Cold atoms Lecture 3. 17. October 2007. BEC for interacting particles Description of the interaction Mean field approximation: GP equation Variational

Cold atoms

Lecture 3.17. October 2007

BEC for interacting particlesDescription of the interactionMean field approximation: GP equationVariational properties of the GP equation

3

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

4

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.

5

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

6

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

7


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

minimum

vdW radius 1

62

8


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

9





6TRUE 6


U rr

6 2 26 6

1/ 4

For the 6 - 12 potential

4 4 2C m

26

2 66

26

6

6

1/ 4


2

2

"

1

m

C

m

C

10





6TRUE 6


U rr

26

2 66

26

6

6

1/ 4


2

2

"

1

m

C

m

C

rough estimate of the last bound state energy

B collision energy of the

condensate a m

to sCk T

11

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 ("pseudopotential")

3

0sin( )kr

r

... the scattering leng t, hs sr a a

2

( ) ( )

4 s

U r g

ag

m

r

12

Experimental data

as

13

Experimental data

nm

3.44.7

6.88.78.7

10.4

nm

-1.44.1

-1.7-1.95.6

127.2

1 a.u. = 1 bohr 0.053 nm

as

14

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

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

as

Mean-field treatment of interacting atoms

16

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

Most of the interactions is 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

ADDITIONAL NOTES

17

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

Most of the interactions is 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.

2Hartree

2

2

1ˆ ( ) ( )2

'( ) d ( ) ( ), ( )

( )

a a H aa

H a b a b b

E

H p V Vm

eV U n U r

r

n

r r

r r r r r

r r

21( ) ( )

2 Hp V V Em

r r r r

ELECTRONS

18

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.

19

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

single self-consistent equation

for a single orbital

20



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



21



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??



22



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??



23



Putting



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

• concerns a macroscopic quantity • suitable for numerical solution.

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??



24

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

25




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



26




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

27

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

ADDITIONAL NOTES

28

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

29



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

30



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!! !

EN

Interacting atoms in a constant potential

32

The simplest case of all: a homogeneous gas

2 ( )

2 ( )

n

m n

r

r

( ) ( )

V g

V

n

g n

r r

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

the GP equation simplifies

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

33


2 ( )

2 ( )

n

m n

r

r

( ) ( )

V g n

g n V

r r




34


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


35


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

36


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

212N

EV

37


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

212N

EV

38


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

212N

EV

impossible!!!

Interacting atoms in a parabolic trap

40

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

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

41

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

42




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 0 020

230 3

0

22

0

41

22

d ( 1)

8

s

s

am a r a a a

mma

Na

a

a Nr

a

r r r

r r r r

r r r

43




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 0 020

230 3

0

22

0

41

22

d ( 1)

8

s

s

am a r a a a

mma

Na

a

a Nr

a

r r r

r r r r

r r r

a single dimensonless

para e

2

met r

44

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

45

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 compete, the condensate 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

46


Qualitative



Quantitative


2 3INT 0

2KIN 0 0 4

s sN a a NagNn

N Na a

EE

47


Qualitative



Quantitative


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 NagNn

N Na a

EE

48


Qualitative



Quantitative


weak individual collisions

collective effect weak or strong depending on N

can vary in a wide range

realistic value

02 3INT 0

2KIN 0 0 4

s sN a a NagNn

N Na a

EE

1

The end

50

On the way to the mean-field Hamiltonian

ADDITIONAL NOTES

51

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

52


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'

53


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

substitute back

and integrate

(ˆ( )) aa

n r rr HV r

BACK

54

ADDITIONAL NOTES

Variational approach to the condensate ground state

55

Variational estimate of the condensate properties

ADDITIONAL NOTES

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 for non-interacting particles (Hartree Ansatz, here identical with the symmetrized Hartree-Fock)

Sfor allˆ ˆ' ' ' , ' ' 1NE H H H'

equivalent with the SR ˆ ˆ0 H 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

56

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


ADDITIONAL NOTES

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

Explicit calculation yields

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

BACK

57


ADDITIONAL NOTES

ANNEX Interpretation of the Lagrange multiplier The idea is to identify it with the chemical potential. First, we modify the notation to express the particle number dependence

The first result is that is not the average energy per particle:

2 3 40 0 0 0 0 0

2 3 40 0 0 0 0from

1 1/ ]/ 1 d

2 21

the GPE 1 d2

N N N N N N N N

N N N N N N

E N N p V N gm

p V N gm

r

r

E

2 3 4

220 0 0 0

1 1] 1 d

2 2

1], ( )

2

N

N N N N N N

N p V N gm

E p V N gm

-1

r

r r r r

E

E

58


ADDITIONAL NOTES

Compare now systems with N and N -1 particles:

0 1 0 1 0 1, 1] ] ]N N N N N N N NNN NE E E E E

N … energy to remove a particle without relaxation of the condensate

use of the variational principle for GPE

In the "thermodynamic"asymptotics of large N, the inequality tends to equality. This only makes sense, and can be proved, for g > 0.

Reminescent of the Koopmans’ theorem in the HF theory of atoms. Derivation:

1 12 3 42 2

1 12 3 41 2 2

1 12 3 41 2 2

0

] 1 d

] 1 1 1 2 d

1 1 2 d

o f r

N m

N m

N N m

N N

N p N V N N g

N p N V N N g

p V N N N N g

r

r

r

E

E

E E

BACK

59

2 2 2 2 2 20 0

1 12 2V m r m x y z r


ADDITIONAL NOTES

SCALING ANSATZ FOR A SPHERICAL PARABOLIC TRAP

The potential energy has the form

Without interactions, the GPE reduces to the SE for isotropic oscillator

The solution (for the ground state orbital) is

We (have used and) will need two integrals:

2 2 20 0 0 0

312 2

1

2p m r

m

r r

2

20

2 1/ 43 200 0 0 0 0 02

0 0

12

e , ,

r

aA a A am ma

r

2 2

2 2 2 31 2

12d e , d e

u u

I u I u u

60

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.

Next, the total energy is calculated for this orbital

The solution (for) is

2

2 1/ 43 20

12e ,

r

bA A b

r


ADDITIONAL NOTES

SCALING ANSATZ

2 2 2

2 2 2

22 2 43 3 3

0 0 0 0

2 26 3 2 3 2 6 30 0

0 4 2 20

2

1] d d 1 d

2 2

1 1d e d e 1 d e

2

r r r

b b b

N N V N N gm

a maNA r r N A g

b a

r r r r r r r

r r r

E

61


ADDITIONAL NOTES

22

2 2 200 0 0 2

10

41 1, , , saa m A g

m I b ma b

0

32 2 2 2 112 0 0

0 2 4 2 3 3 / 2 2 301 1

]

/ 23 41 11

22sI bI bI b a ma a

N Nmb a bb I b I b

E

2 320 0

0 0 02 2 30 0

22 3

0

13]

4 2

3 1 1

4

sNa a abN N E

b a a b

bE

a

E

dimension-less energy per particle

dimension-less orbital size

For an explicit evaluation, we (have used and) will use the identities:

The integrals, by the Fubini theorem, are a product of three:

Finally,

This expression is plotted in the figures in the main lecture.

BACK

The end

Documents

Cold atoms Lecture 3. 17. October 2007. BEC for interacting particles Description of the interaction Mean field approximation: GP equation Variational