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QUANTUM DEGENERATE
BOSE SYSTEMS
IN LOW DIMENSIONS
G. Astrakharchik
S. Giorgini
Trento, 14 March 2003
Istituto Nazionale per la Fisica della Materia
Research and Development Center on
Bose-Einstein Condensation
Dipartimento di Fisica – Università di Trento
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 overview
Homogeneous 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
N 
V 
N
n
V
fixed density
• Normalization condition
D
d k
N  N0  
n
D k
(2 )
momentum
distribution
d Dk
1
N  N0  
D (  2 k 2 / 2 m   ) / k BT
(2 ) e
1
T  T3 D
2 2  n 



k B m  2.61
D=3
if
D2
for any T >0
If =0
0
chemical
potential
2/3
 0
 0
N0  0
N0  0
infrared divergence in nk
k BT
nk k
 2 2
0
 k / 2m
D=3 converges
D2 diverges
Interacting case
T0
If
Hohenberg theorem (1967)
“per absurdum argumentatio”
Bogoliubov 1/k2 theorem
N0
n0 
0
N
1 2mk BT
nk   2 2 n0
2
 k
Rules out BEC in 2D and 1D at finite temperature
Thermal fluctuations destroy BEC in 2D and 1D
quantum fluctuations?
T=0
Uncertainty principle
fluctuations of
particle operator
If
But
n0 
N0
0
N
S (k ) 
k
2mc
(Stringari-Pitaevskii 1991)
fluctuations of
density operator
n0
1
nk  
2 S (k )
static structure factor
sum rules result
1 2mc
nk  
n0
2 k
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 ?
T  n
T  2 2 / mk BT
1 / D
Thermal wave-length
One-body density matrix :
central quantity to investigate the coherence properties of the system
d Dk
ik s

 ( s)  
n
e


ˆ
( s )ˆ (0)
D k
(2 )
1.0
0.8
 ( s ) s


N0
V
long-range order
(r)/
condensate density
0.6
0.4
0.2
0.0
0
2
4
6
8
r (angstrom)
liquid 4He at equilibrium density
2D
low-T
from hydrodynamic theory (Kane – Kadanoff 1967)
 (T )
 ( s ) s   1 / s
high-T
k BTm
 (T ) 
2 2 n
classical gas
 (s) s  e
s 2 / T2
Something happens at intermediate temperatures
Berezinskii-Kosterlitz-Thouless transition temperature TBKT
(Berezinskii 1971 --- Kosterlitz – Thouless 1972)
T<TBKT
system is superfluid
T>TBKT
system is normal
Thermally excited vortices destroy superfluidity
Defect-mediated phase transition
• Universal jump (Nelson – Kosterlitz 1977)

 s (TBKT
)
TBKT
2m 2 k B

 2
• Dilute gas in 2D: Monte Carlo calculation (Prokof’ev et al. 2001)
TBKT
1
2 2 n


log( 2 / mg 2 D )   380
mk B
Torsional oscillator experiment on 2D 4He films
(Bishop – Reppy 1978)
Dynamic theory by
Ambegaokar et al. 1980
1D
From hydrodynamic theory (Reatto – Chester 1967)

 ( s ) s   1 / s
T=0
T 0
 ( s ) s   e  s / r ( T )
0
mc

2n
2 n 2
r0 (T ) 
k BTm
k BT   2 n 2 / m degeneracy temperature in 1D
4He
adsorbed in carbon nanotubes
Cylindrical graphitic tubes:
Yano et al. 1998
Teizer et al. 1999
1 nm diameter
103 nm long
superfluid behavior
1D behavior of binding energy
BEC in low-D: trapped systems
z


m2 2 m z2 2
Vext (r ) 
r 
z
2
2
anisotropy
parameter
a)
•) kBT  z
kBT  
E / N  z / 2  z
•) kBT  
motion is frozen along z
kinematically the gas is 2D
kBT  z
E / N    
z
2 1/ 2
 az   / m z
motion is frozen in the x,y plane
kinematically the gas is 1D
2 1/ 2

r
 a   / m
Goerlitz et al. 2001
3D 2D
3D 1D
b)
Finite size of the system
R , Rz
cut-off for long-range fluctuations
BEC in 2D
fluctuations are strongly quenched
(Bagnato – Kleppner 1991)
k BT2 D   (N / 1.64)1/ 2
k BT3 D   (N / 1.20)1/ 3
  (2 z )1/ 3
Thermodynamic limit
N 
  0
 N 1/ 2 fixed
But density of thermal atoms
n(r )   log(1  e
2
T
 m2 r2 / 2 k BT
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 ?
) r
 
 0
1D systems
• No BEC in the thermodynamic limit N
• For finite N macroscopic occupation of lowest single-particle state
kBT1D  z N / log(2N )
If
k BT3 D    k BT1D
2-step condensation
(Ketterle – van Druten 1996)
Effects of interaction (Petrov - Holzmann – Shlyapnikov 2000)
(Petrov – Shlyapnikov – Walraven 2000)
Characteristic radius of phase fluctuations
2D
1D
R  T eT / T
k BT  N ( ) 2 / 
R  Rz (T / T )
k BT  N ( z ) 2 / 
T  T
 R  R

 R  Rz
2D
T  T
 R  R

 R  Rz
2D
1D
1D
true condensate
(quasi-condensate)
condensate with
fluctuating phase
Dettmer et al. 2001
Richard et al. 2003
Beyond mean-field effects in 1D at T=0
• Lieb-Liniger Hamiltonian
Exactly solvable model with repulsive zero-range force
H LL
2 N 2

 g1D   ( zi  z j )

2
2m i 1 zi
i j
Girardeau 1960 --- Lieb – Liniger 1963 --- Yang – Yang 1969
g1D
2 2

0
m a1D
at T=0 one parameter n|a1D|
a1D scattering length
Equation of state
10
3
10
2
10
1
10
0
-1
10
-2
10
-3
1x10
-4
1x10
-5
10
-6
E/N
10
MF
TG
10
-3
-2
10
10
-1
10
0
10
1
10
2
10
3
n|a1D|
n a1D  1
E / N  g1D n / 2
n a1D  1
 2 2 n 2
E/N 
6m
mean-field
Tonks-Girardeau
fermionization
One-body density matrix
Quantum Monte-Carlo (Astrakharchik – Giorgini 2002)
z     / 2mc
10
1
 ( z )  1 / z
30
(z)/
1 / 2 TG
 

2n  0 MF
mc
1
0.3
10
k  1 / 
nk  1 / k
1
-3
0.1
0.1
1
10
zn
100
3
Momentum distribution
0.8
0.6
k n(k)/n
10
-3
0.3
0.4
1
0.2
30
10
0.0
-3
10
10
-2
3
10
-1
k/n
10
0
10
1
Lieb-Liniger + harmonic confinement
N
2 N 2
mωz2 2
H 
 g1D   ( zi  z j )  
zi

2
2m i 1 zi
2
i j
i 1
Exactly solvable in the TG regime (Girardeau - Wright - Triscari 2001)
Local density approximation (LDA) (Dunjko - Lorent - Olshanii 2001)
If
E / N  z
m z2 2
  local n( z ) 
z
2
1D behavior is assumed from the beginning
3D-1D crossover
Quantum Monte-Carlo (Blume 2002 --- Astrakharchik – Giorgini 2002)
N
2 N 2
H 
 i  V ( ri  r j )  Vext (ri )

2m i 1
i j
i 1
Harmonic confinement
highly anistropic traps
z

 1

m
Vext (r )  2 r2   z2 z 2 
2
Interatomic potential (a s-wave scattering length)
hard-sphere model
 
V (r )  
0
(r  a)
(r  a)
soft-sphere model (R=5a)
V0  0 (r  R)
V (r )  
(r  R)
0
Compare DMC results with
• Mean-field – Gross-Pitaevskii equation
 2 2
2
  Vext (r )  g 3 D  N  1 (r ) (r )  (r )

 2m

with
g3D
4 2 a

m
• 1D Lieb-Liniger
with
(Olshanii 1998)
g1D
2 2 a

ma2
a1D
a2

a
10
0
TG
-1
E/N  
()
10
GP
10
-2
LL+LDA
10
-3
IG
10
-4
10
-3
10
-2
=z/
N=5
10
-1
a/a=0.2
10
0
E/N  
()
10
0
TG
GP
10
-1
10
-2
IG
LL+LDA
10
-3
N=5 a/a=1
10
-4
10
-3
10
-2
=z/
10
-1
10
0
E/N  
()
N=100
a/a=0.2
GP
0
10
TG
IG
10
-1
LL+LDA
10
-3
10
-2
-1
10
=z/
0
10
Possible experimental evidences of TG regime
• size of the cloud (Dunjko-Lorent-Olshanii 2001)
z
2
 az
N
2
1/3
TG
z
2
az 
a

 3 N 
5 
a 
• collective compressional mode (Menotti-Stringari 2002)
  2z TG
  3 z
MF
• momentum distribution (Bragg scattering – TOF)
nk 
1
k
TG
1
nk 
k
MF
MF
Infrared behavior k<<1/ --- Finite-size cutoff k>>1/Rz
a / a  0.2
1
12
10
  103
n(k)/N
8
0
N=100
10
n(k)/N
10
-1
10
6
-2
10
4
N=20
1/
1/Rz
-1
0
10
N=5
2
0
0.0
1
10
10
k (1/az)
0.5
1.0
k
(1/az)
1.5
2.0
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
a  a
a  1 / n
confinement induced resonance (Olshanii 1998 - Bergeman et al. 2003)
g1D
2 2 a
1

ma2 1  1.03a / a