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 1 / 5

Polaron ◮ Electron moving in an ionic crystal 1 / 5

Polaron ◮ Electron moving in an ionic crystal ◮ Electron induces polarization field 1 / 5

Polaron ◮ Electron moving in an ionic crystal ◮ Electron induces polarization field ◮ Polaron = electron + polarization field 1 / 5

Polaron ◮ Electron moving in an ionic crystal ◮ Electron induces polarization field ◮ Polaron = electron + polarization field 1 / 5

Microscopic Description The polaron is described on the Hilbert space L 2 ( R 3 ) ⊗ F . Here, F denotes the bosonic Fock space equipped with the creation and annihilation operators a ∗ k resp. a k satisfying the canonical commutation relations ∀ k , k ′ ∈ R 3 . [ a k , a ∗ k ′ ] = δ ( k − k ′ ) , [ a k , a k ′ ] = [ a ∗ k , a ∗ k ′ ] = 0 , The quantum mechanical description of the polaron is based on the Fröhlich Hamiltonian (1937) � � dk | k | − 1 � � H F = − ∆ + √ α e − ik · x a ∗ k + e ik · x a k dk a ∗ + k a k R 3 R 3 acting on L 2 � R 3 � ⊗ F . Here, α > 0 denotes the coupling constant . 2 / 5

Microscopic Description The polaron is described on the Hilbert space L 2 ( R 3 ) ⊗ F . Here, F denotes the bosonic Fock space equipped with the rescaled creation and annihilation operators b ∗ k resp. b k satisfying the canonical commutation relations ∀ k , k ′ ∈ R 3 . [ b k , b ∗ k ′ ] = α − 2 δ ( k − k ′ ) , [ b k , b k ′ ] = [ b ∗ k , b ∗ k ′ ] = 0 , The quantum mechanical description of the polaron is based on the Fröhlich Hamiltonian (1937) in strong coupling units � � dk | k | − 1 � � e − ik · x b ∗ k + e ik · x b k dk b ∗ H α = − ∆ + + k b k R 3 R 3 acting on L 2 � R 3 � ⊗ F . Here, α > 0 denotes the coupling constant . 2 / 5

Effective Dynamics We consider the time evolution of the polaron determined through the Schrödinger equation i ∂ t Ψ t = H α Ψ t , with initial data Ψ 0 = ψ 0 ⊗ W ( α 2 ϕ 0 )Ω where W ( f ) = e b ∗ ( f ) − b ( f ) for all f ∈ L 2 ( R 3 ). Question: Ψ t = e − iH α t Ψ 0 ≈ ψ t ⊗ W ( α 2 ϕ t )Ω ? Let ( ψ t , ϕ t ) ∈ H 1 ( R 3 ) × L 2 ( R 3 ) satisfy the Landau-Pekar equations � i ∂ t ψ t = h ϕ t ψ t , i α 2 ∂ t ϕ t = ϕ t + σ ψ t with initial data ( ψ 0 , ϕ 0 ) ∈ H 1 ( R 3 ) × L 2 ( R 3 ). The Hamiltonian of the electron is given by h ϕ = − ∆ + V ϕ with � σ ψ ( k ) = | k | − 1 � dk | k | − 1 ϕ ( k ) e ik · x , V ϕ ( x ) = 2 Re | ψ | 2 ( k ) . R 3 3 / 5

Effective Dynamics σ ess ( h ϕ 0 ) Initial data: 0 Let ϕ 0 ∈ L 2 ( R 3 ) such that e ( ϕ 0 ) := inf ψ � ψ, h ϕ 0 ψ � < 0. Let ψ ϕ 0 denote the unique positive corresponding ground state. e ( ϕ 0 ) Leopold-R.-Schlein-Seiringer (2019): There exists C > 0 such that � e − iH α t ψ ϕ 0 ⊗ W ( α 2 ϕ 0 )Ω − e i ω ( t ) ψ t ⊗ W ( α 2 ϕ t )Ω � 2 L 2 ( R 3 ) ⊗F ≤ C α − 2 (1 + | t | ) Remark: approximation for times | t | ≪ α 2 Earlier Results: ◮ Frank-Schlein (2014): e − iH α t Ψ 0 ≈ ψ t ⊗ W ( α 2 ϕ 0 )Ω for times | t | ≪ α ◮ Frank-Gang (2017): e − iH α t Ψ 0 ≈ ψ t ⊗ W ( α 2 ϕ t )Ω for times | t | ≪ α ◮ Griesemer (2017): e − iH α t ψ P ⊗ W ( α 2 ϕ P ) ≈ e iE P t ψ P ⊗ W ( α 2 ϕ P ) for times | t | ≪ α 2 4 / 5

Adiabatic Theorem for LP equations Let ( ψ t , ϕ t ) satisfy � σ ess ( h ϕ 0 ) = h ϕ t ψ t , i ∂ t ψ t 0 i α 2 ∂ t ϕ t = ϕ t + σ ψ t . e 1 ( ϕ 0 ) Λ with initial data ( ψ ϕ 0 , ϕ 0 ) where e ( ϕ 0 ) ϕ 0 ∈ L 2 ( R 3 ) such that e ( ϕ 0 ) < 0. Let Λ denote the spectral gap of h ϕ 0 . Leopold-R.-Schlein-Seiringer (2019): There exist C , C Λ > 0 such that � ψ t − e − i � ω ( t ) ψ ϕ t � 2 ≤ C α − 2 , ∀ | t | ≤ C Λ α 2 . Remark: ◮ The theorem restricts to times | t | ≤ C Λ α 2 to ensure the persistence of a spectral gap of order one . ◮ Similar results in dimension one are due to Frank-Gang (2017). 5 / 5

Adiabatic Theorem for LP equations Let ( ψ t , ϕ t ) satisfy � σ ess ( h ϕ 0 ) = h ϕ t ψ t , i ∂ t ψ t 0 i α 2 ∂ t ϕ t = ϕ t + σ ψ t . e 1 ( ϕ 0 ) Λ with initial data ( ψ ϕ 0 , ϕ 0 ) where e ( ϕ 0 ) ϕ 0 ∈ L 2 ( R 3 ) such that e ( ϕ 0 ) < 0. Let Λ denote the spectral gap of h ϕ 0 . Leopold-R.-Schlein-Seiringer (2019): There exist C , C Λ > 0 such that � ψ t − e − i � ω ( t ) ψ ϕ t � 2 ≤ C α − 2 , ∀ | t | ≤ C Λ α 2 . Remark: ◮ The theorem restricts to times | t | ≤ C Λ α 2 to ensure the persistence of a spectral gap of order one . ◮ Similar results in dimension one are due to Frank-Gang (2017). 5 / 5

Adiabatic Theorem for LP equations Let ( ψ t , ϕ t ) satisfy � σ ess ( h ϕ 0 ) = h ϕ t ψ t , i ∂ t ψ t 0 i α 2 ∂ t ϕ t = ϕ t + σ ψ t . e 1 ( ϕ 0 ) Λ( t ) Λ with initial data ( ψ ϕ 0 , ϕ 0 ) where e ( ϕ 0 ) ϕ 0 ∈ L 2 ( R 3 ) such that e ( ϕ 0 ) < 0. Let t C Λ α 2 Λ denote the spectral gap of h ϕ 0 . Leopold-R.-Schlein-Seiringer (2019): There exist C , C Λ > 0 such that � ψ t − e − i � ω ( t ) ψ ϕ t � 2 ≤ C α − 2 , ∀ | t | ≤ C Λ α 2 . Remark: ◮ The theorem restricts to times | t | ≤ C Λ α 2 to ensure the persistence of a spectral gap of order one . ◮ Similar results in dimension one are due to Frank-Gang (2017). 5 / 5