Derivation of 1d and 2d GrossPitaevskii equations for strongly - PowerPoint PPT Presentation

Derivation of 1d and 2d Gross–Pitaevskii equations for strongly confined 3d bosons Lea Boßmann University of T¨ ubingen Venice, 20 August 2019

Problem Results Limiting sequences Strategy of Proof In a nutshell Consider N interacting bosons in a BEC which are in two or one spatial directions confined by a trap of diameter ε .

Problem Results Limiting sequences Strategy of Proof In a nutshell Consider N interacting bosons in a BEC which are in two or one spatial directions confined by a trap of diameter ε . Let simultaneously N → ∞ and ε → 0.

Problem Results Limiting sequences Strategy of Proof In a nutshell Consider N interacting bosons in a BEC which are in two or one spatial directions confined by a trap of diameter ε . Let simultaneously N → ∞ and ε → 0. We show that the dynamics of this system are effectively described by a one-/two-dimensional nonlinear equation.

Problem Results Limiting sequences Strategy of Proof In a nutshell Consider N interacting bosons in a BEC which are in two or one spatial directions confined by a trap of diameter ε . Let simultaneously N → ∞ and ε → 0. We show that the dynamics of this system are effectively described by a one-/two-dimensional nonlinear equation. Joint work with Stefan Teufel. References: 1d: J. Math. Phys. 60:031902; Ann. Henri Poincar´ e 20(3):1003 2d: arXiv:1907.04547

Problem Results Limiting sequences Strategy of Proof Microscopic model Coordinates: z = ( x , y ) ∈ R 3 x ∈ R d y ∈ R 3 − d N -body Hamiltonian N � � ε 2 V ⊥ ( y j − ∆ j + 1 � + � H = ε ) w N ,ε ( z i − z j ) j =1 i < j • V ⊥ : confining potential; rescaled by ε

Problem Results Limiting sequences Strategy of Proof Microscopic model Coordinates: z = ( x , y ) ∈ R 3 x ∈ R d y ∈ R 3 − d N -body Hamiltonian N � � ε 2 V ⊥ ( y j − ∆ j + 1 � + � H = ε ) w N ,ε ( z i − z j ) j =1 i < j • V ⊥ : confining potential; rescaled by ε w N ,ε ( z ) := µ 1 − 3 β w � µ − β z � Pair interaction: β ∈ (0 , 1] • w ≥ 0 spherically symmetric, bounded, supp w ⊆ B 1 (0) � − 1 → µ β : effective range of the interaction • � N µ = ε 3 − d

Problem Results Limiting sequences Strategy of Proof Microscopic model Coordinates: z = ( x , y ) ∈ R 3 x ∈ R d y ∈ R 3 − d N -body Hamiltonian N � � ε 2 V ⊥ ( y j − ∆ j + 1 � + � H = ε ) w N ,ε ( z i − z j ) j =1 i < j • V ⊥ : confining potential; rescaled by ε w N ,ε ( z ) := µ 1 − 3 β w � µ − β z � Pair interaction: β ∈ (0 , 1] • w ≥ 0 spherically symmetric, bounded, supp w ⊆ B 1 (0) � − 1 → µ β : effective range of the interaction • � N µ = ε 3 − d Limit: ( N , ε ) → ( ∞ , 0) with suitable restrictions

Problem Results Limiting sequences Strategy of Proof Assumptions on the initial data � γ (1) � = 0 � − | ϕ ε 0 � � ϕ ε � 1 BEC: ( N ,ε ) → ( ∞ , 0) Tr lim 0 | 0 • γ (1) 0 : one-particle reduced density matrix of ψ N ,ε 0

Problem Results Limiting sequences Strategy of Proof Assumptions on the initial data � γ (1) � = 0 � − | ϕ ε 0 � � ϕ ε � 1 BEC: ( N ,ε ) → ( ∞ , 0) Tr lim 0 | 0 • γ (1) 0 : one-particle reduced density matrix of ψ N ,ε 0 • Condensate wave function: ϕ ε 0 ( z ) = Φ 0 ( x ) χ ε ( y ) • transverse GS: � − ∆ y + 1 ε 2 V ⊥ ( y � χ ε ( y ) = E 0 ε 2 χ ε ( y ) ε )

Problem Results Limiting sequences Strategy of Proof Assumptions on the initial data � γ (1) � = 0 � − | ϕ ε 0 � � ϕ ε � 1 BEC: ( N ,ε ) → ( ∞ , 0) Tr lim 0 | 0 • γ (1) 0 : one-particle reduced density matrix of ψ N ,ε 0 • Condensate wave function: ϕ ε 0 ( z ) = Φ 0 ( x ) χ ε ( y ) • transverse GS: � − ∆ y + 1 ε 2 V ⊥ ( y � χ ε ( y ) = E 0 ε 2 χ ε ( y ) ε ) • Φ 0 ∈ H 2 d ( R d ) → evolves in time

Problem Results Limiting sequences Strategy of Proof Assumptions on the initial data � γ (1) � = 0 � − | ϕ ε 0 � � ϕ ε � 1 BEC: ( N ,ε ) → ( ∞ , 0) Tr lim 0 | 0 • γ (1) 0 : one-particle reduced density matrix of ψ N ,ε 0 • Condensate wave function: ϕ ε 0 ( z ) = Φ 0 ( x ) χ ε ( y ) • transverse GS: � − ∆ y + 1 ε 2 V ⊥ ( y � χ ε ( y ) = E 0 ε 2 χ ε ( y ) ε ) • Φ 0 ∈ H 2 d ( R d ) → evolves in time � E ( ψ N ,ε � = 0 � � 2 Energy per particle: lim ) − E b β (Φ 0 ) 0 ( N ,ε ) → ( ∞ , 0) • E ( ψ ) := 1 N � ψ, H ψ � − E 0 ε 2 • E b β (Φ) := � Φ , − ∆ x + 1 2 b β | Φ | 2 � � Φ �

Problem Results Limiting sequences Strategy of Proof Effective d -dimensional Gross–Pitaevskii dynamics Theorem Under assumptions (1) and (2) and for any t ∈ R , � γ (1) ( t ) − | ϕ ε ( t ) � � ϕ ε ( t ) | � = 0 , � � ( N ,ε ) → ( ∞ , 0) Tr lim where ϕ ε ( t ) = Φ( t ) χ ε and Φ( t ) is the solution of − ∆ x + b β | Φ( t , x ) | 2 � i ∂ � ∂ t Φ( t , x ) = Φ( t , x ) with Φ(0) = Φ 0 and where | χ ( y ) | 4 d y  � 8 π a β = 1 (GP)  b β = | χ ( y ) | 4 d y � � w � 1 β ∈ (0 , 1) (NLS)  χ : ground state of − ∆ y + V ⊥ ( y ) a : scattering length of w ,

Problem Results Limiting sequences Strategy of Proof Related results • X. Chen, J. Holmer. ARMA 2013. d = 2 → β ∈ (0 , 2 5 ), repulsive interactions • X. Chen, J. Holmer. APDE 2017. d = 1 → β ∈ (0 , 3 7 ), attractive interactions • J. v. Keler, S. Teufel. AHP 2016. d = 1 → β ∈ (0 , 1 3 ), repulsive interactions

Problem Results Limiting sequences Strategy of Proof Simultaneous limit ( N , ε ) → ( ∞ , 0) 1 ε 0 0 1 N − 1

Problem Results Limiting sequences Strategy of Proof Simultaneous limit ( N , ε ) → ( ∞ , 0) 1 ε 0 0 1 N − 1 • admissibility condition: ε must shrink fast enough → upper bound on ε

Problem Results Limiting sequences Strategy of Proof Simultaneous limit ( N , ε ) → ( ∞ , 0) 1 ε 0 0 1 N − 1 • admissibility condition: ε must shrink fast enough → upper bound on ε • moderate confinement: ε must not shrink too fast → lower bound on ε

Problem Results Limiting sequences Strategy of Proof Parameter range for β ∈ (0 , 1) d=2 1 1 β = 1 β = 2 3 3 0 0 0 1 1 0 1 1 N N 1 1 β = 5 β = 11 6 12 0 0 0 1 0 1 1 1 N N

Problem Results Limiting sequences Strategy of Proof d = 2, β ∈ (0 , 2 Comparison with [ChHo2013] 5 ) 1 1 3 β = 1 β = 11 3 0 0 0 1 0 1 N 1 N 1 1 1 β = 11 β = 23 30 60 0 0 0 1 1 0 1 1 N N

Problem Results Limiting sequences Strategy of Proof Limiting sequences for β = 1 d=2 1 0 0 1 1 N

Problem Results Limiting sequences Strategy of Proof Strategy of proof • General strategy: method from [Pickl2015]

Problem Results Limiting sequences Strategy of Proof Strategy of proof • General strategy: method from [Pickl2015] • Adaptation to strong confinement: • 3d micro dynamics ↔ 1d/2d effective dynamics • split interaction into quasi-1d/2d interaction + remainders • remainders controllable with admissibility condition

Problem Results Limiting sequences Strategy of Proof Strategy of proof • General strategy: method from [Pickl2015] • Adaptation to strong confinement: • 3d micro dynamics ↔ 1d/2d effective dynamics • split interaction into quasi-1d/2d interaction + remainders • remainders controllable with admissibility condition Thank you very much for your attention!