QUANTUM DEGENERATEQUANTUM DEGENERATE

BOSE SYSTEMSBOSE SYSTEMS

IN LOW DIMENSIONSIN LOW DIMENSIONS

G. Astrakharchik

S. Giorgini

Istituto Nazionale per la Fisica della Materia Research and Development Center on

Bose-Einstein CondensationDipartimento di Fisica – Università di Trento

Trento, 14 March 2003

Bose – Einstein condensates of alkali atoms

• dilute systems na3<<1

• 3D mean-field theory works

• low-D role of fluctuations is enhanced• 2D thermal fluctuations• 1D quantum fluctuations

beyond mean-field effects

many-body correlations

Summary

• General overviewHomogeneous systems

Systems in harmonic traps

• Beyond mean-field effects in 1D

• Future perspectives

BEC in low-D: homogeneous systems

Textbook exercise: Non-interacting Bose gas in a box

• Thermodynamic limit

• Normalization condition

momentumdistribution

fixed density

V

Nn

VN

kD

D

nkd

NN)2(0

D=3 converges

D=3 if

D2 for any T >0

If =0 infrared divergence in nk

D2 diverges

1

1)2( /)2/(0 22 TkmkD

D

Be

kdNN

0chemicalpotential

3/22

3 61.22

n

mkTT

BD

0 0 0 N

0 0 0 N

mkTk

n Bkk 2/220

Interacting case

T0 Hohenberg theorem (1967) Bogoliubov 1/k2 theorem

“per absurdum argumentatio”

If

Rules out BEC in 2D and 1D at finite temperature

Thermal fluctuations destroy BEC in 2D and 1D

quantum fluctuations?

000

N

Nn

022

221

nk

Tmkn B

k

T=0 Uncertainty principle (Stringari-Pitaevskii 1991)

If

But

fluctuations ofparticle operator

fluctuations ofdensity operator

000

N

Nn

)(21 0

kS

nnk static structure factor

mck

kS2

)( sum rules result

0

221

nk

mcnk

Rules out BEC in 1D systems even at T=0

Quantum fluctuations destroy BEC in 1D(Gavoret – Nozieres 1964 ---- Reatto – Chester 1967)

Are 2D and 1D Bose systems trivial as they

enter the quantum degenerate regime ?

DT n /1

TmkBT /2 2 Thermal wave-length

One-body density matrix :central quantity to investigate the coherence properties of the system

)0(ˆ)(ˆ)2(

)(

senkd

s ikD

D sk

V

Ns

s0)(

condensate density

liquid 4He at equilibrium density

0 2 4 6 80.0

0.2

0.4

0.6

0.8

1.0

(r)

/

r (angstrom)

long-range order

2D

Something happens at intermediate temperatures

)(/1)( Ts ss

22 /)( Tss es

low-T from hydrodynamic theory (Kane – Kadanoff 1967)

high-T classical gas

nTmk

T B22

)(

Berezinskii-Kosterlitz-Thouless transition temperature TBKT

(Berezinskii 1971 --- Kosterlitz – Thouless 1972)

• Universal jump (Nelson – Kosterlitz 1977)

• Dilute gas in 2D: Monte Carlo calculation (Prokof’ev et al. 2001)

T<TBKT system is superfluid

T>TBKT system is normal

Thermally excited vortices destroy superfluidityDefect-mediated phase transition

2

22)(

B

BKT

BKTs kmT

T

380 )/log(2 1

22

2

DB

BKT mgmk

nT

Torsional oscillator experiment on 2D 4He films(Bishop – Reppy 1978)

Dynamic theory by

Ambegaokar et al. 1980

1DFrom hydrodynamic theory (Reatto – Chester 1967)

T=0

T0

4He adsorbed in carbon nanotubesCylindrical graphitic tubes: 1 nm diameter 103 nm long

Yano et al. 1998 superfluid behavior

Teizer et al. 1999 1D behavior of binding energy

ss s /1)(

)(/ 0)( Trss es

nmc

2

Tmkn

TrB

2

0

2)(

mnTkB /22 degeneracy temperature in 1D

BEC in low-D: trapped systems

a)

•)

•)

22

22

22)( z

mr

mV z

ext

r

z

anisotropyparameter

TkTk BzB motion is frozen along zkinematically the gas is 2D

zBB TkTk motion is frozen in the x,y planekinematically the gas is 1D

zzNE 2//

NE / mar /

2/12

zz maz /2/12

Goerlitz et al. 2001

3D 2D

3D 1D

b) Finite size of the system

cut-off for long-range fluctuations fluctuations are strongly quenched

BEC in 2D (Bagnato – Kleppner 1991)

Thermodynamic limit

zRR ,

2/12 )64.1/(NTk DB

3/13 )20.1/(NTk DB 3/12 )( z

0

N

fixed 2/1N

But density of thermal atoms

Perturbation expansion in terms of g2D n breaks down

Evidence of 2D behavior in Tc

(Burger et al. 2002)

• BKT-type transition ?

• Crossover from standard BEC to BKT ?

02/2 )1log()(

22

rTkrm

TBern

1D systems

• No BEC in the thermodynamic limit N• For finite N macroscopic occupation of lowest single-particle state

If

)2log(/1 NNTk zDB (Ketterle – van Druten 1996)

DBDB TkTk 13

2-step condensation

Effects of interaction (Petrov - Holzmann – Shlyapnikov 2000)

(Petrov – Shlyapnikov – Walraven 2000)

Characteristic radius of phase fluctuations

2D

1D

TTT eR /

)/( TTRR z

/)( 2 NTkB

/)( 2zB NTk

TT

TT

1D

2D

zRR

RR

1D

2D

zRR

RR

true condensate

(quasi-condensate)condensate withfluctuating phase

Dettmer et al. 2001

Richard et al. 2003

Beyond mean-field effects in 1D at T=0

• Lieb-Liniger HamiltonianExactly solvable model with repulsive zero-range force

Girardeau 1960 --- Lieb – Liniger 1963 --- Yang – Yang 1969

at T=0 one parameter n|a1D|

N

i jijiD

iLL zzg

zmH

112

22

)(2

02

1

2

1 D

D amg

a1D scattering length

10-3 10-2 10-1 100 101 102 10310-6

1x10-5

1x10-4

10-3

10-2

10-1

100

101

102

103

TG

MF

E/N

n|a1D

|

11 Dan 2// 1 ngNE D

11 Danm

nNE

6/

222 Tonks-Girardeaufermionization

Equation of state

mean-field

One-body density matrix Quantum Monte-Carlo (Astrakharchik – Giorgini 2002)

0.1 1 10 100

0.1

1

10-3

0.3

1

30

103

(z)

/

z n

zz /1)(

mcz 2/

MF 0

TG 2/1

2 n

mc

/1k

1/1 knk

Momentum distribution

10-3 10-2 10-1 100 1010.0

0.2

0.4

0.6

0.8

10-3

0.3

1

30

103

k

n(k)

/n

k/n

Lieb-Liniger + harmonic confinement

Exactly solvable in the TG regime (Girardeau - Wright - Triscari 2001)

Local density approximation (LDA) (Dunjko - Lorent - Olshanii 2001)

If

1D behavior is assumed from the beginning

N

ii

zN

i jijiD

i

zmω

zzgzm

H1

22

112

22

2)(

2

zNE /

22

2)( z

mzn z

local

3D-1D crossoverQuantum Monte-Carlo (Blume 2002 --- Astrakharchik – Giorgini 2002)

Harmonic confinement

Interatomic potential (a s-wave scattering length)

)()(2 11

22

i

N

iext

N

i jijii VV

mH rrr

2222

2)( zr

mV zext r 1

z

highly anistropic traps

)( 0

)( )(

ar

arrV

)( 0

)( 0)( 0

Rr

RrVrV

hard-sphere model soft-sphere model (R=5a)

Compare DMC results with

• Mean-field – Gross-Pitaevskii equation

• 1D Lieb-Liniger

(Olshanii 1998)

)()()(1)(2

2

32

2

rrrr

NgV

m Dext

ma

g D

2

3

4 with

2

2

1

2

ma

ag D

with

aa

a D

2

1

10-4 10-3 10-2 10-1 100

10-3

10-2

10-1

100

N=5 a/a=0.2IG

TG

GP

LL+LDA

E

/N

(

)

=z/

10-4 10-3 10-2 10-1 100

10-3

10-2

10-1

100

N=5 a/a=1

IG

TG

LL+LDA

GP

E

/N

(

)

=z/

10-3 10-2 10-1 100

10-1

100

LL+LDA

IG

GPTG

N=100 a/a=0.2

E/N

(

)

=z/

Possible experimental evidences of TG regime

• size of the cloud (Dunjko-Lorent-Olshanii 2001)

• collective compressional mode (Menotti-Stringari 2002)

• momentum distribution (Bragg scattering – TOF)

MF 35

1/3

2

aa

Na

z z

TG 2

2 Naz z

TG 2 z MF 3 z

TG k

1kn MF 1k

nk

Infrared behavior k<<1/ --- Finite-size cutoff k>>1/Rz

0.0 0.5 1.0 1.5 2.00

2

4

6

8

10

12

10-1 100 101

10-2

10-1

100

101

1/1/Rz

n(k)

/N

k (1/az)

N=100

N=20

N=5

n(k)

/N

k (1/az)

2.0/ aa

310

Future perspectives

• Low-D and optical lattices– many-body correlations

superfluid – Mott insulator quantum phase transition (in 3D Greiner et al. 2002)

– Thermal and quantum fluctuations

low-D effects

Investigate coherence and superfluid properties

• Tight confinement and Feshbach resonances(Astrakharchik-Blume-Giorgini)

Quasi-1D system

confinement induced resonance (Olshanii 1998 - Bergeman et al. 2003)

aa

na /1

aamaa

g D /03.1112

2

2

1