Mathematical Analysis and Numerical Simulation for Bose-Einstein Condensation Weizhu Bao Department of Mathematics & Center of Computational Science and Engineering National University of Singapore URL:

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.

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

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

Part I Predication & Mathematical modeling

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

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

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

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

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)

Experimental difficulties Low temperatures  absolutely zero (nK) Low density in a gas

Experimental techniques Laser cooling Magnetic trapping Evaporative Cooling ($100k—300k)

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 Giant vortices Vortex lattice dynamics

Mathematical modeling N-body problem –(3N+1)-dim linear Schroedinger equation Mean field theory: –Gross-Pitaevskii equation (GPE): –(3+1)-dim nonlinear Schroedinger equation (NLSE) Quantum kinetic theory –High temperature: QBME (3+3+1)-dim –Around critical temperature: QBME+GPE –Below critical temperature: GPE

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

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

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

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

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

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

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

Gross-Pitaevskii equation Typical parameters ( ) – Used in JILA – Used in MIT

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 ( )

Numerical Verification

Numerical Results Bao, Y. Ge, P. Markowich & R. Weishaupl, 06’

Gross-Pitaevskii equation General form of GPE ( ) with Normalization condition

Gross-Pitaevskii equation Two kinds of interaction –Repulsive (defocusing) interaction –Attractive (focusing) interaction Four typical interaction regimes: –Linear regime: one atom in the condensation –Weakly interacting condensation

Gross-Pitaevskii equation –Strongly repulsive interacting condensation –Strongly attractive interaction in 1D Other potentials –Box potential –Double-well potential –Optical lattice potential –On a ring or torus

Gross-Pitaevskii equation Conserved quantities –Normalization of the wave function –Energy Chemical potential

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

Comparison of two scaling

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

Part II Analysis & Computation for Ground states

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

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

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, Bull. Institute of Math., Acad. Scinica, 07’

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’)

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

Energies revisited Total energy: –Kinetic energy: –Potential energy: –Interaction energy: Chemical potential

Box Potential in 1D The potential: The nonlinear eigenvalue problem Case I: no interaction, i.e. –A complete set of orthonormal eigenfunctions

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

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

Box Potential in 1D –Matched asymptotic approximation Consider near x=0, rescale We get The inner solution Matched asymptotic approximation for ground state

Box Potential in 1D Approximate energy Asymptotic ratios: Width of the boundary layer:

Numerical observations:

Box Potential in 1D Matched asymptotic approximation for excited states Approximate chemical potential & energy

Fifth excited states

Energy & Chemical potential

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

Harmonic Oscillator Potential in 1D The potential: The nonlinear eigenvalue problem Case I: no interaction, i.e. –A complete set of orthonormal eigenfunctions

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

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:

Harmonic Oscillator Potential in 1D –A new way to define the energy –Asymptotic ratios

Numerical observations:

Harmonic Oscillator Potential in 1D –Thomas-Fermi approximation for first excited state Jump at x=0! Interior layer at x=0

Harmonic Oscillator Potential in 1D –Matched asymptotic approximation –Width of interior layer: –Ordering:

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

BEC on a ring The potential: The nonlinear eigenvalue problem For linear case, i.e. –A complete set of orthonormal eigenfunctions

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

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’)

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

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: ???

Mathematical justification

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

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

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

Backward Euler Spectral method Discretization Spectral order in space; efficient & accurate

Ground states Numerical results (Bao&W. Tang, JCP, 03’; Bao, F. Lim & Y. Zhang, TTSP, 06’) –In 1d Box potential: –Ground state; excited states: first fifthGround statefirstfifth Harmonic oscillator potential: –ground & first excited & Energy and chemical potentialgroundfirst excited Energy and chemical potential Double well potential : –Ground & first excited stateGroundfirst excited state Optical lattice potential: –Ground & first excited state with potentialGroundfirst excited statewith potential next

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Ground states Numerical results (Bao&W. Tang, JCP, 03’; Bao, F. Lim & Y. Zhang, BIM, 07’) –In 2d Harmonic oscillator potentials: –groundground Optical lattice potential: –Ground & excited statesGround & excited states –In 3D Optical lattice potential: ground excited statesgroundexcited states next

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Part III Analysis & Computation for Dynamics in BEC

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

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 Numerical test next

back Angular momentum expectation Energy

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

back Symmetric trap Anisotropic trap

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

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 solutionTrajectory 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

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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: 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’ )

Time-splitting spectral method

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 –2 nd, 4 th or higher order accuracy in time –Time transverse invariant –‘Optimal’ resolution in semicalssical regime

Dynamics of Ground states 1d dynamics: 2d dynamics of BEC (Bao, D. Jaksch & P. Markowich, J. Comput. Phys., 03’) –Defocusing:Defocusing –Focusing (blowup):Focusing (blowup): 3d collapse and explosion of BEC (Bao, Jaksch & Markowich,J. Phys B, 04’) –Experiment setup leads to three body recombination lossExperiment setup –Numerical results: Number of atoms, central density & MovieNumber of atoms central density Movie next

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Collapse and Explosion of BEC back

Number of atoms in condensate back

Central density back

Central quantized vortices Central vortex states in 2D: with Vortex Dynamics –Dynamical stability –Interaction Pattern I Pattern II

Central Vortex states

Vortex stability & interaction Dynamical stability (Bao & Y. Zhang, Math. Mod. Meth. Appl. Sci., 05’) –m=1: stable velocityvelocity –m=2: unstable velocityvelocity Interaction (Bao & Y. Zhang, Math. Mod. Meth. Appl. Sci., 05’) –N=2: Pair: velocity trajectory phase phase2velocitytrajectoryphasephase2 Anti-pair: phase trajectory angular trajectory2phasetrajectoryangular trajectory2 –N=3: velocity trajectoryvelocitytrajectory –Pattern II: Linear nonlinearLinear nonlinear Interaction laws: –On-going with Prof. L. Fu & Miss Y. Zhang next

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Linear case

back Noninear case: BEC

back Linear case

back Linear case

back Linear case

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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 ?

Part IV Rotating BEC & multi-component BEC

Rotating BEC The Schrodinger equation ( ) –The Hamiltonian: –The interaction potential is taken as in Fermi form

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

Rotating BEC General form of GPE ( ) with Normalization condition

Rotating BEC Conserved quantities –Normalization of the wave function –Energy Chemical potential

Semiclassical scaling When, re-scaling With Leading asymptotics

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

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

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

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’

Numerical results –Ground states: in 2D in 3D isosurfacein 2D in 3D isosurface –Quantized vortex generation in 2D surface contoursurface contour –Vortex lattice Symmetric trapping anisotropic trappingSymmetric trapping anisotropic trapping –Giant vortex generation in 2D surface contoursurfacecontour –Giant vortex In 2D In 3DIn 2D In 3D next

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Numerical & Asymptotical results Critical angular frequencyCritical angular frequency: symmetric state vs quantized vortex state Asymptotics of the energyAsymptotics of the energy: RatiosRatios 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. NextNext

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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

Conservation laws Conserved quantities –Normalization of the wave function –Energy Chemical potential

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

back Angular momentum expectation Energy

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

back Symmetric trap Anisotropic trap

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

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

Solution of the center of mass In a rotating BEC with a symmetric trap: –Trajectory of the centerTrajectory of the center –Distance between the center and trapping centerDistance –Motion of the solution: –Pattern Classification:Pattern Classification: next

back 1/5, 4/5, 1 3/2, 6, Pi

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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

Solution of the center of mass In a rotating BEC with an anisotropic trap –When resultsresults The trajectory is a spiral coil to infinity The trajectory is an ellipse –Otherwise result1 result2result1result2 The center moves chaotically & graph is a bounded set The center moves along a straight line to infinity next

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Total density with dissipation Time-dependent Gross-Pitaevskii equation Lemma The dynamics of total density satisfies –The total density decreases when density function energy nextdensity function energynext

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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’)

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

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

Dynamics of ground state Choose initial data as: : ground state Change the frequency in the external potential: –Case 1: symmetric: surface contoursurface contour –Case 2: non-symmetric: surface contoursurfacecontour –Case 3: dynamics of a vortex lattice with 45 vortices: image contour nextimage contournext

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Interaction of two vortices in linear

Interaction of vortices in nonlinear

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?

Two-component BEC The 3D coupled Gross-Pitaevskii equations Normalization conditions Intro- & inter-atom Interactions

Two-component BEC Nondimensionalization Normalization conditions – There is external driven field –No external driven field

Two-component BEC Energy Reduction to one-component:

Two-Component BEC Semiclassical scaling Semiclassical limit –No external field: WKB expansion, two-fluid model –With external field: WKB expansion doesn’t work, Winger transform

Ground state No external field: Nonlinear eigenvalue problem Existence & uniqueness of positive solution Numerical methods can be extended

Ground states crater

Ground state With external field: Nonlinear eigenvalue problem Existence & uniqueness of positive solution ??? Numerical methods can be extended????

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

Dynamics

Spinor BEC Spinor F=1 BEC With

Spinor BEC Total mass conservation Total magnetization conservation Energy conservation

Spinor BEC Dimension reduction Ground state –Existence & uniqueness of positive solution?? –Numerical methods ??? Dynamics –Dynamical laws –Numerical methods: TSSP Semiclassical limit & hydrodynamics equation??

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

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

Mean-field Theory Generalized GPE for condensate wavefunction Temperature-dependent fluctuation field for non-condensate

Hartree-Fock Bogoliubov Theory Ignore the three-field correlation function Bogoliubov transformation where creates (annihilates) a Bogoliubov quasiparticle of energy ε j The quasiparticles are non-interacting

Hartree-Fock Bogoliubov Theory Bogoliubov equations for non-condensate where

Time-independent Hartree-Fock Bogoliubov Theory Stationary states Time-independent generalized GPE and Bogoliubov equations

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)

Hartree-Fock Approximation Approximate Bogoliubov excitations with single-particle excitations, i.e. let Restricted to finite temperature close to T c, where the non- condensed particles have higher energies

ZGN Theory Mean-field theory deals with BEC in collisionless region (low density thermal cloud): l >> l is the collisonal mean-free-path of excited particles  is the wavelength of excitations In collision-dominated region 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 k B T >> ħ      : trap frequency) k B T >> gn

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

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

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 T c ) For lower temperature, the spectrum of excited atoms should be described by Bogoliubov approximation

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 QBE+GPE –Comparison with experiments –Rotational frame

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

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, ….

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