Mathematical Analysis and Numerical
Simulation for Bose-Einstein Condensates
Weizhu Bao
Department of Mathematics
& Center of Computational Science and Engineering
National University of Singapore
Email: bao@math.nus.edu.sg
URL: http://www.math.nus.edu.sg/~bao
Outline
Motivation & theoretical predication
Gross-Pitaevskii equation (GPE)
Stationary, ground & central vortex states
Methods & results for ground states
Methods & results for dynamics
Extension to rotation frame & multi-component
Conclusions & Future challenges
Motivation
•
Bose-Einstein condensation:
–
–
–
–
•
Bosons at nano-Kevin temperature
many atoms occupy in one obit (at quantum mechanical ground state)
`super-atom’
new matter of wave. i.e., the fifth matter of state
Theoretical predication: Bose & Einstein
–
–
•
Bose, Z. Phys., 26 (1924) 82
Einstein, Sitz. Ber. Kgl. Preuss. Adad., Wiss. 22 (1924) 261
Experimental realization: JILA 1995
–
–
Anderson et al., Science, 269 (1995), 198: JILA Group; Rb
Davis et al., Phys. Rev. Lett., 75 (1995), 3969: MIT Group; Rb
–
Bradly et al., Phys. Rev. Lett., 75 (1995), 1687, Rice Group; Li
Experimental Results
JILA (95’,Rb,5,000)
ETH (02’,Rb, 300,000)
Motivation
•
2001 Nobel prize in physics:
–
•
C. Wiemann: U. Colorado; E. Cornell: NIST & W. Ketterle: MIT
Mathematical models:
–
–
•
Gross-Pitaevskii equation (mean field theory)
Quantum Boltzmann master equation (kinetic)
Mathematical analysis
–
•
Existence, dynamical laws, soliton-like solution, damping effect, etc.
Numerical Simulations
–
–
Numerical methods
Guiding and predicting outcome of new experiments
Possible applications
Quantized vortex for studying superfluidity
Square Vortex
lattices in spinor
BECs
Giant
vortices
Vortex
lattice
dynamics
Test quantum mechanics theory
Bright atom laser: multi-component
Quantum computing
Atom tunneling in optical lattice trapping, …..
Gross-Pitaevskii equation
Gross-Pitaevskii Equation (GPE)



 2 

1 2 
i  ( x , t )     ( x , t )  Vd ( x ) ( x , t )   d |  ( x , t ) |  ( x , t )
t
2
Normalization condition
 2 
 d | ( x, t ) | dx  1.
R
Two extreme regimes:
– Weakly interacting condensation |  d |  1
– Strongly repulsive interacting condensation  d  1
Gross-Pitaevskii equation
Conserved quantities
– Normalization of the wave function
N ( (t )) 
– Energy



| ( x , t ) | 2 dx  N ( (0))  1
Rd
d
1
2
2
4
[
|


(
x
,
t
)
|

V
(
x
)|

(
x
,
t
)
|

|

(
x
,
t
)
|
] dx
d
Rd
2
2
 E  ( (0))
E  ( (t ))  
Chemical potential


( (t ))  
Rd
 2

 2
 4 
1
[ |  ( x , t ) | Vd ( x ) |  ( x , t ) |   d | ( x , t ) | ] dx
2
Semiclassical scaling
When
, re-scaling
 d  1
x  x  1/ 2      d / 4
  1/  d2 /( d  2)




  
2 2  
i   ( x , t )     ( x , t )  Vd ( x )  ( x , t )  |   ( x , t ) |2   ( x , t )
t
2
With
E  (  )  
Rd
[
2


1
|   |2 Vd ( x ) |  |2  |  |4 ] dx  O(1)
2
2
Leading asymptotics (Bao & Y. Zhang, Math. Mod. Meth. Appl. Sci., 05’)
E ( )   1 E  (  )  O   1   O   d2 /( d  2) 
  ( )   1  (  )  O  1   O   d2 /( d  2) 
Quantum Hydrodynamics
Set


   e
iS  / 




, v  S , J    v  ,   1 /  d2 /(d  2 )
Geometrical Optics: (Transport + Hamilton-Jacobi)
 t      (   S  )  0,
1
2
 2

 t S  S  Vd ( x )   
2
2

1

 
Quantum Hydrodynamics (QHD): (Euler +3rd dispersion)
t      (   v  )  0

t ( J )    (
J  J




)  P(  )   Vd 
P(  )   2 / 2
2
4
(    ln   )
Stationary states
Stationary solutions of GPE


i  t
 ( x, t )  e  ( x )
Nonlinear eigenvalue problem with a constraint

 
 2 
1 2 
  ( x )     ( x )  Vd ( x ) ( x )   d |  ( x ) |  ( x ),
2
 2 
 d |  (x) | dx  1

x Rd
R
Relation between eigenvalue and eigenfunction
    ( )  E ( ) 
d
2

Rd
 4 
|  (x) | dx
Ground state
Ground state:
E (g )  min E ( ),  g    (g )  E (g ) 
|| || 1
d
2

R
d
| g ( x ) |4 dx
Existence and uniqueness of positive solution :
d  0
– Lieb et. al., Phys. Rev. A, 00’
Uniqueness up to a unit factor
g
 g ei0
with any constant 0
Boundary layer width & matched asymptotic expansion
– Bao, F. Lim & Y. Zhang, Trans. Theory Stat. Phys., 06’
Numerical methods for ground states
Runge-Kutta method: (M. Edwards and K. Burnett, Phys. Rev. A, 95’)
Analytical expansion: (R. Dodd, J. Res. Natl. Inst. Stan., 96’)
Explicit imaginary time method: (S. Succi, M.P. Tosi et. al., PRE, 00’)
Minimizing E ( ) by FEM: (Bao & W. Tang, JCP, 02’)
Normalized gradient flow: (Bao & Q. Du, SIAM Sci. Comput., 03’)
– Backward-Euler + finite difference (BEFD)
– Time-splitting spectral method (TSSP)
Gauss-Seidel iteration method: (W.W. Lin et al., JCP, 05’)
Spectral method + stabilization: (Bao, I. Chern & F. Lim, JCP, 06’)
Imaginary time method
Idea: Steepest decent method + Projection

1  E ( )
t ( x , t )  

2 
 
 ( x , t n 1)

 ( x , t n 1 ) 
,
 
||  ( x , t n 1) ||


 ( x ,0)  0 (x) with

1 2
   V ( x )   |  |2  , t n  t  t n 1
2

0
n  0,1,2, 
1
2
ˆ1
E (ˆ1 )  E (0 )
E (ˆ )  E ( )
1

|| 0 ( x ) || 1.
1
E (1 )  E (0 ) ??
g
– The first equation can be viewed as choosing t  i  in GPE
– For linear case: (Bao & Q. Du, SIAM Sci. Comput., 03’)
E0 (  (., tn1 ) )  E0 (  (., tn ) )    E0 (  (., 0) )
– For nonlinear case with small time step, CNGF
Normalized gradient glow
Idea: letting time step go to 0 (Bao & Q. Du, SIAM Sci. Comput., 03’)
 ( (., t ))
1 2
2
t ( x , t )     V ( x )    |  |  
 , t  0,
2
2
||  (., t ) ||
 ( x , 0)  0 ( x )
with
|| 0 ( x ) || 1.
– Energy diminishing
||  (., t ) |||| 0 || 1,
d
E ( (., t ))  0,
dt
t0
– Numerical Discretizations
• BEFD: Energy diminishing & monotone (Bao & Q. Du, SIAM Sci. Comput., 03’)
• TSSP: Spectral accurate with splitting error (Bao & Q. Du, SIAM Sci. Comput., 03’)
• BESP: Spectral accuracy in space & stable (Bao, I. Chern & F. Lim, JCP, 06’)
Ground states
Numerical results (Bao&W. Tang, JCP, 03’; Bao, F. Lim & Y. Zhang, TTSP, 06’)
– In 1d
• Box potential: V ( x)  0 0  x  1;  otherwise
2
V(x)

x
/2
• Harmonic oscillator potential:
– In 2d
• In a rotational frame
• With a fast rotation
– In 3d
• With a fast rotation
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Dynamics of BEC
Time-dependent Gross-Pitaevskii equation



 2 

1 2 
i  ( x , t )     ( x , t )  Vd ( x ) ( x , t )   d |  ( x , t ) |  ( x , t )
t
2


 ( x ,0)   0 ( x )
Dynamical laws
–
–
–
–
–
Time reversible & time transverse invariant
Mass & energy conservation
Angular momentum expectation
Condensate width
Dynamics of a stationary state with its center shifted
Angular momentum expectation
Definition:


Lz (t ) :  * Lz dx  i  * ( y x  x y ) dx , t  0
Rd
Lemma
Dynamical laws
d Lz (t )
dt
Rd
(Bao & Y. Zhang, Math. Mod. Meth. Appl. Sci, 05’)
 2 
 (   )  xy |  ( x , t ) | dx , t  0
2
x
2
y
Rd
For any initial data, with symmetric trap, i.e.  x   y , we have
Lz (t )  Lz (0),
E ,0 ( )  E ,0 ( 0 ),
Numerical test
next
t  0.
Angular momentum
expectation
Energy
back
Dynamics of condensate width
Definition:




 r (t )   ( x 2  y 2 ) | ( x , t |2 dx ,  (t )    2 | ( x , t |2 dx
Rd
Rd
Dynamic laws (Bao & Y. Zhang, Math. Mod. Meth. Appl. Sci, 05’)
– When
d  2& x   y
for any initial data:
d 2 r (t )
2

4
E
(

)

4

 r (t ),

0
x
2
dt
– When
d  2& x   y
1
2
with initial data
 x (t )   y (t )   r (t ),
next
t 0
Numerical Test
t0
– For any other cases:
d 2 (t )
2

4
E
(

)

4

 (t )  f (t ),

0

2
dt
 0 ( x, y )  f (r ) eim
t 0
Symmetric trap
Anisotropic trap
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Dynamics of Stationary state with a shift

 
Choose initial data as:  0 ( x )  s ( x  x0 )
The analytical solutions is: (Garcia-Ripoll el al., Phys. Rev. E, 01’)
 ( x , t )  ei t s ( x  x (t )) eiw( x ,t ) ,
s
– In 2D:
x(t )   x2 x(t )  0,
w( x , t )  0, x (0)  x0
example
y (t )   y (t )  0,
2
y
x(0)  x0 ,
y (0)  y0 , x(0)  0,
y (0)  0
– In 3D, another ODE is added
z (t )   z2 z (t )  0,
z (0)  z0 , z (0)  0
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Numerical methods for dynamics
Lattice Boltzmann Method (Succi, Phys. Rev. E, 96’; Int. J. Mod. Phys., 98’)
Explicit FDM (Edwards & Burnett et al., Phys. Rev. Lett., 96’)
Particle-inspired scheme (Succi et al., Comput. Phys. Comm., 00’)
Leap-frog FDM (Succi & Tosi et al., Phys. Rev. E, 00’)
Crank-Nicolson FDM (Adhikari, Phys. Rev. E 00’)
Time-splitting spectral method (Bao, Jaksch&Markowich, JCP, 03’)
Runge-Kutta spectral method (Adhikari et al., J. Phys. B, 03’)
Symplectic FDM (M. Qin et al., Comput. Phys. Comm., 04’)
Time-splitting spectral method (TSSP)
Time-splitting:
Step 1:
Step 2:
1
i  t ( x, t )   2  ,
2
i  t ( x , t )  Vd ( x ) ( x , t )   d | ( x, t ) |2  ( x, t )
 | ( x, t ) || ( x , tn ) |
 ( x , tn1 )  e
 i (Vd ( x )   d | ( x ,tn )|2 ) t
 ( x , tn )
For non-rotating BEC
– Trigonometric functions (Bao, D. Jaksck & P. Markowich, J. Comput. Phys., 03’)
– Laguerre-Hermite functions (Bao & J. Shen, SIAM Sci. Comp., 05’)
Properties of TSSP
–
–
–
–
–
–
Explicit, time reversible & unconditionally stable
Easy to extend to 2d & 3d from 1d; efficient due to FFT
Conserves the normalization
Spectral order of accuracy in space
2nd, 4th or higher order accuracy in time
Time transverse invariant


Vd ( x )  Vd ( x )  


|  ( x , t ) |2
unchanged
– ‘Optimal’ resolution in semicalssical regime
h  O   ,
k  O   ,
  1 / d 2 /(2d )
Dynamics of Ground states
1d dynamics: 1  100 at t  0, x  4x
2d dynamics of BEC (Bao, D. Jaksch & P. Markowich, J. Comput. Phys., 03’)
– Defocusing:  2  20, at t  0  x  2 x ,  y  2 y
– Focusing (blowup): At t  0 2  40  50
3d collapse and explosion of BEC (Bao, Jaksch & Markowich,J. Phys B, 04’)
– Experiment setup leads to three body recombination loss



1 2
i  ( x , t )      V ( x )   |  |2   i 0  2 |  |4 
t
2
– Numerical results:
• Number of atoms , central density & Movie
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Collapse and Explosion of BEC
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Number of atoms in condensate
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Central density
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Extension
GPE with damping term (Bao & D. Jaksch, SIAM J. Numer. Anal., 04’)

1 2
i  ( x , t )      V ( x )   |  |2   ig ( |  |2 )
t
2
Two-component BEC



1
 ( x , t )    2  V1 ( x )  ( 11 |  |2  12 |  |2 )  f (t )
t
2



1
i
 ( x , t )    2  V2 ( x )  (  21 | |2   22 |  |2 )  f (t )
t
2
i
– Methods for ground state & dynamics (Bao, Multiscale Mod. Sim., 04’)
– Dynamics laws (Bao & Y. Zhang, 06’)
Extension
GPE in a rotational frame
2



2

i   ( x , t )  [
  V ( x )   Lz  N U 0 | |2 ]
t
2m
   
Lz : xpy  ypx  i( x y  y x )  i , L  x  P, P  i
– For ground state (Bao, H. Wang & P. Markowich, Commun. Math. Sci., 04’)
– Dynamical laws (Bao,Du&Zhang, SIAM Appl. Math., 06’;Appl. Numer. Math. 06’)
– Numerical methods
• Time-splitting +polar coordinate (Bao,Du&Zhang, SIAM Appl. Math., 06’)
• Time-splitting + ADI in space (Bao & H. Wang, J. Comput. Phys., 06’)
Conclusions & Future Challenges
Conclusions:
–
–
–
–
–
Mathematical results for ground & excited states
Dynamical laws in BEC
Efficient methods for ground state & dynamics
Comparison with experimental resutls
Vortex stability & interaction in 2D
Future Challenges
– Multi-component BEC
– Quantized vortex states & dynamics in 3D
– Coupling GPE & QBE
Collaborators
• External
–
–
–
–
–
–
–
P.A. Markowich, Institute of Mathematics, University of Vienna, Austria
D. Jaksch, Department of Physics, Oxford University, UK
Q. Du, Department of Mathematics, Penn State University, USA
J. Shen, Department of Mathematics, Purdue University, USA
L. Pareschi, Department of Mathematics, University of Ferarra, Italy
W. Tang & L. Fu, IAPCM, Beijing, China
I-Liang Chern, Department of Mathematics, National Taiwan University, Taiwan
• External
– Yanzhi Zhang, Hanquan Wang, Fong Ying Lim, Ming Huang Chai
– Yunyi Ge, Fangfang Sun, etc.