Cold atoms Lecture 3. 18 th October, 2006. Non-interacting bosons in a trap

Cold atoms

Lecture 3.18th October, 2006

Non-interacting bosons in a trap

3

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

4

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

5

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

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

• characteristic energy

• characteristic length

7


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



8


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



9

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

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

x

y

x E

10

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


11

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


12

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

13

How good is the thermodynamic limit1D illustration (almost doable)

0( ) )(

1 11 1d

e 1 e 1 e 1jjN

?

14

How good is the thermodynamic limit1D illustration

0( ) )(

1 11 1d

e 1 e 1 e 1jjN

?

15Bk T

15


0( ) )(

1 11 1d

e 1 e 1 e 1jjN

?

/

red bars… the sum

firstterm

continuousBE step function,

whose integralequals the rest

of the sum

15Bk T

16


0( ) )(

1 11 1d

e 1 e 1 e 1jjN

?

/

red bars… the sum

firstterm

continuousBE step function,

whose integralequals the rest

of the sum

15Bk T

The quantitative criterion for the thermodynamic limit

1B Ck T

17

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

18

Seeing the condensate – reminescence of L2

double average, quantum and thermal

= insert unit operator

chang

de

e the summation

fine the one-particle density mat

rix

order

|

Tr

X n

X n

n X

X

X

X

= n

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:

19

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

20

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

21

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

22


THERM

BEC THERM

(000)

22

000(000)

BEC

( )

( )

known laborious

( )

1 1= 000 000

e 1 e 1

1 1( ) ( )

e 1 e 1

( ) ( )

E

E

n

n n

r r r r r r r

r r r r

r r

r r


23


THERM

BEC THERM

(000)

22

000(000)

BEC

( )

( )

known laborious

( )

1 1= 000 000

e 1 e 1

1 1( ) ( )

e 1 e 1

( ) ( )

E

E

n

n n

r r r r r r r

r r r r

r r

r r


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

rThe characteristic lengths directly observable

24

Particle Density in Space: Boltzmann Limit

We approximate the thermal distribution by its classical limit.

Boltzmann distribution in an external field:

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

25




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:

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

26



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

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

27



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

28

Real space Image of an Atomic Cloud

29


30


31


• 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

32

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

r r

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

33

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


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

34

BEC observed by TOF in the velocity distribution

Qualitative features: all Gaussians

wide vz.narrow

isotropic vs. anisotropic

The end

36

Documents

Cold atoms Lecture 3. 18 th October, 2006. Non-interacting bosons in a trap