The Friedrichs-Lee model and its singular coupling limit Davide - PowerPoint PPT Presentation

The Friedrichs-Lee model and its singular coupling limit Davide Lonigro University of Bari & INFN Joint work with Paolo Facchi and Marilena Ligab` o Toru´ n, June 17, 2019

Outline 1. The Friedrichs-Lee model 2. The singular coupling problem 3. Spectral properties of the Friedrichs-Lee model 4. Multi-atom extension 1

The Friedrichs-Lee model

Lee field theory Mathematical model for two-level system and field: • Atom with excitation energy ε a , ground state |↓� and excited state |↑� ; • Bosonic field: measure space ( X , µ ) as momentum space, and ω : X → R as dispersion relation. Lee field theory: 1 H Lee = ( H atom ⊗ I ) + ( I ⊗ H field ) + V g , with • H atom = ε a |↑� �↑| ; � • H field = ω ( k ) a ∗ ( k ) a( k ) d µ ; X � � σ + ⊗ g ( k ) a( k ) + σ − ⊗ g ( k ) a ∗ ( k ) � • V g = d µ , X where the form factor g ∈ L 2 µ ( X ) weights the coupling. 1 T. Lee (1954), Phys. Rev., 95(5), pp. 1329–1334. 2

Lee field theory Mathematical model for two-level system and field: • Atom with excitation energy ε a , ground state |↓� and excited state |↑� ; • Bosonic field: measure space ( X , µ ) as momentum space, and ω : X → R as dispersion relation. Lee field theory: 1 H Lee = ( H atom ⊗ I ) + ( I ⊗ H field ) + V g , with • H atom = ε a |↑� �↑| ; � • H field = ω ( k ) a ∗ ( k ) a( k ) d µ ; X � � σ + ⊗ g ( k ) a( k ) + σ − ⊗ g ( k ) a ∗ ( k ) � • V g = d µ , X where the form factor g ∈ L 2 µ ( X ) weights the coupling. 1 T. Lee (1954), Phys. Rev., 95(5), pp. 1329–1334. 2

The Friedrichs-Lee model The single-excitation sector is C ⊕ L 2 µ ( X ). Its Physical interpretation generic normalized state is | x | 2 is the probability of measuring � � the atom in its excited state Ψ 0 , x ξ ∈ L 2 Ψ = , x ∈ C , µ ( X ) , and ξ ( k ) is the boson wavefunction ξ in the momentum representation. � with | x | 2 + | ξ ( k ) | 2 d µ = 1. X Defining (Ω ξ )( k ) = ω ( k ) ξ ( k ), here our Hamiltonian (Friedrichs-Lee model) acts as follows: 2 �� � � � � � g , ·� x ε a Dom H g = : x ∈ C , ξ ∈ Dom Ω H g = , . Ω ξ g Physical interpretation A state atom+field has finite mean value (resp. variance) of the total energy if and only if its field component has finite mean value (resp. variance) of the field energy. 2 K. O. Friedrichs (1948), Comm. Pure Appl. Math., 1(4), pp. 361–406. 3

The Friedrichs-Lee model The single-excitation sector is C ⊕ L 2 µ ( X ). Its Physical interpretation generic normalized state is | x | 2 is the probability of measuring � � the atom in its excited state Ψ 0 , x ξ ∈ L 2 Ψ = , x ∈ C , µ ( X ) , and ξ ( k ) is the boson wavefunction ξ in the momentum representation. � with | x | 2 + | ξ ( k ) | 2 d µ = 1. X Defining (Ω ξ )( k ) = ω ( k ) ξ ( k ), here our Hamiltonian (Friedrichs-Lee model) acts as follows: 2 �� � � � � � g , ·� x ε a Dom H g = : x ∈ C , ξ ∈ Dom Ω H g = , . Ω ξ g Physical interpretation A state atom+field has finite mean value (resp. variance) of the total energy if and only if its field component has finite mean value (resp. variance) of the field energy. 2 K. O. Friedrichs (1948), Comm. Pure Appl. Math., 1(4), pp. 361–406. 3

The Friedrichs-Lee model The single-excitation sector is C ⊕ L 2 µ ( X ). Its Physical interpretation generic normalized state is | x | 2 is the probability of measuring � � the atom in its excited state Ψ 0 , x ξ ∈ L 2 Ψ = , x ∈ C , µ ( X ) , and ξ ( k ) is the boson wavefunction ξ in the momentum representation. � with | x | 2 + | ξ ( k ) | 2 d µ = 1. X Defining (Ω ξ )( k ) = ω ( k ) ξ ( k ), here our Hamiltonian (Friedrichs-Lee model) acts as follows: 2 �� � � � � � g , ·� x ε a Dom H g = : x ∈ C , ξ ∈ Dom Ω H g = , . Ω ξ g Physical interpretation A state atom+field has finite mean value (resp. variance) of the total energy if and only if its field component has finite mean value (resp. variance) of the field energy. 2 K. O. Friedrichs (1948), Comm. Pure Appl. Math., 1(4), pp. 361–406. 3

The singular coupling problem

Singular coupling The model is well-defined provided that g ∈ L 2 µ ( X ), Problem since � � � � Can we generalize this model to ε a x + � g , ξ � x = H g . include a singular (i.e. not Ω ξ + xg ξ normalizable) coupling? Example: exponential decay of the survival probability of the atom’s excited state Ψ 0 is ⇒ Zeno evolution 3 at small times. On the other hand, by prohibited since Ψ 0 ∈ Dom H g = formal calculations, an exponential decay may be obtained by choosing ω ( k ) = k , g ( k ) = const ., but obviously this form factor is not normalizable! The idea If we want to consider a broader class of form factors, we need to change the domain. 3 See e.g. H. Nakazato, M. Namiki, and S. Pascazio (1996), Int. J. Mod. Phys. B, 10(3), pp. 247–295. 4

Singular coupling The model is well-defined provided that g ∈ L 2 µ ( X ), Problem since � � � � Can we generalize this model to ε a x + � g , ξ � x = H g . include a singular (i.e. not Ω ξ + xg ξ normalizable) coupling? Example: exponential decay of the survival probability of the atom’s excited state Ψ 0 is ⇒ Zeno evolution 3 at small times. On the other hand, by prohibited since Ψ 0 ∈ Dom H g = formal calculations, an exponential decay may be obtained by choosing ω ( k ) = k , g ( k ) = const ., but obviously this form factor is not normalizable! The idea If we want to consider a broader class of form factors, we need to change the domain. 3 See e.g. H. Nakazato, M. Namiki, and S. Pascazio (1996), Int. J. Mod. Phys. B, 10(3), pp. 247–295. 4

Singular coupling The model is well-defined provided that g ∈ L 2 µ ( X ), Problem since � � � � Can we generalize this model to ε a x + � g , ξ � x = H g . include a singular (i.e. not Ω ξ + xg ξ normalizable) coupling? Example: exponential decay of the survival probability of the atom’s excited state Ψ 0 is ⇒ Zeno evolution 3 at small times. On the other hand, by prohibited since Ψ 0 ∈ Dom H g = formal calculations, an exponential decay may be obtained by choosing ω ( k ) = k , g ( k ) = const ., but obviously this form factor is not normalizable! The idea If we want to consider a broader class of form factors, we need to change the domain. 3 See e.g. H. Nakazato, M. Namiki, and S. Pascazio (1996), Int. J. Mod. Phys. B, 10(3), pp. 247–295. 4

Singular coupling Ω An easy trick: the generic state in Dom Ω can be equivalently written as ξ − x Ω 2 +1 g for some ξ ∈ Dom Ω. We have: � � � � x ε x + � g , ξ � = H g , Ω 1 ξ − x Ω 2 +1 g Ω ξ + x Ω 2 +1 g with � � Ω ε = ε a − g , Ω 2 + 1 g . Physical interpretation ε may be interpreted as a “dressed” (coupling-dependent) excitation energy of the atom, with ε a being the “bare” one. The two quantities differ by a Lamb shift. In this expression, our model can be generalized. How? 5

Singular coupling Ω An easy trick: the generic state in Dom Ω can be equivalently written as ξ − x Ω 2 +1 g for some ξ ∈ Dom Ω. We have: � � � � x ε x + � g , ξ � = H g , Ω 1 ξ − x Ω 2 +1 g Ω ξ + x Ω 2 +1 g with � � Ω ε = ε a − g , Ω 2 + 1 g . Physical interpretation ε may be interpreted as a “dressed” (coupling-dependent) excitation energy of the atom, with ε a being the “bare” one. The two quantities differ by a Lamb shift. In this expression, our model can be generalized. How? 5

Singular coupling For s ≥ 0, define H s and H − s as the spaces of functions g that are bounded w.r.t. the norms � s := � ( | Ω | + 1) s / 2 g � 2 = ( | ω ( k ) | + 1) s | g ( k ) | 2 d µ ; � g � 2 X | g ( k ) | 2 � − s := � ( | Ω | + 1) − s / 2 g � 2 = � g � 2 ( | ω ( k ) | + 1) s d µ. X A scale of normed spaces 4 is obtained: . . . ⊂ H 2 ⊂ H 1 ⊂ H ≡ H 0 ⊂ H − 1 ⊂ H − 2 ⊂ . . . , with H − s and H s being dual spaces. In particular, H 2 = Dom Ω. The point is: ...hence g ∈ H − 2 implies � � � � x ε x + � g , ξ � Ω 1 • Ω 2 +1 g ∈ H ; = Ω 2 +1 g , H g Ω 1 ξ − x Ω ξ + x Ω 2 +1 g Ω 2 +1 g • � g , ξ � is well-defined for every ξ ∈ Dom Ω... is well-defined up to g ∈ H − 2 ! 4 See e.g. S. Albeverio and P. Kurasov, Singular Perturbations of Differential Operators (2000). 6

Singular coupling For s ≥ 0, define H s and H − s as the spaces of functions g that are bounded w.r.t. the norms � s := � ( | Ω | + 1) s / 2 g � 2 = ( | ω ( k ) | + 1) s | g ( k ) | 2 d µ ; � g � 2 X | g ( k ) | 2 � − s := � ( | Ω | + 1) − s / 2 g � 2 = � g � 2 ( | ω ( k ) | + 1) s d µ. X A scale of normed spaces 4 is obtained: . . . ⊂ H 2 ⊂ H 1 ⊂ H ≡ H 0 ⊂ H − 1 ⊂ H − 2 ⊂ . . . , with H − s and H s being dual spaces. In particular, H 2 = Dom Ω. The point is: ...hence g ∈ H − 2 implies � � � � x ε x + � g , ξ � Ω 1 • Ω 2 +1 g ∈ H ; = Ω 2 +1 g , H g Ω 1 ξ − x Ω ξ + x Ω 2 +1 g Ω 2 +1 g • � g , ξ � is well-defined for every ξ ∈ Dom Ω... is well-defined up to g ∈ H − 2 ! 4 See e.g. S. Albeverio and P. Kurasov, Singular Perturbations of Differential Operators (2000). 6