Duncan O'Dell et al- Rotons in Bose-Einstein Condensates: Engineering the Correlations of a Quantum Gas Using Light

8/3/2019 Duncan O'Dell et al- Rotons in Bose-Einstein Condensates: Engineering the Correlations of a Quantum Gas Using Light

1/43

Rotons in Bose-Einstein Condensates:

Duncan ODell University of Sussex, England

Stefano Giovanazzi University of St Andrews, Scotland

Gershon Kurizki Weizmann Institute, Israel

Engineering the correlations of a QuantumGas Using Light

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laser


2/43

Rotons in a BEC: plan of the talk1. Introduction to BEC: 1st order coherence

2. The current paradigm (dogma?) for BEC: short-range interactions

3. Laser-induced dipole-dipole interactions4. Electrostriction of a BEC by dipole-dipole forces

5. Superfluid helium: rotons

6. Rotons in a gaseous BEC: a quantum gas vs. a

quantum fluid7. Squeezing

8. Conclusions


3/43

1. Introduction to BEC:1st order coherence


4/43

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BEC is a phase transition:


5/43

1st order coherence: ','',G1 rrrrrr V!==!

Field operator: rarar ii

iJJ {

!=0

00

Macroscopic

occupation of

ground state:000

Naa }} (C-number)

Single particle

density matrix

NN }0

rNrr 000 )( J!=p= Condensate wavefunction/orderparameter

'',0

*

00

1 rrNrrG JJ!

GIANT MACROSCOPIC MATTER-WAVE

BEC is a phase transition of the

first order correlation function:

Off-diagonal long-range order

(O. Penrose & L. Onsager 1956)


6/43

and when two BECs overlap

Double BEC

(centre images)

M.R. Andrews et al,

Science 275, 637 (1997)

After expansion and

overlap matter wave

interference!


7/43

Wolfgang Ketterle

MIT, January 1997

Interference

1+1 = 0

and1+1 = 4


8/43

2nd order coherence:

rrrrrr ====! ''',G2

2nd order correlation = particle-particle correlation

= density-density correlation=entanglement

??

Answer: for a conventional BEC 2nd order coherence isgenerally very poor

Reason: very short range interactions


9/43

2. The current paradigm:short-range interactions


10/43

Low energy scattering between atoms with short range interactions

rkf

kririk

scate),(e~ U]

For any FINITE range potential:

UHU H cossine12

),( li

0

ll

l

Pk

lkf

g

!

!

Low energy limit :ka

k ll

!

! {

0

12

0

H

H0pk akf p@ ),( U

Scattering length a gives the range of the potential

4T

kr

kRkrsincf. hard-sphere scattering:

kR!0

H

R=a


11/43

S-wave scattering at low temperatures is a universalfeature

offinite ranged potentials

van der Waals potential falls off as 1/r6

Alkalis typically have a~nm

interatomic separation in a BEC l~db~100nm

Atomic BECs are very dilute, nearly ideal, quantum GASES

Pseudo-potential approximation: )'(4

)'(2

rr

g

m

arrV ! H

T

J

! )'()'()('][ 21 rrrVrdrdrE VVV

)()(4

)(2

)(2

22

2

rrNm

ar

mr ]]

T]Q]

JJ! Gross-Pitaevskii eqn.

Interaction energy functional

CURRENT PARADIGM FOR INTERACTIONS IN A BEC:


12/43

Current paradigm continued: Excitation spectrum

Theory (N.N. Bogoliubov 1947) Experiment (J. Steinhauer, R. Ozeri,

N. Katz and N. Davidson, PRL 88,

120407 (2002))

JJ

J

J

Q

[

[

Q

QVT

[

!

!

!!

!!

!

m

p

p

p

pck

p

mm

ac

cpm

pp

2)(

:largeFor

(phonons))(

:smallFor

energyfield-mean

potentialchemical

4where

2)(

2

s

2

2

s

2

s

2

2

2


13/43

General comments concerning correlations in systems with

short-range vs long-range interactions

Dilute system:1/3-n:spacingcleinterparti-mean

:lengthscattering a

Long-range interaction(Coulombic) :

1/3-

22

0

n:spacingcleinterparti-mean

/4:radiusBohr meaB

JTI!

Correlations propagated by particles:e.g. density wave (phonons).

Interaction plays minor role.

G(r-r)

Correlations propagated by

interaction:

Particle motion plays minor role.

V(r-r)


14/43

3. Laser-induced dipole-dipole

interactions

j

jj

ij

ii

d

rErrGErE0

dd

ret0 )()()( IE! {


15/43

Dipole-dipole interaction is a 4th order QED process:

forward scattering of a laser photon by an atom pair.

Energy shift of pair: E

I2


16/43

5S1/2

5P3/2

F

2

1

5D5/2

MF

1234

0 1 2 3 4-4 -3 -2 -1

3

2

1

0

5P1/2

1

2

377.11 THz (D1)

384.23 THz (D2)

Rubidium 87, I=3/2


17/43

? AqrrqrrqrqrqrrrrV

qVeec

IU

jiijjiijji

jijidd

cossincos3

1

cos,4

22

3

2

0

2

!

!

HH

IT

Erqrr

E(q) = dynamical polarizability of atoms

I = laser field intensity

q = laser field wave vector

e = laser field polarization

Vij = retarded dipole-dipole interaction tensor

ray+J4

3 rate of single atom spontaneousRayleigh scattering

Craig & Thirunamachandran,Molecular Quantum

Electrodynamics (Academic Press, London, 1984)

Treat atom-field interaction

classically (a very large

detuning from all atomic

resonance and very small

saturation parameter) whilstmaintaining quantum nature

of the external atomic state.

Fully retarded (laser-induced) dipole-dipole interaction


18/43

Insert Udd into mean-field equationfor atomic order parameter (r,t)

*=!x

=xH

H totH

tiJ

kinetic energy (negligible in the Thomas-Fermi limit)

22 ,2/ trdrmHkin =!

J

external potentialenergy (harmonic trap)

2/;, 222

rmVtrVdrH rhohoho [!=! dipo

le-dipo

lein

ter

actio

nene

rgy

22

,'','2/1 trrrUtrdrdrH dddd ==! s-wavescattering 42 ,/2 trdrmaH

s=! JT


19/43

*=!

x=x

H

H totH

tiJ

')'()'()(

)()(4

)()(2

)(

32dd

22

trap2

2

rdrrrUrN

rrNm

a

rVrm

r

!

]]

]]T

]]Q]

J

J

Generalised Gross-Pitaevskii Equation

(mean-field equation for the condensate)


20/43

4. Electrostriction and self-binding of aBEC by dipole-dipole forces


21/43

Single off-resonant linearly polarized laser

Gaussian ansatz for the

condensate wave function

Light polarization along z axis: Rayleigh

scattering forbidden in z direction.

`Superradiant collective Rayleigh

scattering and hence collective atomic

recoil suppressed.

Dipolar interaction causes a

compression of the condensatealong the z axis, and can cause

its self-trapping in the z direction

compression

self-trapping

A harmonically trapped cigar-shaped BEC is

tightly confined in the radial plane (the radial

size is equal or less than a wavelength)

x(

z(

? AkrrkkrkrrV

kyVc

IrU

zz

zz

cossinsinkrcoscos311

cos),(4

)(

22223

2

0

2

UU

IT

E

!

! rk


22/43

')'()'()()()(4

)()(2)(32

dd

22

trap2

2

rdrrrUrNrrNm

arVrmr ! ]]]]

T]]Q]

JJ

Parameter determining ratio

of dipole-dipole interaction

to s-wave scattering

(dimensionless intensity):

Generalized Gross-Pitaevskii eqn.

Collapse forI> 3/2 due to

instability caused by static r-3part

of dipole-dipole interactionac

mI

22

0

2

8

I

JTI

E!

Single off-resonant linearly polarized laser

Gaussian ansatz for the

condensate wave function

compression

self-trapping

A harmonically trapped cigar-shaped BEC is

tightly confined in the radial plane (the radial

size is equal or less than a wavelength)

x(

z(


23/43

Radial dependence of the

angularly averaged laser-

induced potential

r

u

rV !)(

More laser beams: Electromagnetically induced `gravity(at least 3 orthogonal beams) to force an angular average in the NEAR ZONE

r

u

Self-gravity

Bose star

White dwarfam

c

II 2

2

0

2

0 7

48

E

IT J

!"

m

nu

p

T[

42 !

D. OD, S. Giovanazzi,

G. Kurizki and V. Akulin

PRL 84, 5687 (2000)


24/43

5. Superfluid helium: rotons


25/43

Crucially, the range of the

interaction in helium is

approximately the same as

interatomic separation.

Superfluid helium:

a quantum fluid .

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26/43

The static structure factorS(q)

BEC

Scattering of light or a massive

particle (neutron)

NqSM

EEMr

r

mm

cqm

cq

qkkiHfM

kpkp

f

ifif

fiqf

f

ffii

)(

)condensateofstatefinalaboutcaret(don'

)(2

:rateTransition

)(ofF.T.where

E,E,bosonsN

,,ParticlestateFinalstateInitial

22

2

2

i

22

int

2

fii

fi

O

[HT

VV

HJVJO

JJ

II

!

!

!

!!

!!

JJ

JJ

Scattering experiment

measures S(q)directly

(total intensity scattered

in direction given by q)

mkqm

qmk

qk aaH

,,

in t cc

! O

ii

2

i

1)( JVVJ

JVJ

!|

qq

f

qf

NN

qS


27/43

The Feynman dispersion relation

)(2)(

22

kmS

k

k

JJ

![

Conventional wisdom:

no rotons in gaseous BEC

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cf.


28/43

6. Rotons in a gaseous BEC: aquantum gas vs. a quantum fluid


29/43

Setup:

1. No collective (`superradiant) Rayleigh scattering

2. Tightly trapped in radial direction (e.g. wr=1.5Laser)

effectively 1D system (radial excitations frozen out)

Ioffe-Pritchard trap + light

polarization-polarized atoms

Effective 1D interatomic potential

Assume radial wavefunction is a Gaussian (ground state oftrap) and integrate out radial coordinate.


30/43


1D reduced potential:

laserlaserrecoil /4),(

PakEkwU

zr

z

3.0,5.1 !! Iwr


31/43

Static structure function (a measure

of pair correlation) for various laser

intensitiestunable correlations!

Roton minimum in dispersion

relation due to atom-atom

correlations induced by the

dipole-dipole interactions.

D. OD, S. Giovanazzi and G. Kurizki

PRL 90

, 110

40

2 (200

3)


32/43

7. Squeezing


33/43

Bogoliubov ground state contains pairing correlations (depletion)

-

-

-

-

-

!!!!!

!!!!!

!!!!!

!!!!!!

!!!!!!!

!!!!!!

!!!!!!

!!!!!

1,1,1,1,4

0,0,2,2,4

0,0,1,1,2

0,0,0,0,0

2211021

22110

2

1

22110

1

22110B

kkkkkkk

kkkkkk

kkkkkk

kkkkk

nnnnNn

nnnnNn

nnnnNn

nnnnNn

FF

F

F

2

s

222s 2)(mc

mkmckUV

k

kk

JJ !! [Fwhere

This is perfect two-mode squeezing due to kpairing.

| kkkkk bVbUc :transf.BogoliubovN.B.

M M


34/43

? AU^ iexp)()( krk !Squeezing parameter:

Two-mode squeezing operator:

? A

s | kkkkk cckcckS

)()(exp)( *

^^^

? A ? A ? A TUU^^ p!! ss ,r(k)sinhiexpr(k)cosh)()( kkkkkk ccScSb

0,)(000

0B !g!4! {!s

{kkk

knnS ^Bogoliubov ground state

? A

1

1tanh

2

0

22

0recoil

!

!!z

z

z

zzz

kzkS

kS

kk

kkkEkkr

z

[F

JThen:

And using 122 !kk

VU

)(2

1)(

kS

kSUk

!

)(2

1)(

kS

kSVk

!

,


35/43

Squeezing parameter

? Azk kz ^F tanh!


36/43

Application: sub-quantum noise atom interferometry

rnrnn

NN

n

N2211

1 1

;;2

1

1

JJ!

!=


1122z2112

S,, aaaaaaSaaS

!!!

2-state system:

n1=0 n1=N

N

1}(N

N

1

!(N

Uncorrelated:

Correlated:


37/43

8. Static dipole-dipole interactions

S i Di l Di l I i


38/43

Magnetic dipole-dipole interaction:

the magnetic moments of the atoms

are aligned with a strong magnetic

field [Goral, Rzazewski, and Pfau, 2000]

Electrostatic dipole-dipole interaction:

(i) permanent electric moments (polar

molecules); (ii) electric moments

induced by a strong electric fieldE[Yi and You 2000; Santos, Shlyapnikov, Zoller

and Lewenstein 2000]

-

!

3

2

0

22 cos31

4)(

r

ErU

dd

U

TI

E

-

!

3

22

0cos31

4)(

rrUdd

U

T

QQ

tunability

+

-

+

-

+

-favourable

un-

favourable

long-range +

anisotropic

the atomic cloud likes

to be cigar-shaped

Static Dipole-Dipole Interactions

Two current experiments with

chromium BEC:

Tilman Pfau at Stuttgart

John Doyle at Harvard

EorH


39/43

Controlling dipole-dipole interactions by rapidly

rotating the external field

[Giovanazzi, Gorlitz & Pfau PRL 89, 130401 (2002)]

The sign of the interaction

can be reversed, or theinteraction can even be

averaged out completely

when

_ a? A

trapLarmor

0

where

t)sin(t)cos()sin()cos()(

[[

NN

"";""

;;! yxzBtB3

222

0cos31

2

1cos3

4 rU

tdd

.NT

QQ

!

angle'`magicthe7.54 r!N r.

N

z


40/43

-

!

3

2

dddd

cos31

4

)(

r

CrU

U

T

long-range:

short-range: )()(4

)(2

rgrm

arUs HH

T|!

J

gC3

dddd |I

collapse1dd "I

007.0dd !I

004.0dd !I

089.0dd !I

360.0dd !I

Magnetic dipole-dipole:87Rb

Na

52Cr

50Cr

Dipole-dipole vs s-wave

MOLECULES?.dipole moment 1 Debye.


41/43

Coulomb-like interactions in quasi 1-D cigar

-

!

22

2dd2

z

tot

2

13121

2)(

R

kRR

gkU

z

zI

T

Effective 1-D potential in

cigar with dipole-dipole

interactions

z

Rr

r

/2e

4

1

TF.T. of

R

m

pgn

m

pE

22

2

22

2

Bog

!

Bogoliubov

dispersion relation

(s-wave only)

2

4

k

u

r

u T

m

nu

p

T[

42 !Coulombinteraction:

Plasmon gap

Choose sign ofIddby rotating field

2

22

Bog2

pm

pE [

!

Trap cutoff2

1

k

1

12k

E

k

p[

Reduced phase noise


42/43

`Roton-minimum in quasi 1-D BEC with static

dipole-dipole interactionsz

R

Rotons in a pancake shaped BEC with static dipole-dipole interactions

Santos, Shlyapnikov, and Lewenstein PRL 90, 250403 (2003)

See also:

E[h[trap]

k [1/R]

Repulsive dipole-dipole:

5.4dd !I [J!)0(gn

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liquid helium)(2

)(22

kmS

kk

JJ ![


43/43

Conclusions

Induced dipole-dipole interactions are very

long-range, which is novel in a BEC

Easily tunable via laser intensity andpolarisation, spatial configuration of a number

of lasers etc.

Introduce long-range (laser wavelength)correlations

Quantum gas quantum liquid : rotons

Documents

Duncan O'Dell et al- Rotons in Bose-Einstein Condensates: Engineering the Correlations of a Quantum Gas Using Light