（顾 强）
Department of Physics,
University of Science and Technology Beijing
（北京科技大学 物理系）
KITPC, Beijing, August 24, 2012

Cold atoms in the synthetic magnetic field
Charged Bose gas in magnetic field
BKT transition of 2D Bose gas
Trapped Bose gas

Charged Bose gas in magnetic field
1949 Possibly earliest study on charged bosons
Phys. Rev. 76 , 400(1949)
• Landau diamagnetism: due to the quantization of orbital motions of charged particles

Charged Bose gas in magnetic field
1954 Schafroth-Blatt-Butler superconductor
Phys. Rev. 96 , 1149(1954), Letters to the Editor

Charged Bose gas in magnetic field
1955 Meissner-Ochsenfeld effect
Phys. Rev. 100 , 463 (1955)

Charged Bose gas in magnetic field
 j k z

1

2 j
 
2 2 k z
D
L
 qBL L x y
2
 c where
Phys. Rev. 100 , 463 (1955)
  qB infinity
Phys. Rev. E 60 , 5275 (1999)

Charged Bose gas in magnetic field
PRL 96 , 147003 (2006)

PM vs DM in Fermi gas
• Ideal fermi gas: Pauli paramagnetism 1927 where is the density of state at Fermi surface is the Bohr magneton due to the intrinsic magnetic moment of electrons
• Ideal Fermi gas:
Landau diamagnetism 1930 due to the quantization of orbital motions of charged particles

PM vs DM in spin-1 Bose gas
The Hamiltonian
： where  : chemical potential; D
L
 qBL L y
2
 c
 jk z
: Landau level j

0,1, 2,...
  qB
 j k z

1

2 j
: cyclotron frequency
: degeneracy;
 
2 2 k z


: Zeeman energy
  
1, 0; g


: Lande factor
  g q

B
J. Phy.:Condens. Matt. 23, 026003(2011)

PM vs DM in spin-1 Bose gas
Grand thermodynamical potential
：

T

0
 

V
 m
2
*
2



2
3


 
1

D , 0
 where


   l



1

 lx
   

1
 e
 lx
 
D

3
  
/(

)

(1/ 2
   x
   particle number and magnetization ：
M
T

0
 q
 m *
* 2

2


3
2 

  
1

D , 0


 x g
 n
 x

 m
2

*
2

1
2

1

2

D , 0
 x
2

2

D ,1
 


2
3



1

2

D , 0


PM vs DM in spin-1 Bose gas g

0.30
g

0.45

= 10 (dash-dot-dotted) ，
3 (dotted) ，
0.3 (dashed) ，
0.05(solid) g

0.50

PM vs DM in spin-1 Bose gas
For S=1 particles:
Bose gas, dotted line
Fermi gas, dashed line
Boltzmann gas, solid line
Phy. Lett. A 374 , 2580 (2010)
J. Phys.: Condens. Matter 23 , 026003 (2011)

PM vs DM in spin-1 Bose gas
Grand thermal potential of Maxwell-Boltzmann gas ：

T

0
 
1

 j k z
,

D e
L

 jkz
 
 g c is determined by
1
2

1 x

 e x

1

1
 e
2 c g x

2
 
1)
 e
 
1
In the limit cases, g c t 

1/ 8

0.35355
g c t 
0

1/ 2

Cold atoms in the synthetic magnetic field
Charged Bose gas in magnetic field
BKT transition of 2D Bose gas
Trapped Bose gas

BKT transition of 2D Bose gas
Berezinskii, JETP 34 , 610 (1972)
Kosterlitz,Thouless, J. Phys. C 6 , 1181 (1973)

BKT transition of 2D Bose gas
Experimental systems: liquid 4 He, SC films, SC Josephson array…
Phys. Rev. Lett. 40 , 1727 (1978).
Phys. Rev. Lett. 42 , 1165 (1979).
Phys. Rev. Lett. 47 , 1542 (1981).

BKT transition of 2D Bose gas
From SC Josephson array to lattice of Bose-Einstein condensates
The transition occurs when T c
=πJ/2

BKT transition of 2D Bose gas
Experimental system in cold atoms:
ENS(2006): Nature 441, 1118 (2006)
Phys. Rev. A 81, 023623 (2010)
At low-T, the interference fringes are straight. Just below the transition temperature, the fringes become wavy due to decreased spatial phase coherence. Phase dislocations become common at temperatures above the transition. These “zipper patterns” indicate the presence of free vortices.

BKT transition of 2D Bose gas
JILA(2007): Phys. Rev. Lett. 99, 030401 (2007)

BKT transition of 2D Bose gas
NIST(2009):
Thermal —— Quasicondensate —— BKT Superfluid
Phys. Rev. Lett. 102, 170401 (2009)

BKT transition of 2D Bose gas
Rotating frame
Synthetic magnetic field: Easier to construct optical lattices

BKT transition of 2D Bose gas
The frustrated XY model: Phys. Rev. A 82, 063625 (2010)
Hofstadter butterfly
Phys. Rev. B 14 , 2239 (1976).

BKT transition of 2D Bose gas
The origin of frustration:
Using the relationship: the model can be mapped into the “frustrated” quantum phase model:
The frustrated XY model was used to describe the superconducting Josephson arrays in transverse magnetic field by Teitel and Jayaprakash, 1983.
Phys. Rev. B 27 , 598 (1983); Phys. Rev. Lett. 51 , 1999 (1983).

BKT transition of 2D Bose gas
U(1) gauge symmetry breaking
 The physical quantities of a system described by the frustrated XY model can be gauge dependent, although observable quantities are usually gauge invariant in conventional systems.
 The imaging of density of the expanding condensates in cold-atom experiments in fact measures the canonical momentum of the original model.
 That is why the vector gauge potentials can be detected in the momentum distribution of the density matrix.

BKT transition of 2D Bose gas gauge dependent
Rev. Mod. Phys. 80, 885 (2008)
We can get the density profile by solving the frustrated XY model using the standard Metropolis Monte Carlo method.

BKT transition of 2D Bose gas
Metropolis Monte Carlo step
First of all, we choose a initial state that each site holds a phase zero. Every Monte
Carlo step can be performed as follows:
1. Pick a random site i from the lattices
2. Choose a random phase
3. Calculate the energy shift with
4. Accept the random walk with the probability
5. Calculate the new energy of the system and pick a new site j, go to step 1.
(over-relaxation, cluster algorithm, etc.)

BKT transition of 2D Bose gas
Illustration of the expansion image of the system at different temperatures T for the fully frustrated case ( f =1 / 2). The BKT transition takes place at about
T = 0 .
5 J/kB where the peaks decay to nearly zero in (b). The color represents the relative magnitude of the density which increases from purple to white.

BKT transition of 2D Bose gas
The central peak G 0 ( G ( kx = 0 , ky = 0)) of expansion image as a function of temperature T for four different fractional frustration f =1/2, 2/5, 1/3, 1/5. The square points are numerical results with the error bar obtained using the standard deviation. The circle in each figure guides the estimated critical transition temperature. The insert shows the corresponding expansion image close to the ground state.

BKT transition of 2D Bose gas
We are searching for experimental possibilities to identify the transition diagram. For example, to find the T
C at f=1/3 or 1/2.
The superconducting Josephson arrays in transverse magnetic field.
Phys. Rev. B 27 , 598 (1983); Phys. Rev. Lett. 51 , 1999 (1983).

BKT transition of 2D Bose gas the Raman detuning gradient
Nature 462 , 628 (2009).
For the case that
So we get:
Realized detuning in experiments:

BKT transition of 2D Bose gas
The transition temperature
For:
Higher transition temperature can be get by increasing the average particle number at each site N of the lattice potential V
0
。
0 or reducing depth

Cold atoms in the synthetic magnetic field
Charged Bose gas in magnetic field
BKT transition of 2D Bose gas
Trapped Bose gas

Trapped Bose gas
Neutral atoms in rotating frame
Charged bosons in magnetic field / Neutral bosons in synthetic field
Hamiltonian in direction:
  denote the frequencies of harmonic potential in the x, y plane and in the z direction.

Trapped Bose gas
The energy spectrum:
 z
,

, m
 n z
1
2
(2 n
ρ
 m

1)
α 2 
B
2  mB where
   l
2  
0
2 with
  l qB
2 Mc
B
   l z
  
0 z
    
T 0
T z

,  , m
 ln 1
 e

 z
,  , m
 
T

  
T

2
4 g
4

0
 
T with
 
0
D

1
 z

3 d r 

D

2 M
2
 q i A c

 2    2 

T
 k T
B
(
 z
)

Trapped Bose gas
To determine the BEC temperature:
N
 

 exp
 z
,

, m
 
T

1



1

T

3
2 g
3

0
 
T
Chin. Phys. Lett. 28, 060306 (2011)
T
N

N
0

N
T

N
0
 
'

 exp
 z
,

, m
 
0
T

1




1

N
0

T
3

2 g
3
(0)
The BEC temperature:
T
Homogeneous gas in magnetic field
Kling and Pelster, Phys. Rev. A 76 , 023609 (2007)
The Landau diamagnetization:
M
 


B
0



B
T
  
'
(2 n


) exp




2 z

B
2

,

, m

0

T

 m

1

N B
0

2

B
2
 
NB

2

B
2

Trapped Bose gas in in the synthetic magnetic field
Beyond SCA:
10
0
9
8
NUM(

=1)
SCA(

=1)
-300
NUM(

B=0)
SCA (
 B=0)
NUM(

B=0.5)
SCA (

B=0.5)
NUM(
 B=1)
SCA (

B=1)
-600
7
6
0 1 2
B
3 4
-900
0 10 20
T
30 40 50
0
-300
-600
-900
-1200
0
NUM(
 B=0)
SCA (

B=0)
NUM(
 B=0.5)
SCA (
 B=0.5)
NUM(

B=1)
SCA (
 B=1)
10 20
T
30 40 50

Trapped Bose gas in rotating frame
Beyond SCA:
10 150
0 8
NUM(

)
SCA(

) 6 -150
-300 4
-450 2
0
0.0
0.2
0.8
1.0
-600
0 0.4

0.6
0
-300
-600
NUM(
 =0)
SCA (
 =0)
NUM(
 =0.25)
SCA (

=0.25)
NUM(

=0.5)
SCA (
 =0.5)
10
-900
-1200
0 10 20
T
30 40 50
NUM(
 =0)
SCA (
 =0)
NUM(
 =-0.25)
SCA (

=-0.25)
NUM(
 =-0.5)
SCA (

=-0.5)
20
T
30 40 50

Beijing, August 24, 2012