Dynamics of Sound Waves in Interacting Bose Gases Dirk - Andr e - PowerPoint PPT Presentation

Dynamics of Sound Waves in Interacting Bose Gases Dirk - Andr´ e Deckert Department of Mathematics University of California Davis March 19, 2014 Joint work with J. Fr¨ ohlich (ETH), P. Pickl (LMU), and A. Pizzo (UCD). D.-A. Deckert (UC Davis) Sound Waves in BECs @ Warwick March 19, 2014 1 / 20

Dynamics of Sound Waves D.-A. Deckert (UC Davis) Sound Waves in BECs @ Warwick March 19, 2014 2 / 20

Model of the Bose Gas i ∂ t Ψ t ( x 1 , . . . , x N ) = H Ψ t ( x 1 , . . . , x N )   � N �  − ∆ x j  H = + α U ( x j − x k ) 2 j =1 k < j with U ∈ C ∞ c , and initially the gas particles are quite regularly arranged: Ψ 0 is “close” to a product state N � Λ − 1 / 2 ϕ 0 ( x j ) , Ψ 0 ( x 1 , . . . , x N ) = � Ψ 0 � 2 = 1 j =1 for a smooth one-particle wave function ϕ 0 with: � ϕ 0 � ∞ = 1 and supported in a box of volume Λ ⊂ R 3 . D.-A. Deckert (UC Davis) Sound Waves in BECs @ Warwick March 19, 2014 3 / 20

Model of the Bose Gas i ∂ t Ψ t ( x 1 , . . . , x N ) = H Ψ t ( x 1 , . . . , x N )   � N �  − ∆ x j  H = + α U ( x j − x k ) 2 j =1 k < j with U ∈ C ∞ c , and initially the gas particles are quite regularly arranged: Ψ 0 is “close” to a product state N � Λ − 1 / 2 ϕ 0 ( x j ) , Ψ 0 ( x 1 , . . . , x N ) = � Ψ 0 � 2 = 1 j =1 for a smooth one-particle wave function ϕ 0 with: � ϕ 0 � ∞ = 1 and supported in a box of volume Λ ⊂ R 3 . D.-A. Deckert (UC Davis) Sound Waves in BECs @ Warwick March 19, 2014 3 / 20

A physically relevant scaling Gas density: ρ = N Λ . For the product state Ψ 0 one finds � � � � � N � ϕ 0 2 � � ( x ) = ρ α U ∗ | ϕ 0 | 2 ( x ) . Ψ 0 , α U ( x − x k )Ψ 0 = N α U ∗ � Λ 1 / 2 k =1 For U ∈ C ∞ c ( R 3 ) one formally has: � − ∆ x j + α U ( x j − x k ) = O (1) + O ( αρ ) 2 k < j Hence, for α ∼ 1 ρ, one can expect non-trivial dynamics for ρ ≫ 1. D.-A. Deckert (UC Davis) Sound Waves in BECs @ Warwick March 19, 2014 4 / 20

A physically relevant scaling Gas density: ρ = N Λ . For the product state Ψ 0 one finds � � � � � N � ϕ 0 2 � � ( x ) = ρ α U ∗ | ϕ 0 | 2 ( x ) . Ψ 0 , α U ( x − x k )Ψ 0 = N α U ∗ � Λ 1 / 2 k =1 For U ∈ C ∞ c ( R 3 ) one formally has: � − ∆ x j + α U ( x j − x k ) = O (1) + O ( αρ ) 2 k < j Hence, for α ∼ 1 ρ, one can expect non-trivial dynamics for ρ ≫ 1. D.-A. Deckert (UC Davis) Sound Waves in BECs @ Warwick March 19, 2014 4 / 20

A physically relevant scaling Gas density: ρ = N Λ . For the product state Ψ 0 one finds � � � � � N � ϕ 0 2 � � ( x ) = ρ α U ∗ | ϕ 0 | 2 ( x ) . Ψ 0 , α U ( x − x k )Ψ 0 = N α U ∗ � Λ 1 / 2 k =1 For U ∈ C ∞ c ( R 3 ) one formally has: � − ∆ x j + α U ( x j − x k ) = O (1) + O ( αρ ) 2 k < j Hence, for α ∼ 1 ρ, one can expect non-trivial dynamics for ρ ≫ 1. D.-A. Deckert (UC Davis) Sound Waves in BECs @ Warwick March 19, 2014 4 / 20

Microscopic dynamics:   N � �  − ∆ x j + 1  i ∂ t Ψ( x 1 , . . . , x N ) = H Ψ( x 1 , . . . , x N ) , H = U ( x j − x k ) 2 ρ j =1 k < j Macroscopic dynamics : h x [ ϕ t ] = − ∆ x 2 + 1 · U ∗ | ϕ t | 2 ( x ) . i ∂ t ϕ t ( x ) = h [ ϕ t ] ϕ t ( x ) , For ρ ≫ 1 one can hope to control the micro- with the macro-dynamics in a sufficiently strong sense, e.g., � � � � � � � � � � ϕ t ϕ t � � � � � ≤ C ( t ) ρ − γ , � Tr x 2 ,..., x N | Ψ t �� Ψ t | − for γ > 0 , ρ ≫ 1 . � � � ϕ t � 2 � ϕ t � 2 For fixed volume Λ, i.e., ρ = O ( N ), many results are available, e.g., Hepp ’74, Spohn ’80, Rodniaski & Schlein ’09, Fr¨ ohlich & Knowles & Schwarz ’09, Pickl ’10, Erd˜ os & Schlein & Yau ’10, . . . Spectrum for large volume: Derezi´ nski & Napi´ orkowski ’13 D.-A. Deckert (UC Davis) Sound Waves in BECs @ Warwick March 19, 2014 5 / 20

Microscopic dynamics:   N � �  − ∆ x j + 1  i ∂ t Ψ( x 1 , . . . , x N ) = H Ψ( x 1 , . . . , x N ) , H = U ( x j − x k ) 2 ρ j =1 k < j Macroscopic dynamics : h x [ ϕ t ] = − ∆ x 2 + 1 · U ∗ | ϕ t | 2 ( x ) . i ∂ t ϕ t ( x ) = h [ ϕ t ] ϕ t ( x ) , For ρ ≫ 1 one can hope to control the micro- with the macro-dynamics in a sufficiently strong sense, e.g., � � � � � � � � � � ϕ t ϕ t � � � � � ≤ C ( t ) ρ − γ , � Tr x 2 ,..., x N | Ψ t �� Ψ t | − for γ > 0 , ρ ≫ 1 . � � � ϕ t � 2 � ϕ t � 2 For fixed volume Λ, i.e., ρ = O ( N ), many results are available, e.g., Hepp ’74, Spohn ’80, Rodniaski & Schlein ’09, Fr¨ ohlich & Knowles & Schwarz ’09, Pickl ’10, Erd˜ os & Schlein & Yau ’10, . . . Spectrum for large volume: Derezi´ nski & Napi´ orkowski ’13 D.-A. Deckert (UC Davis) Sound Waves in BECs @ Warwick March 19, 2014 5 / 20

Microscopic dynamics:   N � �  − ∆ x j + 1  i ∂ t Ψ( x 1 , . . . , x N ) = H Ψ( x 1 , . . . , x N ) , H = U ( x j − x k ) 2 ρ j =1 k < j Macroscopic dynamics : h x [ ϕ t ] = − ∆ x 2 + 1 · U ∗ | ϕ t | 2 ( x ) . i ∂ t ϕ t ( x ) = h [ ϕ t ] ϕ t ( x ) , For ρ ≫ 1 one can hope to control the micro- with the macro-dynamics in a sufficiently strong sense, e.g., � � � � � � � � � � ϕ t ϕ t � � � � � ≤ C ( t ) ρ − γ , � Tr x 2 ,..., x N | Ψ t �� Ψ t | − for γ > 0 , ρ ≫ 1 . � � � ϕ t � 2 � ϕ t � 2 For fixed volume Λ, i.e., ρ = O ( N ), many results are available, e.g., Hepp ’74, Spohn ’80, Rodniaski & Schlein ’09, Fr¨ ohlich & Knowles & Schwarz ’09, Pickl ’10, Erd˜ os & Schlein & Yau ’10, . . . Spectrum for large volume: Derezi´ nski & Napi´ orkowski ’13 D.-A. Deckert (UC Davis) Sound Waves in BECs @ Warwick March 19, 2014 5 / 20

Open Key Questions 1 Can the control be maintained for large volume Λ? 2 Is the approximation good enough to be able to see O Λ ,ρ (1) excitations of the gas, e.g., sound waves, for large Λ? 3 Can the thermodynamic limit, Λ → ∞ for fixed ρ , and the mean-field limit, ρ → ∞ , be decoupled? We demonstrate how to control the microscopic dynamics of excitations in the following large volume regime: Λ Λ , ρ ≫ 1 such that ρ ≪ 1 and N = ρ Λ . D.-A. Deckert (UC Davis) Sound Waves in BECs @ Warwick March 19, 2014 6 / 20

Open Key Questions 1 Can the control be maintained for large volume Λ? 2 Is the approximation good enough to be able to see O Λ ,ρ (1) excitations of the gas, e.g., sound waves, for large Λ? 3 Can the thermodynamic limit, Λ → ∞ for fixed ρ , and the mean-field limit, ρ → ∞ , be decoupled? We demonstrate how to control the microscopic dynamics of excitations in the following large volume regime: Λ Λ , ρ ≫ 1 such that ρ ≪ 1 and N = ρ Λ . D.-A. Deckert (UC Davis) Sound Waves in BECs @ Warwick March 19, 2014 6 / 20

Tracking Excitations for large Λ Coherent excitation of the gas: N � Λ − 1 / 2 ϕ 0 ( x j ) , Ψ 0 ( x 1 , . . . , x N ) = for ϕ 0 = Ω 0 + ǫ 0 , given: j =1 A smooth and flat reference state Ω 0 : supp Ω 0 = O (Λ), � Ω � ∞ = 1 with sufficiently regular tails; A smooth and localized coherent excitation ǫ 0 : supp ǫ 0 = O (1). D.-A. Deckert (UC Davis) Sound Waves in BECs @ Warwick March 19, 2014 7 / 20

Splitting of the dynamics: given the macroscopic dynamics ϕ t use � �� � − ∆ x | Ω t | 2 − 1 i ∂ t Ω t = 2 + U ∗ Ω t . (reference) as reference state and define the excitation by ǫ t = ϕ t e i � U � 1 t − Ω t (excitation) Control of approximation: define � � � � � � ρ ( micro ) = proj ⊥ � Λ 1 / 2 Ψ t Λ 1 / 2 Ψ t � proj ⊥ Ω t Tr x 2 ,..., X N Ω t , t ρ ( macro ) = | ǫ t �� ǫ t | , t � � � � � ρ ( micro ) − ρ ( macro ) and control � for large Λ , ρ . t t . . . which turns out to be not so easy. D.-A. Deckert (UC Davis) Sound Waves in BECs @ Warwick March 19, 2014 8 / 20

Splitting of the dynamics: given the macroscopic dynamics ϕ t use � �� � − ∆ x | Ω t | 2 − 1 i ∂ t Ω t = 2 + U ∗ Ω t . (reference) as reference state and define the excitation by ǫ t = ϕ t e i � U � 1 t − Ω t (excitation) Control of approximation: define � � � � � � ρ ( micro ) = proj ⊥ � Λ 1 / 2 Ψ t Λ 1 / 2 Ψ t � proj ⊥ Ω t Tr x 2 ,..., X N Ω t , t ρ ( macro ) = | ǫ t �� ǫ t | , t � � � � � ρ ( micro ) − ρ ( macro ) and control � for large Λ , ρ . t t . . . which turns out to be not so easy. D.-A. Deckert (UC Davis) Sound Waves in BECs @ Warwick March 19, 2014 8 / 20