Bose-Einstein condensation and limit theorems Kay Kirkpatrick, UIUC - PowerPoint PPT Presentation

Bose-Einstein condensation and limit theorems Kay Kirkpatrick, UIUC 2015

Bose-Einstein condensation: from many quantum particles to a quantum “superparticle” Kay Kirkpatrick, UIUC/MSRI TexAMP 2015

The big challenge: making physics rigorous

The big challenge: making physics rigorous microscopic first principles � zoom out � Macroscopic states Courtesy Greg L and Digital Vision/Getty Images

1925: predicting Bose-Einstein condensation (BEC)

1925: predicting Bose-Einstein condensation (BEC) 1995: Cornell-Wieman and Ketterle experiment Courtesy U Michigan

After the trap was turned off BEC stayed coherent like a single macroscopic quantum particle. Momentum is concentrated after release at 50 nK. (Atomic Lab)

The mathematics of BEC Gross and Pitaevskii, 1961: a good model of BEC is the cubic nonlinear Schr¨ odinger equation (NLS): i ∂ t ϕ = − ∆ ϕ + µ | ϕ | 2 ϕ Fruitful NLS research: competition between two RHS terms

The mathematics of BEC Gross and Pitaevskii, 1961: a good model of BEC is the cubic nonlinear Schr¨ odinger equation (NLS): i ∂ t ϕ = − ∆ ϕ + µ | ϕ | 2 ϕ Fruitful NLS research: competition between two RHS terms Can we rigorously connect the physics and the math?

The mathematics of BEC Gross and Pitaevskii, 1961: a good model of BEC is the cubic nonlinear Schr¨ odinger equation (NLS): i ∂ t ϕ = − ∆ ϕ + µ | ϕ | 2 ϕ Fruitful NLS research: competition between two RHS terms Can we rigorously connect the physics and the math? Yes!

The outline (w/ G. Staffilani, B. Schlein, G. Ben Arous) microscopic first principles � � Macroscopic states 1. N bosons � mean-field limit � Hartree equation 2. N bosons � localizing limit � NLS 3. Quantum probability and CLTs

A quantum “particle” is really a wavefunction For each t , ψ ( x , t ) ∈ L 2 ( R d ) solves a Schr¨ odinger equation i ∂ t ψ = − ∆ ψ + V ext ( x ) ψ

A quantum “particle” is really a wavefunction For each t , ψ ( x , t ) ∈ L 2 ( R d ) solves a Schr¨ odinger equation i ∂ t ψ = − ∆ ψ + V ext ( x ) ψ =: H ψ ◮ − ∆ = − � d i =1 ∂ x i x i ≥ 0 ◮ external trapping potential V ext ◮ solution ψ ( x , t ) = e − iHt ψ 0 ( x )

A quantum “particle” is really a wavefunction For each t , ψ ( x , t ) ∈ L 2 ( R d ) solves a Schr¨ odinger equation i ∂ t ψ = − ∆ ψ + V ext ( x ) ψ =: H ψ ◮ − ∆ = − � d i =1 ∂ x i x i ≥ 0 ◮ external trapping potential V ext ◮ solution ψ ( x , t ) = e − iHt ψ 0 ( x ) | ψ 0 | 2 = 1 = ⇒ | ψ ( x , t ) | 2 is a probability density for all t . ◮ � Exercise: why?

Particle in a box √ V ext = “ ∞ · 1 [0 , 1] C ” has ground state ψ ( x ) = 2 sin ( π x )

The microscopic N -particle model Wavefunction ψ N ( x , t ) = ψ N ( x 1 , ..., x N , t ) ∈ L 2 ( R dN ) ∀ t solves the N -body Schr¨ odinger equation: N N � � i ∂ t ψ N = − ∆ x j ψ N + U ( x i − x j ) ψ N =: H N ψ N j =1 i < j

The microscopic N -particle model Wavefunction ψ N ( x , t ) = ψ N ( x 1 , ..., x N , t ) ∈ L 2 ( R dN ) ∀ t solves the N -body Schr¨ odinger equation: N N � � i ∂ t ψ N = − ∆ x j ψ N + U ( x i − x j ) ψ N =: H N ψ N j =1 i < j ◮ pair interaction potential U ◮ solution ψ N ( x , t ) = e − iH N t ψ 0 N ( x ) ◮ joint density | ψ N ( x 1 , . . . , x N , t ) | 2

More assumptions For N bosons, ψ N is symmetric (particles are exchangeable): ψ N ( x σ (1) , ..., x σ ( N ) , t ) = ψ N ( x 1 , ..., x N , t ) for σ ∈ S N .

More assumptions For N bosons, ψ N is symmetric (particles are exchangeable): ψ N ( x σ (1) , ..., x σ ( N ) , t ) = ψ N ( x 1 , ..., x N , t ) for σ ∈ S N . Initial data is factorized (particles i.i.d.): N � ψ 0 ϕ 0 ( x j ) ∈ L 2 s ( R 3 N ) . N ( x ) = j =1

More assumptions For N bosons, ψ N is symmetric (particles are exchangeable): ψ N ( x σ (1) , ..., x σ ( N ) , t ) = ψ N ( x 1 , ..., x N , t ) for σ ∈ S N . Initial data is factorized (particles i.i.d.): N � ψ 0 ϕ 0 ( x j ) ∈ L 2 s ( R 3 N ) . N ( x ) = j =1 But interactions create correlations for t > 0.

Mean-field pair interaction U = 1 N V Weak: order 1 / N . Long distance: V ∈ L ∞ ( R 3 ). N N − ∆ x j ψ N + 1 � � i ∂ t ψ N = V ( x i − x j ) ψ N . N j =1 i < j

Mean-field pair interaction U = 1 N V Weak: order 1 / N . Long distance: V ∈ L ∞ ( R 3 ). N N − ∆ x j ψ N + 1 � � i ∂ t ψ N = V ( x i − x j ) ψ N . N j =1 i < j Spohn, 1980: If ψ N is initially factorized and approximately factorized for all t , i.e., ψ N ( x , t ) ≃ � N j =1 ϕ ( x j , t ), then “ ψ N → ϕ ” and ϕ solves the Hartree equation: i ∂ t ϕ = − ∆ ϕ + ( V ∗ | ϕ | 2 ) ϕ.

Convergence “ ψ N → ϕ ” means in the sense of marginals: � � � γ (1) N →∞ N − | ϕ �� ϕ | − − − − → 0 , � � � Tr

Convergence “ ψ N → ϕ ” means in the sense of marginals: � � � γ (1) N →∞ N − | ϕ �� ϕ | − − − − → 0 , � � � Tr where | ϕ �� ϕ | ( x 1 , x ′ 1 ) = ϕ ( x 1 ) ϕ ( x ′ 1 ) and one-particle marginal density γ (1) := Tr N − 1 | ψ N �� ψ N | has kernel N � γ (1) N ( x 1 ; x ′ ψ N ( x 1 , x N − 1 , t ) ψ N ( x ′ 1 , t ) := 1 , x N − 1 , t ) d x N − 1 .

Other mean-field limit theorems Erd¨ os and Yau, 2001: Convergence of marginals for Coulomb interaction, V ( x ) = 1 / | x | , not assuming approximate factorization.

Other mean-field limit theorems Erd¨ os and Yau, 2001: Convergence of marginals for Coulomb interaction, V ( x ) = 1 / | x | , not assuming approximate factorization. Rodnianski-Schlein ’08, Chen-Lee-Schlein, ’11: convergence rate Tr ≤ Ce Kt � � � γ (1) N − | ϕ �� ϕ | . � � N �

Other mean-field limit theorems Erd¨ os and Yau, 2001: Convergence of marginals for Coulomb interaction, V ( x ) = 1 / | x | , not assuming approximate factorization. Rodnianski-Schlein ’08, Chen-Lee-Schlein, ’11: convergence rate Tr ≤ Ce Kt � � � γ (1) N − | ϕ �� ϕ | . � � N � Preview of localizing interactions: ( V N ∗ | ϕ | 2 ) ϕ → ( δ ∗ | ϕ | 2 ) ϕ Erd¨ os, Schlein, Yau, K., Staffilani, Chen, Pavlovic, Tzirakis...

Definition of BEC at zero temperature Almost all particles are in the same one-particle state: { ψ N ∈ L 2 s ( R 3 N ) } N ∈ N exhibits Bose-Einstein condensation into one-particle quantum state ϕ ∈ L 2 ( R 3 ) iff one-particle marginals converge in trace norm: γ (1) = Tr N − 1 | ψ N �� ψ N | N →∞ − − − − → | ϕ �� ϕ | . N

Definition of BEC at zero temperature Almost all particles are in the same one-particle state: { ψ N ∈ L 2 s ( R 3 N ) } N ∈ N exhibits Bose-Einstein condensation into one-particle quantum state ϕ ∈ L 2 ( R 3 ) iff one-particle marginals converge in trace norm: γ (1) = Tr N − 1 | ψ N �� ψ N | N →∞ − − − − → | ϕ �� ϕ | . N Generalizes factorized: ψ N ( x ) = � N j =1 ϕ ( x j ) is BEC into ϕ .

BEC limit theorems with parameter β ∈ (0 , 1] Now localized strong interactions: N d β V ( N β ( · )) → b 0 δ . N N − ∆ x j + 1 � � N d β V ( N β ( x i − x j )) . H N = N j =1 i < j

BEC limit theorems with parameter β ∈ (0 , 1] Now localized strong interactions: N d β V ( N β ( · )) → b 0 δ . N N − ∆ x j + 1 � � N d β V ( N β ( x i − x j )) . H N = N j =1 i < j Theorems (Erd¨ os-Schlein-Yau 2006-2008 d = 3 K.-Schlein-Staffilani 2009 d = 2 plane and rational tori): Systems that are initially BEC remain condensed for all time, and the macroscopic evolution is the NLS: i ∂ t ϕ = − ∆ ϕ + b 0 | ϕ | 2 ϕ.

Our limit theorems make the physics of BEC rigorous N N − ∆ x j + 1 � � N d β V ( N β ( x i − x j )) H N = N j =1 i < j N -body Schrod. ψ 0 micro : − → ψ N N init. BEC ↓ ↓ marg. MACRO : ϕ 0 − → ϕ NLS evolution i ∂ t ϕ = − ∆ ϕ + b 0 | ϕ | 2 ϕ.

A taste of quantum probability ( H , P , ϕ ) Hilbert space H , set of projections P , and state ϕ . Quantum random variables (RVs) or observables: operators on H .

A taste of quantum probability ( H , P , ϕ ) Hilbert space H , set of projections P , and state ϕ . Quantum random variables (RVs) or observables: operators on H . The expectation of an observable A in a pure state is � E ϕ [ A ] := � ϕ | A ϕ � = ϕ ( x ) A ϕ ( x ) dx . Position observable is X ( ϕ )( x ) := x ϕ ( x ) with density | ϕ | 2 .

Only some probability facts have quantum analogues Courtesy of Jordgette

The BEC limit theorems imply quantum LLNs If A is a one-particle observable and A j = 1 ⊗ · · · ⊗ 1 ⊗ A ⊗ 1 ⊗ · · · ⊗ 1 , then for each ǫ > 0, �� N 1 � � lim sup P ψ N A j � N � N →∞ � j =1

The BEC limit theorems imply quantum LLNs If A is a one-particle observable and A j = 1 ⊗ · · · ⊗ 1 ⊗ A ⊗ 1 ⊗ · · · ⊗ 1 , then for each ǫ > 0, �� � N � 1 � � � A j − � ϕ | A ϕ � � ≥ ǫ lim sup P ψ N = 0 . � � N � � N →∞ � j =1

BEC can explode as a bosenova