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