1. Bose-Einstein condensation and long range order

INTERNATIONAL SCHOOL OF PHYSICS "ENRICO FERMI" Varenna, July 1st - July 11th 2008 " QUANTUM COHERENCE IN SOLID STATE SYSTEMS "

Sandro Stringari

University of Trento

Introduction to Bose-Einstein condensation

CNR-INFMCNR-INFM

Lecture 2. Superfluidity and hydrodynamics.

Lecture 3. GP equation for the order parameter and beyond

Lecture 4. Strongly interacting atomic Fermi gases

Plan of the course

Lecture 1. BEC and long range order

Bose-Einstein condensation (JILA 1995)

iSe!=!

- BEC (can be defined at equilibrium) - Superfluidity (mainly related to transport phenomena)

Natural link between BEC and superfluidity provided by order parameter

Non trivial intercorrelated effects:

- Role of interactions (are interactions friends or enemies of BEC and superfluidity? Tunability of interaction) - Non uniform nature of the confinement (harmonic, periodic); BEC in both momentum and coordinate space; - Dimensionality; (1D and 2D configurations now achievable)

Sm

vS

!=h

Bose-EinsteinBose-Einstein condensationcondensation and and superfluiditysuperfluidity

Key concepts in low temperature physics .Key concepts in low temperature physics .Recent major progress in atomic quantum gasesRecent major progress in atomic quantum gases

sqrt of condensate density

Superfluid velocity(irrotationality)

- F. Dalfovo et al. Rev. Mod Phys. 71, 463 (1999) - “Bose-Einstein Condensation in Atomic Gases”, E. Fermi Summer School, M. Inguscio et al. (1999)

- A. Leggett, Rev. Mod. Phys. 73, 307 (2001)

- E. Cornell, W. Ketterle and C. Weiman, Nobel Lectures Rev. Mod. Phys. 74 (2002) - C. Pethick and H. Smith, “Bose-Einstein Condensation in Dilute Bose Gases”, Cambridge University Press (2002)

- L. Pitaevskii and S. Stringari “Bose-Einstein Condensation”, Oxford University Press (2003)

- “Ultracold Atomic Fermi gases” E. Fermi Summer School, W. Ketterle et al. (2008)

- S. Giorgini et al. cond-mat/0706.3360 (Rev.Mod.Phys. In press)

Usefulrefs.

1-body density matrix and long-range order

)'(ˆ)(ˆ)',()1( rrrrn !!=+

(Bose field operators)

hh

/)1(3 )2/),2/()2()( ipsesRsRndRdspn !! !+= "#

),()( )1(rrnrn =

Relevant observables related to 1-body density:

- Density:

- Momentum distribution:

!==h/)1()1( )(

1)()',( ipsepdpnV

snrrnIn uniform systems

)(~)()( 0 pnpNpn += !

0)()1( =!"ssn

IfV

Nnsn

s

00

)1( )( ==!"

Off-diagonal long range order (Landau, Lifschitz, Penrose, Onsager)

If n(p) is smooth function

Example of calculation of density matrix in highlycorrelated many-body system: liquid He4

(Ceperley, Pollock 1987)

Smooth function

CTT <<

CTT >

'rr ! 0!

For large N the sum can be replaced by integral which tends to zero at large distances. Viceversa contribution from condensate remains finite up to distances fixed by size of

BEC and long range order: consequence of macroscopicoccupation of a single-partice state.Procedure holds also in non uniform and finite size systems.

Long range order and eigenvalues of density matrix

)()'()',(' )1(rnrrrndr

iii!! =" )'()()',( *)1(

rrnrrniiii

!!"=

BEC occurs when . It is then convenient to rewritedensity matrix by separating contribution arising from condensate:

)'()()'()()',( *

0

*

00

)1(rrnrrNrrn

iiiii!!!! " #

+=

10>>! Nn

o

In bulk condensate fraction is 0.1. In the dropletthe fraction is larger because of surface effects.Condensate density is close to 0.1 in the center of the droplet.It increases and reaches 1 at the surface.

(Lewart, Pandharipande and Pieper, Phys. Rev. B (1988))

Diagonalization of 1-body density matrix in a “small” droplet of liquid He4 at T=0

ORDER PARAMETER

Diagonalization of 1-body density matrix permits toidentify single-particle wave functions . In terms of such functions one can write field operator in the form:

iiiararr ˆ)(ˆ)()(ˆ

000 !! " #+=$

i!

If one can use Bolgoliubov approximation

(non commutativity inessential formost physical properties within 1/N approximation).

10>>N

000ˆ,ˆ Naa !+

1]ˆ,ˆ[ 00 =+aa

Bogoliubov approximation is equivalent to treating themacroscopic component of the field operator as a classical field

(true also in liquid helium):

)(ˆ)()(ˆ 0 rrr !+!=! "

)()( 000 rNr !="

iiiarr ˆ)()(ˆ

0!" # $

=%

with

Usually fluctuations are small in dilute gasesat T=0 field operator is classical object(analogy with classical limit of QED, see Lecture 3). In helium quantum fluctuations are instead always crucial

)(ˆ r!"

thermal and quantumfluctuations

From field operator to classical field

Order parameter

• Complex function• Defined up to a constant phase factor• Fixing the phase breaking of gauge symmetry• Corresponds to average where average

means

• For stationary configurations time evolution is hence fixed by chemical potential

)(

00 )()( riSerr !=!

!S>!=<! )(ˆ)( rr

o

>+!>=<!< 1|)(ˆ|)(ˆ NrNr

>>=!

NetNtNiE |)(| /)( h

N

E

!

!=µ

Chemical potential: fundamental parameter in Bose-Einstein condenstates. Fixes time evolution of the phase

)(),( 0

/

0 retrti!=!

" hµ

Behaviour of BEC in non interacting gas (grand canonical ensemble):

1]/)exp[(

1

!!=

Tkn

Bi

i

µ"

BEC starts when chemical potential takes minimum value, so close to ( ) that occupation number of i=0 state becomes large and comparable to N:

! =i

iNn

number of atoms out of the condensate depends only on T (not on N)

0! Tk

B<<! µ"

0

1

0

0>>

!"

µ#

TkN

B

If for i>0 one can replace with andoccupation number of i-state does not depend any more on N

µ!µ! ">>"0i

µ 0!

Mechanism of BEC: !" ##+=

00

01]/)exp[(

1i

BiTk

NN$$

TN

Condition fixes value of critical temperatureNNT=

00Nn !

value of is fixed bynormalization condition

µ

1!

0!

µ

iiiH !"! =0

3D gas in harmonic potential

)1()(3

3

0

cT

TNTN !=3/1

94.0 NTkhocB

!h=

[ ]! " #++=

0,, 1)(exp

1

zyx nnnzzyyxx

Tnnn

N$$$%h

[ ]222222

2

1zyxmV zyxext !!! ++=

zzyyxxzyxnnnnnn !!!" hhh )

2

1()

2

1()

2

1(),,( +++++=

BEC starts at )0,0,0(!µ =

If one can transform sum into integral (semiclassical approximation)Integration yields:( increases with T, independent of N)

Condition then yields

iBTk !h>>

NNT=

and

)1(3

3

gTk

Nho

BT !!

"

#$$%

&=

'h

3/1)( zyxho !!!! =

TN

CONDENSATE FRACTION (Jila 96)EXPERIMENTAL EVIDENCEFOR PHASE TRANSITION

( )3/1CTT!

ROLE OF INTERACTIONS ON BEC:SOME QUESTIONS

• Do interactions modify shape of order parameter?• Do interactions reinforce or weaken BEC?• Do they enhance or decrease critical temperature?• Can BEC be fragmented? (more than one s.p. state

with macroscopic occupancy?)

No general answerEffects depend on dimensionality, sign of interaction,nature of trapping (harmonic, double well,periodic..)

BEC BEC fragmentationfragmentation and and rolerole of of interactionsinteractions

- Robustness of BEC ensured by (repulsive) two-body interactions in uniform configurations (mean field effect) - If trapping is not uniform (ex.: double well), interactions can work in opposite direction (quantum fluctuations)

)(ˆ)(ˆ)(ˆ)(ˆ2

int rrrrdrg

H ! """"=++

vacaN

becN)(

!

10

+= atoms occupy same sp state (BEC state)

vacaaNN

frgNN 10 )()(

!!

110

10

++=

atoms occupy two orthogonalsp states (fragmented BEC)

int0HHH +=

Since , (kinetic energy) gives negligible contribution

10NNN +=

3/1

10 ,0!

"= Vpp

Compare energy of two different configurations for a gasconfined in uniform box:

0H

BEC BEC fragmentationfragmentation and and NozieresNozieres’’ argumentargument

momentamomenta of of spsp states states

Express field operator in terms of 0- and 1- sp states

1100ˆ)(ˆ)()(ˆ ararr !! +="

]ˆˆ2)ˆ()()ˆ()[(

]ˆˆ2)ˆ()()ˆ()[(

)(ˆ)(ˆ)(ˆ)(ˆ

1010

2

1

2

1

2

0

2

0

10

*

1

*

0

2

1

2*

1

2

0

2*

0

aaaa

aaaa

rrrr

!!!!

!!!!

++

"++

=####

++++

++

!

!!!

+=

++=

4

0

2

10

2

1

2

010

4

1

2

1

4

0

2

0

)(2

1)(

]4[2

1)(

"

""""

NNgbecE

NNNNgfrgE

)1,( 10 >>NN

2

1

2

0!! =Since ( full overlap between s.p. wave functions)

0)()(4

010 >=!=" # $NgNbecEfrgEE

fragmentation is inhibited by repulsive interactions (mean field effect)

By writing field operator as and neglecting higher order terms originating from overlap between wave functions a and b the many-body Hamiltonian

)ˆˆˆˆ(2

)ˆˆˆˆˆˆˆˆ(4

abbabbbbaaaaE

HJC ++++++

+!+="

!=4

2 aC gE "

baba

ˆˆˆ !! +="

with

)(ˆ)(ˆ)(ˆ)(ˆ2

)(ˆ)(2

)(ˆ 22

rrrrdrg

rrVm

rdrH ext !! """"+"##$

%&&'

(+)*"=

+++ h

takes Boson Hubbard form

and )())(2

)((2 22

rrVm

rdrbextaJ

!!" # +$%%=h

In the absence of interaction ( )the eigenstates of H are:

vacba

vacba

)ˆˆ(

)ˆˆ(

1

0

++

++

!=

+=

"

"0=CE

tunneling

If condensates areseparated interactionsfavour fragmentationExample:BEC in double well potential

a b

01!!" #=

J

We are now ready to compare interaction energy between

vacba

N

becN

N

)(2!

1 +++=

vacbaNN

frg NN

ab

2/2/ )()(!2/!2/

1 ++=

(ground state in the absence of two-body interaction)

vacbabaNN

frg NN

N

2/2/

01 )ˆˆ()ˆˆ(2!2/!2/

1 ++++!+=

Nozieres’ argument applies to configurations and bec 01frg

since wave functions fully overlap in space: interactions make robust against

ba!! ±

bec 01frg

)1(8

)(int != NNE

becEC

)()2(8

)]1()1([4

)(

int

int

becENNE

NNNNE

frgE

C

bbaaC

ab

<!=

=!+!=

interactions favour BEC fragmentation ( ) 0>CE

In general competition between interaction and tunneling

Different behaviour if one considersconfigurations and ( and do not overlap!)bec

abfrgComparison between interaction energy in and bec

abfrg a!

b!

!!!!

Assume, for simplicity:

)()(

)()(

)()(

/

pnpn

pep

dzz

ba

b

ipd

a

ba

=

=

+=

! ""

""

h

With BEC

)())/cos(1(2)(

)()()(

pnpdpn

znznzn

a

ba

h+=

+=

With BEC fragmentation

)(2)(

)()()(

pnpn

znznzn

a

ba

=

+=

Look at interference fringes in density after expansion (Lecture 3)

or in ‘in situ’ momentum distribution

and do not overlap incoordinate space; they fullyoverlap in momentum space

a!

b!

bpapp

pppn

baˆ)(ˆ)()(ˆ

)(ˆ)(ˆ)(

!! +="

>""=<+

vacba

N

becN

N

)(2!

1 +++=

vacbaNN

frg NN

ab

2/2/ )()(!2/!2/

1 ++=

Can Can wewe distinguishdistinguish experimentallyexperimentally betweenbetween and ? and ?bec abfrg

n(p)BEC

fragmentedBEC

Momentum distribution in double well potential

]/[ dp h!

1-body density

]2/)([)(222222 2/)(2/)(2/)1( !!! dzdzz

eeeNzn""+""

++=

22 2/)1( )( !zNezn

"=

!"

=h/)1( )(

1)( ipzepdpnV

zn

With BEC

With BEC fragmentation

widthwidth of of eacheach condensate condensate

!6=d

!

long long rangerange orderorder

MeasurementMeasurement of the of the phasephase

ResultsResults forfor momentummomentum distributiondistribution describeddescribed in inlast last slidesslides correspondcorrespond toto averagingaveraging procedure. procedure.ResultResult of single of single measurementmeasurement ( (via via forfor exampleexampleinelasticinelastic photonphoton scattering scattering)) can can bebe differentdifferent: : -- IfIf the state the state isis coherentcoherent (BEC) (BEC) eacheach measurementmeasurement willwill reproducereproduce samesame positionspositions of of peakspeaks of n(p) of n(p) (the (the phasephase isis reproduciblereproducible). ). -- IfIf condensatescondensates are in are in fragmentedfragmented BEC BEC state the state the measurementmeasurement processprocess willwill ““createcreate”” the relative the relative phasephase and the and the measuredmeasured momentummomentum distributiondistribution willwill exhibitexhibit interferenceinterference fringesfringes accordingaccording toto the the lawlaw In In thisthis case the case the valuevalue of the of the phasephase isis randomrandom and the and the averagingaveraging procedure procedure washeswashes out out interferenceinterference effectseffects in n(p). in n(p).

)()]/cos(1[2)( pnSpdpn ar++= h

rS

- - MeasurementsMeasurements of the of the phasephase are more are more easilyeasily obtainedobtained bybyimagingimaging density density distributiondistribution after after releaserelease of the of the trapstraps..The The twotwo condensatescondensates overlapoverlap in coordinate space in coordinate space givinggivingrise rise toto interferenceinterference fringesfringes ( (LectureLecture 3). 3).- - MeasurementMeasurement of of momentummomentum distributiondistribution hashas the theadvantageadvantage of of determiningdetermining the the phasephase in situ in situ (non (nondestructivedestructive measurementmeasurement).).

unresolvedunresolved condensatescondensates

oscillations in the oscillations in the streamstreamof of outcoupledoutcoupled atomsatoms

In situIn situ measurementmeasurement of the of the phasephase in in momentummomentum distribution distribution(double (double wellwell configuration) configuration)A A twotwo-photon -photon BragBrag pulse causes pulse causes atomatom emissionemission sensitive to the sensitive to the Relative phase of the Relative phase of the twotwo BECBEC’’ssSaba et al., MIT 2005Saba et al., MIT 2005

)( pn!

Lecture 1. BEC and long range order.Long range order and order parameter.BEC fragmentation and role of interactions.

Lecture 2. Superfluidity and hydrodynamics.Landau criterion (Galilean vs rotational). Hydrodynamic theory of superfluids. Collective oscillations and expansion.

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