Transcript Introduction: 1D quantum gas

Some aspects of 1D Bose gases:
evolution from BEC to Tonks gas
陈 澍 (Shu Chen)
中国科学院物理研究所
Institute of Physics
Chinese Academy of Sciences
Aug 15, 2007 KITPC
Outlines
 Introduction:
 Bose
1D quantum gas
gas in hard-wall trap
A solvable example of many-body system exhibiting crossover
from BEC to Tonks gas
 Modified
GP theory for 1D quantum gas
Confinement of atoms by harmonic trap

3D harmonic trap
1
V  m  x2 x 2   y2 y 2   z2 z 2 
2

Quasi-1d: cigar-shape
trap  y  z   
  x ,   kBT
Transverse motion frozen
7Li
6Li
Realization of 1D quantum gas in optical lattice
For a 2D optical lattice, the atoms are confined to an array of tightly
confining 1D potential tubes.
In each tube, radial motion confined to zero point oscillations
effective 1D quantum gas
Experiments with 1D condensates:
A. Goerlitz et al., PRL (2001), F. Schreck et al. PRL (2001), M. Greiner et al. PRL (2001)
more recently:
H. Moritz et al., PRL (2003), Nature 429, 277 (2004), Science, 305, 1125 (2004)
One dimensional Bose gas

Requirements: 1D bosonic quantum gas, tightly confined in two
dimensions and only weakly confined along the axial direction
   

Parameter  governs the crossover from weak to strong interacting
regime
 
mg
 2 0
Interaction energy
Kinetic energy
I ~ ng
 2n2
K ~
m
I

K
Cartoon for the 1D Bose gas in a trap
Weakly interacting TF regime
Strongly interacting Tonks regime
Tonks gas was realized experimentally
B. Paredes et al., Nature 429, 277 (2004),
T. Kinoshita et al., Science, 305, 1125 (2004)
Effective 1D Hamiltonian
Olshanii PRL 81 (1998) 938

Transverse motion frozen
^
^
^
  x, y, z     x  0  y, z 

Projection onto transverse ground state yields
2
^

 ^
g ^
2
H   dx  x   
 x  V  x, t     x   x    x 
2
 2m

^ 
g 
2
2ma1D
a1D
d 2
2a s

as 
1

1
.
46


d 

d   2 / m
V(x)=0  Lieb-Liniger model

Exactly solved by Bethe ansatz
 N 2

  2  2c    xi  x j    E
1i  j  N
 i 1 xi

GS energy density given by

 g0 /2
 ( 0 )   2 2 2
  0 / 6m
  1
  1
For harmonic trap, no exact solution, however, one can
work in the modified GP theory
works even in strongly interacting regime!

For hard-wall trap, exact solution is available.
Outlines
 Introduction:
 Bose
1D quantum gas
gas in hard-wall trap
A solvable example of many-body system exhibiting crossover
from BEC to Tonks gas
 Modified
GP theory for 1D quantum gas
Experimental realization of 1D Bose gas in hard-wall trap
Phys. Rev. A 71,
041604 (2005).
Model: Lieb-Linger model with open boundary condition
 N 2

  2  2c    xi  x j    E
1i  j  N
 i 1 xi

Boundary condition:
  0, x2 ,
xN     x1 ,
xN  L   0
Model for1D interacting Bose gas in hard-wall trap
 N 2

  2  2c    xi  x j    E
1i  j  N
 i 1 xi

R : 0  x1 , x2 ,
M. Gaudin, Phys. Rev. A,
4, 386 (1971). for c >0
, xN  L
  c  
We will study the full physical regime:
Wave function:
  x1 , x2 ,

, xN    x p1 , x p2 ,
P

According to the symmetry condition,  x p1 , x p2 ,
permutation of   x1 , xN 
 
, x pN  x p1  x p2 
, x pN

 x pN

can be obtained by
Exact solution of 1D Bose gas in hard-wall trap

Bethe ansatz wave function:
  x1 ,




xN     AP exp  i  rj k p j x j  
P , r1 , , rN 
 j
 

Bethe ansatz equations (BAE):
exp  i 2k j L  
N

ikl  ik j  c ikl  ik j  c
l 1  j  ikl  ik j  c ikl  ik j  c

c
c
k j L  n j    arctan
 arctan

k j  kl
k j  kl
l 1  j  
N
k j L  nj 

r j  1
.



k j  kl
k j  kl 

arctan

arctan


c
c 
l 1  j  
N
Eigenenergy :
E   j 1 k
N
2
j
GS solutions correspond to:
n j  11  j  N 
nj  j 1  j  N 
Quasi-momentum distribution (c>0)
 GS
density of state in k-space:
N=200 and c=0.1,1,10,100. Inset:N=1000 and c= 10.
 k  k j 1 
1
L  j


2

 k j 1  k j
Density distribution (c>0)
dxN   x, x2 ,
L
  x 
N  dx2
0

L
0
dx1
, xN    x, x2 ,
dxN   x1 , x2 ,
, xN 
, xN 
2
Continuous crossover from
weakly interacting Bose gas
to Tonks gas
N=4
Y. Hao, Y. Zhang, J.Q. Liang and S. Chen, Phys. Rev. A 73, 063617 (2006)
Density distribution in harmonic trap
One body density matrix
Momentum density distribution (c>0)
n(k) for TG gas is different
from that of free Fermi gas
N=4
Y. Hao, Y. Zhang, and S. Chen, (2007) preprint
Attractive Bose gas with c<0
 GP
theory: collapse of BEC (3D)!
 Attractive
Bose gas in hard wall trap:
what a picture can exact results tell us?
1D Bose gas in hard-wall trap (c<0)
Two body problem

BAE
n1  n2
c
  arctan
,
2
2
1 2  c
L  ln
,
2 2  c
L 
k1    i,
k 2    i,

k  2
GS energy
E  k 2  2
2
2
n1  n2  1
(Ground state)
Example: solution for the 10-atom system
1.42  c  0
5 Dimers
1.72  c  1.42
4-string solution + three 2-string
1.88  c  1.72
6-string solution + two 2-string
3.02  c  1.88
8-string solution + one 2-string
c  3.02
10-string solution
Crossover for the N-atom system (c<0)

Weak attractive interaction regime
N/2 Dimers

Intermediate regime
(N-M)-string solution + M/2 2-string solution
(1<M<N )

Strong attractive interaction regime N-string
solution
Density distribution (c<0)
N=2
GS energy:
N=4
the density profile matches the case of c=0.
Formation of a compounded particle with mass
Nm
The second order correlation function
The atoms tend to cluster
together more easily for the
attractive interaction and the
atoms bunch closer as the
interaction becomes stronger.
For the repulsive interactions,
the atoms avoid each other
and the atom-bunching
reduces and vanishes finally
for increasing interactions.
Outlines
 Introduction:
 Bose
1D quantum gas
gas in hard-wall trap
A solvable example of many-body system exhibiting crossover
from BEC to Tonks gas
 Modified
GP theory for 1D quantum gas
1D Gross-Pitaevskii theory

G-P equation in the weakly interacting limit
x
i  t 0  [
 V ( x, t )  g 0 ] 0
2m
2
2
0 
 0 e
0
failure of MFT (G-P thoery for BEC) in the strongly interacting regime!
Local density approximation (LDA)
The LDA assuming that locally the system
behaves like a uniform gas
Modified Gross-Pitaevskii theory

Modified G-P equation
 x2
i t 0  [
 V ( x, t )   ( 0 ( x, t ))] 0
2m
2
0 

0 e
0
 g 0

 ( 0 )  [ 0 ( 0 )]   2 2 2
0
 0 / 2m
Well describe the density profile, but
overestimate the interference
  1
  1
How good is the M-GP theory?

Density distribution in Tonks limit:
Kolomeisky et. al. PRL 85, 1146, 2000
Perfect agreement with the exact result by Bose-Fermi mapping
Tonks gas

Tonks gas = hard-core boson gas
Tonks 1936
strong repulsion  avoiding point-contact occupation
effectively described by the boundary condition

Bose-Fermi mapping
Girardeau 1960
Tonks gas

Density distribution same as the free-fermion’s

Momentum distribution
Comment for the modified GP theory
Describe the density profile well, but
overestimate the interference
If you use GPE to study the interference, you can always get
interference no matter how strong the interaction is, even in the
TG limit.
How to account properly the effect of interaction?
Quantum fluctuations suppress interference
S. Chen & Egger PRA 2003
Density phase representation

Density phase representation
^

with


 ( x)ei  x  
i 0  x   x 

0 ( x)  ( x)e 
[( x), ( x' )]  i x  x'
Dynamics of (x,t) governed by
 2 x
i t   [
 V ( x, t )  g   ]
2m
2

Small parameter ( and ) expansion
zero order  time dependent GP equation
first order  EOMs for  and 
Effective Hamiltonian for quantum
fluctuation operators

Effective Hamiltonian
2
 2 0

1   0  2
2
H   dx 
 
  x  
  x0   x 
2 0
2m
 2m


EOM of  .
b
1  ( 0 )
( D   ) D 
 x [  0  x ]
b
m 0
  1 for TF and   2
S. Chen & Egger PRA 2003
.
for TG
D   t  x (b/ b) x
Formulas for interference signal
Interference signal around meeting point (±L/2)
I ( x, t )  0 ( x, t )  0 ( x ', t )  2 Re W ( x, x ', t )
W ( x, x ', t )    ( x, t ) ( x ', t ) 
x  x ' L
W ( x, x' , t )  W0 ( x, x' , t )e  F ( x , x ',t )
Our task is to evaluate W(x,x',t)
-L/2
x=0
L/2
y=x±L/
2
W0 ( x, x ', t )  0 ( x, t ) 0 ( x ', t )ei[0 ( x ,t ) 0 ( x ',t )]
F ( x, x ', t )  [ ( x, t )   ( x ', t )]2  / 2
-L/2
y=0
Quantum fluctuations suppress interference
L/2
Interference vs interaction


Trapping and expansion: initial
preparation
V  x, t  
1
m x2 x 2   t 
2
Interference affected by interaction
(0)=14.3, T=0
(0)=0.001
Thomas-Fermi regime
S. Chen & Egger PRA 2003
Tonks regime
Interference in Tonks limit
Density profile: same with free-Fermion’s
Interference signal
No interference fringes: phase difference and
cancellation of fringes from different orbits
Some experimental progress
Spin-1 Bose gas (spinor gas with F=1)

Realized in optical trap
spin is not polarized

Spinor symmetric interaction of F=1 atoms
U ij    xi  x j  U 0 P0  U 2 P2     xi  x j  c0  c2 Fi  Fj 
c0  4m
2
a0  2 a2
3
, c2  4m
2
a2 a0
3
Spin-1 Bose gas
The second quantized Hamiltonian of Spin-1 Bose gas:
2
 ^ 
 ^ c0
d2
H   dx   i  
 Vext  x    i 
2
2
m
dx
2
 

2
^
^ 
^ ^ 
c2 ^  ^ 
:   i  i  :   k  i  F   F   j  l 
ij
kl
2



In the mean field approach, the spin-dependent energy functional:
2
 *


c2 * *
d2
E   dx  i  

V
x







F
F


 
ext   
i
k
i   ij    kl
j
l
2
2
m
dx
2

 

where
   i    i
i
2
and      c0  /2.
i
If c2=0, the model is integrable for V(x)=0.
Modified Gross-Pitaevskii Equations (GPEs)
By using the exact BA solution, the interaction effect is properly taken into account.
c0  /2,
  
 e     2 2 2
 / 6m,
2m

2
  1
  1
The spin-dependent term can be expressed as:
*k *i  F   F   j l
ij
kl
  2    2   2 0    2 0    2      2*0 2     2 02**
(*)
(＊) The only processes that change the spin states occur when an atom in the mF  1 state
scatters with another in the mF  1 state giving two atoms in the mF  0 state, or vice versa.
The conservation quantity:
The particle number
Magnetization
N  N  N0  N
M  N  N
Density distribution of the GS of
87
Rb
(FM)
Thomas-Fermi regime
m=0
m=0.2
Tonks-Girardeau regime
Black lines: + component
Red lines: 0 component
Green lines: - component
Y. Hao, Y. Zhang, J.Q. Liang and S. Chen, Phys. Rev. A 73, 053605 (2006)
Phase separation induced by anisotropic spin-spin interaction
87
Rb
(FM)
Y. Hao, Y. Zhang, J.Q. Liang and S. Chen
EPJD (2007)
No phase separation induced for AFM interaction
23
Na (AFM)
Summary

Exact results of the 1D interacting Bose
gases in hard-wall trap

General theory for 1D gas beyond MFT
Acknowledgements
Collaboraters:
Dr. Yajiang Hao
Institute of Physics, Chinese Academy of Sciences
Prof. YunBo Zhang and Prof. J.-Q. Liang
ShanXi University
Prof. R Egger
Duesserdorf University
Financial support: NSF of China, Bairen program of CAS
谢谢大家！
Thank you for your attention!