quantum degenerate bose systems in low dimensions g. astrakharchik s. giorgini istituto nazionale...

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