Transcript Charged Bose gas in magnetic field

Cold atoms in the synthetic magnetic field
Qiang Gu
（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
Phys. Rev. 100, 463 (1955)
2
 jk
z
2
kz
1

   j  
2m *
2

DL 
qB L x L y
2 c
where
 
qB
m*c
infinity
Phys. Rev. E 60, 5275 (1999)
Charged Bose gas in magnetic field
PRL96, 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;
 jk
z
: Landau level
j  0,1, 2, ...
 
qB
m*c
qB L x L y
2 c
: degeneracy;
2
 jk
z
2
kz
1

   j  
2m *
2

: cyclotron frequency
  : Zeeman energy
   1, 0;
DL 
g
   g
q
B
m*c
: Lande factor
J. Phy.:Condens. Matt. 23, 026003(2011)
PM vs DM in spin-1 Bose gas
Grand thermodynamical potential：
3
 T 0  
V  m *  2
2


2



   D , 0
1


where     ,   


l 1
D  3

1  e 

 lx
x  

  g q B /( m c )  (1 / 2   ) 
particle number and magnetization：
3
M T 0
 lx    
l 2e
q  m*


m * c  2 
2
2


    D , 0 
3
 m*
n  x
 2 
1

1

 x  g     1  2  D , 0   x  2  2  D ,1 
2

2
2


  2  D , 0
1

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  
gc
1


D Le

 
jk z
  

j , k z ,
is determined by
1
2

1
x
In the limit cases,

1
x
e 1
gc
t 
g c (e

e
2 gc x
2 gc x
e
 1)
gc x
1
 1 / 8  0.35355
gc
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 4He, 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 Tc=πJ/2
BKT transition of 2D Bose gas
Experimental system in cold atoms:
ENS(2006):
Phys. Rev. A 81, 023623 (2010)
Nature 441, 1118 (2006)
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.
2.
3.
4.
Pick a random site i from the lattices
Choose a random phase
Calculate the energy shift
with
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 = 0.5J/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 G0 (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 TC 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
So we get:
For the case that
Realized detuning in experiments:
Nature 462, 628 (2009).
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 N0 or reducing depth
of the lattice potential V0。
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:
where  0 and  z denote the frequencies of harmonic potential in the x, y plane
and in the z direction.
Trapped Bose gas
The energy spectrum:
n
where
 
z
,n ,m
 nz 
 
2
l
1
 (2 n ρ  m  1) α  B  m B
2
with  l 
2
0
 n ,n ,m  
z 


T
 T  ln  1  e

nz ,n ,m

0 
with
 D
1
3 
d r

z
2M


D     i
q
c
qB
B  l  z
2Mc
  T  0
,
The grand thermodynamic potential
T
2
2
4

T
 0   
   2 g4 



T



2
 V (r ) 
A
2

2





T  k BT (  z )
  0  z
结果与分析
Trapped Bose gas
To determine the BEC temperature:
N 

 nz ,n ,m  


exp

1


T


N  N0  NT  N0 

1

T
3

2
Chin. Phys. Lett. 28, 060306 (2011)
  
g3  0

 T

 nz ,n ,m   0


'
exp

1


T


(above T C )
1
 N0 
T
3

2
g 3 (0)
(bellow T C )
The BEC temperature:
Homogeneous gas in magnetic field
Kling and Pelster, Phys. Rev. A 76, 023609 (2007)
The Landau diamagnetization:
M 
 0
B

 T
B
 
(2 n   m ) B
'
m
 B
  n ,n ,m   0 
2
exp 


2
z
T
  1


N0B
 B
2

2
NB
 B
2
2
Trapped Bose gas in in the synthetic magnetic field
Beyond SCA:
10
0
NUM(B=0)
SCA(B=0)
NUM(B=0.5)
SCA(B=0.5)
NUM(B=1)
SCA(B=1)
NUM(=1)
SCA(=1)
-300
M
8
-600
7
6
0
1
2
3
-900
4
0
10
B
20
30
T
0
NUM(B=0)
SCA(B=0)
NUM(B=0.5)
SCA(B=0.5)
NUM(B=1)
SCA(B=1)
-300

Tc
9
-600
-900
-1200
0
10
20
30
T
40
50
40
50
Trapped Bose gas in rotating frame
Beyond SCA:
10
150
8
0
NUM()
SCA()
-150
NUM(=0)
SCA(=0)
NUM(=-0.25)
SCA(=-0.25)
NUM(=-0.5)
SCA(=-0.5)
M
Tc
6
4
-300
2
-450
-600
0.2
0.4
0.6
0.8
1.0
0
10

20
30
T
0
NUM(=0)
SCA(=0)
NUM(=0.25)
SCA(=0.25)
NUM(=0.5)
SCA(=0.5)
-300

0
0.0
-600
-900
-1200
0
10
20
30
T
40
50
40
50
Thanks for your attention!
Beijing, August 24, 2012