Derivation of the time dependent Gross-Pitaevskii equation for the dynamics of the Bose-Einstein condensate

Transcription

1 Derivation of the time dependent Gross-Pitaevskii equation for the dynamics of the Bose-Einstein condensate László Erdős University of Munich Evian-les-Bains, 2007 June Joint work with B. Schlein and H.T. Yau

2 INTERACTING MANY-BODY QUANTUM SYSTEMS x = (x, x 2,..., x N ) R 3N position of the particles. Wave function: Ψ N (x,..., x N ) L 2 symm(r 3N ) (bosons) H N = N i t Ψ N,t = H N Ψ N,t [ xj + U(x j ) ] + N j= i<j V (x i x j ) U is a one-body background ( trapping ) potential V 0 is the (repulsive) interaction potential 2

3 HARTREE EQUATION H N = N j= [ xj + U(x j ) ] + 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 Each particle: subject to the same mean-field pot. x N N j= V (x x j ) ϕ(x j ) 2 (V ϕ 2 )(x) (Law of large numbers, if the state is indeed a product) Hepp, Spohn, Ginibre Velo, Bardos Golse Mauser, E-Yau GOAL: V = (approx) Dirac delta interaction = Cubic NLS 3

4 (COMPLETE) BOSE-EINSTEIN CONDENSATION 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 ( dim. projection). One-particle marginal density of a general N-body state γ N γ () N (x; x ) := γ N (x, Y ; x, Y )dy, Y = (x 2,... x N ) Operator on the one-particle space, 0 γ () N, Tr γ() N =. DEFINITION: γ N is in a complete condensate state if lim N γ() N = φ φ (much weaker than full factorization: γ N = φ φ N ) Expected for the ground state of weakly interacting systems. 4

5 RESULTS IN TRAPPED INTERACTING BOSE GAS [Lieb-Seiringer-Yngvason, Lieb-Seiringer] H N = N j= [ xj + U(x j ) ] + N i<j N 3 V (N(x i x j )) Approx Dirac delta interaction with range /N ( hard core ) Ground state energy is given by the Gross-Pitaevskii functional E GP (ϕ) := lim N N inf Ψ, H NΨ = ϕ 2 + U ϕ 2 + 4πa 0 ϕ 4, inf E GP(ϕ) ϕ, ϕ = a 0 = scatt. length of V Complete condensation in ground state: γ () N (x; x ) φ(x)φ(x ), Condensation without trap is open. φ = minimizer of E GP 5

6 O(/N) N /3 N particles Density = O(N) O() (trap) The system is at high density on the O() scale but it is in the dilute regime viewed on the O(/N) scale. 6

7 SCATTERING LENGTH Let supp V be compact. Consider the zero energy scattering eq. [ + 2 V ] f = 0, f(x), x Then f = w with w(x) = a 0 x, for some a 0. a 0 is called the scattering length of V For hard-core V, the scattering length is the hard core radius. In a dilute gas of neutral bosons, the scattering length is the only characteristic of the interaction. The scattering length also satisfies ( w) V ( w)2 = 8πa 0 if f = w minimizes the energy under f(x), f 0. 7

8 Very short range interaction V in a low energy many body state creates a two body correlation described by Ψ(x) = j<k ( w(x j x k )) N j= ϕ(x j ) where w is short range, and ϕ is smooth ( const) on this scale. The kinetic and the interaction energies of this Ansatz: j j Ψ 2 N j j ϕ 2 + w(x j x k ) 2 ϕ(x j ) 2 ϕ(x k ) 2 k j ϕ 2 + N 2 w 2 ϕ 4 Ψ, V (x j x k )Ψ N2 2 j<k N2 2 V (x y) [ w(x y) ] 2 ϕ(x) 2 ϕ(y) 2 dxdy V ( w) 2 φ 4 8

9 This can be made rigorous in the LSY-scaling that effectively introduces a scale separation as N. /N V (x y) N ϕ (x) w (x y) N y Two-scale structure of the wavefn: Ψ = ( w N (x i x j )) i<j j ϕ(x j ) 9

10 H N = N j= [ xj + U(x j ) ] + N Ground state energy is given by: E GP (ϕ) := i<j N 3 V (N(x i x j )) ϕ 2 + U ϕ 2 + 4πa 0 ϕ 4, a 0 = scatt. length of V (Stationary) Gross-Pitaevskii theory postulates: (i) Single orbital theory (Wavefunction product) (ii) All interactions are described by an on-site nonlinear term (iii) Emergence of the scattering length is a correlation effect, it implicitly postulates a short scale two-body correlation structure despite (i). The state lives on two different scales. Does GP theory hold for the time evolution? 0

11 TIME DEPENDENT GROSS-PITAEVSKII (GP) THEORY The GP energy functional also describes the evolution: γ () N,0 ϕ ϕ = γ() N,t ϕ t ϕ t Propagation of chaos = (weak) factorization persists 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 postulates of GP theory hold away from the ground state MAIN RESULT: Many-body Schr. Eq = GP Eq. (N )

12 MAIN THEOREM V 0, smooth, spherical, cpct supp, small. V N (x) := N N3 V (Nx). Initial state Ψ N (x) = N j= Let Ψ N,t solve the N-body Schrödinger eq. i t Ψ N,t = N ϕ(x j ), ϕ H (R 3 ) xj + V N (x i x j ) j= i<j Ψ N,t Then for k and t > 0 fixed, for the k-point marginal of Ψ N,t, γ (k) N,t ϕ t ϕ t k N (weakly and in trace norm) where ϕ t is the solution of the GP equation [ ] i t ϕ t = + 8πa 0 ϕ t 2 ϕ t, ϕ t=0 = ϕ 2

13 The theorem also holds if Ψ N has a short scale structure Ψ N (x) = W N (x) N j= ϕ(x j ), W N (x) = j<k [ w(n(xj x k )) ] For example, our theorem also holds if Ψ N is the ground state with a trap U, then we remove the trap and observe the dynamics of the condensate (typical experiment). 3

14 A SURPRISE FOR THE PRODUCT INITIAL STATE NL energy predicts the wrong NL evolution for Ψ N = ϕ N [ N Ψ N, H N Ψ N ϕ 2 + b ] 0 2 ϕ 4, b 0 := V since, recalling V N (x) = N N3 V (Nx), N i<j V N (x i x j ) N j= ϕ(x j ) 2 b 0 2 ϕ 4 but the marginals of Ψ N,t factorize, γ () N,t ϕ t ϕ t, with i ϕ t = ϕ t + 8πa 0 ϕ t 2 ϕ t (b 0 > 8πa 0!!) Energy lost? No. The limit holds in (weak) L 2 and not in H. The product state instantenously builds up a short scale correlation to minimize its local energy. This short scale correlation then drives the orbitals according to a 0. The excess energy is (probably) diffused into incoherent modes on scales /N l and does not influence the evolution of the condensate. 4

15 Fundamental difficulty of N-particle analysis (N ) There is no good norm. The conserved L 2 -norm is too strong. Ψ(x,... x N ) carries info of all particles (too detailed). 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,... x k ). It monitors only k particles. Good news: Most physical observables involve only k =, 2- particle marginals. Enough to control them. Bad news: there is no closed equation for them. 5

16 BASIC TOOL: BBGKY HIERARCHY N H = j + j= N j<k V (x j x k ) V = V N may depend on N so that V N = O(). 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= [ ] j, γ (k) N,t + N k [ i<j V (x i x j ), γ (k) N,t ] + N k N k [ ] Tr k+ V (x j x k+ ), γ (k+) j= N,t A system of N coupled equations. (k =,2,..., N) Closure? Wish: Propag. of chaos: γ (2) γ () γ () (N ) 6

17 i t γ (k) N,t = k j= [ GENERAL SCHEME OF THE PROOF ] j, γ (k) N,t + N k [ i<j V (x i x j ), γ (k) N,t ] + N k N k [ ] Tr k+ V (x j x k+ ), γ (k+) formally converges to the Hartree hierarchy: (k =, 2,...) i t γ (k),t = k j= [ j, γ,t (k) ] + k j= j= [ Tr xk+ V (x j x k+ ), γ (k+),t ] ( ) N,t { γ (k) t = k γ() t } k=,2... If we knew that solves ( ) i t γ t () = [ +V ϱ t, γ t () ] (*) had a unique solution, and lim N γ (k) N,t exists and satisfies (*), then the limit must be the factorized one = Propagation of chaos + convergence to Hartree eq. 7

18 Step : Prove apriori bound on γ (k) N,t uniformly in N. Need a good norm and space H! (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 () solve NLS. Then γ t (k) = γ t () 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 Apriori bound: conservation of H k = mixed Sob. bound Uniqueness: Strichartz via Feynman graphs 8

19 APRIORI BOUNDS Sobolev norm [E-Yau, 0 ] γ (k) H k := Tr... k γ (k) k... Ψ t, H k Ψ t is conserved. We turn it into Sobolev-type norms Ψ, H k Ψ (CN) k... k Ψ 2 = (CN) k γ (k) H k Nontrivial, since H k = ( i i + ) k V (x i x j ) i<j = ( ) i...( ) ik ( )V V ( )( ) +... and V is not bounded by (too singular). 9

20 Ψ, H k Ψ (CN) k... k Ψ 2 = (CN) k γ (k) H k For GP it is incorrect, the short scale is too singular; w N 2 ( w N (x x 2 )) 2 a x +a a 2 ( x + a) 6dx = O(a ) = O(N) After removing the singular part: Proposition: Define Φ 2 (x) := ( w N (x x 2 )) Ψ(x). Then Ψ, H 2 Ψ (CN) 2 2 Φ 2 2 Weak limit of Ψ and Φ 2 are equal, but Φ 2 can be controlled in Sobolev space. Use compactness for Φ 2! Similarly for k > 2. 20

21 IDEA OF THE H 2 -APRIORI BOUND Work in one particle setting, i.e. in R 3. H = + V, V (x) = N N3 V 0 (Nx) Let f = w be the scattering solution ( + V )f = 0 By scaling, { a 0 f(x) = f 0 (Nx) O() N x (here a 0 is the scattering length of V 0 ). x N x N Let Ψ be any wavefunction, factorize f out (0 < f ) Ψ = fφ LEMMA: If V 0 is sufficiently small (e.g. a 0 is small), then (Ψ, H 2 Ψ) c Φ 2 2

22 (Ψ, H 2 Ψ) c Φ 2 Ψ = fφ HΨ = ( + V )Ψ = fl [ Ψ/f ] with L := + 2( log f) FACT: L is self-adjoint with respect to f 2 (x)dx: ΦLΩ f 2 = L Ω Φ f 2 = Φ Ω f 2 (Ψ, H 2 Ψ) = = = HΨ 2 = LΦ 2 f 2 = Φ (LΦ) f 2 ΦL( Φ) f 2 + Φ[, L]Φ f 2 2 Φ 2 f 2 [ 2 f + Φ + ( f)2 ] f f 2 Φ f 2 22

23 (Ψ, H 2 Ψ) = 2 Φ 2 f 2 + [ 2 f Φ f + ( f)2 ] f 2 Φ f 2 From the scaling of f: thus 2 f a 0 N x 3 a 0 x 2, [ Φ ] Φ f 2 Ca 0 ( f)2 ( ) a0 2 a 0 N x 2 x 2, x 2 Φ 2 Ca 0 thus, after estimating f 2 C > 0, we have (Ψ, H 2 Ψ) C 2 Φ 2 Ca 0 2 Φ 2 C if a 0 is small enough. 2 Φ 2 2 Φ 2 23

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

25 γ (k) t = U(t)γ (k) t 0 ds U(t s)b(k) U(s)γ k ds k s...ds n U(s )B (k) U(s 2 )B (k+)... B (k+n ) γs k+n n k=t Stricharz inequality? Space-time smoothing of e it. e it ψ = L p (L q (dx)dt) (2 p, 2/p + 3/q = 3/2) ( dt dx e it ψ q ) p/q /p C ψ 2 Problem 2. B (k) B (k+)... B (k+n ) n!, because B (k) = k [... ]. This can destroy convergence. Gain back from time integral k s k=t ds...ds n n! Here L (ds) was critically used, Stricharz destroys convergence. 25

26 We expand it into Feynman graphs, use combinatorial identities and do multiple integrals carefully. An example for combinatorics: 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! Number of graphs on m vertices without time ordering = C m The resummation reduced m! to C m. The factorial was fake! 26

27 CONCLUSIONS We derived the GP equation from many-body Ham. with interaction on scale /N. Coupling const. = scattering length. A specific short scale correlation structure is preserved or even emerges along the dynamics. In the N limit, this structure is negligible in L 2 sense (ensuring a closed eq. for the orbitals) but not in energy sense, thus it influences the dynamics via the emergence of the scatt. length. For interaction on scale /N β, β <, the coupling constant is the Born approximation to scattering length. [Not explained] Conservation of H k can imply bounds in Sobolev space Stricharz can be strengthened with Feynman diagrams in many body problems 27

28 DIFFERENT APPROX. OF DIRAC DELTA INTERACTION THEOREM: [E-Schlein-Yau, 2006] V N (x) := N N3β V (N β x). V as before, β <. Consider the family of initial states Then Ψ N (x) = N j= ϕ(x j ), ϕ H (R 3 ) γ (k) N,t ϕ t ϕ t k N where ϕ t is the solution of the NLS [ ] 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. Why β = is the borderline? 28

29 The interaction energy in a typical state Ψ = j<k ( w jk ) N j= ϕ j = Ψ, V (x j x k )Ψ N2 2 k<j V ( w) V (x) V N (x) := N N3β V (N β x) N δ(x), w w N /N β= β< /N β V (x) N -w(x) -w(x) V (x) N V N ( w N ) = 8πa 0 N V N ( w N ) V N = b 0 N 29

30 Special case: k = : i t γ () N,t (x ; x ) = ( x + x )γ () N,t (x ; x ) + dx 2 ( V (x x 2 ) V (x x 2) ) γ (2) N,t (x, x 2 ; x, x 2) + o(). To get a closed equation for γ () N,t, we need some relation between γ () N,t and γ(2) N,t. Most natural: independence Propagation of chaos: No production of correlations If initially γ (2) N,0 = γ() N,0 γ() N,0, then hopefully γ(2) N,t γ() N,t γ() 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, x 2 ; x, x 2 ) = γ(),t (x, x )γ(),t (x 2; x 2 ). 30

31 i t γ () N,t (x ; x ) = ( x + x )γ () N,t (x ; x ) + dx 2 ( V (x x 2 ) V (x x 2) ) γ (2) N,t (x, x 2 ; x, x 2) }{{} γ (),t (x,x )γ(),t (x 2;x 2 ) With the notation ϱ t (x) := γ,t () (x; x), it converges, to i t γ (),t (x ; x ) = ( x + x )γ (),t (x ; x ) i t γ (),t = [ +V ϱ t, γ,t () ] + ( V ϱ t (x ) V ϱ t (x )) γ (),t (x ; x ) Hartree eq for density matrix If V = V N scaled, then the short scale structure can be relevant. For γ (2) N,t (x, x 2 ; x, x 2) = ( w N (x x 2 ))γ () N,t (x, x )γ() N,t (x 2, x 2 ), +o() = V N (x x 2 )( w N (x x 2 )) [ Smooth ] = { 8πa0 if β = b 0 if β < 3

32 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 m=0 Ξ (k) m (t) + Ω (k) n (t) ds ds 2...ds n U t s BU s s 2 B... U sn 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. 32

33 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 ]: x j s t Time axis =e i(s t) j Vertices represent B, e.g. V (x x 2 )γ(x, x 2 ; x, x 2 )δ(x 2 x 2 ) x x x x x 2 x 2 33

34 Ξ (k) m =... ds ds 2...ds m U t s BU s s 2 B... U sm s m BU sm γ (k+m),0 corresponds to summation over all graphs Γ of the form: Roots:... Leaves: 2k outgoing variables of Ξ (k) m m vertices 2(k+m) incoming variables of γ (k+m)... t s s s 2 3 m s 0 Time axis Tr OΞ (k) m = Γ Val(Γ) 34

35 Value of a graph Γ in momentum space Val(Γ) = e E dα e dp e e 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() 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? 35

36 Combinatorics reduced by resummation: m! is artificial Let k = 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 ) earlier ones) Number of graphs on m vertices without time ) ordering = Number of binary trees = Catalan numbers C m m+( 2m m 36

37 Val(Γ) UV regime: Finiteness of Val(Γ) ( ( ) 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+) guarantees a p e 5/2 decay on each leaf. Power counting (k =, 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, 2 types of vertex integrations that form a closed system. 37