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

Transcription

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

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

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4 The big challenge: making physics rigorous

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

6 1925: predicting Bose-Einstein condensation (BEC)

7 1925: predicting Bose-Einstein condensation (BEC) 1995: Cornell-Wieman and Ketterle experiment Courtesy U Michigan

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9 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)

10 The mathematics of BEC Gross and Pitaevskii, 1961: a good model of BEC is the cubic nonlinear Schrödinger equation (NLS): i t ϕ = ϕ + µ ϕ 2 ϕ Fruitful NLS research: competition between two RHS terms

11 The mathematics of BEC Gross and Pitaevskii, 1961: a good model of BEC is the cubic nonlinear Schrödinger 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?

12 The mathematics of BEC Gross and Pitaevskii, 1961: a good model of BEC is the cubic nonlinear Schrödinger 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!

13 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

14 A quantum particle is really a wavefunction For each t, ψ(x, t) L 2 (R d ) solves a Schrödinger equation i t ψ = ψ + V ext (x)ψ

15 A quantum particle is really a wavefunction For each t, ψ(x, t) L 2 (R d ) solves a Schrödinger equation i t ψ = ψ + V ext (x)ψ =: Hψ = d i=1 x i x i 0 external trapping potential V ext solution ψ(x, t) = e iht ψ 0 (x)

16 A quantum particle is really a wavefunction For each t, ψ(x, t) L 2 (R d ) solves a Schrödinger equation i t ψ = ψ + V ext (x)ψ =: Hψ = 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?

17 Particle in a box V ext = 1 [0,1] C has ground state ψ(x) = 2 sin (πx)

18 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 Schrödinger equation: i t ψ N = N xj ψ N + j=1 N U(x i x j )ψ N =: H N ψ N i<j

19 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 Schrödinger equation: i t ψ N = N xj ψ N + j=1 N U(x i x j )ψ N =: H N ψ N i<j pair interaction potential U solution ψ N (x, t) = e ih Nt ψ 0 N (x) joint density ψ N (x 1,..., x N, t) 2

20 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.

21 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 ψn 0 (x) = ϕ 0 (x j ) L 2 s (R 3N ). j=1

22 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 ψn 0 (x) = ϕ 0 (x j ) L 2 s (R 3N ). j=1 But interactions create correlations for t > 0.

23 Mean-field pair interaction U = 1 N V Weak: order 1/N. Long distance: V L (R 3 ). i t ψ N = N xj ψ N + 1 N j=1 N V (x i x j )ψ N. i<j

24 Mean-field pair interaction U = 1 N V Weak: order 1/N. Long distance: V L (R 3 ). i t ψ N = N xj ψ N + 1 N j=1 N V (x i x j )ψ N. 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.

25 Convergence ψ N ϕ means in the sense of marginals: γ (1) N ϕ ϕ Tr N 0,

26 Convergence ψ N ϕ means in the sense of marginals: γ (1) N ϕ ϕ Tr N 0, one-particle marginal density γ (1) N where ϕ ϕ (x 1, x 1 ) = ϕ(x 1)ϕ(x 1 ) and := Tr N 1 ψ N ψ N has kernel γ (1) N (x 1; x 1, t) := ψ N (x 1, x N 1, t)ψ N (x 1, x N 1, t)dx N 1.

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

28 Other mean-field limit theorems Erdös 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 γ (1) N ϕ ϕ Tr CeKt N.

29 Other mean-field limit theorems Erdös 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 γ (1) N ϕ ϕ Tr CeKt N. Preview of localizing interactions: (V N ϕ 2 )ϕ (δ ϕ 2 )ϕ Erdös, Schlein, Yau, K., Staffilani, Chen, Pavlovic, Tzirakis...

30 Definition of BEC at zero temperature Almost all particles are in the same one-particle state: {ψ N L 2 s (R 3N )} 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) N = Tr N 1 ψ N ψ N N ϕ ϕ.

31 Definition of BEC at zero temperature Almost all particles are in the same one-particle state: {ψ N L 2 s (R 3N )} 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) N = Tr N 1 ψ N ψ N N ϕ ϕ. Generalizes factorized: ψ N (x) = N j=1 ϕ(x j) is BEC into ϕ.

32 BEC limit theorems with parameter β (0, 1] Now localized strong interactions: N dβ V (N β ( )) b 0 δ. H N = N xj + 1 N j=1 N N dβ V (N β (x i x j )). i<j

33 BEC limit theorems with parameter β (0, 1] Now localized strong interactions: N dβ V (N β ( )) b 0 δ. H N = N xj + 1 N j=1 N N dβ V (N β (x i x j )). i<j Theorems (Erdös-Schlein-Yau 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.

34 Our limit theorems make the physics of BEC rigorous H N = N xj + 1 N j=1 N N dβ V (N β (x i x j )) i<j N-body Schrod. micro : ψ 0 N ψ N init. BEC marg. MACRO : ϕ 0 ϕ NLS evolution i t ϕ = ϕ + b 0 ϕ 2 ϕ.

35 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.

36 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 Aϕ = ϕ(x)aϕ(x)dx. Position observable is X (ϕ)(x) := xϕ(x) with density ϕ 2.

37 Only some probability facts have quantum analogues Courtesy of Jordgette

38 The BEC limit theorems imply quantum LLNs If A is a one-particle observable and then for each ɛ > 0, A j = 1 1 A 1 1, { 1 lim sup P ψn N N N j=1 A j

39 The BEC limit theorems imply quantum LLNs If A is a one-particle observable and then for each ɛ > 0, A j = 1 1 A 1 1, { 1 lim sup P ψn N N } N A j ϕ Aϕ ɛ = 0. j=1

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41 BEC can explode as a bosenova

42 BEC can explode as a bosenova We need a control theory of BEC Central limit theorem for BEC (Ben Arous-K.-Schlein, 2013) Our quantum CLT has correlations coming from interactions CLT for quantum groups (Brannan-K., 2015)

43 Our CLT for interacting quantum many-body systems Theorem (Ben Arous, K., Schlein, 2013): Under suitable assumptions on the initial state ψ 0 N, ϕ 0, A, and V, then for t R

44 Our CLT for interacting quantum many-body systems Theorem (Ben Arous, K., Schlein, 2013): Under suitable assumptions on the initial state ψ 0 N, ϕ 0, A, and V, then for t R A t := 1 N N j=1 (A j E ϕt A) distrib. as N N (0, σ 2 t ).

45 Our CLT for interacting quantum many-body systems Theorem (Ben Arous, K., Schlein, 2013): Under suitable assumptions on the initial state ψ 0 N, ϕ 0, A, and V, then for t R A t := 1 N N j=1 (A j E ϕt A) distrib. as N N (0, σ 2 t ). The variance that we would guess is correct at t = 0 only: σ 2 t has ϕ 0 ϕ t... σ 2 0 = E ϕ0 [A 2 ] (E ϕ0 A) 2

46 Our CLT for interacting quantum many-body systems Theorem (Ben Arous, K., Schlein, 2013): Under suitable assumptions on the initial state ψ 0 N, ϕ 0, A, and V, then for t R A t := 1 N N j=1 (A j E ϕt A) distrib. as N N (0, σ 2 t ). The variance that we would guess is correct at t = 0 only: σ 2 0 = E ϕ0 [A 2 ] (E ϕ0 A) 2 σ 2 t has ϕ 0 ϕ t... and twisted by the Bogoliubov transform.

47 We studied freely independent RVs via quantum groups (instead of random matrices) with Michael Brannan (Texas A&M)

48 Theorem (Brannan, K. 2015): Deformed quantum groups have an action ASK MIKE FOR HIS ACTION FIGURE TEX CODE on Free Araki-Woods factors

49 Theorem (Brannan, K. 2015): Deformed quantum groups have an action ASK MIKE FOR HIS ACTION FIGURE TEX CODE on Free Araki-Woods factors Γ = Γ(R n, U t ) := {l(ξ) + l(ξ) : ξ H R } with free quasi-free state ϕ Ω, α(c i ) = u ij c j, U t = A it, some A > 0. Usually a full type III λ factor for λ [0, 1]. Best case: λ = 1!

50 Theorem (Brannan, K. 2015): Deformed quantum groups have an action ASK MIKE FOR HIS ACTION FIGURE TEX CODE on Free Araki-Woods factors Γ = Γ(R n, U t ) := {l(ξ) + l(ξ) : ξ H R } with free quasi-free state ϕ Ω, α(c i ) = u ij c j, U t = A it, some A > 0. Usually a full type III λ factor for λ [0, 1]. Best case: λ = 1! (exciting new development from MSRI...)

51 Theorem (Brannan, K. 2015): For all almost-periodic representations U t on H R, there is a sequence of quantum groups { } O + F (n) that has Haar distributional limit n 1 (Γ, ϕ Ω ).

52 Theorem (Brannan, K. 2015): For all almost-periodic representations U t on H R, there is a sequence of quantum groups { } O + F (n) that has Haar distributional limit (Γ, ϕ Ω ). n 1 Exciting idea: Our new Weingarten-type calculus

53 Theorem (Brannan, K. 2015): For all almost-periodic representations U t on H R, there is a sequence of quantum groups { } O + F (n) that has Haar distributional limit (Γ, ϕ Ω ). n 1 Exciting idea: Our new Weingarten-type calculus is similar to Martin Hairer s new Feynman-type calculus

54 Theorem (Brannan, K. 2015): For all almost-periodic representations U t on H R, there is a sequence of quantum groups { } O + F (n) that has Haar distributional limit (Γ, ϕ Ω ). n 1 Exciting idea: Our new Weingarten-type calculus is similar to Martin Hairer s new Feynman-type calculus MH: quantum to classical; B-K: classical to quantum

55 How do physics, the world, and the universe work? Physics Analysis

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57 Thanks NSF DMS , OISE , CAREER DMS arxiv: (AJM), (CPAM), (CMP), (PJM)

58 Why do interactions become the cubic nonlinearity? i t ψ N = xj ψ N + 1 N V (xi x j )ψ N Particle 1 sees 1 N N V (x 1 x j ) 1 N j=2 N j=2 = N 1 N N (V ϕ 2 )(x 1 ) V (x 1 y) ϕ(y) 2 dy V (x 1 y) ϕ(y) 2 dy