The excitation spectrum of the Bose gas in the Gross-Pitaevskii - - PowerPoint PPT Presentation

SLIDE 1

The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime

Serena Cenatiempo - Gran Sasso Science Institute, L’Aquila joint work with Chiara Boccato, Christian Brennecke and Benjamin Schlein Quantissima in the Serenissima III Venice - August 22, 2019

SLIDE 2

Introduction and results Strategy of the proof The Gross-Pitaevskii regime Results

The Gross-Pitaevskii regime

Consider N bosons in a cubic box Λ described by HN = −

N

• i=1

∆xi +

N

• i<j

N2V

• N(xi − xj)
• ,

|Λ| = 1 ◮ If V (x) has scattering length a, then N2V (Nx) has scattering length a/N ◮ States with small energy are characterized by a correlation structure on length scales of a ∼ N−1 − → understand role of correlations

The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime

• S. Cenatiempo

Quantissima III, Venice 2/14

SLIDE 3

Introduction and results Strategy of the proof The Gross-Pitaevskii regime Results

The Gross-Pitaevskii regime

Consider N bosons in a cubic box Λ described by HN = −

N

• i=1

∆xi +

N

• i<j

N2V

• N(xi − xj)
• ,

|Λ| = 1 ◮ If V (x) has scattering length a, then N2V (Nx) has scattering length a/N ◮ States with small energy are characterized by a correlation structure on length scales of a ∼ N−1 − → understand role of correlations Relevance: ◮ effective description for the strong and short range interactions among atoms in BEC experiments ◮ scaling regime leading to a rigorouns derivation of the Gross-Pitaevskii equation i∂tϕ(t) = −∆ϕ(t) + 8πa |ϕ(t)|2ϕ(t) ◮ HN equivalent to the Hamiltonian for N bosons in a box with L = N interacting through a fixed potential V , i.e. ρ = N/L3 = N−2

The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime

• S. Cenatiempo

Quantissima III, Venice 2/14

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Introduction and results Strategy of the proof The Gross-Pitaevskii regime Results

Condensation in the Gross-Pitaevskii regime

N bosons in Λ = [0; 1]×3, periodic boundary conditions HN =

• p∈Λ∗

p2a∗

pap + 1

2N

• p,q,r∈Λ∗
• V (r/N) a∗

p+ra∗ q−rapaq ,

Λ∗ = 2πZ3 [Lieb-Seiringer-Yngvason ‘00] The ground state energy of HN is given by EN = 4πa N + o(N) [Lieb-Seiringer ‘02, ’06; Nam-Rougerie-Seiringer, ’16] Any ΨN ∈ L2

s(ΛN) with

• ΨN, HNΨN
• ≤ 4πaN + o(N) exhibits Bose-Einstein condensation, i.e.

γ(1)

N

− − − − →

N→∞ |ϕ0ϕ0|

where ϕ0(x) = 1 for all x ∈ Λ.

The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime

• S. Cenatiempo

Quantissima III, Venice 3/14

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Introduction and results Strategy of the proof The Gross-Pitaevskii regime Results

Condensation in the Gross-Pitaevskii regime

N bosons in Λ = [0; 1]×3, periodic boundary conditions HN =

• p∈Λ∗

p2a∗

pap + 1

2N

• p,q,r∈Λ∗
• V (r/N) a∗

p+ra∗ q−rapaq ,

Λ∗ = 2πZ3 [Lieb-Seiringer-Yngvason ‘00] The ground state energy of HN is given by EN = 4πa N + o(N) Note that

• ϕ⊗N

HNϕ⊗N

• = (N−1)

V (0) 2

≫ 4πaN [Lieb-Seiringer ‘02, ’06; Nam-Rougerie-Seiringer, ’16] Any ΨN ∈ L2

s(ΛN) with

• ΨN, HNΨN
• ≤ 4πaN + o(N) exhibits Bose-Einstein condensation, i.e.

γ(1)

N

− − − − →

N→∞ |ϕ0ϕ0|

where ϕ0(x) = 1 for all x ∈ Λ. 8πa =

• dx f (x)V (x)

The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime

• S. Cenatiempo

Quantissima III, Venice 3/14

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Introduction and results Strategy of the proof The Gross-Pitaevskii regime Results

Bogoliubov theory in the Gross-Pitaevskii regime

[Boccato-Brennecke-C.-Schlein ‘19] The ground state energy of HN is

EN = 4πa(N − 1) + eΛa2 − 1 2

• p∈Λ∗

+

• p2 + 8πa −
• |p|4 + 16πap2 − (8πa)2

2p2

• + O(N− 1

4 )

where Λ∗

+ = 2πZ3 \ {0} and

eΛ = 2 − lim

M→∞

• p∈Z3\{0}:

|p1|,|p2|,|p3|≤M

cos(|p|) p2

The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime

• S. Cenatiempo

Quantissima III, Venice 4/14

SLIDE 7

Introduction and results Strategy of the proof The Gross-Pitaevskii regime Results

Bogoliubov theory in the Gross-Pitaevskii regime

[Boccato-Brennecke-C.-Schlein ‘19] The ground state energy of HN is

EN = 4πa(N − 1) + eΛa2 − 1 2

• p∈Λ∗

+

• p2 + 8πa −
• |p|4 + 16πap2 − (8πa)2

2p2

• + O(N− 1

4 )

where Λ∗

+ = 2πZ3 \ {0} and

eΛ = 2 − lim

M→∞

• p∈Z3\{0}:

|p1|,|p2|,|p3|≤M

cos(|p|) p2 Remark (1) For small potentials κV : 4πa(N − 1) + eΛa2 = 4πaN(N − 1) with 8πaN = κ V (0) − 1 2N

• p1∈Λ∗

+

κ2 V 2(p1/N) 2p2

1

+ . . .

The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime

• S. Cenatiempo

Quantissima III, Venice 4/14

SLIDE 8

Introduction and results Strategy of the proof The Gross-Pitaevskii regime Results

Bogoliubov theory in the Gross-Pitaevskii regime

[Boccato-Brennecke-C.-Schlein ‘19] The ground state energy of HN is

EN = 4πa(N − 1) + eΛa2 − 1 2

• p∈Λ∗

+

• p2 + 8πa −
• |p|4 + 16πap2 − (8πa)2

2p2

• + O(N− 1

4 )

where Λ∗

+ = 2πZ3 \ {0} and

eΛ = 2 − lim

M→∞

• p∈Z3\{0}:

|p1|,|p2|,|p3|≤M

cos(|p|) p2 Remark (2) Replace V by VR(x) = R−2V (x/R) with scattering length aR = aR: letting R → ∞ the finite size effect becomes subleading w.r.t. Bogoliubov sum. The result for EN is the analog in the GP regime of the Lee-Huang-Yang formula, valid in the thermodynamic limit: [..., Yau-Yin ’13, ... , Fournais-Solovej ’19]

The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime

• S. Cenatiempo

Quantissima III, Venice 4/14

SLIDE 9

Introduction and results Strategy of the proof The Gross-Pitaevskii regime Results

Bogoliubov theory in the Gross-Pitaevskii regime

[Boccato-Brennecke-C.-Schlein ‘19] The ground state energy of HN is

EN = 4πa(N − 1) + eΛa2 − 1 2

• p∈Λ∗

+

• p2 + 8πa −
• |p|4 + 16πap2 − (8πa)2

2p2

• + O(N− 1

4 )

where Λ∗

+ = 2πZ3 \ {0} and

eΛ = 2 − lim

M→∞

• p∈Z3\{0}:

|p1|,|p2|,|p3|≤M

cos(|p|) p2 The spectrum of HN − EN below an energy ζ consists of eigenvalues

• p∈Λ∗

+

np

• |p|4 + 16πa|p|2 + O(N−1/4(1 + ζ3)) ,

np ∈ N

The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime

• S. Cenatiempo

Quantissima III, Venice 4/14

SLIDE 10

Introduction and results Strategy of the proof The Gross-Pitaevskii regime Results

Bogoliubov theory in the Gross-Pitaevskii regime

[Boccato-Brennecke-C.-Schlein ‘19] The ground state energy of HN is

EN = 4πa(N − 1) + eΛa2 − 1 2

• p∈Λ∗

+

• p2 + 8πa −
• |p|4 + 16πap2 − (8πa)2

2p2

• + O(N− 1

4 )

where Λ∗

+ = 2πZ3 \ {0} and

eΛ = 2 − lim

M→∞

• p∈Z3\{0}:

|p1|,|p2|,|p3|≤M

cos(|p|) p2 The spectrum of HN − EN below an energy ζ consists of eigenvalues

• p∈Λ∗

+

np

• |p|4 + 16πa|p|2 + O(N−1/4(1 + ζ3)) ,

np ∈ N Remark (3) Linear dispersion relation of low energy excitations of the Bose gas Previous results in the mean field scaling [Seiringer ’11, Grech-Seiringer ’13, Lewin-Nam-Serfaty-Solovej ’14, Derezinski-Napiorkovski ’14, Pizzo ’16] and for singular interactions [Boccato-Brennecke-C. -Schlein ’17]

The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime

• S. Cenatiempo

Quantissima III, Venice 4/14

SLIDE 11

Introduction and results Strategy of the proof The Gross-Pitaevskii regime Results

Bogoliubov theory in the Gross-Pitaevskii regime

[Boccato-Brennecke-C.-Schlein ‘19] The ground state energy of HN is

EN = 4πa(N − 1) + eΛa2 − 1 2

• p∈Λ∗

+

• p2 + 8πa −
• |p|4 + 16πap2 − (8πa)2

2p2

• + O(N− 1

4 )

where Λ∗

+ = 2πZ3 \ {0} and

eΛ = 2 − lim

M→∞

• p∈Z3\{0}:

|p1|,|p2|,|p3|≤M

cos(|p|) p2 The spectrum of HN − EN below an energy ζ consists of eigenvalues

• p∈Λ∗

+

np

• |p|4 + 16πa|p|2 + O(N−1/4(1 + ζ3)) ,

np ∈ N Remark (4) Condensate depletion bound: for any ψN ∈ L2

s(ΛN)

s.t

• ψN, HNψN
• ≤ 4πaN + ζ we have

N(1 −

• ϕ0, γ(1)

N ϕ0

• ) ≤ C(ζ + 1)

i.e. condensation holds with optimal rate.

The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime

• S. Cenatiempo

Quantissima III, Venice 4/14

SLIDE 12

Introduction and results Strategy of the proof Excitation Hamiltonian Correlation structure

Removing particles in the Bose-Einstein condensate

For ψN ∈ L2

s(ΛN) and ϕ0 ∈ L2(Λ)

[Lewin-Nam-Serfaty-Solovej ‘12] ψN = α0 ϕ⊗N + α1 ⊗s ϕ⊗N−1 + . . . + αj ⊗s ϕ⊗N−j + . . . + αN , where αj ∈ L2(Λ)⊗sj and αj ⊥ ϕ0 ; ϕ0(x) = 1 for all x ∈ Λ.

The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime

• S. Cenatiempo

Quantissima III, Venice 5/14

SLIDE 13

Introduction and results Strategy of the proof Excitation Hamiltonian Correlation structure

Removing particles in the Bose-Einstein condensate

For ψN ∈ L2

s(ΛN) and ϕ0 ∈ L2(Λ)

[Lewin-Nam-Serfaty-Solovej ‘12] ψN = α0 ϕ⊗N + α1 ⊗s ϕ⊗N−1 + . . . + αj ⊗s ϕ⊗N−j + . . . + αN , where αj ∈ L2(Λ)⊗sj and αj ⊥ ϕ0 ; ϕ0(x) = 1 for all x ∈ Λ. Unitary map: UN(ϕ0) : L2

s(ΛN) −

→ F ≤N

+

=

N

• n=0

L2

⊥ϕ0(Λ)⊗sn

ψN − → UN(ϕ0)ψN = {α0, α1, . . . , αN, 0, 0, . . .}

The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime

• S. Cenatiempo

Quantissima III, Venice 5/14

SLIDE 14

Introduction and results Strategy of the proof Excitation Hamiltonian Correlation structure

Removing particles in the Bose-Einstein condensate

For ψN ∈ L2

s(ΛN) and ϕ0 ∈ L2(Λ)

[Lewin-Nam-Serfaty-Solovej ‘12] ψN = α0 ϕ⊗N + α1 ⊗s ϕ⊗N−1 + . . . + αj ⊗s ϕ⊗N−j + . . . + αN , where αj ∈ L2(Λ)⊗sj and αj ⊥ ϕ0 ; ϕ0(x) = 1 for all x ∈ Λ. Unitary map: UN(ϕ0) : L2

s(ΛN) −

→ F ≤N

+

=

N

• n=0

L2

⊥ϕ0(Λ)⊗sn

ψN − → UN(ϕ0)ψN = {α0, α1, . . . , αN, 0, 0, . . .} Conjugation with UN reminds of Bogoliubov approximation UN a∗

0a0 U∗ N = N − N+

UN a∗

0ap U∗ N =

• N − N+ ap

UN a∗

pa0 U∗ N = a∗ p

• N − N+

UN a∗

paq U∗ N = a∗ paq

N+ =

• p∈Λ∗\{0}

a∗

pap

The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime

• S. Cenatiempo

Quantissima III, Venice 5/14

SLIDE 15

Introduction and results Strategy of the proof Excitation Hamiltonian Correlation structure

Removing particles in the Bose-Einstein condensate

HN =

• p∈Λ∗

p2a∗

pap + 1

2N

• p,q,r∈Λ∗
• V (r/N)a∗

p+ra∗ q−rapaq ,

Λ∗ = 2πZ3 Excitation Hamiltonian: LN = UNHNU∗

N : F ≤N +

→ F ≤N

+

LN = N − 1 2N

• V (0)(N − N+) +
• V (0)

2N N+(N − N+) +

• p∈Λ∗

+

p2a∗

pap +

• p∈Λ∗

+

• V (p/N) a∗

p

• N−1−N+

N

• ap

+1 2

• p∈Λ∗

+

• V (p/N)
• a∗

p (N−N+)(N−1−N+) N2

a∗

−p + h.c.

• + 1

√ N

• p,q∈Λ∗

+:p+q=0

• V (p/N)
• a∗

p+qa∗ −paq

• N−N+

N

+ h.c.

• + 1

2N

• p,q∈Λ∗

+,r∈Λ∗:r=−p,−q

• V (r/N) a∗

p+ra∗ qapaq+r

The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime

• S. Cenatiempo

Quantissima III, Venice 6/14

SLIDE 16

Introduction and results Strategy of the proof Excitation Hamiltonian Correlation structure

A simplified sketch of the strategy: V (p/N) → κ V (p) with κ sufficiently small

Lmf

N = N − 1

2N κ V (0)(N − N+) + κ V (0) 2N N+(N − N+) +

K

• p∈Λ∗

+

p2a∗

pap

+

• p∈Λ∗

+

κ V (p) a∗

p

N−1−N+

N

• ap+ 1

2

• p∈Λ∗

+

κ V (p)

• a∗

p (N−N+)(N−1−N+) N2

a∗

−p + h.c.

• + 1

√ N

• p,q∈Λ∗

+:p+q=0

κ V (p)

• a∗

p+qa∗ −paq

• N−N+

N

+ h.c.

• + 1

2N

• p,q∈Λ∗

+,r∈Λ∗:r=−p,−q

κ V (r) a∗

p+ra∗ qapaq+r

The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime

• S. Cenatiempo

Quantissima III, Venice 7/14

SLIDE 17

Introduction and results Strategy of the proof Excitation Hamiltonian Correlation structure

A simplified sketch of the strategy: V (p/N) → κ V (p) with κ sufficiently small

Lmf

N = N − 1

2N κ V (0)(N − N+) + κ V (0) 2N N+(N − N+) +

K

• p∈Λ∗

+

p2a∗

pap

+

• p∈Λ∗

+

κ V (p) a∗

p

N−1−N+

N

• ap+ 1

2

• p∈Λ∗

+

κ V (p)

• a∗

p (N−N+)(N−1−N+) N2

a∗

−p + h.c.

• + 1

√ N

• p,q∈Λ∗

+:p+q=0

κ V (p)

• a∗

p+qa∗ −paq

• N−N+

N

+ h.c.

• + 1

2N

• p,q∈Λ∗

+,r∈Λ∗:r=−p,−q

κ V (r) a∗

p+ra∗ qapaq+r

◮ Upper bound: E mf

N ≤

• Ω, Lmf

N Ω

• = N

2 κ

V (0) + C

The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime

• S. Cenatiempo

Quantissima III, Venice 7/14

SLIDE 18

Introduction and results Strategy of the proof Excitation Hamiltonian Correlation structure

A simplified sketch of the strategy: V (p/N) → κ V (p) with κ sufficiently small

Lmf

N = N − 1

2N κ V (0)(N − N+) + κ V (0) 2N N+(N − N+) +

K

• p∈Λ∗

+

p2a∗

pap

+

• p∈Λ∗

+

κ V (p) a∗

p

N−1−N+

N

• ap+ 1

2

• p∈Λ∗

+

κ V (p)

• a∗

p (N−N+)(N−1−N+) N2

a∗

−p + h.c.

• + 1

√ N

• p,q∈Λ∗

+:p+q=0

κ V (p)

• a∗

p+qa∗ −paq

• N−N+

N

+ h.c.

• + 1

2N

• p,q∈Λ∗

+,r∈Λ∗:r=−p,−q

κ V (r) a∗

p+ra∗ qapaq+r ≥ N

2 κ V (0) + K + VN − Cκ(N+ + 1) ◮ Upper bound: E mf

N ≤

• Ω, Lmf

N Ω

• = N

2 κ

V (0) + C

The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime

• S. Cenatiempo

Quantissima III, Venice 7/14

SLIDE 19

Introduction and results Strategy of the proof Excitation Hamiltonian Correlation structure

A simplified sketch of the strategy: V (p/N) → κ V (p) with κ sufficiently small

Lmf

N = N − 1

2N κ V (0)(N − N+) + κ V (0) 2N N+(N − N+) +

K

• p∈Λ∗

+

p2a∗

pap

+

• p∈Λ∗

+

κ V (p) a∗

p

N−1−N+

N

• ap+ 1

2

• p∈Λ∗

+

κ V (p)

• a∗

p (N−N+)(N−1−N+) N2

a∗

−p + h.c.

• + 1

√ N

• p,q∈Λ∗

+:p+q=0

κ V (p)

• a∗

p+qa∗ −paq

• N−N+

N

+ h.c.

• + 1

2N

• p,q∈Λ∗

+,r∈Λ∗:r=−p,−q

κ V (r) a∗

p+ra∗ qapaq+r ≥ N

2 κ V (0) + K + VN − Cκ(N+ + 1) ◮ Upper bound: E mf

N ≤

• Ω, Lmf

N Ω

• = N

2 κ

V (0) + C ◮ Lower bound: Lmf

N ≥ N 2 κ

V (0) +

• (2π)2 − Cκ
• N+ − C ≥ N

2 κ

V (0) − C

The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime

• S. Cenatiempo

Quantissima III, Venice 7/14

SLIDE 20

Introduction and results Strategy of the proof Excitation Hamiltonian Correlation structure

A simplified sketch of the strategy: V (p/N) → κ V (p) with κ sufficiently small

Lmf

N = N − 1

2N κ V (0)(N − N+) + κ V (0) 2N N+(N − N+) +

K

• p∈Λ∗

+

p2a∗

pap

+

• p∈Λ∗

+

κ V (p) a∗

p

N−1−N+

N

• ap+ 1

2

• p∈Λ∗

+

κ V (p)

• a∗

p (N−N+)(N−1−N+) N2

a∗

−p + h.c.

• + 1

√ N

• p,q∈Λ∗

+:p+q=0

κ V (p)

• a∗

p+qa∗ −paq

• N−N+

N

+ h.c.

• + 1

2N

• p,q∈Λ∗

+,r∈Λ∗:r=−p,−q

κ V (r) a∗

p+ra∗ qapaq+r ≥ N

2 κ V (0) + K + VN − Cκ(N+ + 1) ◮ Upper bound: E mf

N ≤

• Ω, Lmf

N Ω

• = N

2 κ

V (0) + C ◮ Lower bound: Lmf

N ≥ N 2 κ

V (0) +

• (2π)2 − Cκ
• N+ − C ≥ N

2 κ

V (0) − C |E mf

N

− N

2 κ

V (0)| ≤ C

The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime

• S. Cenatiempo

Quantissima III, Venice 7/14

SLIDE 21

Introduction and results Strategy of the proof Excitation Hamiltonian Correlation structure

A simplified sketch of the strategy: V (p/N) → κ V (p) with κ sufficiently small

Lmf

N = N − 1

2N κ V (0)(N − N+) + κ V (0) 2N N+(N − N+) +

K

• p∈Λ∗

+

p2a∗

pap

+

• p∈Λ∗

+

κ V (p) a∗

p

N−1−N+

N

• ap+ 1

2

• p∈Λ∗

+

κ V (p)

• a∗

p (N−N+)(N−1−N+) N2

a∗

−p + h.c.

• + 1

√ N

• p,q∈Λ∗

+:p+q=0

κ V (p)

• a∗

p+qa∗ −paq

• N−N+

N

+ h.c.

• + 1

2N

• p,q∈Λ∗

+,r∈Λ∗:r=−p,−q

κ V (r) a∗

p+ra∗ qapaq+r ≥ N

2 κ V (0) + K + VN − Cκ(N+ + 1) ◮ Upper bound: E mf

N ≤

• Ω, Lmf

N Ω

• = N

2 κ

V (0) + C ◮ Lower bound: Lmf

N ≥ N 2 κ

V (0) +

• (2π)2 − Cκ
• N+ − C ≥ N

2 κ

V (0) − C ◮ Number of excitations in low energy states:

• U∗ψN, Lmf

N U∗ψN

• =
• ψN, Hmf

N ψN

• ≤ N

2 κ

V (0) + ζ |E mf

N

− N

2 κ

V (0)| ≤ C

The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime

• S. Cenatiempo

Quantissima III, Venice 7/14

SLIDE 22

Introduction and results Strategy of the proof Excitation Hamiltonian Correlation structure

A simplified sketch of the strategy: V (p/N) → κ V (p) with κ sufficiently small

Lmf

N = N − 1

2N κ V (0)(N − N+) + κ V (0) 2N N+(N − N+) +

K

• p∈Λ∗

+

p2a∗

pap

+

• p∈Λ∗

+

κ V (p) a∗

p

N−1−N+

N

• ap+ 1

2

• p∈Λ∗

+

κ V (p)

• a∗

p (N−N+)(N−1−N+) N2

a∗

−p + h.c.

• + 1

√ N

• p,q∈Λ∗

+:p+q=0

κ V (p)

• a∗

p+qa∗ −paq

• N−N+

N

+ h.c.

• + 1

2N

• p,q∈Λ∗

+,r∈Λ∗:r=−p,−q

κ V (r) a∗

p+ra∗ qapaq+r ≥ N

2 κ V (0) + K + VN − Cκ(N+ + 1) ◮ Upper bound: E mf

N ≤

• Ω, Lmf

N Ω

• = N

2 κ

V (0) + C ◮ Lower bound: Lmf

N ≥ N 2 κ

V (0) +

• (2π)2 − Cκ
• N+ − C ≥ N

2 κ

V (0) − C ◮ Number of excitations in low energy states:

N 2 κ

V (0) + cN+ ≤

• U∗ψN, Lmf

N U∗ψN

• =
• ψN, Hmf

N ψN

• ≤ N

2 κ

V (0) + ζ |E mf

N

− N

2 κ

V (0)| ≤ C

The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime

• S. Cenatiempo

Quantissima III, Venice 7/14

SLIDE 23

Introduction and results Strategy of the proof Excitation Hamiltonian Correlation structure

A simplified sketch of the strategy: V (p/N) → κ V (p) with κ sufficiently small

Lmf

N = N − 1

2N κ V (0)(N − N+) + κ V (0) 2N N+(N − N+) +

K

• p∈Λ∗

+

p2a∗

pap

+

• p∈Λ∗

+

κ V (p) a∗

p

N−1−N+

N

• ap+ 1

2

• p∈Λ∗

+

κ V (p)

• a∗

p (N−N+)(N−1−N+) N2

a∗

−p + h.c.

• + 1

√ N

• p,q∈Λ∗

+:p+q=0

κ V (p)

• a∗

p+qa∗ −paq

• N−N+

N

+ h.c.

• + 1

2N

• p,q∈Λ∗

+,r∈Λ∗:r=−p,−q

κ V (r) a∗

p+ra∗ qapaq+r ≥ N

2 κ V (0) + K + VN − Cκ(N+ + 1) ◮ Upper bound: E mf

N ≤

• Ω, Lmf

N Ω

• = N

2 κ

V (0) + C ◮ Lower bound: Lmf

N ≥ N 2 κ

V (0) +

• (2π)2 − Cκ
• N+ − C ≥ N

2 κ

V (0) − C ◮ Number of excitations in low energy states:

N 2 κ

V (0) + cN+ ≤

• U∗ψN, Lmf

N U∗ψN

• =
• ψN, Hmf

N ψN

• ≤ N

2 κ

V (0) + ζ ⇒ N+ ≤ C(1 + ζ)

The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime

• S. Cenatiempo

Quantissima III, Venice 7/14

SLIDE 24

Introduction and results Strategy of the proof Excitation Hamiltonian Correlation structure

A simplified sketch of the strategy: V (p/N) → κ V (p) with κ sufficiently small

Lmf

N = N − 1

2N κ V (0)(N − N+) + κ V (0) 2N N+(N − N+) +

K

• p∈Λ∗

+

p2a∗

pap

+

• p∈Λ∗

+

κ V (p) a∗

p

N−1−N+

N

• ap+ 1

2

• p∈Λ∗

+

κ V (p)

• a∗

p (N−N+)(N−1−N+) N2

a∗

−p + h.c.

• + 1

√ N

• p,q∈Λ∗

+:p+q=0

κ V (p)

• a∗

p+qa∗ −paq

• N−N+

N

+ h.c.

• + 1

2N

• p,q∈Λ∗

+,r∈Λ∗:r=−p,−q

κ V (r) a∗

p+ra∗ qapaq+r ≥ N

2 κ V (0) + K + VN − Cκ(N+ + 1) ◮ Upper bound: E mf

N ≤

• Ω, Lmf

N Ω

• = N

2 κ

V (0) + C ◮ Lower bound: Lmf

N ≥ N 2 κ

V (0) +

• (2π)2 − Cκ
• N+ − C ≥ N

2 κ

V (0) − C ◮ Number of excitations in low energy states:

N 2 κ

V (0) + cN+ ≤

• U∗ψN, Lmf

N U∗ψN

• =
• ψN, Hmf

N ψN

• ≤ N

2 κ

V (0) + ζ ⇒ N+ ≤ C(1 + ζ) ◮ Under the stronger assumption ψN = χ(HN ≤ N

2 κ

V (0) + ζ)ψN we have: ξN, (K + VN + 1)(N+ + 1)kξN ≤ C(1 + ζ)k

The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime

• S. Cenatiempo

Quantissima III, Venice 7/14

SLIDE 25

Introduction and results Strategy of the proof Excitation Hamiltonian Correlation structure

Include correlations between condensate and excitation pairs

States with small energy in the Gross-Pitaevskii limit are characterized by a correlation structure on length scales of a ∼ N−1 which we model by the solution of the Neumann problem

• − ∆ + 1

2N2V (Nx)

• fN(x) = λNfN(x)
• n the ball |x| ≤ ℓ = 1/2, with

fN(x) = 1 and ∂|x|fN(x) = 0 for |x| = ℓ One has

• N3V (Nx)fN(x)dx
• V (·/N)⋆

fN

• − 8πa
• ≤ C

N

The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime

• S. Cenatiempo

Quantissima III, Venice 8/14

SLIDE 26

Introduction and results Strategy of the proof Excitation Hamiltonian Correlation structure

Include correlations between condensate and excitation pairs

We include correlations in F ≤N

+

defining, T(η) = exp 1 2

• |p|>µ

ηp

• b∗

pb∗ −p − bpb−p

• : F ≤N

+

→ F ≤N

+

S(η) = exp 1 √ N

• |r|>µ,|v|<ν

ηr

• b∗

r+va∗ −rav − h.c.

• : F≤N

+

→ F ≤N

+

with ηp = − 1

N2

• (1 − fN)(p/N)

and b∗

p = a∗ p

• N − N+

N , bp =

• N − N+

N ap : F ≤N

+

− → F ≤N

+

U∗

Nb∗ pUN = a∗ p

a0 √ N , U∗

NbpUN = a∗

√ N ap : L2(ΛN) − → L2(ΛN) The operators b∗

p and bp create and annihilate excitations, but do not change

the total number of particles.

The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime

• S. Cenatiempo

Quantissima III, Venice 9/14

SLIDE 27

Introduction and results Strategy of the proof Excitation Hamiltonian Correlation structure

Include correlations between condensate and excitation pairs

We include correlations in F ≤N

+

defining, T(η) = exp 1 2

• |p|>µ

ηp

• b∗

pb∗ −p − bpb−p

• : F ≤N

+

→ F ≤N

+

S(η) = exp 1 √ N

• |r|>µ,|v|<ν

ηr

• b∗

r+va∗ −rav − h.c.

• : F≤N

+

→ F ≤N

+

with ηp = − 1

N2

• (1 − fN)(p/N)

and b∗

p = a∗ p

• N − N+

N , bp =

• N − N+

N ap : F ≤N

+

− → F ≤N

+

U∗

Nb∗ pUN = a∗ p

a0 √ N , U∗

NbpUN = a∗

√ N ap : L2(ΛN) − → L2(ΛN) The operators b∗

p and bp create and annihilate excitations, but do not change

the total number of particles. |ηp| ≤ C e−|p|/N p2 η2 ≤ C, ηH1 ≤ C √ N

The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime

• S. Cenatiempo

Quantissima III, Venice 9/14

SLIDE 28

Introduction and results Strategy of the proof Excitation Hamiltonian Correlation structure

Include correlations between condensate and excitation pairs

With T(η) = exp 1

2

• |p|>µ ηp(b∗

pb∗ −p − bpb−p)

• define

GN = T ∗(η)UN HN U∗

NT(η) : F ≤N +

→ F ≤N

+

Then GN = 4πaN +

K

• p∈Λ∗

+

p2a∗

pap +

• |p|≤µ
• V (0)a∗

pap + 4πa

• |p|≤µ
• bpb−p + b∗

pb∗ −p

• +

1 √ N

• p,q∈Λ∗

+, p=q

• V (p/N)
• b∗

p+qa∗ −paq + h.c.

• + 1

2N

• p,q∈Λ∗

+,r∈Λ∗:

r=−p,−q

• V (r/N)a∗

p+ra∗ qapaq+r

• VN

+ EN with ±EN ≤

C µα (K + VN) + Cµβ , for some α, β > 0.

Key fact: T(η) renormalizes the constant and the non-diagonal quadratic term.

The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime

• S. Cenatiempo

Quantissima III, Venice 10/14

SLIDE 29

Introduction and results Strategy of the proof Excitation Hamiltonian Correlation structure

Add correlations due to triplets

With S(η) = exp

• 1

√ N

• |r|>µ, |v|≤ν ηr(b∗

r+va∗ −rav − h.c. )

• define

RN = S∗(η)T ∗(η)UN HN U∗

NT(η)S(η) : F ≤N +

→ F ≤N

+

Then RN = 4πaN +

• p∈Λ∗

+

p2a∗

pap + 8πa

• |p|≤µ

a∗

pap + 4πa

• |p|≤µ
• bpb−p + b∗

pb∗ −p

• + 8πa

√ N

• |p|<µ, q∈Λ∗

+

• b∗

p+qa∗ −paq + h.c.

• + 1

2N

• p,q∈Λ∗

+,r∈Λ∗:

r=−p,−q

• V (r/N)a∗

p+ra∗ qapaq+r + ˜

EN with ± ˜ EN ≤

C µα (K + VN) + Cµβ , for some α, β > 0.

RN is almost excitation Hamiltonian for mean field potential 8πa χ(|p| ≤ µ)

The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime

• S. Cenatiempo

Quantissima III, Venice 11/14

SLIDE 30

Introduction and results Strategy of the proof Excitation Hamiltonian Correlation structure

Bounds on excitation vectors

With results for mean-field interactions [Seiringer ’11] plus exployting localization techniques [Lewin-Nam-Serfaty-Solovej ’14] we obtain RN ≥ 4πaN + cN+ − C . Let ψN ∈ L2

s(ΛN) with ψN, HNψN ≤ 4πaN + ζ. Then, the excitation vector

ξN = S∗(η)T ∗(η)UNψN is such that 4πaN + ζ ≥ ξN, RNξN ⇒ ξN, N+ξN ≤ C(ζ + 1)

The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime

• S. Cenatiempo

Quantissima III, Venice 12/14

SLIDE 31

Introduction and results Strategy of the proof Excitation Hamiltonian Correlation structure

Bounds on excitation vectors

With results for mean-field interactions [Seiringer ’11] plus exployting localization techniques [Lewin-Nam-Serfaty-Solovej ’14] we obtain RN ≥ 4πaN + cN+ − C . Let ψN ∈ L2

s(ΛN) with ψN, HNψN ≤ 4πaN + ζ. Then, the excitation vector

ξN = S∗(η)T ∗(η)UNψN is such that 4πaN + ζ ≥ ξN, RNξN ⇒ ξN, N+ξN ≤ C(ζ + 1)

The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime

• S. Cenatiempo

Quantissima III, Venice 12/14

SLIDE 32

Introduction and results Strategy of the proof Excitation Hamiltonian Correlation structure

Bounds on excitation vectors

With results for mean-field interactions [Seiringer ’11] plus exployting localization techniques [Lewin-Nam-Serfaty-Solovej ’14] we obtain RN ≥ 4πaN + cN+ − C . Let ψN ∈ L2

s(ΛN) with ψN, HNψN ≤ 4πaN + ζ. Then, the excitation vector

ξN = S∗(η)T ∗(η)UNψN is such that 4πaN + ζ ≥ ξN, RNξN ⇒ ξN, N+ξN ≤ C(ζ + 1) Stronger bounds: if ψN = χ(HN ≤ 4πaN + ζ)ψN we find ξN, (K + VN + 1)(N+ + 1)kξN ≤ C(1 + ζ)k for any k ∈ N.

The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime

• S. Cenatiempo

Quantissima III, Venice 12/14

SLIDE 33

Introduction and results Strategy of the proof Excitation Hamiltonian Correlation structure

Bogoliubov spectum

Conjugation with suitably unitary operators leads to a quadratic excitation Hamiltonian up to error terms which are small on low energy states: JN = ˜ S∗(η) ˜ T ∗(η) UNHNU∗

N ˜

T(η) ˜ S(η) = CJN + QJN

• determine the

low energy spectrum

+ VN + EJN where ± EJN ≤ C N−1/4(N+ + 1)2(HN + 1)

The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime

• S. Cenatiempo

Quantissima III, Venice 13/14

SLIDE 34

Introduction and results Strategy of the proof Excitation Hamiltonian Correlation structure

Bogoliubov spectum

Conjugation with suitably unitary operators leads to a quadratic excitation Hamiltonian up to error terms which are small on low energy states: JN = ˜ S∗(η) ˜ T ∗(η) UNHNU∗

N ˜

T(η) ˜ S(η) = CJN + QJN

• determine the

low energy spectrum

+ VN + EJN where ± EJN ≤ C N−1/4(N+ + 1)2(HN + 1) and QJN =

• p∈Λ∗

+

• Fpb∗

pbp + 1

2Gp( b∗

pb∗ −p + bpb−p )

• with

Fp = p2(sinh2ηp + cosh2ηp) + V (·/N) ⋆ fN

• p (sinhηp + coshηp)2

Gp = 2p2 sinhηp coshηp + V (·/N) ⋆ fN

• p (sinhηp + coshηp)2

The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime

• S. Cenatiempo

Quantissima III, Venice 13/14

SLIDE 35

Introduction and results Strategy of the proof Excitation Hamiltonian Correlation structure

Bogoliubov spectum

Conjugation with suitably unitary operators leads to a quadratic excitation Hamiltonian up to error terms which are small on low energy states: JN = ˜ S∗(η) ˜ T ∗(η) UNHNU∗

N ˜

T(η) ˜ S(η) = CJN + QJN

• determine the

low energy spectrum

+ VN + EJN where ± EJN ≤ C N−1/4(N+ + 1)2(HN + 1) and QJN =

• p∈Λ∗

+

• Fpb∗

pbp + 1

2Gp( b∗

pb∗ −p + bpb−p )

• with

Fp = p2(sinh2ηp + cosh2ηp) + V (·/N) ⋆ fN

• p (sinhηp + coshηp)2 ≃ p2

Gp = 2p2 sinhηp coshηp + V (·/N) ⋆ fN

• p (sinhηp + coshηp)2 ≃ 1

p2 The operator QJN may be diagonalized using T(τ) = exp 1 2

• p∈Λ∗

+

τp(b∗

pb∗ −p − bpb−p)

• ,

tanh(2τp) = −Gp Fp |τp| ≃ |p|−4

The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime

• S. Cenatiempo

Quantissima III, Venice 13/14

SLIDE 36

Introduction and results Strategy of the proof Excitation Hamiltonian Correlation structure

Summary

◮ Conjugating the Gross-Pitaevskii Hamiltonian with suitable unitary maps we are able to extract the large contributions to the energy neglected in Bogoliubov approximation ◮ The results extend to non-translation-invariant bosonic systems trapped by confining external fields [Brennecke-Schlein-Schraven, in preparation] ◮ Perspectives: statical properties of low dimensional bosons two dimensional bosons interacting through singular potentials

• f the form N2β−1V (Nβx) with β > 0, or Gross-Pitaevskii

interaction e2NV (eNx) (with C. Caraci) strongly confined 3d bosons (with L. Bossmann)

The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime

• S. Cenatiempo

Quantissima III, Venice 14/14

SLIDE 37

Finite size effect

Replace V by VR(x) = R−2V (x/R) with scattering length aR = aR. For large R the order one contributions to the ground state energy scale as eΛa2R2 and −1 2

• p∈2πZ3\{0}
• p2 + 8πaR −
• |p|4 + 16πaRp2 − (8πaR)2

p2

• = R

2

• p∈ 2π

√ R Z3\{0}

• p2 + 8πa −
• |p|4 + 16πap2 − (8πa)2

p2

• ≃

R5/2 2(2π)3

• R3
• p2 + 8πa −
• |p|4 + 16πap2 − (8πa)2

p2

• dp

= 4πR5/2(16πa)5/2 15(2π)3 = 4πa · 128 15√π a3/2R5/2 Letting R → ∞ (independently of N), i.e. making the effective density larger, the finite volume correction becomes subleading, w.r.t. Bogoliubov sum.

SLIDE 38

Action of the quadratic conjugation

Renormalized excitation Hamiltonian: define GN = T ∗LNT = T ∗UHNU∗T : F ≤N

+

→ F ≤N

+

Action of T: with K = p2a∗

pap (kinetic energy) and

VN = (2N)−1 V (r/N)a∗

p+ra∗ qaq+rap (potential energy), we have

T ∗KT ≃ K +

• p2η2

p +

• p2ηp
• a∗

pa∗ −p + a−pap

• T ∗VNT ≃ VN + 1

2N V (r/N)ηq+rηq + 1 2N V (r/N)ηr+p

• a∗

pa∗ −p + apa−p

• Combine with

T ∗ 1 2 V (p/N)

• a∗

pa∗ −p + apa−p

• T

≃ 1 2 V (p/N)ηp + 1 2 V (p/N)

• a∗

pa∗ −p + apa−p

• to get rid of off-diagonal quadratic term.

SLIDE 39

Action of the cubic phase

Renormalized excitation Hamiltonians:

GN = CGN + QGN + CN + VN + EGN , ± EGN ≤ C N−1/2(HN + N 2 + 1)(N+ + 1) JN = CJN + QJN + VN + EJN , ± EJN ≤ C N−1/4(HN + N 2 + 1)(N+ + 1) Expanding to second order, we find JN = S∗GNS = e−AGNeA ≃ GN + [GN, A] + 1 2[[GN, A], A] + . . . ◮ [K, A] is cubic in creation and annihilation operators [VN, A] is quintic in creation and annihilation operators, and contains terms which are not in normal order. Restoring normal order generates an additional cubic term. We choose A so that the sum of these cubic terms renormalize the cubic operator in GN, making it small on low energy states. ◮ The commutators [CN, A] and [[HN, A], A] produce constant and quadratic contributions that transform CGN and QGN into CJN and QJN .

SLIDE 40

Localization techniques

With results for mean-field interactions [Seiringer ’11]

RN ≥ U

• i<j

ν(xi − xj)U∗ + (K + VN)(1 − C µα ) − 4πa N

• |r|<µ,

p,q∈Λ∗

+

• V (r/N)a∗

p+ra∗ qapaq+r − C

≥ 4πaN + 1 2 (K + VN) −N 2

+/N − C

Localization technique: [Lieb-Solovej, ’04], [Lewin-Nam-Serfaty-Solovej, ’12] RN ≥ fMRNfM + gMRNgM − C N/M2 fM =

• 1

if N+ ≤ M f 2

M + g2 M = 1

if N+ > 2M

Pick M = εN, for sufficiently small ε > 0. Then fMRNfM ≥ f 2

M4πaN + Cf 2 MN+ − Cf 2 M

On gM use [Lieb-Seiringer, ’06], [Nam-Rougerie-Seiringer, ’16] gMRNgM ≥ g 2

M

• 4πaN + cN
• ≥ g 2

M

• 4πaN + cN+

SLIDE 41

Localization techniques

With results for mean-field interactions [Seiringer ’11]

RN ≥ U

• i<j

ν(xi − xj)U∗ + (K + VN)(1 − C µα ) − 4πa N

• |r|<µ,

p,q∈Λ∗

+

• V (r/N)a∗

p+ra∗ qapaq+r − C

≥ 4πaN + 1 2 (K + VN) −N 2

+/N − C ≥ 4πaN + cN+ − C

Localization technique: [Lieb-Solovej, ’04], [Lewin-Nam-Serfaty-Solovej, ’12] RN ≥ fMRNfM + gMRNgM − C N/M2 fM =

• 1

if N+ ≤ M f 2

M + g2 M = 1

if N+ > 2M

Pick M = εN, for sufficiently small ε > 0. Then fMRNfM ≥ f 2

M4πaN + Cf 2 MN+ − Cf 2 M

On gM use [Lieb-Seiringer, ’06], [Nam-Rougerie-Seiringer, ’16] gMRNgM ≥ g 2

M

• 4πaN + cN
• ≥ g 2

M

• 4πaN + cN+

SLIDE 42

Bose-Einstein condensation

We obtained RN ≥ 4πaN + cN+ − C . Let ψN ∈ L2

s(ΛN) with ψN, HNψN ≤ 4πaN + ζ

Then, the excitation vector ξN = S∗T ∗UψN is such that 4πaN + ζ ≥ ξN, RNξN ⇒ ξN, N+ξN ≤ C(ζ + 1) This bound implies BEC since 1 − ϕ0, γNϕ0 = 1 − 1 N ψN, a∗

0a0ψN

= 1 − 1 N U∗TSξN, a∗

0a0U∗TSξN

= 1 − 1 N TSξN, (N − N+)TSξN = 1 N TSξN, N+TSξN ≤ C N ξN, N+ξN ≤ C(ζ + 1) N

SLIDE 43

Proof of Bogoliubov theory: unitary operators

To prove Bogoliubov prediction we use ˜ T(η) = exp 1 2

• p∈Λ∗

+

ηp

• b∗

pb∗ −p − bpb−p

˜ S(η) = exp 1 √ N

• r∈PH ,v∈PL

ηr

• b∗

r+va∗ −r (cosh(η)vav + sinh(η)va∗ −v

• − h.c.
• with

PL = {p ∈ Λ∗

+ : |p| ≤ N1/2}

PH = Λ∗

+/PL

(16πa)1/2 N N1/2

(1/R)

|p|

free particle regime linear spectrum

PL PH Then, the excitation hamiltonian JN = ˜ S∗(η) ˜ T ∗(η)UN HN U∗

N ˜

T(η) ˜ S(η) can be decomposed as JN = CJN + QJN

• determine the

low energy spectrum

+ VN + EJN , ± EJN ≤ C N−1/4(HN + N 2

+ + 1)(N+ + 1)

SLIDE 44

Proof of Bogoliubov spectrum: diagonalization

Let MN = T ∗(τ)JN T(τ) : F ≤N

+

→ F ≤N

+ , then

MN = EN +

p∈Λ∗

+

• |p|4 + 16πa|p|2a∗

pap + EMN

with EN = 4πa(N − 1) + eΛa2 − 1

2

• p∈Λ∗

+

• p2 + 8πa −
• |p|4 + +16πa|p|2 + (8πa)2

2p2

• and

EMN ≤ CN−1/4(HN + N 2

+ + 1)(N+ + 1).

Finally, we use of the min-max principle to compare the eigenvalues λm of MN − EN (i.e. the eigenvalues of HN − EN) with the eigenvalues ˜ λm of DN =

p∈Λ∗

+

• |p|4 + +16πa|p|2a∗

pap ,

showing that below an energy ζ |λm − ˜ λm| ≤ C N−1(1 + ζ3)