On Quantum Electrodynamics of atomic resonances of the proof J er - PowerPoint PPT Presentation

QED of atomic resonances J´ er´ emy Faupin The model Results Ingredients On Quantum Electrodynamics of atomic resonances of the proof J´ er´ emy Faupin Institut Elie Cartan de Lorraine Universit´ e de Lorraine Joint work with M. Ballesteros, J. Fr¨ ohlich, B. Schubnel

QED of atomic resonances Outline of the talk J´ er´ emy Faupin The model Results 1 The model Ingredients A simple model of an atom of the proof The quantized electromagnetic field Total physical system 2 Results Main results Related results 3 Ingredients of the proof Mathematical tools Strategy of the proof

QED of atomic resonances J´ er´ emy Faupin The model A simple model of an atom The quantized electro- magnetic field Part I Total physical system The model Results Ingredients of the proof

QED of atomic resonances The atom (1) J´ er´ emy Faupin Assumptions The model • The atom is non-relativistic A simple model of an atom • The atom is assumed to have only finitely many excited states The quantized electro- magnetic field Internal degrees of freedom Total physical system • Internal degrees of freedom described by an N -level system Results • Hilbert space: C N Ingredients of the proof • Hamiltonian: N × N matrix given by 0 E N · · · 0 1 . ... H is := . A , E N > · · · > E 1 B C . 0 @ 0 · · · E 1 • The energy scale of transitions between internal states of the atom is measured by the quantity δ 0 := min i � = j | E i − E j |

QED of atomic resonances The atom (2) J´ er´ emy Faupin External degrees of freedom The model • Usual Hilbert space of orbital wave functions: L 2 ( R 3 ) A simple model of x ∈ R 3 • Position of the (center of mass of the) atom: � an atom The quantized • Kinetic energy of the free center of mass motion: − 1 2 ∆ electro- magnetic field Total physical Atomic Hamiltonian system Results • Hilbert space Ingredients H at := L 2 ( R 3 ) ⊗ C N of the proof • Hamiltonian: H at := − 1 2∆ + H is , with domain D ( H at ) = H 2 ( R 3 ) ⊗ C N Electric dipole moment Represented by � d = ( d 1 , d 2 , d 3 ) , where, for j = 1 , 2 , 3 , d j ≡ I ⊗ d j is an N × N hermitian matrix

QED of atomic resonances The quantized electromagnetic field (1) J´ er´ emy Faupin Fock space The model • Wave vector of a photon: � k ∈ R 3 A simple model of an atom • Helicity of a photon: λ ∈ { 1 , 2 } The quantized electro- • Notation: magnetic field R 3 := R 3 × { 1 , 2 } = Total k := ( � k , λ ) | � k ∈ R 3 , λ ∈ { 1 , 2 } ˘ ¯ physical system Moreover, R 3 n := ( R 3 ) × n , and, for B ⊂ R 3 , Results Ingredients of the proof Z Z X d � B := B × { 1 , 2 } , dk := k B B λ =1 , 2 • Hilbert space of states of photons given by H f := F + ( L 2 ( R 3 )) , where F + ( L 2 ( R 3 )) is the symmetric Fock space over the space L 2 ( R 3 ) of one-photon states: M L 2 s ( R 3 n ) H f = C ⊕ n ≥ 1

QED of atomic resonances The quantized electromagnetic field (2) J´ er´ emy Faupin The model Photon creation- and annihilation operators A simple model of Denoted by an atom The quantized a ∗ ( k ) ≡ a ∗ λ ( � a ( k ) ≡ a λ ( � for all k = ( � k , λ ) ∈ R 3 electro- k ) , k ) , magnetic field Total physical system Fock vacuum Results Fock space H f contains a unit vector, Ω, called “vacuum (vector)” and Ingredients of the proof unique up to a phase, with the property that a ( k )Ω = 0 , for all k Hamiltonian Hamiltonian of the free electromagnetic field given by Z R 3 | � k | a ∗ ( k ) a ( k ) dk H f =

QED of atomic resonances Total physical system (1) J´ er´ emy Faupin Hilbert space The model Total Hilbert space: A simple model of an atom H = H at ⊗ H f The quantized electro- magnetic Interaction of the atom with the quantized electromagnetic field field Total physical Interaction Hamiltonian: system H I := − � d · � E ( � x ) , Results Ingredients where � of the proof E denotes the quantized electric field: Z x ⊗ a ( k ) − e − i � x ⊗ a ∗ ( k ) 1 “ e i � ” � R 3 Λ( � k ) | � k · � k · � E ( � x ) := − i k | 2 � ǫ ( k ) dk ǫ ( k ) ∈ R 3 represents the polarization vector: • k �→ � ǫ ( k ) · � ǫ (( r � ǫ (( � k , λ )) , ∀ r > 0 , ∀ k ∈ R 3 | � ǫ ( k ) | = 1 , � k = 0 , � k , λ )) = � • Λ : R 3 �→ R is an ultraviolet cut-off: k ) = e −| � k | 2 / (2 σ 2 Λ( � Λ ) , σ Λ ≥ 1

QED of atomic resonances Total physical system (2) J´ er´ emy Faupin The model Total Hamiltonian A simple model of an atom Total Hamiltonian of the system: The quantized electro- H := H at + H f + λ 0 H I , λ 0 ∈ R magnetic field Total physical system Translation invariance Results • Photon momentum operator: Ingredients of the proof Z � � ka ∗ ( k ) a ( k ) dk P f := R 3 • Total momentum operator: P tot := − i � � ∇ + � P f • [ H , � P tot , j ] = 0 , j = 1 , 2 , 3

QED of atomic resonances The fibre Hamiltonian J´ er´ emy Faupin Direct integrals The model A simple • Isomorphism model of an atom H = L 2 ( R 3 ) ⊗ C N ⊗ H f ∼ = L 2 ( R 3 ; C N ⊗ H f ) The quantized electro- magnetic field • Direct integral decomposition Total physical system Z ⊕ Z ⊕ Results H = R 3 H � p d � p , H = R 3 H ( � p ) d � p , Ingredients of the proof where the fibre space is p := C N ⊗ H f , H � and the fibre Hamiltonian is p ) := H is + 1 P f ) 2 + H f + λ 0 H I , 0 , p − � H ( � 2( � where Z 1 “ ” R 3 Λ( � k ) | � ǫ ( k ) · � ǫ ( k ) · � d ⊗ a ∗ ( k ) H I , 0 := i k | � d ⊗ a ( k ) − � dk 2

QED of atomic resonances Spectrum of H 0 ( P ) J´ er´ emy Faupin Simplification The model A simple p 2 / 2, we obtain the Hamiltonian Subtracting the trivial term � model of an atom The p ) := H is + 1 quantized P 2 � p · � H ( � f − � P f + H f + λ 0 H I , 0 electro- magnetic 2 field Total physical system Non-interacting Hamiltonian Results p ) := H is + 1 Ingredients P 2 � p · � H 0 ( � f − � P f + H f of the proof 2 Spectrum • ( [ E 1 , ∞ ) if | � p | ≤ 1 , σ ( H 0 ( � p )) = p 2 p | − 1 2 − � [ E 1 + | � 2 , ∞ ) if | � p | ≥ 1 . • Pure point spectrum p ∈ R 3 σ pp ( H 0 ( � p )) = { E 1 , E 2 , . . . E N } for all �

QED of atomic resonances J´ er´ emy Faupin The model Results Main results Related results Part II Ingredients of the proof Results

QED of atomic resonances Complex dilatations in Fock space J´ er´ emy Faupin Dilatation operator in the 1-photon space The model (Unitary) dilatation operator: for θ ∈ R , Results Main γ θ ( φ )( � k , λ ) := e − 3 θ/ 2 φ ( e − θ � for φ ∈ L 2 ( R 3 ) results k , λ ) , Related results Ingredients Second quantization of the proof Second quantization of γ θ : Γ θ := Γ( γ θ ) operator on H f defined by: Γ θ (Φ)( k 1 , . . . , k n ) := e − 3 n θ/ 2 Φ( e − θ � k 1 , λ 1 , . . . , e − θ � k n , λ n ) Dilated Hamiltonian θ = H is + 1 p )Γ ∗ 2 e − 2 θ � P 2 f − e − θ � p · � P f + e − θ H f + λ 0 H I ,θ , H θ ( � p ) := Γ θ H ( � where Z 1 “ ” H I ,θ := ie − 2 θ R 3 Λ( e − θ � k ) | � ǫ ( k ) · � ǫ ( k ) · � d ⊗ a ∗ ( k ) k | � d ⊗ a ( k ) − � dk . 2 Analytically extended to D (0 , π/ 4) := { θ ∈ C : | θ | < π/ 4 } .

QED of atomic resonances Spectrum of the non-interacting dilated J´ er´ emy Hamiltonian Faupin The model Non-interacting dilated Hamiltonian Results Main results p ) := H is + e − 2 θ � P 2 Related 2 − e − θ � p · � P f + e − θ H f f H θ, 0 ( � results Ingredients of the proof Spectrum For δ 0 > 0, E 1 , . . . , E N are simple eigenvalues of H θ, 0 ( � p ). For | � p | < 1 and θ = i ϑ , ϑ ∈ R , the spectrum of H θ, 0 ( � p ) is included in a region of the following form: E N E 1 E 2 E 3 ... ϑ 2 ϑ p ∈ R 3 , | � Figure: Shape of the spectrum of H θ, 0 ( � p ) for � p | < 1.

QED of atomic resonances Main results J´ er´ emy Faupin The model Results Main results Theorem (Ballesteros, F, Fr¨ ohlich, Schubnel) Related results Let 0 < ν < 1. There exists λ c ( ν ) > 0 such that, for all | λ 0 | < λ c ( ν ) and Ingredients p ∈ R 3 , | � � p | < ν , the following properties are satisfied: of the proof a) E ( � p ) := inf σ ( H ( � p )) is a non-degenerate eigenvalue of H ( � p ), b) For all i 0 ∈ { 1 , · · · , N } and θ ∈ C with 0 < Im ( θ ) < π/ 4 large enough, p ) has an eigenvalue, z ( ∞ ) ( � p ), such that z ( ∞ ) ( � H θ ( � p ) → E i 0 as λ 0 → 0. For i 0 = 1, z ( ∞ ) ( � p ) = E ( � p ). Moreover, for | � p | < ν , | λ 0 | small enough and 0 < Im ( θ ) < π/ 4 large enough, the ground state energy, E ( � p ), its associated eigenprojection, π ( � p ), and resonances energies, z ( ∞ ) ( � p ), are analytic in � p , λ 0 and θ . In particular, they are independent of θ