The excitation spectrum of the Bose gas in the Gross-Pitaevskii - PowerPoint PPT Presentation

The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime Serena Cenatiempo - Gran Sasso Science Institute, L’Aquila joint work with Chiara Boccato, Christian Brennecke and Benjamin Schlein Quantissima in the Serenissima III Venice - August 22, 2019

Introduction and results The Gross-Pitaevskii regime Strategy of the proof Results The Gross-Pitaevskii regime Consider N bosons in a cubic box Λ described by � N � N � � N 2 V H N = − ∆ x i + N ( x i − x j ) , | Λ | = 1 i =1 i < j ◮ If V ( x ) has scattering length a , then N 2 V ( Nx ) has scattering length a / N ◮ States with small energy are characterized by a correlation structure on length scales of a ∼ N − 1 − → understand role of correlations The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime S. Cenatiempo Quantissima III, Venice 2/14

Introduction and results The Gross-Pitaevskii regime Strategy of the proof Results The Gross-Pitaevskii regime Consider N bosons in a cubic box Λ described by � N � N � � N 2 V H N = − ∆ x i + N ( x i − x j ) , | Λ | = 1 i =1 i < j ◮ If V ( x ) has scattering length a , then N 2 V ( Nx ) has scattering length a / N ◮ States with small energy are characterized by a correlation structure on length scales of a ∼ N − 1 − → understand role of correlations Relevance: ◮ effective description for the strong and short range interactions among atoms in BEC experiments ◮ scaling regime leading to a rigorouns derivation of the Gross-Pitaevskii equation i ∂ t ϕ ( t ) = − ∆ ϕ ( t ) + 8 π a | ϕ ( t ) | 2 ϕ ( t ) ◮ H N equivalent to the Hamiltonian for N bosons in a box with L = N interacting through a fixed potential V , i.e. ρ = N / L 3 = N − 2 The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime S. Cenatiempo Quantissima III, Venice 2/14

Introduction and results The Gross-Pitaevskii regime Strategy of the proof Results Condensation in the Gross-Pitaevskii regime N bosons in Λ = [0; 1] × 3 , periodic boundary conditions � � p a p + 1 p 2 a ∗ V ( r / N ) a ∗ p + r a ∗ Λ ∗ = 2 π Z 3 � H N = q − r a p a q , 2 N p ∈ Λ ∗ p , q , r ∈ Λ ∗ [Lieb-Seiringer-Yngvason ‘00] The ground state energy of H N is given by E N = 4 π a N + o ( N ) [Lieb-Seiringer ‘02, ’06; Nam-Rougerie-Seiringer, ’16] Any Ψ N ∈ L 2 s (Λ N ) with � � Ψ N , H N Ψ N ≤ 4 π a N + o ( N ) exhibits Bose-Einstein condensation, i.e. γ (1) − N →∞ | ϕ 0 �� ϕ 0 | − − − → N where ϕ 0 ( x ) = 1 for all x ∈ Λ. The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime S. Cenatiempo Quantissima III, Venice 3/14

Introduction and results The Gross-Pitaevskii regime Strategy of the proof Results Condensation in the Gross-Pitaevskii regime N bosons in Λ = [0; 1] × 3 , periodic boundary conditions � � p a p + 1 p 2 a ∗ V ( r / N ) a ∗ p + r a ∗ Λ ∗ = 2 π Z 3 � H N = q − r a p a q , 2 N p ∈ Λ ∗ p , q , r ∈ Λ ∗ [Lieb-Seiringer-Yngvason ‘00] The ground state energy of H N is given by E N = 4 π a N + o ( N ) � � � = ( N − 1) � ϕ ⊗ N H N ϕ ⊗ N V (0) Note that ≫ 4 π a N 8 π a = d x f ( x ) V ( x ) 0 0 2 [Lieb-Seiringer ‘02, ’06; Nam-Rougerie-Seiringer, ’16] Any Ψ N ∈ L 2 s (Λ N ) with � � Ψ N , H N Ψ N ≤ 4 π a N + o ( N ) exhibits Bose-Einstein condensation, i.e. γ (1) − N →∞ | ϕ 0 �� ϕ 0 | − − − → N where ϕ 0 ( x ) = 1 for all x ∈ Λ. The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime S. Cenatiempo Quantissima III, Venice 3/14

Introduction and results The Gross-Pitaevskii regime Strategy of the proof Results Bogoliubov theory in the Gross-Pitaevskii regime [Boccato-Brennecke-C.-Schlein ‘19] The ground state energy of H N is � � � � | p | 4 + 16 π a p 2 − (8 π a ) 2 E N = 4 π a ( N − 1) + e Λ a 2 − 1 + O ( N − 1 p 2 + 8 π a − 4 ) 2 p 2 2 p ∈ Λ ∗ + + = 2 π Z 3 \ { 0 } and where Λ ∗ � cos( | p | ) e Λ = 2 − lim p 2 M →∞ p ∈ Z 3 \{ 0 } : | p 1 | , | p 2 | , | p 3 |≤ M The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime S. Cenatiempo Quantissima III, Venice 4/14

Introduction and results The Gross-Pitaevskii regime Strategy of the proof Results Bogoliubov theory in the Gross-Pitaevskii regime [Boccato-Brennecke-C.-Schlein ‘19] The ground state energy of H N is � � � � | p | 4 + 16 π a p 2 − (8 π a ) 2 E N = 4 π a ( N − 1) + e Λ a 2 − 1 + O ( N − 1 p 2 + 8 π a − 4 ) 2 p 2 2 p ∈ Λ ∗ + + = 2 π Z 3 \ { 0 } and where Λ ∗ � cos( | p | ) e Λ = 2 − lim p 2 M →∞ p ∈ Z 3 \{ 0 } : | p 1 | , | p 2 | , | p 3 |≤ M Remark (1) For small potentials κ V : 4 π a ( N − 1) + e Λ a 2 = 4 π a N ( N − 1) with κ 2 � � V 2 ( p 1 / N ) V (0) − 1 8 π a N = κ � + . . . 2 p 2 2 N 1 p 1 ∈ Λ ∗ + The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime S. Cenatiempo Quantissima III, Venice 4/14

Introduction and results The Gross-Pitaevskii regime Strategy of the proof Results Bogoliubov theory in the Gross-Pitaevskii regime [Boccato-Brennecke-C.-Schlein ‘19] The ground state energy of H N is � � � � | p | 4 + 16 π a p 2 − (8 π a ) 2 E N = 4 π a ( N − 1) + e Λ a 2 − 1 + O ( N − 1 p 2 + 8 π a − 4 ) 2 p 2 2 p ∈ Λ ∗ + + = 2 π Z 3 \ { 0 } and where Λ ∗ � cos( | p | ) e Λ = 2 − lim p 2 M →∞ p ∈ Z 3 \{ 0 } : | p 1 | , | p 2 | , | p 3 |≤ M Remark (2) Replace V by V R ( x ) = R − 2 V ( x / R ) with scattering length a R = a R : letting R → ∞ the finite size effect becomes subleading w.r.t. Bogoliubov sum. The result for E N is the analog in the GP regime of the Lee-Huang-Yang formula, valid in the thermodynamic limit: [..., Yau-Yin ’13, ... , Fournais-Solovej ’19] The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime S. Cenatiempo Quantissima III, Venice 4/14

Introduction and results The Gross-Pitaevskii regime Strategy of the proof Results Bogoliubov theory in the Gross-Pitaevskii regime [Boccato-Brennecke-C.-Schlein ‘19] The ground state energy of H N is � � � � | p | 4 + 16 π a p 2 − (8 π a ) 2 E N = 4 π a ( N − 1) + e Λ a 2 − 1 + O ( N − 1 p 2 + 8 π a − 4 ) 2 p 2 2 p ∈ Λ ∗ + + = 2 π Z 3 \ { 0 } and where Λ ∗ � cos( | p | ) e Λ = 2 − lim p 2 M →∞ p ∈ Z 3 \{ 0 } : | p 1 | , | p 2 | , | p 3 |≤ M The spectrum of H N − E N below an energy ζ consists of eigenvalues � � | p | 4 + 16 π a | p | 2 + O ( N − 1 / 4 (1 + ζ 3 )) , n p ∈ N n p p ∈ Λ ∗ + The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime S. Cenatiempo Quantissima III, Venice 4/14

Introduction and results The Gross-Pitaevskii regime Strategy of the proof Results Bogoliubov theory in the Gross-Pitaevskii regime [Boccato-Brennecke-C.-Schlein ‘19] The ground state energy of H N is � � � � | p | 4 + 16 π a p 2 − (8 π a ) 2 E N = 4 π a ( N − 1) + e Λ a 2 − 1 + O ( N − 1 p 2 + 8 π a − 4 ) 2 p 2 2 p ∈ Λ ∗ + + = 2 π Z 3 \ { 0 } and where Λ ∗ � cos( | p | ) e Λ = 2 − lim p 2 M →∞ p ∈ Z 3 \{ 0 } : | p 1 | , | p 2 | , | p 3 |≤ M The spectrum of H N − E N below an energy ζ consists of eigenvalues � � | p | 4 + 16 π a | p | 2 + O ( N − 1 / 4 (1 + ζ 3 )) , n p ∈ N n p p ∈ Λ ∗ + Remark (3) Linear dispersion relation of low energy excitations of the Bose gas Previous results in the mean field scaling [Seiringer ’11, Grech-Seiringer ’13, Lewin-Nam-Serfaty-Solovej ’14, Derezinski-Napiorkovski ’14, Pizzo ’16] and for singular interactions [Boccato-Brennecke-C. -Schlein ’17] The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime S. Cenatiempo Quantissima III, Venice 4/14

Introduction and results The Gross-Pitaevskii regime Strategy of the proof Results Bogoliubov theory in the Gross-Pitaevskii regime [Boccato-Brennecke-C.-Schlein ‘19] The ground state energy of H N is � � � � | p | 4 + 16 π a p 2 − (8 π a ) 2 E N = 4 π a ( N − 1) + e Λ a 2 − 1 + O ( N − 1 p 2 + 8 π a − 4 ) 2 p 2 2 p ∈ Λ ∗ + + = 2 π Z 3 \ { 0 } and where Λ ∗ � cos( | p | ) e Λ = 2 − lim p 2 M →∞ p ∈ Z 3 \{ 0 } : | p 1 | , | p 2 | , | p 3 |≤ M The spectrum of H N − E N below an energy ζ consists of eigenvalues � � | p | 4 + 16 π a | p | 2 + O ( N − 1 / 4 (1 + ζ 3 )) , n p ∈ N n p p ∈ Λ ∗ + L 2 s (Λ N ) Remark (4) Condensate depletion bound: for any ψ N ∈ s.t � � ψ N , H N ψ N ≤ 4 π a N + ζ we have � � ϕ 0 , γ (1) N (1 − N ϕ 0 ) ≤ C ( ζ + 1) i.e. condensation holds with optimal rate. The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime S. Cenatiempo Quantissima III, Venice 4/14

Introduction and results Excitation Hamiltonian Strategy of the proof Correlation structure Removing particles in the Bose-Einstein condensate For ψ N ∈ L 2 s (Λ N ) and ϕ 0 ∈ L 2 (Λ) [Lewin-Nam-Serfaty-Solovej ‘12] ψ N = α 0 ϕ ⊗ N + α 1 ⊗ s ϕ ⊗ N − 1 + . . . + α j ⊗ s ϕ ⊗ N − j + . . . + α N , 0 0 0 where α j ∈ L 2 (Λ) ⊗ s j and α j ⊥ ϕ 0 ; ϕ 0 ( x ) = 1 for all x ∈ Λ. The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime S. Cenatiempo Quantissima III, Venice 5/14