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 ⊂ Rd HN =

N

• j=1
• − ∆xj + V (xj)
• + λ
• 1≤k<ℓ≤N

w(xk − xℓ), λ ≪ 1

• n HN := L2

s(ΩN) = {Ψ ∈ L2(ΩN) symmetric}

Mathieu LEWIN (CNRS / Cergy) Derivation of Hartree Aarhus, April 6, 2013 2 / 12

Assumptions

HN =

N

• j=1
• − ∆xj + V (xj)
• +

1 N − 1

• 1≤k<ℓ≤N

w(xk − xℓ) w = w1 + w2 and V = V1 + V2 + V+ wi, Vi ∈ Lpi (Ω) with max(1, d/2) < pi < ∞ or pi = ∞ but → 0 at ∞ V+ ≥ 0, V+ ∈ Ld/2

loc

Confined case: Ω is bounded (with chosen b.c.), or V+ → ∞ at ∞ Unconfined case: Ω = Rd and V+ ≡ 0 We can deal with much more complicated models, e.g. in Ω = Rd −∆ + V (x)

• m2 + |A(x) − i∇|2s − m2s + V (x)

Mathieu LEWIN (CNRS / Cergy) Derivation of Hartree Aarhus, April 6, 2013 3 / 12

Hartree theory

Restrict HN to uncorrelated functions Ψ = u(x1) · · · u(xN), with ´

Ω |u|2 = 1:

• u⊗N, HNu⊗N

N = ˆ

Ω

• |∇u(x)|2 + V (x)|u(x)|2

dx + 1 2 ˆ

Ω

ˆ

Ω

w(x − y)|u(x)|2|u(y)|2dx dy := EV

H (u)

Theorem (Validity of Hartree theory [LewNamRou-13])

Under the previous assumptions on V and w, we have: lim

N→∞

inf σ(HN) N = eH := inf

• EV

H (u) :

ˆ

Ω

|u|2 = 1

• 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 ΨN is usually not close to u⊗N in norm! Note that Ψ, HNΨ = trHN

• HN|ΨΨ|
• .

Definition (Density matrices)

The k-particle density matrix of Ψ is the operator γ(k)

Ψ

acting on Hk defined by the partial trace γ(k)

Ψ

= trk+1→N |ΨΨ|. Equivalently γ(k)

Ψ (X, X ′) =

ˆ

ΩN−k Ψ(X, Y )Ψ(X ′, Y ) dY

where X, X ′ ∈ Ωk. We have γ(k)

Ψ

≥ 0 and trHk γ(k)

Ψ = 1 for all k ≥ 1.

Hartree state: γ(k)

u⊗N = |u⊗ku⊗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 (ΨN) such that ΨN, HNΨN = N eH + o(N). In the confined case, there exists a subsequence Nj and a Borel probability measure µ on the unit sphere SH of H = L2(Ω), supported on the set M of minimizers for eH, such that lim

j→∞ γ(k) ΨNj =

ˆ

M

dµ(u) |u⊗ku⊗k| strongly in the trace-class for any fixed k. In particular, if eH admits a unique minimizer u0 (up to a phase factor), then there is complete Bose-Einstein condensation on u0: lim

N→∞ γ(k) ΨN = |u⊗k 0 u⊗k 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

Particles can escape to infinity. eV

H (λ) := inf

• EV

H (u) :

´

Ω |u|2 = λ

• Theorem (Bose-Einstein condensation [LewNamRou-13])

Consider any sequence (ΨN) such that ΨN, HNΨN = N eH + o(N). In the unconfined case, there exists a subsequence Nj and a Borel probability measure µ on the unit ball BH of H = L2(Rd), supported on the set M =

• u ∈ BH : EV

H (u) = eV H (|

|u| |2) = eV

H (1) − e0 H(1 − |

|u| |2)

• ,

such that γ(k)

ΨNj ⇀∗

ˆ

M

dµ(u) |u⊗ku⊗k| weakly-∗ in the trace-class for any fixed k. If furthermore eV

H (1) < eV H (λ) + e0 H(1 − λ),

∀0 ≤ λ < 1, then supp(µ) ⊂ M ⊂ SH and the limit for γ(k)

Nj 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 HN =

N

• j=1
• −∆xj −

1 t|xj|

• +

1 N − 1

• 1≤k<ℓ≤N

1 |xk − xℓ|, eH(t) := inf

´

R3 |u|2=1

ˆ

R3 |∇u|2 − |u|2

t|x| + 1 2 ˆ

R6

|u(x)|2|u(y)|2 |x − y| dx dy

• eH(t) admits a unique minimizer ut iff t ≤ tc ≃ 1.21

t ≤ tc ⇒ γ(k)

ΨN → |u⊗k t

u⊗k

t

| strongly (Benguria-Lieb ’83) t > tc ⇒ γ(k)

ΨN ⇀∗ |˜

u⊗k

t

˜ u⊗k

t

| weakly-∗, where ˜ ut = (tc/t)3/2utc(tc · /t)

Mathieu LEWIN (CNRS / Cergy) Derivation of Hartree Aarhus, April 6, 2013 8 / 12

Quantum de Finetti

ΨN, HNΨN N = 1 2 trH2 H2γ(2)

ΨN ⇒ what is the set of γ(2) ΨN’s in the limit N → ∞?

Mathieu LEWIN (CNRS / Cergy) Derivation of Hartree Aarhus, April 6, 2013 9 / 12

Quantum de Finetti

ΨN, HNΨN N = 1 2 trH2 H2γ(2)

ΨN ⇒ what is the set of γ(2) ΨN’s in the limit N → ∞?

Theorem (Quantum de Finetti [Størmer-69, Hudson-Moody-75])

Let H be any separable Hilbert space and Hk := k

s H. Consider a hierarchy

{γ(k)}∞

k=0 of non-negative self-adjoint operators, where each γ(k) acts on Hk and

satisfies trHk γ(k) = 1. If the hierarchy is consistent trk+1 γ(k+1) = γ(k), ∀k ≥ 0, then there exists a Borel probability measure µ on the sphere SH of H such that, for all k ≥ 1, γ(k) = ˆ

SH

|u⊗ku⊗k| dµ(u). 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)

ΨN ⇀∗ γ(k), ∀k ≥ 1

system is confined ⇒ strong CV ⇒ consistent hierarchy ◮ Step 2: de Finetti. γ(k) = ˆ

SH

|u⊗ku⊗k| dµ(u), ∀k ≥ 1, for some Borel probability measure µ ◮ Step 3: Conclusion. lim inf

N→∞

ΨN, HNΨN N = lim inf

N→∞

tr(H2γ(2)

ΨN)

2 ≥ 1 2 tr(H2γ(2)) = 1 2 ˆ

SH

• u⊗2, H2u⊗2

dµ(u) = ˆ

SH

EV

H (u) dµ(u) ≥ eH

Since λ1(HN) ≤ N eH, 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 ΨN ∈ HN of normalized states, such that (for a subsequence) γ(k)

ΨNj ⇀∗ γ(k),

∀k ≥ 0. Then there exists a Borel probability measure µ on the unit ball BH of H such that, for all k ≥ 1, γ(k) = ˆ

BH

|u⊗ku⊗k| dµ(u).

• Ex. ΨN = (uN)⊗N with uN ⇀ v =

⇒ µ = δv 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 eV

H (1) has a unique, non-degenerate minimizer u0, then

lim

N→∞ (λj(HN) − N eH) = λj(H)

where H is the second-quantization of Hess EV

H (u0)/2 on the bosonic Fock

space Γs(H+) built on H+ = {u0}⊥. Furthermore,

• Ψj

N − N

• n=0

u⊗(N−n) ⊗s ϕj

n

• HN

→ 0 where Φj = (ϕj

n)n≥0 ∈ Γs(H+) solves H Φj = λj(H) Φj, with n≥0

• ϕj

n

• 2

Hn = 1. without interactions with interactions

Mathieu LEWIN (CNRS / Cergy) Derivation of Hartree Aarhus, April 6, 2013 12 / 12