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 The Analysis of Complex Quantum Systems: Large Coulomb Systems and Related Matters CIRM - October 21, 2019
Intro & results Bose-Einstein condensation Proofs The Gross-Pitaevskii regime System of interest: N interacting bosons N trapped bosons, described by N N � � � � � � L 2 sym ( R 3 N ) H N = − ∆ x j + V ext ( x j ) + x i − x j on V j =1 i < j The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime S. Cenatiempo CIRM - October 21, 2019 2/22
Intro & results Bose-Einstein condensation Proofs The Gross-Pitaevskii regime System of interest: N interacting bosons N trapped bosons, described by N N � � � � � � L 2 sym ( R 3 N ) H N = − ∆ x j + V ext ( x j ) + x i − x j on V j =1 i < j The interaction is characterized by the scattering length a , defined through the zero energy scattering function f � � − ∆ + V / 2 f = 0 f ( x ) − | x |→∞ 1 − − − → For short range potentials f ( x ) = 1 − a for | x | > R | x | The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime S. Cenatiempo CIRM - October 21, 2019 2/22
Intro & results Bose-Einstein condensation Proofs The Gross-Pitaevskii regime System of interest: N interacting bosons N trapped bosons, described by N N � � � � � � L 2 sym ( R 3 N ) H N = − ∆ x j + V ext ( x j ) + x i − x j on V j =1 i < j The interaction is characterized by the scattering length a , defined through the zero energy scattering function f � � − ∆ + V / 2 f = 0 f ( x ) − | x |→∞ 1 − − − → � 8 π a = d x f ( x ) V ( x ) For short range potentials f ( x ) = 1 − a for | x | > R | x | The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime S. Cenatiempo CIRM - October 21, 2019 2/22
Intro & results Bose-Einstein condensation Proofs The Gross-Pitaevskii regime System of interest: N interacting bosons N trapped bosons, described by N N � � � � � � L 2 sym ( R 3 N ) H N = − ∆ x j + V ext ( x j ) + x i − x j on V j =1 i < j The interaction is characterized by the scattering length a , defined through the zero energy scattering function f � � − ∆ + V / 2 f = 0 f ( x ) − | x |→∞ 1 − − − → � 8 π a = d x f ( x ) V ( x ) For short range potentials f ( x ) = 1 − a for | x | > R | x | Experimentally a can be measured via the zero energy cross section: σ 0 = 4 π a 2 The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime S. Cenatiempo CIRM - October 21, 2019 2/22
Intro & results Bose-Einstein condensation Proofs The Gross-Pitaevskii regime Physical phenomenon: Bose-Einstein condensation N -particle wave function Condensate wave function − − − − → large N ψ N ( x 1 , . . . , x N ) ∈ L 2 ( R 3 N ) ϕ ∈ L 2 ( R 3 ) Reduced one-particle density matrix � γ (1) ψ N ( x ; y ) = dx 2 . . . dx N ψ N ( x , x 2 , . . . , x N ) ψ N ( y , x 2 , . . . , x N ) For every compact operator A on L 2 ( R 3 ) � ψ N , ( A ⊗ 1) ψ N � = Tr A γ (1) ψ N Complete condensation in the state ψ N : γ (1) ψ N − − N →∞ | ϕ �� ϕ | − − → The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime S. Cenatiempo CIRM - October 21, 2019 3/22
Intro & results Bose-Einstein condensation Proofs The Gross-Pitaevskii regime Physical phenomenon: Bose-Einstein condensation N -particle wave function Condensate wave function − − − − → large N ψ N ( x 1 , . . . , x N ) ∈ L 2 ( R 3 N ) ϕ ∈ L 2 ( R 3 ) Reduced one-particle density matrix � γ (1) ψ N ( x ; y ) = dx 2 . . . dx N ψ N ( x , x 2 , . . . , x N ) ψ N ( y , x 2 , . . . , x N ) For every compact operator A on L 2 ( R 3 ) � ψ N , ( A ⊗ 1) ψ N � = Tr A γ (1) ψ N Complete condensation in the state ψ N : γ (1) ψ N − − N →∞ | ϕ �� ϕ | − − → N →∞ | ϕ �� ϕ | ⊗ k i.e. the expectation of any For bosons this also implies: γ ( k ) ψ N − − − − → k -particle observable in the state ψ N can be computed using ϕ ⊗ k . The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime S. Cenatiempo CIRM - October 21, 2019 3/22
Intro & results Bose-Einstein condensation Proofs The Gross-Pitaevskii regime Mathematical problems prove the appearance of Statics: condensation in gas of interacting bosons at low temperature & inve- stigate low energy states Expectation: Bogoliubov theory Anderson et al., BEC in a vapor of Rb-87 (1995) Dynamics: after cooling the gas to very low temperatures the traps are switched off and the evolution of the condensate is observed. a Effective non-linear one-body equation vs many-body Schr¨ odinger dynamics cond-mat/0503044 (2005) The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime S. Cenatiempo CIRM - October 21, 2019 4/22
Intro & results Bose-Einstein condensation Proofs The Gross-Pitaevskii regime Mathematical problems Statics: prove the appearance of condensation in gas of interacting bosons at low temperature & inve- stigate low energy states Expectation: Bogoliubov theory Anderson et al., BEC in a vapor of Rb-87 (1995) Dynamics: after cooling the gas to very low temperatures the traps are switched off and the evolution of the condensate is observed. a Effective non-linear one-body equation vs many-body Schr¨ odinger dynamics cond-mat/0503044 (2005) The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime S. Cenatiempo CIRM - October 21, 2019 4/22
Intro & results Bose-Einstein condensation Proofs The Gross-Pitaevskii regime Homogeneous dilute Bose gases N bosons enclosed in a cubic box of side length L , periodic b.c. N � � � � ρ a 3 ≪ 1 H N = − ∆ x j + V x i − x j , j =1 1 ≤ i < j ≤ N Results in the thermodynamic limit i.e. N , L → ∞ and ρ = N / L 3 fixed ◮ occurrence of condensation - hard-core bosons at half filling [Dyson-Lieb-Simon,‘78] - renormalization group ongoing program: [Benfatto ‘94], [Balaban-Feldman-Kn¨ orrer-Trubowitz ‘08-‘16] ◮ thermodynamic functions - ground state energy: [Dyson‘57], [Lieb-Yngvason ‘98], [Erd¨ os-Schlein-Yau ‘08], [Giuliani-Seiringer ‘09], [Yau-Yin ‘13], [Brietzke-Solovej, Brietzke-Fournais- Solovej, Fournais- Solovej ‘19] ◮ low lying excitation spectrum � superfluidity The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime S. Cenatiempo CIRM - October 21, 2019 5/22
Intro & results Bose-Einstein condensation Proofs The Gross-Pitaevskii regime The Gross-Pitaevskii regime 2 ] 3 , with periodic b.c. and Consider N bosons in the box Λ = [ − 1 2 , 1 N N � � � � N 2 V H N = − ∆ x i + N ( x i − x j ) i =1 i < j ◮ If V ( x ) has scattering length a , then N 2 V ( Nx ) has → dilute regime ρ a 3 GP = O ( N − 2 ) scattering length a GP = a / N − ◮ States with small energy are characterized by a correlation structure on length scales of a GP ∼ N − 1 − → understand role of correlations The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime S. Cenatiempo CIRM - October 21, 2019 6/22
Intro & results Bose-Einstein condensation Proofs The Gross-Pitaevskii regime The Gross-Pitaevskii regime 2 ] 3 , with periodic b.c. and Consider N bosons in the box Λ = [ − 1 2 , 1 N N � � � � N 2 V H N = − ∆ x i + N ( x i − x j ) i =1 i < j ◮ If V ( x ) has scattering length a , then N 2 V ( Nx ) has → dilute regime ρ a 3 GP = O ( N − 2 ) scattering length a GP = a / N − ◮ States with small energy are characterized by a correlation structure on length scales of a GP ∼ N − 1 − → understand role of correlations Relevance: ◮ effective description of the strong and short range interactions among atoms in typical Bose-Einstein condensation experiments The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime S. Cenatiempo CIRM - October 21, 2019 6/22
Intro & results Bose-Einstein condensation Proofs The Gross-Pitaevskii regime The Gross-Pitaevskii regime 2 ] 3 , with periodic b.c. and Consider N bosons in the box Λ = [ − 1 2 , 1 N N � � � � N 2 V H N = − ∆ x i + N ( x i − x j ) i =1 i < j ◮ If V ( x ) has scattering length a , then N 2 V ( Nx ) has → dilute regime ρ a 3 GP = O ( N − 2 ) scattering length a GP = a / N − ◮ States with small energy are characterized by a correlation structure on length scales of a GP ∼ N − 1 − → understand role of correlations Relevance: ◮ effective description of the strong and short range interactions among atoms in typical Bose-Einstein condensation 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 ) The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime S. Cenatiempo CIRM - October 21, 2019 6/22
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