Mathematical Analysis and Numerical Simulation for Bose-Einstein Condensation

1
Mathematical Analysis and Numerical
Simulation for Bose-Einstein Condensation

• Weizhu Bao
• Department of Mathematics
• Center of Computational Science and Engineering
• National University of Singapore
• Email bao_at_math.nus.edu.sg
• URL http//www.math.nus.edu.sg/bao

2
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
• I-Liang Chern, Department of Mathematics,
  National Taiwan University, Taiwan
• C. Schmeiser R.M. Weishaeupl, University of
  Vienna, Austria
• W. Tang L. Fu, Beijing Institute of Appl. Phys.
  Comput. Math., China
• Internal
• Yanzhi Zhang, Hanquan Wang, Fong Ying Lim, Ming
  Huang Chai
• Yunyi Ge, Fangfang Sun, etc.

3
Outline

• Part I Predication Mathematical modeling
• Theoretical predication
• Physical experiments and results
• Applications
• Gross-Pitaevskii equation
• Part II Analysis Computation for Ground states
  
• Existence uniqueness
• Energy asymptotics asymptotic approximation
• Numerical methods
• Numerical results

4
Outline

• Part III Analysis Computation for Dynamics in
  BEC
• Dynamical laws
• Numerical methods
• Vortex stability interaction
• Part IV Rotating BEC multi-component BEC
• BEC in a rotational frame
• Two-component BEC
• Spinor BEC
• BEC at finite temperature
• Conclusions Future challenges

5
Part I

• Predication
• Mathematical modeling

6
Theoretical predication

• Particles be divided into two big classes
• Bosons photons, phonons, etc
• Integer spin
• Like in same state many can occupy one obit
• Sociable gregarious
• Fermions electrons, neutrons, protons etc
• Half-integer spin each occupies a single obit
• Loners due to Pauli exclusion principle

7
Theoretical predication

• For atoms, e.g. bosons
• Get colder
• Behave more like waves less like particles
• Very cold
• Overlap with their neighbors
• Extremely cold
• Most atoms behavior in the same way, i.e
  gregarious
• quantum mechanical ground state,
• super-atom new matter of wave fifth state

8
Theoretical predication

• S.N. Bose Z. Phys. 26 (1924)
• Study black body radiation object very hot
• Two photons be counted up as either identical or
  different
• Bose statistics or Bose-Einstein statistics
• A. Einstein Sitz. Ber. Kgl. Preuss. Adad. Wiss.
  22 (1924)
• Apply the rules to atoms in cold temperatures
• Obtain Bose-Einstein distribution in a gas

9
(No Transcript)
10
(No Transcript)
11
Experimental results

• JILA (95, Rb, 5,000) Science 269 (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

12
Experimental results

• Experimental implementation
• JILA (95) First experimental realization of BEC
  in a gas
• NIST (98) Improved experiments
• MIT, ENS, Rice,
• ETH, Oxford,
• Peking U.,
• 2001 Nobel prize in physics
• C. Wiemann U. Colorado
• E. Cornell NIST
• W. Ketterle MIT

ETH (02, Rb, 300,000)
13
Experimental difficulties

• Low temperatures ? absolutely zero (nK)
• Low density in a gas

14
Experimental techniques

• Laser cooling
• Magnetic trapping
• Evaporative Cooling

(100k300k)
15
Possible applications

• Quantized vortex for studying superfluidity
• Test quantum mechanics theory
• Bright atom laser multi-component
• Quantum computing
• Atom tunneling in optical lattice trapping, ..

Square Vortex lattices in spinor BECs
Vortex latticedynamics
Giant vortices
16
Mathematical modeling

• N-body problem
• (3N1)-dim linear Schroedinger equation
• Mean field theory
• Gross-Pitaevskii equation (GPE)
• (31)-dim nonlinear Schroedinger equation (NLSE)
• Quantum kinetic theory
• High temperature QBME (331)-dim
• Around critical temperature QBMEGPE
• Below critical temperature GPE

17
Gross-Pitaevskii equation (GPE)

• Physical assumptions
• At zero temperature
• N atoms at the same hyperfine species (Hartree
  ansatz)
• The density of the trapped gas is small
• Interatomic interaction is two-body elastic and
  in Fermi form

18
Second Quantization model

• The second quantized Hamiltonian
• A gas of bosons are condensed into the same
  single-particle state
• Interacting by binary collisions
• Contained by an external trapping potential

19
Second quantization model

• Crucial Bose commutation rules
• Atomic interactions are low-energy two-body
  s-wave collisions, i.e. essentially elastic
  hard-sphere collisions
• The second quantized Hamiltonian

20
Second quantization model

• The Heisenberg equation for motion
• For a single-particle state with macroscopic
  occupation
• Plugging, taking only the leading order term
• neglecting the fluctuation terms (i.e., thermal
  and quantum depletion of the condensate)
• Valid only when the condensate is
  weakly-interacting low tempertures

21
Gross-Pitaevskii equation

• The Schrodinger equation (Gross, Nuovo. Cimento.,
  61 Pitaevskii, JETP,61 )
• The Hamiltonian
• The interaction potential is taken as in Fermi
  form

22
Gross-Pitaevskii equation

• The 3d Gross-Pitaevskii equation (
  )
• V is a harmonic trap potential
• Normalization condition

23
Gross-Pitaevskii equation

• Scaling (w.l.o.g. )
• Dimensionless variables
• Dimensionless Gross-Pitaevskii equation
• With

24
Gross-Pitaevskii equation

• Typical parameters ( )
• Used in JILA
• Used in MIT

25
Gross-Pitaevskii equation

• Reduction to 2d (disk-shaped condensation)
• Experimental setup
• Assumption No excitations along z-axis due to
  large energy
• 2d Gross-Pitaevskii equation (
  )
• 
  

26
Numerical Verification
27
Numerical Results
Bao, Y. Ge, P. Markowich R. Weishaupl, 06
28
Gross-Pitaevskii equation

• General form of GPE ( )
• with
• Normalization condition

29
Gross-Pitaevskii equation

• Two kinds of interaction
• Repulsive (defocusing) interaction
• Attractive (focusing) interaction
• Two extreme regimes
• Weakly interacting condensation
• Strongly repulsive interacting condensation

30
Gross-Pitaevskii equation

• Conserved quantities
• Normalization of the wave function
• Energy
• Chemical potential

31
Semiclassical scaling

• When , re-scaling
• With
• Leading asymptotics (Bao Y. Zhang, Math. Mod.
  Meth. Appl. Sci., 05)

32
Quantum Hydrodynamics

• Set
• Geometrical Optics (Transport
  Hamilton-Jacobi)
• Quantum Hydrodynamics (QHD) (Euler 3rd
  dispersion)

33
Part II

• Analysis Computation
• for
• Ground states

34
Stationary states

• Stationary solutions of GPE
• Nonlinear eigenvalue problem with a constraint
• Relation between eigenvalue and eigenfunction

35
Stationary states

• Equivalent statements
• Critical points of over the
  unit sphere
• Eigenfunctions of the nonlinear eigenvalue
  problem
• Steady states of the normalized gradient
  flow(Bao Q. Du, SIAM J. Sci. Compu., 03)
• Minimizer/saddle points over the unit sphere
• For linear case (Bao Y. Zhang,
  Math. Mod. Meth. Appl. Sci., 05)
• Global minimizer vs saddle points
• For nonlinear case
• Global minimizer, local minimizer (?) vs saddle
  points

36
Ground state

• Ground state
• Existence and uniqueness of positive solution
• Lieb et. al., Phys. Rev. A, 00
• Uniqueness up to a unit factor
• Boundary layer width matched asymptotic
  expansion
• Bao, F. Lim Y. Zhang, Trans. Theory Stat.
  Phys., 06

37
Excited central vortex states

• Excited states
• Central vortex states
• Central vortex line states in 3D
• Open question (Bao W. Tang, JCP, 03 Bao, F.
  Lim Y. Zhang, TTSP, 06)

38
Approximate ground states

• Three interacting regimes
• No interaction, i.e. linear case
• Weakly interacting regime
• Strongly repulsive interacting regime
• Three different potential
• Box potential
• Harmonic oscillator potential
• BEC on a ring or torus

39
Energies revisited

• Total energy
• Kinetic energy
• Potential energy
• Interaction energy
• Chemical potential

40
Box Potential in 1D

• The potential
• The nonlinear eigenvalue problem
• Case I no interaction, i.e.
• A complete set of orthonormal eigenfunctions

41
Box Potential in 1D

• Ground state its energy
• j-th-excited state its energy
• Case II weakly interacting regime, i.e.
• Ground state its energy
• j-th-excited state its energy

42
Box Potential in 1D

• Case III Strongly interacting regime, i.e.
• Thomas-Fermi approximation, i.e. drop the
  diffusion term
• Boundary condition is NOT satisfied, i.e.
• Boundary layer near the boundary

43
Box Potential in 1D

• Matched asymptotic approximation
• Consider near x0, rescale
• We get
• The inner solution
• Matched asymptotic approximation for ground state

44
Box Potential in 1D

• Approximate energy
• Asymptotic ratios
• Width of the boundary layer

45

• Numerical observations

46
Box Potential in 1D

• Matched asymptotic approximation for excited
  states
• Approximate chemical potential energy

47
Fifth excited states
48
Energy Chemical potential
49
Box potential in 1D

• Boundary layers interior layers with width
• Observations energy chemical potential are in
  the same order
• Asymptotic ratios
• Extension to high dimensions

50
Harmonic Oscillator Potential in 1D

• The potential
• The nonlinear eigenvalue problem
• Case I no interaction, i.e.
• A complete set of orthonormal eigenfunctions

51
Harmonic Oscillator Potential in 1D

• Ground state its energy
• j-th-excited state its energy
• Case II weakly interacting regime, i.e.
• Ground state its energy
• j-th-excited state its energy

52
Harmonic Oscillator Potential in 1D

• Case III Strongly interacting regime, i.e.
• Thomas-Fermi approximation, i.e. drop the
  diffusion term
• Characteristic length
• It is NOT differentiable at
• The energy is infinite by direct definition

53
Harmonic Oscillator Potential in 1D

• A new way to define the energy
• Asymptotic ratios

54

• Numerical observations

55
Harmonic Oscillator Potential in 1D

• Thomas-Fermi approximation for first excited
  state
• Jump at x0!
• Interior layer at x0

56
Harmonic Oscillator Potential in 1D

• Matched asymptotic approximation
• Width of interior layer
• Ordering

57
Harmonic Oscillator Potential

• Extension to high dimensions
• Identity of energies for stationary states in
  d-dim.
• Scaling transformation
• Energy variation vanishes at first order in

58
BEC on a ring

• The potential
• The nonlinear eigenvalue problem
• For linear case, i.e.
• A complete set of orthonormal eigenfunctions

59
BEC on a ring

• Ground state its energy
• j-th-excited state its energy
• Some properties
• Ground state its energy
• With a shift
• Interior layer can be happened at any point in
  excited states

60
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 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)

61
Imaginary time method

• Idea Steepest decent method Projection
• Physical institutive in linear case
• Solution of GPE
• Imaginary time dynamics

62
Mathematical justification

• For gradient flow (Bao Q. Du, SIAM Sci.
  Comput., 03)
• For linear case (Bao Q. Du, SIAM Sci.
  Comput., 03)
• For nonlinear case ???

63
Mathematical justification
64
Normalized gradient glow

• Idea (Bao Q. Du, SIAM Sci. Comput., 03)
• The projection step is equivalent to solve an ODE
• Gradient flow with discontinuous coefficients
• Letting time step go to 0
• Mass conservation Energy diminishing

65
Fully discretization

• Consider in 1D
• Different Numerical Discretizations
• Physics literatures Crank-Nicolson FD or Forward
  Euler FD
• 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)
• Crank-Ncolson FD for normalized gradient flow

66
Backward Euler Finite Difference

• Mesh and time steps
• BEFD discretization
• 2nd order in space unconditional stable at each
  step, only a linear system with sparse matrix to
  be solved!

67
Backward Euler Spectral method

• Discretization
• Spectral order in space efficient accurate

68
Ground states

• Numerical results (BaoW. Tang, JCP, 03 Bao, F.
  Lim Y. Zhang, TTSP, 06)
• In 1d
• Box potential
• Ground state excited states first fifth
• Harmonic oscillator potential
• ground first excited Energy and chemical
  potential
• Double well potential
• Ground first excited state
• Optical lattice potential
• Ground first excited state with potential

next
69
back
70
back
71
back
72
back
73
back
74
back
75
back
76
back
77
back
78
back
79
back
80
Ground states

• Numerical results (BaoW. Tang, JCP, 03 Bao, F.
  Lim Y. Zhang, TTSP, 06)
• In 2d
• Harmonic oscillator potentials
• ground
• Optical lattice potential
• Ground excited states
• In 3D
• Optical lattice potential
• ground excited states

next
81
back
82
back
83
back
84
back
85
Part III

• Analysis Computation
• for
• Dynamics in BEC

86
Dynamics of BEC

• Time-dependent Gross-Pitaevskii equation
• Dynamical laws
• Time reversible time transverse invariant
• Mass energy conservation
• Angular momentum expectation
• Condensate width
• Dynamics of a stationary state with its center
  shifted

87
Angular momentum expectation

• Definition
• Lemma Dynamical laws (Bao Y. Zhang, Math.
  Mod. Meth. Appl. Sci, 05)
• For any initial data, with symmetric trap, i.e.
  , we have
• Numerical test next

88
Angular momentum expectation
Energy
back
89
Dynamics of condensate width

• Definition
• Dynamic laws (Bao Y. Zhang, Math. Mod. Meth.
  Appl. Sci, 05)
• When for any initial data
• 
  
• When with initial data
• 
  Numerical Test
• For any other cases
• 
  

next
90
Symmetric trap
Anisotropic trap
back
91
Dynamics of Stationary state with a shift

• Choose initial data as
• The analytical solutions is (Garcia-Ripoll el
  al., Phys. Rev. E, 01)
• In 2D
• In 3D, another ODE is added

92
Solution of the center of mass

• Center of mass Bao Y. Zhang, Appl. Numer.
  Math., 2006
• In a non-rotating BEC
• Trajectory of the center Motion
  of the solution
• Pattern Classification
  
• Each component of the center is a periodic
  function
• In a symmetric trap, the trajectory is a straight
  segment
• If is a rational , the center
  moves periodically with period
• If is an irrational , the
  center moves chaotically, envelope is a rectangle

next
93
back
94
back
95
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,
  JakschMarkowich, JCP, 03)
• Runge-Kutta spectral method (Adhikari et al., J.
  Phys. B, 03)
• Symplectic FDM (M. Qin et al., Comput. Phys.
  Comm., 04)

96
Time-splitting spectral method (TSSP)

• Time-splitting
• 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)

97
Time-splitting spectral method
98
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
• Optimal resolution in semicalssical regime

99
Dynamics of Ground states

• 1d dynamics
• 2d dynamics of BEC (Bao, D. Jaksch P.
  Markowich, J. Comput. Phys., 03)
• Defocusing
• Focusing (blowup)
• 3d collapse and explosion of BEC (Bao, Jaksch
  Markowich,J. Phys B, 04)
• Experiment setup leads to three body
  recombination loss
• Numerical results
• Number of atoms , central density Movie

next
100
back
101
back
102
back
103
Collapse and Explosion of BEC
back
104
Number of atoms in condensate
back
105
Central density
back
106
back
107
Central quantized vortices

• Central vortex states in 2D
• with
• Vortex Dynamics
• Dynamical stability
• Interaction
• Pattern I
• Pattern II

108
Central Vortex states
109
Central Vortex states
110
Vortex stability interaction

• Dynamical stability (Bao Y. Zhang, Math. Mod.
  Meth. Appl. Sci., 05)
• m1 stable velocity
• m2 unstable velocity
• Interaction (Bao Y. Zhang, Math. Mod. Meth.
  Appl. Sci., 05)
• N2 Pair velocity trajectory phase
  phase2
• Anti-pair phase trajectory
  angular trajectory2
• N3 velocity trajectory
• Pattern II Linear nonlinear
• Interaction laws
• On-going with Prof. L. Fu Miss Y. Zhang

next
111
back
112
back
113
back
114
back
115

back
116

Linear case
back
117

Noninear case BEC
back
118

Linear case
back
119

back
Linear case
120

back
Linear case
121

back
122

back
123

back
124
back
125
Some Open Questions

• Dynamical laws for vortex interaction
• With a quintic damping, mass goes to constant
• Convergence error estimate of the TSSP?
• Energy diminishing of the gradient flow in
  nonlinear case error estimate ?

126
Part IV

• Rotating BEC
• multi-component BEC

127
Rotating BEC

• The Schrodinger equation (
  )
• The Hamiltonian
• The interaction potential is taken as in Fermi
  form

128
Rotating BEC

• The 3D Gross-Pitaevskii equation (
  )
• Angular momentum rotation
• V is a harmonic trap potential
• Normalization condition

129
Rotating BEC

• General form of GPE ( )
• with
• Normalization condition

130
Rotating BEC

• Conserved quantities
• Normalization of the wave function
• Energy
• Chemical potential

131
Semiclassical scaling

• When , re-scaling
• With
• Leading asymptotics

132
Quantum Hydrodynamics

• Set
• Geometrical Optics (Transport
  Hamilton-Jacobi)
• Quantum Hydrodynamics (QHD) (Euler 3rd
  dispersion)

133
Stationary states

• Stationary solutions of GPE
• Nonlinear eigenvalue problem with a constraint
• Relation between eigenvalue and eigenfunction

134
Stationary states

• Equivalent statements
• Critical points of over the
  unit sphere
• Eigenfunctions of the nonlinear eigenvalue
  problem
• Steady states of the normalized gradient flow
• Minimizer/saddle points over the unit sphere
• For linear case
• Global minimizer vs saddle points
• For nonlinear case
• Global minimizer, local minimizer (?) vs saddle
  points

135
Ground state

• Ground state
• Existence
• Seiringer (CMP, 02)
• Uniqueness of positive solution
• Lieb et al. (PRA, 00)
• Energy bifurcation
• Aftalion Du (PRA, 01) B., Markowich Wang 04

136
Numerical results

• Ground states
• in 2D in 3D
  isosurface
• Quantized vortex generation in 2D
• surface contour
• Vortex lattice
• Symmetric trapping anisotropic trapping
• Giant vortex generation in 2D
• surface contour
• Giant vortex
• In 2D In 3D

next
137
back
138
back
139
back
140
back
141
back
142
back
143
back
144
back
145
back
146
back
147
back
148
Numerical Asymptotical results

• Critical angular frequency symmetric state vs
  quantized vortex state
• Asymptotics of the energy
• Ratios between energies of different states
• Rank according to energy and chemical potential
• Stationary states are ranked according to their
  energy, then their chemical potential are in the
  same order. Next

149
back
150
back
151
back
152
Dynamical laws of rotating BEC

• Time-dependent Gross-Pitaevskii equation
• Dynamical laws
• Time reversible time transverse invariant
• Conservation laws
• Angular momentum expectation
• Condensate width
• Dynamics of a stationary state with its center
  shifted

153
Conservation laws

• Conserved quantities
• Normalization of the wave function
• Energy
• Chemical potential

154
Angular momentum expectation

• Definition
• Lemma The dynamics of satisfies
• For any initial data, with symmetric trap, i.e.
  , we have
• Numerical test
  next
• Bao, Du Zhang, SIAM J. Appl. Math., 66 (2006),
  758

155
Angular momentum expectation
Energy
back
156
Dynamics of condensate width

• Definition Bao, Du Zhang, SIAM J. Appl.
  Math., 66 (2006), 758
• Dynamic laws
• When for any initial data
• 
  
• When with initial data
• 
  Numerical Test
• For any other cases
• 
  

next
157
Symmetric trap
Anisotropic trap
back
158
Dynamics of Stationary state with a shift

• Choose initial data as
• The analytical solutions is Bao, Du Zhang,
  SIAM J. Appl.Math., 2006
• In 2D
• In 3D, another ODE is added

159
Solution of the center of mass

• Center of mass Bao Zhang, Appl. Numer. Math.,
  2006
• In a non-rotating BEC
• Pattern Classification
  
• Each component of the center is a periodic
  function
• In a symmetric trap, the trajectory is a straight
  segment
• If is a rational , the center
  moves periodically with period
• If is an irrational , the
  center moves chaotically, envelope is a rectangle

160
Solution of the center of mass

• In a rotating BEC with a symmetric trap
• Trajectory of the center
• Distance between the center and trapping center
  
• Motion of the solution 0.5 1 2
  4
• Pattern Classification

next
161
1/5, 4/5, 1
3/2, 6, Pi
back
162
back
163
back
164
back
165
back
166
back
167
Pattern Classification

• Pattern Classification Bao Zhang, Appl. Numer.
  Math., 2006
• The distance between the center and trap center
  is periodic function
• When is a rational
• The center moves periodically
• The graph of the trajectory is unchanged under a
  rotation
• When is an irrational ,
• The center moves chaotically
• The envelope of the trajectory is a circle
• The solution of GPE agrees very well with those
  from the ODE system

back
168
Solution of the center of mass

• In a rotating BEC with an anisotropic trap
• When
  results
• The trajectory is a spiral coil to infinity
• The trajectory is an ellipse
• Otherwise result1
  result2
• The center moves chaotically graph is a bounded
  set
• The center moves along a straight line to
  infinity

next
169
back
170
back
171
back
172
Total density with dissipation

• Time-dependent Gross-Pitaevskii equation
• Lemma The dynamics of total density satisfies
• The total density decreases when
• density function
  energy next

173
back
174
back
175
Numerical Methods

• Time-splitting pseudo-spectral method (TSSP)
• Use polar coordinates (B., Q. Du Y. Zhang, SIAP
  06)
• Time-splitting ADI technique (B. H. Wang,
  JCP, 06)
• Generalized Laguerre-Hermite functions (B., J.
  Shen H. Wang, 06)

176
Numerical methods for rotating BEC

• Numerical Method one (Bao, Q. Du Y. Zhang,
  SIAM, Appl. Math. 06)
• Ideas
• Time-splitting
• Use polar coordinates angular momentum becomes
  constant coefficient
• Fourier spectral method in transverse direction
  FD or FE in radial direction
• Crank-Nicolson in time
• Features
• Time reversible
• Time transverse invariant
• Mass Conservation in discretized level
• Implicit in 1D efficient to solve
• Accurate unconditionally stable

177
Numerical methods for rotating BEC

• Numerical Method two (Bao H. Wang, J. Comput.
  Phys. 06)
• Ideas
• Time-splitting
• ADI technique Equation in each direction become
  constant coefficient
• Fourier spectral method
• Features
• Time reversible
• Time transverse invariant
• Mass Conservation in discretized level
• Explicit unconditionally stable
• Spectrally accurate in space

178
Dynamics of ground state

• Choose initial data as
• ground state
• Change the frequency in the external potential
• Case 1 symmetric
• surface contour
• Case 2 non-symmetric
• surface contour
• Case 3 dynamics of a vortex lattice with 45
  vortices
• image contour
  next

179
back
180
back
181
back
182
back
183
back
184
back
185
Interaction of two vortices in linear
186
Interaction of two vortices in linear
187
Interaction of two vortices in linear
188
Interaction of vortices in nonlinear
189
Interaction of vortices in nonlinear
190
Interaction of vortices in nonlinear
191
Interaction of vortices in nonlinear
192
Some Open Questions

• Dynamical laws for vortex interaction
• With a quintic damping, mass goes to constant
• Semiclassical limit when initial data has
  vortices???
• Vortex line interaction laws, topological change?
• What is a giant vortex?

193
Two-component BEC

• The 3D coupled Gross-Pitaevskii equations
• Normalization conditions
• Intro- inter-atom Interactions

194
Two-component BEC

• Nondimensionalization
• Normalization conditions
• There is external driven field
• No external driven field

195
Two-component BEC

• Energy
• Reduction to one-component

196
Two-Component BEC

• Semiclassical scaling
• Semiclassical limit
• No external field
• WKB expansion, two-fluid model
• With external field
• WKB expansion doesnt work, Winger transform

197
Ground state

• No external field
• Nonlinear eigenvalue problem
• Existence uniqueness of positive solution
• Numerical methods can be extended

198
Ground states
crater
199
Ground state

• With external field
• Nonlinear eigenvalue problem
• Existence uniqueness of positive solution ???
• Numerical methods can be extended????

200
Dynamics

• Dynamical laws
• Conservation of Angular momentum expectation
• Dynamics of condensate width
• Dynamics of a stationary state with a shift
• Dynamics of mass of each component, they are
  periodic function when
• Vortex can be interchanged!
• Numerical methods
• Time-splitting spectral method

201
Dynamics
202
(No Transcript)
203
Dynamics
204
Spinor BEC

• Spinor F1 BEC
• With

205
Spinor BEC

• Total mass conservation
• Total magnetization conservation
• Energy conservation

206
Spinor BEC

• Dimension reduction
• Ground state
• Existence uniqueness of positive solution??
• Numerical methods ???
• Dynamics
• Dynamical laws
• Numerical methods TSSP
• Semiclassical limit hydrodynamics equation??

207
BEC at Finite Temperature

• Condensate coexists with non-condensed thermal
  cloud
• Coupled equations of motion for condensate and
  thermal cloud
• Mean-field theory in collisionless regime
• ZGN theory in collision dominated regime

208
Mean-field Theory

• Evolution of quantum field operator
• where is the annihilation field
  operator
• and is the creation field
  operator
• Mean-field description
• Condensate wavefunction

209
Mean-field Theory

• Generalized GPE for condensate wavefunction
• Temperature-dependent fluctuation field for
  non-condensate

210
Hartree-Fock Bogoliubov Theory

• Ignore the three-field correlation function
• Bogoliubov transformation
• where creates (annihilates) a
  Bogoliubov quasiparticle of energy ej
• The quasiparticles are non-interacting

211
Hartree-Fock Bogoliubov Theory

• Bogoliubov equations for non-condensate
• where

212
Time-independent Hartree-Fock Bogoliubov Theory

• Stationary states
• Time-independent generalized GPE and Bogoliubov
  equations

213
HFB-Popov Approximation

• HFB produces an energy gap in the excitation
  spectrum
• Solution leave out
• Generalized GPE and Bogoliubov equations within
  Popov approximation (gapless spectrum)

214
Hartree-Fock Approximation

• Approximate Bogoliubov excitations with
  single-particle excitations, i.e. let
• Restricted to finite temperature close to Tc,
  where the non-condensed particles have higher
  energies

215
ZGN Theory

• Mean-field theory deals with BEC in collisionless
  region (low density thermal cloud)
• l gtgt l
• l is the collisonal mean-free-path of excited
  particles
• l is the wavelength of excitations
• In collision-dominated region l ltlt l (higher
  density thermal cloud), the problem becomes
  hydrodynamic in nature
• ZGN theory (E. Zaremba, A. Griffin, T. Nikuni,
  1999) describes finite-T BEC with interparticle
  collisions in the semi-classical limit
• kBT gtgt hw0 (w0 trap frequency)
• kBT gtgt gn

216
ZGN Theory

• Apply Popov approximation (ignore ) but
  include the three-field correlation function
• GPE for condensate wavefunction
• Quantum Boltzmann equation for phase-space
  distribution function of non-condensate

217
ZGN Theory

• Thermal cloud density
• Collision between condensate and non-condensate
• -- transfer atoms from/to the condensate
• Collision between non-condensate particles

218
ZGN Theory

• Energy of condensate atoms
• Local chemical potential
• Superfluid velocity
• Energy of non-condensate atoms Hartree-Fock
  energy
• Limited to high temperature (close to Tc)
• For lower temperature, the spectrum of excited
  atoms should be described by Bogoliubov
  approximation

219
Open questions

• Mathematical theory
• Quantum Boltzmann Master equation (QBE)
• GPE with damping term
• Coupling QBE GPE
• Numerical methods
• For QBE P. Markowich L. Pareschi (Numer.
  Math., 05)
• For QBEGPE
• Comparison with experiments
• Rotational frame

220
Conclusions

• Review of BEC
• Experiment progress
• Mathematical modeling
• Efficient methods for computing ground excited
  states
• Efficient methods for dynamics of GPE
• Comparison with experimental results
• Vortex dynamics
• Quantized vortex stability interaction

221
Future Challenges

• Multi-component BEC for bright laser
• Applications of BEC in science and engineering
• Precise measurement
• Fermions condensation, BEC in solids waveguide
• Dynamics in optical lattice, atom tunneling
• Superfluidity dissipation, quantized vortex
  lattice
• Coupling GPE QBE for BEC at finite temperature
• Mathematical theory for BEC
• Interdisciplinary research experiment,physics,
  mathematics, computation, .