Derivation of Hartree theory for generic mean-field Bose gases - PowerPoint PPT Presentation

Derivation of Hartree theory for generic mean-field Bose gases Mathieu LEWIN mathieu.lewin@math.cnrs.fr (CNRS & Universit´ e de Cergy-Pontoise) joint work with P.T. Nam (Cergy) & N. Rougerie (Grenoble) EMS weekend, Aarhus, April 6, 2013 Mathieu LEWIN (CNRS / Cergy) Derivation of Hartree Aarhus, April 6, 2013 1 / 12

Bose-Einstein condensation (BEC) ◮ Macroscopic number of particles occupy the same quantum state Very low temperature Weak interactions Symmetry breaking , e.g. vortices = ⇒ Superfluidity , e.g. for Helium-4 W. Ketterle et al (MIT) Mathieu LEWIN (CNRS / Cergy) Derivation of Hartree Aarhus, April 6, 2013 2 / 12

Bose-Einstein condensation (BEC) ◮ Macroscopic number of particles occupy the same quantum state Very low temperature Weak interactions Symmetry breaking , e.g. vortices = ⇒ Superfluidity , e.g. for Helium-4 W. Ketterle et al (MIT) ◮ N interacting bosons in a domain Ω ⊂ R d N � � � � H N = − ∆ x j + V ( x j ) + λ w ( x k − x ℓ ) , λ ≪ 1 j =1 1 ≤ k <ℓ ≤ N on H N := L 2 s (Ω N ) = { Ψ ∈ L 2 (Ω N ) symmetric } Mathieu LEWIN (CNRS / Cergy) Derivation of Hartree Aarhus, April 6, 2013 2 / 12

Assumptions N 1 � � � � H N = − ∆ x j + V ( x j ) + w ( x k − x ℓ ) N − 1 j =1 1 ≤ k <ℓ ≤ N w = w 1 + w 2 and V = V 1 + V 2 + V + w i , V i ∈ L p i (Ω) with max(1 , d / 2) < p i < ∞ or p i = ∞ but → 0 at ∞ V + ≥ 0, V + ∈ L d / 2 loc Confined case: Ω is bounded (with chosen b.c.), or V + → ∞ at ∞ Unconfined case: Ω = R d and V + ≡ 0 We can deal with much more complicated models, e.g. in Ω = R d m 2 + | A ( x ) − i ∇| 2 � s − m 2 s + V ( x ) � − ∆ + V ( x ) � Mathieu LEWIN (CNRS / Cergy) Derivation of Hartree Aarhus, April 6, 2013 3 / 12

Hartree theory Ω | u | 2 = 1: Restrict H N to uncorrelated functions Ψ = u ( x 1 ) · · · u ( x N ), with ´ u ⊗ N , H N u ⊗ N � � ˆ |∇ u ( x ) | 2 + V ( x ) | u ( x ) | 2 � � = dx N Ω + 1 ˆ ˆ w ( x − y ) | u ( x ) | 2 | u ( y ) | 2 dx dy := E V H ( u ) 2 Ω Ω Theorem (Validity of Hartree theory [LewNamRou-13]) Under the previous assumptions on V and w, we have: inf σ ( H N ) � ˆ � | u | 2 = 1 E V lim = e H := inf H ( u ) : N N →∞ Ω Raggio-Werner (’89), Petz-Raggio-Verbeure (’89): confined case Benguria-Lieb (’83): bosonic atoms; Lieb-Yau (’87): boson stars Mathieu LEWIN (CNRS / Cergy) Derivation of Hartree Aarhus, April 6, 2013 4 / 12

Density matrices ◮ The many-body wave function Ψ N is usually not close to u ⊗ N in norm! � � Note that � Ψ , H N Ψ � = tr H N H N | Ψ �� Ψ | . Definition (Density matrices) acting on H k defined by The k -particle density matrix of Ψ is the operator γ ( k ) Ψ the partial trace γ ( k ) = tr k +1 → N | Ψ �� Ψ | . Ψ Equivalently ˆ where X , X ′ ∈ Ω k . γ ( k ) Ψ ( X , X ′ ) = Ω N − k Ψ( X , Y )Ψ( X ′ , Y ) dY We have γ ( k ) ≥ 0 and tr H k γ ( k ) Ψ = 1 for all k ≥ 1. Ψ Hartree state: γ ( k ) u ⊗ N = | u ⊗ k �� u ⊗ k | Mathieu LEWIN (CNRS / Cergy) Derivation of Hartree Aarhus, April 6, 2013 5 / 12

Bose-Einstein condensation: confined case Theorem (Bose-Einstein condensation [LewNamRou-13]) Consider any sequence (Ψ N ) such that � Ψ N , H N Ψ N � = N e H + o ( N ) . In the confined case , there exists a subsequence N j and a Borel probability measure µ on the unit sphere S H of H = L 2 (Ω) , supported on the set M of minimizers for e H , such that ˆ j →∞ γ ( k ) d µ ( u ) | u ⊗ k �� u ⊗ k | lim Ψ Nj = M strongly in the trace-class for any fixed k. In particular, if e H admits a unique minimizer u 0 (up to a phase factor), then there is complete Bose-Einstein condensation on u 0 : N →∞ γ ( k ) Ψ N = | u ⊗ k 0 �� u ⊗ k lim 0 | , ∀ k ≥ 1 . Similar result at (small) positive temperature Similar previous results by Raggio-Werner (’89), Petz-Raggio-Verbeure (’89) Mathieu LEWIN (CNRS / Cergy) Derivation of Hartree Aarhus, April 6, 2013 6 / 12

Bose-Einstein condensation: unconfined case Ω | u | 2 = λ Particles can escape to infinity. e V � E V ´ � H ( λ ) := inf H ( u ) : Theorem (Bose-Einstein condensation [LewNamRou-13]) Consider any sequence (Ψ N ) such that � Ψ N , H N Ψ N � = N e H + o ( N ) . In the unconfined case , there exists a subsequence N j and a Borel probability measure µ on the unit ball B H of H = L 2 ( R d ) , supported on the set � � | 2 ) = e V | 2 ) u ∈ B H : E V H ( u ) = e V H (1) − e 0 M = H ( | | u | H (1 − | | u | , such that ˆ γ ( k ) d µ ( u ) | u ⊗ k �� u ⊗ k | Ψ Nj ⇀ ∗ M weakly- ∗ in the trace-class for any fixed k. If furthermore e V H (1) < e V H ( λ ) + e 0 H (1 − λ ) , ∀ 0 ≤ λ < 1 , then supp ( µ ) ⊂ M ⊂ S H and the limit for γ ( k ) N j is strong in the trace-class. Mathieu LEWIN (CNRS / Cergy) Derivation of Hartree Aarhus, April 6, 2013 7 / 12

Example: bosonic atoms ◮ N charged bosons (“electrons”) + 1 nucleus of charge Z , t = ( N − 1) / Z N � 1 � 1 1 � � H N = − ∆ x j − + | x k − x ℓ | , t | x j | N − 1 j =1 1 ≤ k <ℓ ≤ N R 3 |∇ u | 2 − | u | 2 | u ( x ) | 2 | u ( y ) | 2 � ˆ t | x | + 1 � ˆ e H ( t ) := inf dx dy 2 | x − y | ´ R 3 | u | 2 =1 R 6 e H ( t ) admits a unique minimizer u t iff t ≤ t c ≃ 1 . 21 t ≤ t c ⇒ γ ( k ) Ψ N → | u ⊗ k �� u ⊗ k | strongly (Benguria-Lieb ’83) t t t > t c ⇒ γ ( k ) u ⊗ k u ⊗ k u t = ( t c / t ) 3 / 2 u t c ( t c · / t ) Ψ N ⇀ ∗ | ˜ �� ˜ | weakly- ∗ , where ˜ t t Mathieu LEWIN (CNRS / Cergy) Derivation of Hartree Aarhus, April 6, 2013 8 / 12

Quantum de Finetti � Ψ N , H N Ψ N � = 1 2 tr H 2 H 2 γ (2) Ψ N ⇒ what is the set of γ (2) Ψ N ’s in the limit N → ∞ ? N Mathieu LEWIN (CNRS / Cergy) Derivation of Hartree Aarhus, April 6, 2013 9 / 12

Quantum de Finetti � Ψ N , H N Ψ N � = 1 2 tr H 2 H 2 γ (2) Ψ N ⇒ what is the set of γ (2) Ψ N ’s in the limit N → ∞ ? N Theorem (Quantum de Finetti [Størmer-69, Hudson-Moody-75]) Let H be any separable Hilbert space and H k := � k s H . Consider a hierarchy k =0 of non-negative self-adjoint operators, where each γ ( k ) acts on H k and { γ ( k ) } ∞ satisfies tr H k γ ( k ) = 1 . If the hierarchy is consistent tr k +1 γ ( k +1) = γ ( k ) , ∀ k ≥ 0 , then there exists a Borel probability measure µ on the sphere S H of H such that, for all k ≥ 1 , ˆ γ ( k ) = | u ⊗ k �� u ⊗ k | d µ ( u ) . S H Quantum equivalent of the famous Hewitt-Savage thm, which deals with probability measures instead of trace-class operators. Mathieu LEWIN (CNRS / Cergy) Derivation of Hartree Aarhus, April 6, 2013 9 / 12

Proof in the confined case ◮ Step 1: extraction of limits . may assume γ ( k ) Ψ N ⇀ ∗ γ ( k ) , ∀ k ≥ 1 system is confined ⇒ strong CV ⇒ consistent hierarchy ◮ Step 2: de Finetti . ˆ γ ( k ) = | u ⊗ k �� u ⊗ k | d µ ( u ), ∀ k ≥ 1, for some Borel probability measure µ S H ◮ Step 3: Conclusion . tr( H 2 γ (2) Ψ N ) � Ψ N , H N Ψ N � lim inf = lim inf N 2 N →∞ N →∞ ≥ 1 2 tr( H 2 γ (2) ) = 1 ˆ ˆ u ⊗ 2 , H 2 u ⊗ 2 � E V � d µ ( u ) = H ( u ) d µ ( u ) ≥ e H 2 S H S H Since λ 1 ( H N ) ≤ N e H , this concludes the proof Mathieu LEWIN (CNRS / Cergy) Derivation of Hartree Aarhus, April 6, 2013 10 / 12

A weak version of de Finetti Unconfined case: particles can go to infinity ⇒ γ ( k ) not necessarily consistent Theorem (Weak quantum de Finetti [LewNamRou-13]) Consider a sequence Ψ N ∈ H N of normalized states, such that (for a subsequence) γ ( k ) Ψ Nj ⇀ ∗ γ ( k ) , ∀ k ≥ 0 . Then there exists a Borel probability measure µ on the unit ball B H of H such that, for all k ≥ 1 , ˆ γ ( k ) = | u ⊗ k �� u ⊗ k | d µ ( u ) . B H Ex. Ψ N = ( u N ) ⊗ N with u N ⇀ v = ⇒ µ = δ v Proof of weak de Finetti: “geometric localization” in Fock space Proof of main thm: w ≥ 0 (e.g. bosonic atoms): same lines as before general case much more involved (looking at weak limits not sufficient) Mathieu LEWIN (CNRS / Cergy) Derivation of Hartree Aarhus, April 6, 2013 11 / 12

The next order Theorem (Bogoliubov theory [LewNamSerSol-12]) If e V H (1) has a unique, non-degenerate minimizer u 0 , then N →∞ ( λ j ( H N ) − N e H ) = λ j ( H ) lim where H is the second-quantization of Hess E V H ( u 0 ) / 2 on the bosonic Fock space Γ s ( H + ) built on H + = { u 0 } ⊥ . Furthermore, � � N � � � � u ⊗ ( N − n ) ⊗ s ϕ j � � � Ψ j � N − → 0 � � � � n � � � � � n =0 � � H N where Φ j = ( ϕ j n ) n ≥ 0 ∈ Γ s ( H + ) solves H Φ j = λ j ( H ) Φ j , with � � 2 � � ϕ j � � � H n = 1 . � n � n ≥ 0 without interactions with interactions Mathieu LEWIN (CNRS / Cergy) Derivation of Hartree Aarhus, April 6, 2013 12 / 12