Effective equations for two-component Bose-Einstein Condensates - PowerPoint PPT Presentation

Effective equations for two-component Bose-Einstein Condensates Gustavo de Oliveira Departamento de Matem´ atica Universidade Federal de Minas Gerais June 2019

Introduction: An example from classical physics Kinetic theory of a gas of N particles ◮ Microscopic theory. Newtons’s equations for the trajectories ( x 1 , x 2 , . . . , x N ) of N particles: x j = v j ˙ N � v j = − ∇ V ( x j − x i ) . ˙ i � = j Here x j = x j ( t ) and V is a short range potential.

Introduction: An example from classical physics Kinetic theory of gas of N particles ◮ Macroscopic theory. Boltzmann’s equation for the density of particles f = f ( x , v , t ) at time t : � � R 3 dv ′ S 2 d ω B ( v − v ′ , ω ) ∂ t f + v · ∇ x f = × [ f ( x , v out , t ) f ( x , v ′ out , t ) − f ( x , v , t ) f ( x , v ′ , t )] . Incoming particles with v and v ′ collide. Outcoming with v out = v + ω · ( v ′ − v ) ω, out = v ′ − ω · ( v ′ − v ) ω. v ′ Here B ( v − v ′ , ω ) is proportional do the cross section.

Introduction: An example from classical physics Kinetic theory of gas of N particles ◮ Scaling limit. Boltzmann’s equation becomes correct in the Boltzmann-Grad limit: N ρ 2 = const . density ρ → 0 , N → ∞ , ◮ Mathematical derivation. Lanford (’75) proved: In the Boltzmann-Grad limit, Boltzmann’s equation follows from Newton’s equation (at least for short times). ◮ Extensions. Later, to a larger class of potentials V .

As the above example illustrates Typical steps in a derivation program ◮ Microscopic theory. Physical law; Many degrees of freedom; Arbitrary initial data; Detailed solutions: impractical or not very useful. ◮ Scaling limit. Appropriate regime of parameters. ◮ Macroscopic theory. Statistical description; Effective theory (or equation); Restricted initial data (possibly). ◮ Mathematical results. Detailed analysis of the problem. ◮ Extensions. Less regular interactions; More general initial data.

An example from quantum theory ◮ Thomas-Fermi theory for large atoms and molecules. Neutral quantum system of N electrons and M nuclei. Ground state energy: E ( N ) = inf � ψ, H N ψ � . For large N : E ( N ) ≈ E TF ( N ) = inf {E TF ( ρ ) | ∫ dx | ρ ( x ) | = N } , where E TF ( ρ ) is the Thomas-Fermi functional. Theorem (Lieb-Simon ’77). Approximation becomes exact as N → ∞ .

Main background reference for this talk N. Benedikter, M. Porta and B. Schlein (2016). The references for the work that we mention can be found there.

Plan 1. Introduction (completed) 2. One-component Bose gases (easier to explain) 3. Two-component Bose gases (similar)

Wave function for N Bosonic particles ◮ N -particle wave function: x 1 , . . . , x N ∈ R 3 , ψ t ( x 1 , . . . , x N ) ∈ C , t ∈ R . ◮ Square-integrable and normalized: ψ t ∈ L 2 ( R 3 N ) ≃ L 2 ( R 3 ) ⊗ · · · ⊗ L 2 ( R 3 ) , � R 3 N | ψ t | 2 = 1 . ◮ | ψ t | 2 probability density. ◮ ψ t is symmetric in each pair of variables x 1 , . . . , x N .

Density operator N -particle L 2 ( R 3 N ) . γ ψ t = | ψ t �� ψ t | on Tr γ ψ t = 1 , � γ ψ t � := Tr | γ ψ t | . 1-particle γ (1) L 2 ( R 3 ) . ψ t = Tr 2 → N γ ψ t on Tr 2 → N Integrate out N − 1 variables of the integral kernel of γ ψ t . γ (1) 1-particle marginal: Plays the role of 1-particle ψ t wave-function.

Bose-Einstein condensation In experiments, since 1995 (Nobel Prize 2001) Trapped cold ( T ∼ 10 − 9 K ) dilute gas of N ∼ 10 3 Bosons. Heuristically N ϕ t ∈ L 2 ( R 3 ) . � ψ t ( x 1 , . . . , x N ) ≃ ϕ t ( x j ) where j =1 γ ψ t ≃ | ϕ t �� ϕ t | ⊗ · · · ⊗ | ϕ t �� ϕ t | . Mathematically � γ (1) � = 0 . � ψ t − | ϕ t �� ϕ t | � Tr

Models Quantum Hamiltonian in the mean-field regime N N � − ∆ x j + V trap ( x j ) � + 1 H trap � � V ( x i − x j ) , = N N j =1 i < j Quantum Hamiltonian in the Gross-Pitaevskii regime N N � − ∆ x j + V trap ( x j ) � + 1 H trap � � N 3 V ( N ( x i − x j )) , = N N j =1 i < j V trap ( y ) = | y | 2 V ≥ 0 , V ( x ) = V ( | x | ) , compact supp . and

Basic problems Ground state energy E ( N ) = inf � ψ, H trap ψ � = inf spec H trap . N N Initial value problem H N = ( H trap with V trap = 0) N i ∂ t ψ t = H N ψ t ψ t =0 = ψ.

In the mean-field regime Expect: ◮ Approximate factorization of condensate ψ t for large N ⇒ = ◮ Approximate independence of particles ⇒ (by the Law of Large Numbers) = Potential experienced by the j th particle N = 1 � dy V ( x j − y ) | ϕ t ( y ) | 2 � V ( x i − x j ) ≃ N i < j = ( V ∗ | ϕ t | 2 )( x j ) . = ⇒ (separation of variables) ◮ The Schr¨ odinger equation should factor into products i ∂ t ϕ t = − ∆ ϕ t + V ∗ | ϕ t | 2 ϕ t .

In the Gross-Pitaevskii regime Very heuristically N N 3 V ( N · ) ∼ 1 1 N δ ( · ) for large N models rare but strong collisions. In this talk, we focus on mean-field. We may skip the slides about Gross-Pitaevskii.

Time-independent theory Mean-field regime Ground state energy per particle: 1 N inf spec H trap = min {E MF ( ϕ ) | ϕ ∈ L 2 ( R 3 ) , � ϕ � = 1 } lim N N →∞ where |∇ ϕ | 2 + V trap | ϕ | 2 + 1 � � 2( V ∗ | ϕ | 2 ) | ϕ | 2 � E MF ( ϕ ) = . The minimizer ϕ MF of E MF obeys � � γ (1) � ψ gs − | ϕ MF �� ϕ MF | � → 0 N → ∞ . Tr as � � (Modern proof: Lewin-Nam-Rougerie (’14))

Time-independent theory Gross-Pitaevski regime Ground state energy per particle: 1 N inf spec H trap = min {E GP ( ϕ ) | ϕ ∈ L 2 ( R 3 ) , � ϕ � = 1 } lim N N →∞ where � � |∇ ϕ | 2 + V trap | ϕ | 2 + 4 π a | ϕ | 4 � E GP ( ϕ ) = . The minimizer ϕ GP of E GP obeys � � γ (1) � Tr ψ gs − | ϕ GP �� ϕ GP | � → 0 as N → ∞ . � � (Lieb-Seiringer-Yngvason (’00))

Fock space L 2 sym ( R 3 n ) . � F = C ⊕ n ≥ 1 State ψ ∈ F : ψ = ψ 0 ⊕ ψ 1 ⊕ ψ 2 ⊕ · · · ⊕ ψ N ⊕ · · · Vacuum state Ω ∈ F : Ω = 1 ⊕ 0 ⊕ 0 ⊕ · · · N number of particles operator on F : ( N ψ ) n = n ψ n . For example � Ω , N Ω � = 0.

Time evolution of condensates — Initial data Product state in L 2 sym ( R 3 N ) ψ t =0 = ϕ ⊗ N . Coherent state in F √ Ψ t =0 = W ( N ϕ ) Ω 1 ⊕ ϕ ⊕ ϕ ⊗ 2 ⊕ ϕ ⊗ 3 ⊕ · · · ⊕ ϕ ⊗ N � � = e − N � ϕ � 2 / 2 √ √ √ ⊕ · · · 2! 3! N ! We have � Ψ t =0 , N Ψ t =0 � = N .

Schr¨ odinger equation on Fock space Condensate state reached – Traps are turned off H N = ( H trap with V trap = 0) . N Hamiltonian on Fock space H = H 0 ⊕ H 1 ⊕ · · · ⊕ H N ⊕ · · · Time evolution is observed � i ∂ t Ψ t = H Ψ t N → ∞ . as Ψ t =0 = Ψ

Mean-field regime Theorem (Rodnianski-Schlein, CMP ’09) Consider the solution √ Ψ t = e − i H t W ( N ϕ )Ω . Let Γ (1) = one-particle reduced density operator of Ψ t . t Then � ≤ C exp( C | t | ) 1 � Γ (1) � � Tr − | ϕ t �� ϕ t | � � t N for all t and N , where ϕ t solves (time-dep. Hartree eqn.) i ∂ t ϕ t = − ∆ ϕ t + ( V ∗ | ϕ t | 2 ) ϕ t with ϕ 0 = ϕ.

Gross-Pitaevskii regime Theorem (Benedikter–de Oliveira–Schlein, CPAM ’14)] Consider the solution √ Ψ t = e − i H t W ( N ϕ ) T ( k )Ω . Let Γ (1) = one-particle reduced density operator of Ψ t . t Then � ≤ C exp( C exp( C | t | )) 1 � � Γ (1) � Tr − | ϕ t �� ϕ t | √ � � t N for all t and N , where ϕ t solves (time-dep. Gross-Pitaevskii eqn.) i ∂ t ϕ t = − ∆ ϕ t + 8 π a | ϕ t | 2 ϕ t with ϕ 0 = ϕ, a > 0 (scattering length of V ).

Two-component condensate State space L 2 ( R 3 N 1 ) ⊗ L 2 ( R 3 N 2 ) . Hamiltonian (in the mean-field regime) H N 1 , N 2 = h N 1 ⊗ I + I ⊗ h N 2 + V N 1 , N 2 where N p N p − ∆ x j + 1 � � h N p = V p ( x i − x j ) N p j =1 i < j and N 1 N 2 1 � � V N 1 , N 2 = V 12 ( x j − y k ) . N 1 + N 2 j =1 k =1

Two-component condensate (1,1)-particle density operator γ (1 , 1) = Tr N 1 − 1 , N 2 − 1 | ψ t �� ψ t | L 2 ( R 3 ) ⊗ L 2 ( R 3 ) . on We embed our model into F ⊗ F . Hamiltonian H = H 1 + H 2 + V . Initial data � � N 1 u )Ω ⊗ W ( Ψ t =0 = W ( N 2 v )Ω .

Two-component condensate Theorem (de Oliveira-Michelangeli, RMP ’19) Consider the solution Ψ t = e − i H t [ W ( � � N 1 u )Ω ⊗ W ( N 2 v )Ω] . Let Γ (1 , 1) = (1,1)-particle reduced density operator of Ψ t . Then t 1 1 � � � � Γ (1 , 1) � − | u t ⊗ v t �� u t ⊗ v t | � ≤ C exp( C | t | ) √ N 1 √ N 2 Tr + � � t for all t , N 1 and N , where u t and v t solve (time-dep. Hartree sys.) i ∂ t u t = − ∆ u t + ( V 1 ∗ | u t | 2 ) u t + c 2 ( V 12 ∗ | v t | 2 ) u t , i ∂ t v t = − ∆ v t + ( V 2 ∗ | v t | 2 ) v t + c 1 ( V 12 ∗ | u t | 2 ) v t with u t =0 = u and v t =0 = v where c j = lim N 1 , N 2 →∞ N j / ( N 1 + N 2 ).