Effective Dynamics of Solitons I M Sigal

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1 Effective Dynamics of Solitons I M Sigal Porquerolles October, 008 Joint work with Jürg Fröhlich, Stephen Gustafson, Lars Jonsson, Gang Zhou and Walid Abou Salem

2 Bose-Einstein Condensation Consider a system of n bosons in an external potential V(x). Its Hamiltonian is: n ( ) g H n = x + V x i i + v xi x j BEC: In the ground state the "same state": i= 1 i j n acting on S R. Ground state 1 L ( d ) ( T = ) 0 all particles are in j= 1 ( x ) Rigorous proof: Lieb-Seiringer '01 (in Gross-Pitaevskii regime as Evolution? n ψ number of particles n and scattering length a j while na is fixed).

3 BEC: GP Regime n Consider a system of n bosons. ψ ψ ( x ) Denote = j. n i= 1 Erdös - Schlein -Yau: The solution of the Schödinger equation Ψ i = H n Ψ and Ψ = ψ t obeys, as n and a 0 and na is fixed, Ψ ψ n n t= in some weak sense where ψ satisfies the GPE with intial condition ψ ( gn = O ) MF approximation 1 : Hepp '74, Ginibre - Velo '79, Spohn '80, - Fröhlich Tsai - Yau '00, Bardos et al '0-03, Fröhlich - Graffi - Schwarz '07 Ground State (GP approximation) : Lieb - Seiringer - Yngvasson ' 00 0

4 Gross-Pitaevskii Equation In GP approximation ( n, a 0 and na is fixed ) the condensate satisfies the equation Here ψ i = ( + V ) ψ + 4 π a ψ ψ, t ψ with =. n V ( x) is a confining potential of the trap: V x Works well for T 0.4T BEC ( α ) d g ( d ) Generalization: ψ ψ f ψ ψ i.e. ψ ψ, α < V general potentials +

5 GPE: General Features (i) d =,3 global existence for a > 0, blowup for a < 0 (ii) GPE is a Hamiltonian system (iii) Conservation laws for energy 4 E ψ : = ψ + V ψ + π a ψ and the number of particles (iv) N ( ψ ) : = ψ = n Regimes: nearly linear ( n a << 1 ), and ( n a >> ) Thomas-Fermi 1

6 Ground State The ground state is the solution to (GPE) of the form ψ x, t = ϕ x e, ϕ x > 0. iµ t µ µ Theorem (Wein., GSS ): If ϕ > 0, then µ µ ϕ minimizes E ψ under N( ψ ) µ = n (physics definition) µ is the Lagrange multiplier µ is a chemical potential = ϕ µ ( E n). ( = ' energy needed to add one more particle/atom' ).

7 Experimental Pictures Fits with ψˆ to a few % ( k )

8 Existence and Stability Existence: Reasonable nonlinearities and potentials V ( x) Three important cases: (i) + V ( x) has a ground state & n a << 1 (ii) V has a minimum & n a >> 1 & a < 0 (iii) V ( x) confining & a > 0. Stability: V confining orbital stability V (i) Soffer-Weinstein, Tsai-Yau 0 at asymptotic stability (ii) Zhou Gang & IM S

9 Relaxation to Ground State: Radiative Damping Ground state Soliton center moves according to Newton's law + friction Friction is due to radiative damping (Fermi Golden Rule) Qn: Long-time dynamics of the soliton?

10 Hamiltonian Structure 1 ( d R C) NLS is a Hamiltonian system on H, with the Hamiltonian H ψ and 4 H ψ : = ψ + V ψ + π a ψ symplectic form Ω u, v = Im u v.

11 Soliton Manifold ( V = ) ϕ ( x) Key object: Given 0 - soliton, define the manifold: M soliton { d + 1 σϕ σ R R } + : =, ( q p ) = d + 1 where σ,, γ, µ R R+ acts on functions as 1/ α i( p x+ γ ) x = x q e σϕ : µ ϕ µ, α (a symmetry transformation for the nonlinearity g ψ ψ ). Goal: Obtain dynamics on M equivalent to the original NLS dynamics. soliton

12 Group Structure Holmer-Zworskii: The equation 1/ α i( p x+ γ ) x : = ( x q) e σϕ µ ϕ µ defines the group structure on d + 1 R R+ as a semidirect product of the Heisenberg group and the scaling group S: H S = : Σ so that M solit = Σ ϕ. d H d

13 Hamiltonian Reduction Define reduced Ham. system h, ω on M as soliton h : = H and ω : = Ω. TM soliton TM soliton Compute ϕ h p V µ µ σ = + eff q + H ϕµ For how long Hamiltonian flow on H ω = ϕ dp dq + pdµ dq + dγ dµ. µ µ R d (, C ) stays close to M? 1 soliton

14 Initial Conditions Recall: We consider the GP equation ψ α i = + V ψ g ψ ψ t V = 0 Initial conditions: 0 = 0 µ + O 0 L sol where = <<1 lpot l l pot sol = ε l ϕ V = 0 ' width' of ( x) µ = length scale of V ( x) ψ σ ϕ ε 0

15 Dynamics THM (FGJS). For t c log(1/ ε)1/ ε (Ehrenfest time) ( +γ) ψ ϕ + ε V = 0 i p x ( x, t)= µ x q e O 1, H with soliton parameters satisfying mod O ( ε ) ɺ V ( q) qɺ = p & δ µ p =, eff V = 0 V = 0 where δ µ : = ϕ & V : = V ϕ. µ µ eff µ C f. Bronskii - Jerrard '00, Fröhlich - Tsai - Yau '00, Fröhlich Gustafson - Jonsson - Sigal '04,'06, Holmer - Zworski '07, Abou Salem '07, '08, Soliton collisions: Abou Salem Fröhlich Sigal '08, Galina Perelman '08

16 Numerical Simulations (Holmer & Zworski) ψ ( x, t)

17 Finite Temperatures For 0 < T < T BEC the condensate co-exists with thermal (non-condensed) atoms. In the mean-field (and, presumably, Gross-Pitaevskii) regime the system is described by the time-dependent version of the Hartree-Fock-Bogolubov (HFB) equations. The following four slides discuss a simplified (two-gas) model, a part of a joint project with Jürg Fröhlich (work in progress).

18 T > 0: The Two-Gas Model In the simplest (two-gas) approximation the HFB equations become ( ) i ψ = + V + g ψ + gn ψ, t i t = + V + g ψ + gn, with = density operator and n x : = x, x, t.

19 Hamiltonian Structure Two-gas approximation to HFB equations is a Hamiltonian system with the Hamiltonian functional and the Poisson bracket ( ψ, ψ, ) : = ( ψ, ψ ) + ( ) H H Tr h GP {, } = ( ) ψ ( x) ψ ( x) A B i A B A B dx ( ψ ψ ) ( ψ ψ ) i Tr ψ ( x) ψ ( x) ( A B B A). Here H, is the standard Gross-Pitaevskii Hamiltonian GP and h : = h + g ψ + gn, and A,, is the operator - Fréchet derivative.

20 Stationary States Stationary states of the system can be found by minimizing the free energy E( ψ, ) TS( ) with the total number of particles ψ + Tr fixed. Here E( ψ, ) = H ( ψ, ψ, ) is the internal energy and S 1, a non-negative, convex, C function on (0,1] (an entropy).

21 Gibbs States Stationary states satisfy the standard Hartree-Fock-Bogolubov equations. The latter imply h = G ψ, T µ, h V g n where : = + + ( ψ + ) and ψ, G( γ ) is the Legendre transform of S( ). The parameter T can be thought of as temperature. (Cf. Markowich - Rein -Wolansky and Dolbeault - Felmer - Lewin.) For V ( x) a confining potential and S( ) = Tr log, the resulting stationary state combines the ground state for the condensed atoms with the Gibbs state for thermal ones.

22 New Mathematical Questions Formation of condensate ( static - bifurcation - and dynamical picture). ( In the time-dependent case the temperature enters through initial conditions.) Phase diagram N o N Number of condensed atoms Total number of atoms = f ( T ) T T0 1 (correction to the ideal gas law, T is a critical temperature for the ideal Bose gas). 0 Arrest of collapse for a < 0 and sufficiently large initial conditions.

23 Additional Open Problems Long-time asymptotic dynamics? Local minima of V(x): trapping vs tunneling? Negative scattering length: microscopic tunneling of macroscopically stable ground state? l ocal min. Gross-Pit. ground state is unstable under microscopic fluctuations Microscopic corrections to GPE and to soliton dynamics?

24 Thank you for your attention