Dynamics of the strongly coupled Polaron Simone Rademacher joint - - PowerPoint PPT Presentation

Dynamics of the strongly coupled Polaron

Simone Rademacher joint work with Nikolai Leopold, Benjamin Schlein and Robert Seiringer CIRM, October 21, 2019.

Polaron

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Polaron

◮ Electron moving in an

ionic crystal

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Polaron

◮ Electron moving in an

ionic crystal

◮ Electron induces

polarization field

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Polaron

◮ Electron moving in an

ionic crystal

◮ Electron induces

polarization field

◮ Polaron

= electron + polarization field

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Polaron

◮ Electron moving in an

ionic crystal

◮ Electron induces

polarization field

◮ Polaron

= electron + polarization field

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Microscopic Description

The polaron is described on the Hilbert space L2(R3) ⊗ F. Here, F denotes the bosonic Fock space equipped with the creation and annihilation operators a∗

k resp. ak satisfying the canonical commutation

relations [ak, a∗

k′] = δ(k − k′),

[ak, ak′] = [a∗

k , a∗ k′] = 0,

∀ k, k′ ∈ R3. The quantum mechanical description of the polaron is based on the Fröhlich Hamiltonian (1937) HF = −∆ + √α

• R3

dk |k|−1 e−ik·xa∗

k + eik·xak

• +
• R3

dk a∗

k ak

acting on L2 R3 ⊗ F. Here, α > 0 denotes the coupling constant.

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Microscopic Description

The polaron is described on the Hilbert space L2(R3) ⊗ F. Here, F denotes the bosonic Fock space equipped with the rescaled creation and annihilation operators b∗

k resp. bk satisfying the canonical commutation

relations [bk, b∗

k′] = α−2δ(k − k′),

[bk, bk′] = [b∗

k , b∗ k′] = 0,

∀ k, k′ ∈ R3. The quantum mechanical description of the polaron is based on the Fröhlich Hamiltonian (1937) in strong coupling units Hα = −∆ +

• R3

dk |k|−1 e−ik·xb∗

k + eik·xbk

• +
• R3

dk b∗

k bk

acting on L2 R3 ⊗ F. Here, α > 0 denotes the coupling constant.

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Effective Dynamics

We consider the time evolution of the polaron determined through the Schrödinger equation i∂tΨt = HαΨt, with initial data Ψ0 = ψ0 ⊗ W (α2ϕ0)Ω where W (f ) = eb∗(f )−b(f ) for all f ∈ L2(R3). Question: Ψt = e−iHαtΨ0 ≈ ψt ⊗ W (α2ϕt)Ω ? Let (ψt, ϕt) ∈ H1(R3) × L2(R3) satisfy the Landau-Pekar equations

• i∂tψt

= hϕtψt, iα2∂tϕt = ϕt + σψt with initial data (ψ0, ϕ0) ∈ H1(R3) × L2(R3). The Hamiltonian of the electron is given by hϕ = −∆ + Vϕ with Vϕ(x) = 2Re

• R3

dk |k|−1ϕ(k)eik·x, σψ(k) = |k|−1 |ψ|2(k).

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Effective Dynamics

Initial data: Let ϕ0 ∈ L2(R3) such that e(ϕ0) := infψψ, hϕ0ψ < 0. Let ψϕ0 denote the unique positive corresponding ground state.

σess(hϕ0) e(ϕ0)

Leopold-R.-Schlein-Seiringer (2019): There exists C > 0 such that e−iHαtψϕ0 ⊗ W (α2ϕ0)Ω − eiω(t)ψt ⊗ W (α2ϕt)Ω2

L2(R3)⊗F ≤ Cα−2(1 + |t|)

Remark: approximation for times |t| ≪ α2 Earlier Results:

◮ Frank-Schlein (2014): e−iHαtΨ0 ≈ ψt ⊗ W (α2ϕ0)Ω for times |t| ≪ α ◮ Frank-Gang (2017): e−iHαtΨ0 ≈ ψt ⊗ W (α2ϕt)Ω for times |t| ≪ α ◮ Griesemer (2017): e−iHαtψP ⊗ W (α2ϕP) ≈ eiEPtψP ⊗ W (α2ϕP) for

times |t| ≪ α2

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Adiabatic Theorem for LP equations

Let (ψt, ϕt) satisfy

• i∂tψt

= hϕtψt, iα2∂tϕt = ϕt + σψt. with initial data (ψϕ0, ϕ0) where ϕ0 ∈ L2(R3) such that e(ϕ0) < 0. Let Λ denote the spectral gap of hϕ0. σess(hϕ0) e(ϕ0) e1(ϕ0) Λ Leopold-R.-Schlein-Seiringer (2019): There exist C, CΛ > 0 such that ψt − e−i

ω(t)ψϕt 2 ≤ Cα−2,

∀ |t| ≤ CΛα2. Remark:

◮ The theorem restricts to times |t| ≤ CΛα2 to ensure the persistence of a

spectral gap of order one.

◮ Similar results in dimension one are due to Frank-Gang (2017).

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Adiabatic Theorem for LP equations

Let (ψt, ϕt) satisfy

• i∂tψt

= hϕtψt, iα2∂tϕt = ϕt + σψt. with initial data (ψϕ0, ϕ0) where ϕ0 ∈ L2(R3) such that e(ϕ0) < 0. Let Λ denote the spectral gap of hϕ0. σess(hϕ0) e(ϕ0) e1(ϕ0) Λ Leopold-R.-Schlein-Seiringer (2019): There exist C, CΛ > 0 such that ψt − e−i

ω(t)ψϕt 2 ≤ Cα−2,

∀ |t| ≤ CΛα2. Remark:

◮ The theorem restricts to times |t| ≤ CΛα2 to ensure the persistence of a

spectral gap of order one.

◮ Similar results in dimension one are due to Frank-Gang (2017).

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Adiabatic Theorem for LP equations

Let (ψt, ϕt) satisfy

• i∂tψt

= hϕtψt, iα2∂tϕt = ϕt + σψt. with initial data (ψϕ0, ϕ0) where ϕ0 ∈ L2(R3) such that e(ϕ0) < 0. Let Λ denote the spectral gap of hϕ0. σess(hϕ0) e(ϕ0) e1(ϕ0) t CΛα2 Λ Λ(t) Leopold-R.-Schlein-Seiringer (2019): There exist C, CΛ > 0 such that ψt − e−i

ω(t)ψϕt 2 ≤ Cα−2,

∀ |t| ≤ CΛα2. Remark:

◮ The theorem restricts to times |t| ≤ CΛα2 to ensure the persistence of a

spectral gap of order one.

◮ Similar results in dimension one are due to Frank-Gang (2017).

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