Dynamics of Bose Einstein Condensates Benjamin Schlein, University - PowerPoint PPT Presentation

Dynamics of Bose Einstein Condensates Benjamin Schlein, University of Zurich Quantissima in the Serenissima, Venezia August 21, 2017 Based on joint work with Christian Brennecke 1

I. The Gross-Pitaevskii Limit Hamiltonian : consider N bosons described by N N � � � � H trap N 2 V ( N ( x i − x j )) = − ∆ x j + V ext ( x j ) + N j =1 i<j with V ext confining and V ≥ 0, regular, radial, short range. Scattering length : defined by zero-energy scattering equation � � − ∆ + 1 2 V ( x ) f ( x ) = 0 , f ( x ) → 1 For | x | large, f ( x ) = 1 − a 0 ⇒ a 0 = scattering length of V | x | By scaling � � − ∆ + N 2 a 0 N = scatt. length of N 2 V ( N. ) 2 V ( Nx ) f ( Nx ) = 0 ⇒ 2

Ground state energy : [ Lieb-Seiringer-Yngvason, ’00 ] proved E N lim N = min E GP ( x ) N →∞ ϕ ∈ L 2 ( R 3 ): � ϕ � =1 with Gross-Pitaevskii energy functional � � |∇ ϕ | 2 + V ext | ϕ | 2 + 4 πa 0 | ϕ | 4 � E GP ( ϕ ) = dx Bose-Einstein condensation: [ Lieb-Seiringer, ’02 ] showed γ (1) → | ϕ 0 �� ϕ 0 | N where ϕ 0 minimizes E GP . Warning : this does not mean that ψ N ≃ ϕ ⊗ N . In fact 0 � � � � � � 1 V (0) |∇ ϕ 0 | 2 + V ext | ϕ 0 | 2 + ϕ ⊗ N , H N ϕ ⊗ N | ϕ 0 | 4 ≃ dx 0 0 N 2 Correlations are crucial! 3

II. Time-evolution of BEC Theorem [Brennecke - S., ’17]: Let ψ N ∈ L 2 s ( R 3 N ) such that  := 1 − � ϕ 0 , γ (1)  a N N ϕ 0 � → 0   as N → ∞ � �   � � N � ψ N , H trap � 1  := ψ N � − E GP ( ϕ 0 ) � → 0 b N N Let N N � � N 2 V ( N ( x i − x j )) − ∆ x j + H N = j =1 i<j and ψ N,t = e − iH N t ψ N . Then, for all t ∈ R , 1 − � ϕ t , γ (1) N,t ϕ t � ≤ C ( a N + b N + N − 1 ) exp( c exp( c | t | )) where ϕ t solves time-dependent Gross-Pitaevskii equation i∂ t ϕ t = − ∆ ϕ t + 8 πa 0 | ϕ t | 2 ϕ t with initial data ϕ t =0 = ϕ 0 . 4

Remark: result immediately implies � � � � � γ (1) � ≤ C ( a N + b N + N − 1 ) 1 / 2 exp( c exp( c | t | )) � � N,t − | ϕ t �� ϕ t | Tr Remark: if ψ N is ground state of trapped systems we expect (in some cases, we know; see next talk) that a N , b N ≃ N − 1 . Alternative statement: let ψ N ∈ L 2 s ( R 3 N ), ϕ ∈ L 2 ( R 3 ) s.t.  � � � �  � γ (1)  � �  a N := Tr − | ϕ �� ϕ | � → 0  N � � � � |∇ ϕ | 2 + 4 πa 0 | ϕ | 4 � 1 � �   � �  N � ψ N , H N ψ N � − � → 0 b N := dx  � Let ψ N,t = e − iH N t ψ N . Then 1 − � ϕ t γ (1) N,t ϕ t � ≤ C ( a N + b N + N − 1 ) exp( c exp( c | t | )) where ϕ t solves GP equation with data ϕ t =0 = ϕ . 5

Previous works: [ Erd˝ os-S.-Yau, ’06-’08 ]: BBGKY approach, no rate. Simpli- fication of parts of proof due to [ Klainerman-Machedon ’07 ], [ Chen-Hainzl-Pavlovic-Seiringer, ’13 ]. [ Pickl, ’10 ]: alternative approach, uncontrolled rate. [ Benedikter-de Oliveira-S. ’12 ]: precise bounds on rate, ap- proximately coherent initial data in Fock space. Related results on mean-field dynamics, among others by Adami, Ammari, Bardos, Breteaux, T. Chen, X. Chen, Erd˝ os, Falconi, Fr¨ ohlich, Ginibre, Golse, Grillakis, Hepp, Holmer, Kirkpatrik, Knowles, Kuz, Lewin, Liard, Machedon, Margetis, Mauser, Mitrouskas, Nam, Napiorkowski, Nier, Pavlovic, Pawilowski, Petrat, Pickl, Pizzo, Rodnianski, Rougerie, S., Spohn, Staffilani, Teta, Velo, Yau 6

III. Ideas from the proof for ψ N ∈ L 2 s ( R 3 N ) and ϕ ∈ L 2 ( R 3 ), Orthogonal excitations: write ψ N = α 0 ϕ ⊗ N + α 1 ⊗ s ϕ ⊗ ( N − 1) + α 2 ⊗ s ϕ ⊗ ( N − 2) + · · · + α N where α j ∈ L 2 ⊥ ϕ ( R 3 ) ⊗ s j . As in [ Lewin-Nam-Serfaty-Solovej, ’12 ], [ Lewin-Nam-S. ’15 ], we define the unitary map N � s ( R 3 N ) → F ≤ N U ϕ : L 2 L 2 ⊥ ϕ ( R 3 ) ⊗ s j ⊥ ϕ = j =0 ψ N → Uψ N = { α 0 , α 1 , . . . , α N } Remark : ψ N = U ∗ ϕ ξ N exhibits BEC in ϕ ∈ L 2 ( R 3 ) if and only if ξ N ∈ F ≤ N ⊥ ϕ has small number of particles. 7

ξ N,t ∈ F ≤ N Evolution of BEC : define excitation vector � ⊥ ϕ t through e − iH N t U ∗ ϕ 0 ξ N = U ∗ � ξ N,t ϕ t In other words, ξ N,t = � � W N,t ξ N with fluctuation dynamics W N,t = U ϕ t e − iH N t U ∗ ϕ 0 : F ≤ N ⊥ ϕ 0 → F ≤ N � ⊥ ϕ t Need to show W ∗ � � ξ N,t , N � ξ N,t � = � ξ N , � N,t N � W N,t ξ N � ≤ C t Problem : we are neglecting correlations ! Need to modify fluctuation dynamics! 8

Idea from [Benedikter-de Oliveira-S. ’12] : interested in evo- lution of approximately coherent initial data : e − i H N t W 0 ξ N = W t � with W t = Weyl operator ξ N,t , Describe correlations through Bogoliubov transformations � 1 �� � � η t ( x ; y ) a ∗ x a ∗ � T t = exp dxdy y − h.c. 2 Define modified excitation vector ξ N,t through e − i H N t W 0 � T 0 ξ N = W t � T t ξ N,t With choice w = 1 − f and � � a 0 η t ( x ; y ) = − Nw ( N ( x − y )) ϕ t ( x ) ϕ t ( y ) ≃ − | x − y | ϕ t ( x ) ϕ t ( y ) � in was possible to show that � ξ N,t , N ξ N,t � ≤ C t . 9

Goal : apply similar idea for N -particles data. Problem : Bogoliubov transf. do not leave F ≤ N ⊥ ϕ t invariant. Modified fields : on F ≤ N ⊥ ϕ t , we define, for f ∈ L 2 ⊥ ϕ t ( R 3 ), � � N − N N − N b ∗ ( f ) = a ∗ ( f ) , b ( f ) = a ( f ) N N Remark ϕ t b ∗ ( f ) U ϕ t = a ∗ ( f ) a ( ϕ t ) U ∗ √ N Generalized Bogoliubov transformations : define � 1 �� � � η t ( x ; y ) b ∗ x b ∗ y − h.c. T t = exp dxdy 2 Then T t : F ≤ N ⊥ ϕ t → F ≤ N ⊥ ϕ t , if η t orthogonal to ϕ t in both variables. 10

Modified fluctuation dynamics : let t U ϕ t e − iH N t U ∗ ϕ 0 T 0 : F ≤ N ⊥ ϕ 0 → F ≤ N W N,t = T ∗ ⊥ ϕ t Generator : define G N,t such that i∂ t W N,t = G N,t W N,t We have � � G N,t = ( i∂T ∗ t ) T t + T ∗ ( i∂ t U ϕ t ) U ∗ ϕ t + U ϕ t H N U ∗ T t t ϕ t The contribution ( i∂ t T ∗ t ) T t is harmless. We focus on the second term. Using rules U ϕ t a ∗ ( f ) a ( g ) U ∗ ϕ t = a ∗ ( f ) a ( g ) U ϕ t a ∗ ( ϕ t ) a ( ϕ t ) U ∗ ϕ t = N − N √ √ U ϕ t a ∗ ( f ) a ( ϕ t ) U ∗ ϕ t = a ∗ ( f ) N b ∗ ( f ) N − N = √ √ U ϕ t a ∗ ( ϕ t ) a ( f ) U ∗ ϕ t = N − N a ( f ) = N b ( f ) 11

we find 4 � L ( j ) ( i∂ t U ϕ t ) U ∗ ϕ t + U ϕ t H N U ∗ ϕ t = N,t j =1 with (roughly) √ L (1) N b (( N 3 V ( N. ) w ( N. ) ∗ | ϕ t | 2 ) ϕ t ) + h.c. N,t = � � � N 3 V ( N. ) ∗ | ϕ t | 2 � L (2) ∇ x a ∗ ( x ) b ∗ N,t = x ∇ x a x + dx x b x � ϕ t ( y ) b ∗ dxdyN 3 V ( N ( x − y )) ϕ t ( x )¯ + x b y � � � + 1 dxdyN 3 V ( N ( x − y )) ϕ t ( x ) ϕ t ( y ) b ∗ x b ∗ y + h.c. 2 � � � L (3) ϕ t ( y ) b ∗ x a ∗ dxdyN 5 / 2 V ( N ( x − y )) N,t = y a x + h.c. � N,t = 1 L (4) dxdyN 2 V ( N ( x − y )) a ∗ x a ∗ y a y a x 2

We find G N,t = C N,t + H N + E N,t with � � x ∇ x a x + 1 ∇ x a ∗ dxdy N 2 V ( N ( x − y )) a ∗ x a ∗ H N = y a y a x 2 and, for any δ > 0, a C > 0 s.t. ± E N,t ≤ δ H N + C ( N + 1) � � ± i N , E N,t ≤ δ H N + C ( N + 1) ± ˙ E N,t ≤ δ H N + C ( N + 1) 12

Control of N : by Gronwall , we conclude � ξ N , W ∗ N,t N W N,t ξ N � ≤ C t � ξ N , ( N + H N ) ξ N � With assumptions on initial data, theorem follows. � Main challenge : action of Bogoliubov transf. T t is explicit, i.e. T t a ∗ ( f ) � T t = a ∗ (cosh η t ( f )) + a (sinh η t ( ¯ � f )) For generalized Bogoliubov transformations, no explicit formula is available. Instead, we have to expand � 1 T ∗ t a ∗ ( f ) T t = n ! ad ( n ) ( a ∗ ( f )) n ≥ 0 13