quantum degenerate bose systems in low dimensions
DESCRIPTION
QUANTUM DEGENERATE BOSE SYSTEMS IN LOW DIMENSIONS. G. Astrakharchik S. Giorgini Istituto Nazionale per la Fisica della Materia Research and Development Center on Bose-Einstein Condensation Dipartimento di Fisica – Universit à di Trento. Trento, 14 March 2003. - PowerPoint PPT PresentationTRANSCRIPT
QUANTUM DEGENERATEQUANTUM DEGENERATEBOSE SYSTEMSBOSE SYSTEMS
IN LOW DIMENSIONSIN LOW DIMENSIONS
G. AstrakharchikS. 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 systemsSystems 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
VN
nVN
kD
D
nkdNN)2(0
D=3 converges
D=3 if
D2 for any T >0
If =0 infrared divergence in nk
D2 diverges
11
)2( /)2/(0 22 TkmkD
D
BekdNN
0chemicalpotential
3/22
3 61.22
n
mkTT
BD
0 0 0 N
0 0 0 N
mkTkn B
kk 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
NNn
0222
21 n
kTmkn B
k
T=0 Uncertainty principle (Stringari-Pitaevskii 1991)
If
But
fluctuations ofparticle operator
fluctuations ofdensity operator
000
NNn
)(21 0
kSnnk static structure factor
mckkS
2)( sum rules result
02
21 n
kmcnk
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(
)(
senkds ikD
D sk
VNs s
0)(
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
nTmkT B
22)(
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 12
22
D
BBKT mg
mknT
Torsional oscillator experiment on 2D 4He films(Bishop – Reppy 1978)
Dynamic theory byAmbegaokar 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 behaviorTeizer et al. 1999 1D behavior of binding energy
ss s /1)(
)(/ 0)( Trss es
nmc
2
TmknTrB
2
02)(
mnTkB /22 degeneracy temperature in 1D
BEC in low-D: trapped systems
a)•)
•)
22
22
22)( zmrmV 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 ?
0
2/2 )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
zRRRR
1D 2D
zRRRR
true condensate
(quasi-condensate)condensate withfluctuating phase
Dettmer et al. 2001Richard 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
nNE6
/222 Tonks-Girardeau
fermionization
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
zmωzzgzm
H1
22
112
22
2)(
2
zNE /
22
2)( zmzn 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)( zrmV zext r 1
z
highly anistropic traps
)( 0)(
)(arar
rV
)( 0)( 0
)( 0
RrRrV
rV
hard-sphere model soft-sphere model (R=5a)
Compare DMC results with
• Mean-field – Gross-Pitaevskii equation
• 1D Lieb-Liniger
(Olshanii 1998)
)()()(1)(2
23
22
rrrr
NgV
m Dext
mag D
2
34 with
2
2
12
ma
ag D
withaaa 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
1/32
aaNaz z
TG
22 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
aamaag D /03.11
122
2
1