r r

E

220 0 0

1( )

2p V N g

m

r r r r-1

eliminates self-interaction

General introduction to the variational method

Restricted search as upper estimate to Hartree energy

87

Start from the Hartree energy functional

Evaluate for a suitably chosen orbital having the properties

Minimize the resulting energy function with respect to the parameters

This yields an optimized approximate Hartree condensate

2

2 2 43 3 30 0 0 0

1] d d 1 d

2 2N N V N N g

m r r r r r r r

E

0 0

0 1 2

normalization

variational parameters

| 1

( ) ( , , , )pf

r r

0 1 2 1 2

1 2

] ( , , )] ( , , )

( , , ) 0 1, 2 , ,n

p p

p

f E

E n p

E E

Application to the condensate in a parabolic trap

89

Reminescence: The trap potential and the ground state

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

90

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.

2

2 1/ 43 20

12e ,

r

bA A b

r

Rescaling the ground state for non-interacting gas

SCALING ANSATZ

91

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

Rescaling the ground state for non-interacting gas

SCALING ANSATZ

variational parameter b

92

Scaling approximation for the total energy

2

2 1/ 43 20

12e ,

r

bA A b

r

Introduce the variational ansatz into the energy functional

93


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

Introduce the variational ansatz into the energy functional

dimension-less energy per particle

dimension-less orbital size

0N E E 22 3

0

3 1 1

4

bE

a

self-interaction

94



0

1

2sN a

a


ADDITIONAL NOTES

2

2 1/ 43 20

12e ,

r

bA A b

r

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



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

95

Minimization of the scaled energy


0

1

2sN a

a


ADDITIONAL NOTES

2

2 1/ 43 20

12e ,

r

bA A b

r

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



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

96

Importance of the interaction: scaling approximation


0g 0g


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

( )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

0

0

Now we will be interested in dynamics of the BE condensates

described by

Time dependent Gross-Pitaevskii equation

101

Time dependent Gross-Pitaevskii equation

Condensate dynamics is often described to a sufficient precision by the mean field approximation.

It leads to the Time dependent Gross-Pitaevskii equation

Intuitively,

Today, I proceed following the conservative approach using the mean field decoupling as above in the stationary case

Three steps:

1. Mean field approximation by decoupling

2. Assumption of the condensate macrosopic wave function

3. Transformation of the TD Hartree equation to TDGPE

2 *

2i ( , ) ( , ) ( , ) ( , ) ( , ) ( , )t mt V t t g t t t r r r r r r

102

1. Mean-field Hamiltonian by decoupling -- exact part

21 1ˆ ( , ) ( )2 2a a a b

a a a b

H p V t Um

r r r

3 3 31ˆ d ( , ) ˆ ˆ ˆd d ' ' '( ) ( ) ( )ˆ2

nW V U nH nt r r r r r rr r r' rr

ˆ ˆ( , ) ( ) ( )t t ttn t n n r r r

3 3

3 3

3 3

ˆ d ( , ) d ' ( ˆ( )

ˆ ˆ ˆ( , ) ( , ) ( )

, )

1d d ' ' ( , ) ( , )

2

' '

ˆ

1d d '

2

H V t U n t

U n

n

n n

n

t n

t

t

W

t

U

r

r

r r r' r r r'

r r r r r r'

r r r r r r' rr

103

1. Mean-field Hamiltonian by decoupling -- approximation

3 3

3 3

3 3

ˆ d ( , ) d ' ( ˆ( )

ˆ ˆ ˆ( , ) ( , ) ( )

, )

1d d ' ' ( , ) ( , )

2

' '

ˆ

1d d '

2

H V t U n t

U n

n

n n

n

t n

t

t

W

t

U

r

r

r r r' r r r'

r r r r r r'

r r r r r r' rr

3 3

3 3

3 3

ˆ ˆ ˆ( ) ( ) ( )1

d d ' ' '2

ˆ d ( , ) d ' ( , )

1d d ' ' ( , ) ( , )

2

( )ˆ ˆ

n n nU

H V t U n t

U

n

n t n

W

t

r r r rr rr r r'

r r r' r r r'

r r r r r r'

r

,HV t r

2GP

1ˆ ( , ) ( , ) ( )2 a a H a

a

H p V t V t tm

r r C

2. Condensate assumption & 3. order parameter

Assume the Hartree condensate

Then

and a single Hartree equation for a single orbital follows

define the order parameter

The last equation becomes the desired TDGPE:

1( ) ( , )

N

aa

t t

r

2( , ) | ( , ) |n t N tr r

220 0 0

1, ( , ) , ,

2ti t p V t gN t tm

r r r r

½0, ,t N t r r

221, ( , ) , ,

2ti t p V t g t tm

r r r r

105

Physical properties of the TDGPE

2 *

2i ( , ) ( , ) ( , ) ( , ) ( , ) ( , )t mt V t t g t t t r r r r r r

1. In the MFA, the present coincides with the order parameter defined from the one particle density matrix and (at zero temperature) all particles are in the condensate

2. TDGPE is a non-linear equation, hence there is no simple superposition principle

3. It has the form of a self-consistent Schrödinger equation: the same type of initial conditions, stationary states, isometric evolution

4. The order parameter now typically depends on both position and time and is complex. It is convenient to define the phase by

5. The MFA order parameter (macroscopic wave function) satisfies the continuity equation

*( , ; ) ( , ) ( , )t t t r r r r

0i ( , )( , ) ( , ) e tt n t rr r

106

Physical properties of the TDGPE

Stationary state of the condensate for time-independent

Make the ansatzi /( , ) ( ) e tt r r

Then satisfies the usual stationary GPE

This is a real-valued differential equation and has a real solution

All other solutions are trivially obtained as

22

2m V g r r r r

0n r

0i( ) en r r

,V tr

107

Physical properties of the TDGPEContinuity equation

The TDGPE and its conjugate:

2

2

*

* * * *

2

2

i ( , ) ( , ) ( , ) ( , ) ( , ) ( , )

i ( , ) ( , ) ( , ) ( , ) ( , ) ( , )

t

t

m

m

t V t t g t t t

t V t t g t t t

r r r r r r

r r r r r r

Just like for the usual Schrödinger equation, we get

2 * *0 0 ( )2 i( ) ( ) ( )

tmn t t t r,

r, r, j r, 0 0 0tn j

Only complex fields are able to carry a finite current (quite general)

Employ the form

The current density becomes

0i ( , )( , ) ( , ) e tt n t rr r

0 0 0 0( ) ( ) ( ) ( ) ( )mt n t t n t t j r, r, r, r, v r,

108



2

2

*

* * * *

2

2

i ( , ) ( , ) ( , ) ( , ) ( , ) ( , )

i ( , ) ( , ) ( , ) ( , ) ( , ) ( , )

t

t

m

m

t V t t g t t t

t V t t g t t t

r r r r r r

r r r r r r


2 * *0 0 ( )2 i( ) ( ) ( )

tmn t t t r,

r, r, j r, 0 0 0tn j


Employ the form


0i ( , )( , ) ( , ) e tt n t rr r

0 0 0( ) ( ) ( ) ( ) ( )smt n t t n t t j r, r, r, r, v r,

phaser, t dependent

109



2

2

*

* * * *

2

2

i ( , ) ( , ) ( , ) ( , ) ( , ) ( , )

i ( , ) ( , ) ( , ) ( , ) ( , ) ( , )

t

t

m

m

t V t t g t t t

t V t t g t t t

r r r r r r

r r r r r r


2 * *0 0 ( )2 i( ) ( ) ( )

tmn t t t r,

r, r, j r, 0 0 0tn j


Employ the form


0i ( , )( , ) ( , ) e tt n t rr r phase

gradientsuperfluid

velocity field


110



2

2

*

* * * *

2

2

i ( , ) ( , ) ( , ) ( , ) ( , ) ( , )

i ( , ) ( , ) ( , ) ( , ) ( , ) ( , )

t

t

m

m

t V t t g t t t

t V t t g t t t

r r r r r r

r r r r r r


2 * *0 0 ( )2 i( ) ( ) ( )

tmn t t t r,

r, r, j r, 0 0 0tn j


Employ the form


0i ( , )( , ) ( , ) e tt n t rr r


phasegradient

superfluidvelocity field

The superfluid flow is irrotational (potential):

But only in simply connected regions – vortices& quantized circulation!!!

rot 0s v

111

Physical properties of the TDGPEIsometric evolution

Three formulations

The norm of the solution of TDGPE is conserved:

This follows from the continuity eq. 0 0 0tn j

2d ( ) 0t t r r,

2

0d ( ) d ( ) 0t SSt t r r, S j r ,

The condensate particle number is conserved (condensate persistent)

0 0d ( ) const.N n t r r,

Finally, isometric evolution

2 2d ( ) const.

tt r r,

I avoid saying "unitary", because the evolution is non-linear

A glimpse at applications

1. Spreading of a cloud released from the trap

2. Interference of two interpenetrating clouds

113

1. 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

114




and shape




"molasses"




gravity

115




and shape




"molasses"




gravity


fashion

116




and shape




"molasses"




gravity


fashion

117

1. Example of calculation

Even without interactions and at zero temperature, the cloud would spread quantum-mechanically

The result would then be identical with a ballistic expansion, the width increasing linearly in time

The repulsive interaction acts strongly at the beginning and the atoms get "ahead of time". Later, the expansion is linear again

From the measurements combined with the simulations based on the hydrodynamic version of the TDGPE, it was possible to infer the scattering length with a reasonable outcome Castin&Dum,

PRL 77, 5315 (1996)

To an outline of a simplified treatment

118

2. Interference of BE condensate in atomic clouds

Sodium atoms form a macroscopic wave function

Experimental proof:

Two parts of an atomic cloud separated and then interpenetratin again

interfere

Wavelength on the order of hundreth of millimetre

experiment in the Ketterle group.

Waves on water

119

Basic idea of the experiment

1. Create a cloud in the trap

2.Cut it into two pieces by laser light

3. Turn off the trap and the laser4. The two clouds spread, interpenetrate and interfere

120

2. Experimental example of interference of atoms

Two coherent condensates are interpenetrating and interfering. Vertical stripe width …. 15 m

Horizontal extension of the cloud …. 1,5mm

121

2. Calculation for this famous example

It is an easy exercise to calculate the interference of two spherical non-interacting clouds

Surprisingly perhaps, interference fringes are formed even taking into account the non-linearities – weak, to be sure

To account for the interference, either the full Euler eq. including the Bohm quantum potential – OR

TD GPE, which is non-linear: only a numerical solution is possible (no superposition principle)

Theoretical contrast of almost 100 percent seems to be corroborated by the experiments, which appear as blurred mostly by the external optical transfer function

A.Röhrl & al.PRL 78, 4143 (1997)

122

2. Calculation for this famous example

It is an easy exercise to calculate the interference of two spherical non-interacting clouds

Surprisingly perhaps, interference fringes are formed even taking into account the non-linearities – weak, to be sure

To account for the interference, either the full Euler eq. including the Bohm quantum potential – OR

TD GPE, which is non-linear: only a numerical solution is possible (no superposition principle)

Theoretical contrast of almost 100 percent seems to be corroborated by the experiments, which appear as blurred mostly by the external optical transfer function

A.Röhrl & al.PRL 78, 4143 (1997)

Idealized theory

Realistic theory

Experim. profile

Linearized TDGPE

124

Small linear oscillations around equilibrium

2 *2i ( , ) ( ) ( , ) ( , ) ( , ) ( , )t mt V t g t t t r r r r r r

I. Stationary state of the condensate

0i /( , ) ( ) e tt r r

Stationary GPE

All solutions are obtained from the basic one as

I

22

2 ( )m V g r r r r

0i( ) en r r

125


2 *2i ( , ) ( ) ( , ) ( , ) ( , ) ( , )t mt V t g t t t r r r r r r

I. Stationary state of the condensate

0i /( , ) ( ) e tt r r

Stationary GPE

All solutions are obtained from the basic one as

II. Small oscillation about the stationary condensate

Substitute into the TDGPE and linearize (autonomous oscillations, not a linear response)

22

2 ( )m V g r r r r

0i( ) en r r

0( , ) ( , ) ( , )t t t r r r

126

where


2

2 2 *0 0

2i ( , ) ( ) ( , )

2 ( , ) ( , ) ( , ) ( , )

t mt V t

g t t g t t

r r r

r r r r

III. Linearized TD GPE for the small oscillation

0 0DROPTHIS

ii /( , ) e ett n

r r

127

where


2

2 2 *0 0

2i ( , ) ( ) ( , )

2 ( , ) ( , ) ( , ) ( , )

t mt V t

g t t g t t

r r r

r r r r


0 0DROPTHIS

ii /( , ) e ett n

r r

128

where

IV. Ansatz for the solution

Substitute into the linearized TDGPE and its conjugate

to obtain coupled diff. eqs. for (which are just Bogolyubov – de Gennes equations)


2

2 2 *0 0

2i ( , ) ( ) ( , )

2 ( , ) ( , ) ( , ) ( , )

t mt V t

g t t g t t

r r r

r r r r


0 0DROPTHIS

ii /( , ) e ett n

r r

suggestive of Bogolyubov , excit. energy with respect o t

i / i i( , ) e ( )e ( )et t tt u

r r rv

( ), ( )u r rv

129

precisely identical with the eqs. for the Bogolyubov transformation generalized to the inhomogeneous case (a large yet finite system with bound states).

For a large box, go over to the k- representation to recover the famous Bogolyubov solution for the homogeneous gas


V. Bogolyubov – de Gennes equations

i / i i( , ) e ( )e ( )et t tt u r r rv

2 2 *

22 * *

2

2

2

2

( ) 2

( ) 2

m

m

V g u g u

V g g u

r r r r r r

r r r r r r

v

v v

VI. Bogolyubov excitations of a homogeneous condensate

i / i ii i( , ) e e e e et t tt u k k kkr krr v

Excitation spectrum

( )k

k ( ) c k

gnc

m

k sound region

quasi-particles are collective excitations

high energy region

quasi-particles are nearly just GP particles

k

cross-over

defines scale for k in terms of healing length

2 22

2

24 2 1

2mgn

mk gn

asymptoticallymerge

2 22 2 22 2 2

2 2 2( ) (

DISPERSIO

) 2

N LAW

m m mgn gn gn k k k k k

GP

22

2

m gn

k

22

2m k

More about the sound part of the dispersion law

( ) c k

gnc

m

kEntirely dependent on the interactions, both the magnitude of the velocity

and the linear frequency range determined by g

Can be shown to really be a sound:

Even a weakly interacting gas exhibits superfluidity; the ideal gas does not.

The phonons are actually Goldstone modes corresponding to a broken symmetry

The dispersion law has no roton region, contrary to the reality in 4He

The dispersion law bends upwards quasi-particles are unstable, can decay

2

1BULK MODULUS

,2

" "

1: , V

VV

pV V

V gNc

Vm n

p

E

EE

132

VII. Interpretation of the coincidence with Bogolyubov theory

Except for Leggett in review of 2001, nobody seems to be surprised. Leggett means that it is a kind of analogy which sometimes happens.

A PARALLEL


Electrons (Fermions)

Atoms (Bosons)

Jellium Homogeneous gas

Hartree

Stationary GPE

Linearized TD Hartree

Linearized TD GPE

equivalent with RPA

equivalent with Bogol.

plasmons Bogolons (sound)

e , 0J Hn n V 0 0, Hn V gn

The end

134

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

Documents

Statistical methods for bosons Lecture 2. 9 th January, 2012