Effective dynamics of many-body quantum systems

Transcription

1 Effective dynamics of many-body quantum systems László Erdős University of Munich Grenoble, May 30, 2006 A l occassion de soixantiéme anniversaire de Yves Colin de Verdiére Joint with B. Schlein and H.-T. Yau GOAL: Describe the time evolution of N interacting particles by effective one-body (nonlinear) equations. 1

2 INTERACTING MANY-BODY QUANTUM SYSTEMS x = (x 1, x 2,..., x N ) R 3N position of the particles. Wave function: Ψ N (x 1,..., x N ) L 2 (R 3N ) Symmetric (=bosonic) or antisymmetric (=fermionic) H N = N i t Ψ N,t = H N Ψ N,t [ xj + U(x j ) ] + λ i<j V (x i x j ) U is a one-body background ( trapping ) potential V is the interaction potential Coupling λ is scaled s.t. kinetic energy potential energy. Bosons: E kin = O(N), thus λ = 1 N. Fermions: E kin = O(N 5/3 ), thus λ = 1 N 1/3. 2

3 BOSONS = NONLINEAR HARTREE EQUATION H N = N [ xj + U(x j ) ] + 1 N i<j V (x i x j ) THEOREM: If Ψ 0 = j ϕ 0 (x j ), then Ψ t j ϕ t (x j ) as N ) where i t ϕ t = ( + U)ϕ t + (V ϕ t 2 ϕ t Limit equation is quantum. Each particle: subject to the same mean-field pot. (LLN for x j ) 1 N N V (x x j ) ϕ(x j ) 2 (V ϕ 2 )(x) Kac, McKean, Dobrushin, Hepp, Spohn, Ginibre-Velo, etc. GOAL: V = Dirac delta interaction = Cubic NLS 3

4 FERMIONS = NONLINEAR VLASOV EQUATION Trapped fermions have energy/particle N 2/3 Time scale N 1/3 Wavelength N 1/3 potential lengthscale O(1) = SC iε t Ψ = ε 2 j j + 1 N k<j Typical semiclassical fermionic state Ψ = j V (x k x j ) Ψ, ε := N 1/3 ϕ j, ϕ j (x) = e ik jx g(x), k j N 1/3 1-pdm: γ (1) (x; x ) := Ψ(x, Y )Ψ(x, Y )dy, Y := (x 2,... x N ). Supported near x x ε. Wigner transform of γ (1) at scale ε W (1) ε (x, v) := γ (1) (x + εη, x εη)e iηv dv Similarly for k-particle density matrices, γ (k) (x 1,... x k ; x 1,... x k ) 4

5 iε t Ψ = [ ε 2 j j + 1 N ] V (x k x j ) Ψ, ε := N 1/3 k<j THEOREM: If the initial state asymptotically factorizes, W (k) N,ε k (k) 1W, then W N,ε (t) k 1 W t (propagation of chaos) The weak limit W t = lim W (1) N,ε (t) satisfies t W t (x, v) + v x W t (x, v) = x (V ϱ t ) v W t (x, v) ϱ t (x) := W t (x, v)dv Limit equation is classical (nonlinear Vlasov equation) [Narnhofer-Sewell]: V is analytic, [Spohn]: V C 2 5

6 NL Vlasov equation is the SC limit of the Hartree eq. ) iε t ϕ ε t = ε 2 ϕ ε t + (V ϕ ε t 2 ϕ ε t iε t γ ε t = [ ε 2 + V ϱ ε t, γ ε t ], ϱ ε t(x) := γ ε t (x, x) Let W ε (t, x, v) be the rescaled Wigner transform of γ ε t. THEOREM [Elgart-E-Schlein-Yau]: V analytic Suppose for k 2 log N and bounded k-body observables O (k), Then for short time O (k), W (k) N,ε (0) O (k), W (k) N,ε (t) k k W (1) (0) 1 N W ε (t) 1 N Hartree eq. exact up to O(ε 3 ). Open: remove analyticity. 6

7 BOSE-EINSTEIN CONDENSATION (BEC) 1. Condensation in noninteracting system. H = N ( j + U(x j ) ) Ground state (bosons): Ψ N (x) = j ϕ(x j ) Complete condensation: all particles in the collective mode. 2. Condensation in interacting system. H = N ( j + U(x j ) ) + λ k<j V (x k x j ) For very low energy states (ground state or low temperature Gibbs states), a macroscopic number of particles are in the same orbital state. 7

8 MATHEMATICAL DEFINITION OF BEC Equivalent description of Schrödinger dynamics: i t γ N,t = [ H, γ N,t ] with γ N,t := ψ N,t ψ N,t density matrix (1 dim. projection). Density matrix formalism is more general: allows mixed states One-particle marginal density of a general N-body state γ N γ (1) N (x; x ) := γ N (x, Y ; x, Y )dy, Y = (x 2,... x N ) Operator on the one-particle space, 0 γ (1) N Spectral decomposition: γ (1) N = j λ j φ j φ j. DEFINITION: γ N is a condensate state if 1, Tr γ(1) N = 1. lim inf N max j λ j > 0 8

9 RESULTS IN TRAPPED INTERACTING BOSE GAS [Lieb-Seiringer-Yngvason, Lieb-Seiringer] H N = N [ xj + U(x j ) ] + 1 N i<j N 3 V (N(x i x j )) Approx Dirac delta interaction with range 1/N ( hard core ) Ground state energy is given by the Gross-Pitaevskii functional E GP (ϕ) := lim N 1 N inf Ψ, H NΨ = ϕ 2 + U ϕ 2 + 4πa 0 ϕ 4, inf E GP (ϕ) ϕ, ϕ =1 a 0 = scatt. length of V Complete condensation in ground state: γ (1) N (x; x ) φ(x)φ(x ), φ = minimizer of E GP Condensation without trap is a major open challenge. 9

10 TIME DEPENDENT GROSS-PITAEVSKII (GP) THEORY The GP energy functional also describes the evolution: γ (1) N,0 ϕ ϕ = γ(1) N,t ϕ t ϕ t The limit orbital satisfies the Gross-Pitaevskii equation i t ϕ t = δe [ ] GP δϕ (ϕ t) = + U + 8πa 0 ϕ t 2 ϕ t, ϕ t=0 = ϕ The condensate wave fn. evolves according to a NLS Many-body effects & corr non-linear on-site self-interaction Experiments of Bose-Einstein Condensation: Trap Bose gas and observe its evolution after the trap removed. GP is expected to be correct for weak interactions as N MAIN RESULT: Many-body Schr. Eq = GP Eq. 10

11 THEOREM: [E-Schlein-Yau, 2006] V N (x) := 1 N N 3 V (Nx). Assume V 0, smooth, spherical, V 1 + V is small. Consider the family of initial states Ψ N (x) = N Then for every k 1 and t > 0 fixed ϕ(x j ), ϕ H 1 (R 3 ) γ (k) N,t ϕ t ϕ t k N where ϕ t is the solution of the GP equation [ ] i t ϕ t = + U + 8πa 0 ϕ t 2 ϕ t, ϕ t=0 = ϕ The theorem also holds if Ψ N has a short scale structure Ψ N (x) = W N (x) N ϕ(x j ), W N (x) = [ 1 w(n(xj x k )) ] j<k Certain correlations on scale 1/N are allowed 11

12 THEOREM: [E-Schlein-Yau, 2006] V N (x) := 1 N N 3β V (N β x). V as before, β < 1. Consider the family of initial states Then Ψ N (x) = N γ (k) N,t ϕ t ϕ t k ϕ(x j ), ϕ H 1 (R 3 ) N where ϕ t is the solution of the GP equation [ ] i t ϕ t = + U + b 0 ϕ t 2 ϕ t, ϕ t=0 = ϕ with b 0 = V 8πa 0 b 0 is the Born approximation to the scattering length. This can be viewed as the SC version of the previous theorem. 12

13 SCATTERING LENGTH Let supp V be compact. Consider the zero energy scattering eq. [ V ] f = 0, f(x) 1, x Then f = 1 w with w(x) = a 0 x, for some a 0. a 0 is called the scattering length of V and it also satisfies V (1 w) = 8πa 0 Actually, f = 1 w minimizes this integral if f(x) 1, f 0. 13

14 Short range interaction V in a many body state creates a two body correlation described by Ψ(x) = j<k (1 w(x j x k )) N ϕ(x j ) where w is short range, and ϕ is smooth on this scale. The interaction energy is Ψ, V (x j x k )Ψ = N 2 V (x y) [ 1 w(x y) ] ϕ(x) 2 ϕ(y) 2 dxdy k<j N 2 V (1 w) if ϕ is smooth 14

15 If V (x) V N (x) := N 1 N 3 V (Nx) N 1 δ(x), then a = a 0 N, 1 w N(x) = 1 a 0 N x, V N (x) := 1 N N 3β V (N β x) for β < 1, V N (1 w N ) = 8πa 0 N V N (1 w N ) V N = b 0 N Interaction effect per particle = { 8πa0 if β = 1 b 0 if β < 1 Short scale correlations vanish as N, but they influence the energy, hence the dynamics by an O(1) effect. 15

16 Fundamental difficulty of N-particle analysis (N 1) There is no good norm. The conserved L 2 -norm is too strong. 1) Let Ψ = N 1 f, Φ = N 1 g, then Ψ Φ 2 = 2 2 f, g N 2. 2) Sobolev estimate is weak in high dimension. 3) Lack of stability in L 2 -norm of the Schrödinger evolution Suppose V V ε and ψ N,t is solution with V. Then t ψ t ψ t 2 ψ t ψ t, 1 N l<j (V V )(x l x j )ψ t Nε But ψ t ψ t ψ t + ψ t = 2 anyway Problem: Ψ(x 1,... x N ) carries info of all particles (too detailed). 16

17 Keep only information about the k-particle correlations: γ (k) Ψ (X k, X k ) := Ψ(X k, Y N k )Ψ(X k, Y N k)dy N k where X k = (x 1,... x k ). It monitors only k particles. Quantum analog of the marginals of a probability density. It is an operator acting on the k-particle space. γ (N) Ψ γ (k) Ψ is the projection operator onto the space spanned by Ψ. is its partial trace on the k-particle space. Good news: Most physical observables involve only k = 1, 2- particle marginals. Enough to control them. Bad news: there is no closed equation for them. 17

18 BASIC TOOL: BBGKY HIERARCHY N H = j + 1 N j<k V (x j x k ) V = V N may depend on N so that V N = O(1). Take the k-th partial trace of the Schrödinger eq. i t γ N,t = [H, γ N,t ] = i t γ (k) N,t = k [ ] j, γ (k) N,t + 1 N k [ V (x i x j ), γ (k) N,t i<j ] + N k N k [ ] Tr k+1 V (x j x k+1 ), γ (k+1) N,t A system of N coupled equations. (k = 1, 2,..., N) The last one is just the original N-body Schr. eq. Seems tautological. 18

19 Special case: k = 1: i t γ (1) N,t (x 1; x 1 ) = ( x 1 + x )γ (1) 1 N,t (x 1; x 1 ) + dx 2 ( V (x1 x 2 ) V (x 1 x 2) ) γ (2) N,t (x 1, x 2 ; x 1, x 2) + o(1). To get a closed equation for γ (1) N,t, we need some relation between γ (1) N,t and γ(2) N,t. Most natural: independence Propagation of chaos: No production of correlations If initially γ (2) N,0 = γ(1) N,0 γ(1) N,0, then hopefully γ(2) N,t γ(1) N,t γ(1) N,t No exact factorization for finite N, but maybe it holds for N. Suppose γ (k),t is a (weak) limit point of γ(k) N,t with γ (2),t (x 1, x 2 ; x 1, x 2 ) = γ(1),t (x 1, x 1 )γ(1),t (x 2; x 2 ). 19

20 i t γ (1) N,t (x 1; x 1 ) = ( x 1 + x )γ (1) 1 N,t (x 1; x 1 ) + dx 2 ( V (x1 x 2 ) V (x 1 x 2) ) γ (2) N,t (x 1, x 2 ; x 1, x 2) }{{} γ (1),t (x 1,x 1 )γ(1),t (x 2;x 2 ) With the notation ϱ t (x) := γ,t (1) (x; x), it converges, to i t γ (1),t (x 1; x 1 ) = ( x 1 + x 1 )γ (1),t (x 1; x 1 ) i t γ (1),t = [ + V ϱ t, γ,t (1) ] + ( V ϱ t (x 1 ) V ϱ t (x 1 )) γ (1),t (x 1; x 1 ) Hartree eq for density matrix If V = V N scaled, then the short scale structure can be relevant. For γ (2) N,t (x 1, x 2 ; x 1, x 2) = (1 w N (x 1 x 2 ))γ (1) N,t (x 1, x 1 )γ(1) N,t (x 2, x 2 ), +o(1) = V N (x 1 x 2 )(1 w N (x 1 x 2 )) [ Smooth ] = { 8πa0 if β = 1 b 0 if β < 1 20

21 i t γ (k) N,t = k GENERAL SCHEME OF THE PROOFS [ ] j, γ (k) N,t + 1 N k [ V (x i x j ), γ (k) N,t i<j ] + N k N k [ ] Tr k+1 V (x j x k+1 ), γ (k+1) formally converges to the Hartree hierarchy: (k = 1, 2,...) i t γ (k),t = k [ Easy fact: if γ (k) { γ (k) t } k=1,2... If we knew that j, γ,t (k) ] + t k factorizes, i.e. γ (k) t [ Tr xk+1 V (x j x k+1 ), γ (k+1) = k 1 γ(1) t, then,t solves ( ) i t γ t (1) = [ + V ϱ t, γ t (1) (*) had a unique solution, and lim N γ (k) N,t exists and satisfies (*), ] ( ) ] N,t then the limit must be the factorized one = Propagation of chaos + convergence to NL Hartree eq. 21

22 Step 1: Prove apriori bound on γ (k) N,t uniformly in N. Need a good norm and space! (Sobolev) Step 2: Choose a convergent subsequence: γ (k) N,t γ(k),t in H Step 3: γ (k),t satisfies the infinite hierarchy (need regularity) Step 4: Let γ t (1) solve NLHE/NLS. Then γ t (k) = γ t (1) the -hierarchy in H. [Trivial] solves Step 5: Show that the -hierarchy has a unique solution in H. Key mathematical steps: Apriori bound and uniqueness 22

23 APRIORI BOUNDS Sobolev norm [E-Yau,01] γ (k) H k := Tr 1... k γ (k) k... 1 Ψ t, H k Ψ t is conserved. We turn it into Sobolev-type norms Ψ, H k Ψ (CN) k 1... k Ψ 2 = (CN) k γ (k) H k For β = 1 incorrect, the short scale is too singular; w N a x 1 2 (1 w N (x 1 x 2 )) 2 After removing the singular part: a 2 ( x + a) 6dx = O(a 1 ) = O(N) Proposition: Define Φ 12 (x) := (1 w N (x 1 x 2 )) 1 Ψ(x). Then Ψ, H 2 Ψ (CN) Φ 12 2 Weak limit of Ψ and Φ 12 are equal, but Φ 12 can be controlled in Sobolev space. Use compactness for Φ 12! Similarly for k > 2. 23

24 UNIQUENESS OF THE -HIERARCHY IN SOBOLEV SPACE i t γ (k) t = k [ j, γ t (k) ] Iterate it in integral form: γ (k) t = U(t)γ (k) t iσ k [ Tr xk+1 δ(x j x k+1 ), γ (k+1) 0 ds U(t s)b(k) U(s)γ k ds k s 1... ds n U(s 1 )B (k) U(s 2 )B (k+1)... B (k+n 1) γs k+n n k=t t ] B (k) γ (k+1) := iσ k Tr k+1 [ δ(xj x k+1 ), γ (k+1)] U(t)γ (k) := e it k j γ (k) e it k j Problem 1. B (k) γ (k+1) H k C γ (k+1) H k+1 is wrong because δ(x) (1 ). Need smoothing from U 0!! 24

25 γ (k) t = U(t)γ (k) t 0 ds U(t s)b(k) U(s)γ k ds k s 1... ds n U(s 1 )B (k) U(s 2 )B (k+1)... B (k+n 1) γs k+n n k=t Stricharz inequality? Space-time smoothing of e it. e it ψ = L p (L q (dx)dt) ( dt dx e it ψ q ) p/q 1/p C ψ 2 Problem 2. B (k) B (k+1)... B (k+n 1) n!, because B (k) = k 1 [... ]. This can destroy convergence. Gain back from time integral k s k=t ds 1... ds n 1 n! Here L 1 (ds) was critically used, Stricharz destroys convergence. We expand it into Feynman graphs and do all integrals carefully. 25

26 CONCLUSIONS 1) Bosons with mean field interaction NL Hartee eq. Fermions with mean field interaction NL Vlasov eq. 3) We derived the Gross-Pitaevskii equation from many-body Bose dynamics with interaction on scale 1/N. Coupling constant is the scattering length. 4) For interaction on scale 1/N β, β < 1, the coupling constant is the Born approx. to scattering length. 5) Conservation of H k can imply bounds in Sobolev space 6) Stricharz can be strengthened with Feynman diagrams in many body problems 26

27 Feynman graphs Iteration the -hierarchy: γ,t = U t γ 0 + t 0 ds U t s Bγ,s Ω (k) n =... γ (k),t = n Ξ (k) m=0 m (t) + Ω (k) n (t) ds 1 ds 2... ds n U t s1 BU s1 s 2 B... U sn 1 s n Bγ (k+n),s n Ξ (k) m are similar but with the initial condition γ 0 at the end. Feynman graphs: convenient representation of Ξ and Ω. 27

28 Lines represent free propagators. E.g. the propagator line of the j-th particle between times s and t represent exp[ i(s t) j ]: Vertices represent B, e.g. V (x 1 x 2 )γ(x 1, x 2 ; x 1, x 2 )δ(x 2 x 2 ) 28

29 Ξ (k) m =... ds 1 ds 2... ds m U t s1 BU s1 s 2 B... U sm 1 s m BU sm γ (k+m),0 corresponds to summation over all graphs Γ of the form: Tr O Ξ (k) m = Γ Val(Γ) 29

30 Value of a graph Γ in momentum space Val(Γ) = e E dα e dp e e 1 α e p 2 e + iη e v V ( ( ) δ α e )δ p e e v e v e it e Root (α e iη e ) O(pe : e Root) γ 0 (p e : e Leaves) p e R 3 is the momentum on edge e α e R variable dual to time running on the edge e. η e = O(1) regularizations satisfying certain compatibitility cond. Two main issues to look at What happens to the m! problem (combinatorial complexity of the BBGKY hiearchy)? What happens to the singular interaction = large p problem In other words: why is Val(Γ) UV-finite? 30

31 Combinatorics reduced by resummation: m! is artificial Let k = 1 for simplicity, i.e. we have a tree (not a forest). The Duhamel expansion keeps track of the full time ordering and it counts the following two graphs separately: Number of graphs on m vertices with time ordering: m! (the j-th new vertex can join each of the (j 1) earlier ones) Number of graphs on m vertices without time ) ordering = Number of binary trees = Catalan numbers C m 1 m+1( 2m m 31

32 Val(Γ) UV regime: Finiteness of Val(Γ) 1 ( ( ) dα e dp e e E e α e p 2 δ α e )δ p e e v V e v e v O(p e : e Root) γ 0 (p e : e Leaves) γ 0 H (m+1) guarantees a p e 5/2 decay on each leaf. Power counting (k = 1, one root case). # of edges = 3m + 2, no. of leaves = 2m + 2 # of effective p e (and α e ) variables: (3m + 2) m = 2m + 2 2m + 2 propagators are used for the convergence of α e integrals Remaining m propagators give p 2 decay each. Total p-decay: 5 2 (2m + 2) + 2m = 7m + 5 in 3(2m + 2) dim. There is some room, but each variable must be checked. We follow the momentum decay on legs as we successively integrate out each vertex. There are 7 types of edges, 12 types of vertex integrations that form a closed system. 32