Pauli Fierz Model Research Articles

Light-induced electron dynamics of molecules in cavities: comparison of model Hamiltonians

The rapidly expanding field of polaritonic chemistry requires accurate theoretical simulations to understand new phenomena at the atomic scale. Computing the optoelectronic properties of molecules using established electronic structure methods is a careful balance of accuracy and computational expense, and expanding these methods to quantum electrodynamics to describe coupled cavity-molecule systems is an active topic of development. Key to these methods are the Hamiltonian operators representing the photon cavity modes. The recently introduced quantum electrodynamics time-dependent configuration interaction (QED-TDCI) method allows for the combination of electron dynamics simulations with quantum electrodynamics, enabling the simulation of time-dependent optoelectronic properties of cavity-molecule systems. Using this method, a comparison of two many-state QED Hamiltonians—the Pauli-Fierz and quantum Rabi model Hamiltonians—is presented, with a particular focus on time-dependent properties in applied electric fields.

Ground states for translationally invariant Pauli-Fierz models at zero momentum

We consider the translationally invariant Pauli-Fierz model describing a charged particle interacting with the electromagnetic field. We show under natural assumptions that the fiber Hamiltonian at zero momentum has a ground state.

Spectrum of the semi-relativistic Pauli–Fierz model II

We consider the ground state of the semi-relativistic Pauli–Fierz Hamiltonian $$ H = |\textbf{p} - \textbf{A(x)}| + H\_f + V\textbf{(x)}. $$ Here $\textbf{A(x)}$ denotes the quantized radiation field with an ultraviolet cutoff function and $H\_f$ the free field Hamiltonian with dispersion relation $|\textbf{k}|$. The Hamiltonian $H$ describes the dynamics of a massless and semi-relativistic charged particle interacting with the quantized radiation field with an ultraviolet cutoff function. In 2016, the first two authors proved the existence of the ground state $\Phi\_m$ of the massive Hamiltonian $H\_m$ is proven. Here, the massive Hamiltonian $H\_m$ is defined by $H$ with dispersion relation $\sqrt{\textbf{k}^2+m^2}$ $(m>0)$. In this paper, the existence of the ground state of $H$ is proven. To this aim, we estimate a singular and non-local pull-through formula and show the equicontinuity of ${a(k)\Phi\_m}{0\<m\<m\_0}$ with some $m\_0$, where $a(k)$ denotes the formal kernel of the annihilation operator. Showing the compactness of the set ${\Phi\_m}{0\<m\<m\_0}$, the existence of the ground state of $H$ is shown.

Non-relativistic Pauli–Fierz Hamiltonian for less than two photons

We consider the Pauli–Fierz model, which describes a particle (an electron) coupled to the quantized electromagnetic field and limit the number of photons to less than 2. By computing the resolvent explicitly, we located the spectrum of the Hamiltonian mass. Our results do not depend on the coupling constant $e$ nor on the infrared cutoff parameter $R$.

Explicit diagonalization of pair interaction models

We provide a general method for constructing bosonic Bogoliubov transformations that diagonalize a general class of quadratic Hamiltonians. These Hamiltonians describe the pair interaction models. Bogoliubov transformations are constructed algebraically, and the resulting Hamiltonians become the second quantizations of explicit one-particle Hamiltonians. Moreover, an explicit formula for the ground state energies is given. Our method systematically diagonalizes various models of quantum field theory, including a model of a harmonic oscillator coupled to a Bose field and the Pauli-Fierz models in the dipole approximation.

Magnetic Schrödinger operators as the quasi-classical limit of Pauli–Fierz-type models

We study the quasi-classical limit of the Pauli–Fierz model: the system is composed of finitelymany non-relativistic charged particles interactingwith a bosonic radiation field. We trace out the degrees of freedom of the field, and consider the classical limit of the latter. We prove that the partial trace of the full Hamiltonian converges, in resolvent sense, to an effective Schrödinger operator with magnetic field and a corrective electric potential that depends on the field configuration. Furthermore, we prove the convergence of the ground state energy of the microscopic systemto the infimum over all possible classical field configurations of the ground state energy of the effective Schrödinger operator.

Spectrum of the semi-relativistic Pauli–Fierz model I

A HVZ type theorem for the semi-relativistic Pauli–Fierz Hamiltonian,H=(p⊗1−A)2+M2⊗1+V⊗1+1⊗Hf,M≥0, in quantum electrodynamics is studied. Here H is a self-adjoint operator in Hilbert space L2(Rd)⊗F≅∫Rd⊕Fdx, A=∫Rd⊕A(x)dx is a quantized radiation field and Hf is the free field Hamiltonian defined by the second quantization of a dispersion relation ω:Rd→R. It is emphasized that massless case, M=0, is included. Let E=inf⁡σ(H) be the bottom of the spectrum of H. Suppose that the infimum of ω is m>0. Then it is shown that σess(H)=[E+m,∞). In particular the existence of the ground state of H can be proven.

Self-adjointness of the semi-relativistic Pauli–Fierz Hamiltonian

The spinless semi-relativistic Pauli–Fierz Hamiltonian [Formula: see text] in quantum electrodynamics is considered. Here p denotes a momentum operator, A a quantized radiation field, M ≥ 0, Hf the free Hamiltonian of a Boson Fock space and V an external potential. The self-adjointness and essential self-adjointness of H are shown. It is emphasized that it includes the case of M = 0. Furthermore, the self-adjointness and the essential self-adjointness of the semi-relativistic Pauli–Fierz model with a fixed total momentum P ∈ ℝd: [Formula: see text] is also proven for arbitrary P.

Quantitative estimates on the binding energy for hydrogen in non-relativistic QED, II: The spin case

The hydrogen binding energy in the Pauli–Fierz model with the spin Zeeman term is determined up to the order α3, where α denotes the fine-structure constant.

Functional integral approach to semi-relativistic Pauli–Fierz models

By means of functional integrations spectral properties of semi-relativistic Pauli–Fierz HamiltoniansH=(p−αA)2+m2−m+V+Hrad in quantum electrodynamics are considered. Here p is the momentum operator, A a quantized radiation field on which an ultraviolet cutoff is imposed, V an external potential, Hrad the free field Hamiltonian and m⩾0 describes the mass of electron. Two self-adjoint extensions of a semi-relativistic Pauli–Fierz Hamiltonian are defined. The Feynman–Kac type formula of e−tH is given. A self-adjointness, a spatial decay of bound states, a Gaussian domination of the ground state and the existence of a measure associated with the ground state are shown. All the results are independent of values of coupling constant α, and it is emphasized that m=0 is included.

Generalized interaction in multigravity

We consider a general approach to describing the interaction in multigravity models in a D-dimensional space-time. We present various possibilities for generalizing the invariant volume. We derive the most general form of the interaction potential, which becomes a Pauli-Fierz-type model in the bigravity case. Analyzing this model in detail in the (3+1)-expansion formalism and also requiring the absence of ghosts leads to this bigravity model being completely equivalent to the Pauli-Fierz model. We thus in a concrete example show that introducing an interaction between metrics is equivalent to introducing the graviton mass.

On Enhanced Binding and Related Effectsin the Non- and Semi-Relativistic Pauli-Fierz Models

We prove enhanced binding and increase of binding energies in the non- and semi-relativistic Pauli-Fierz models, for arbitrary values of the fine-structure constant and the ultra-violet cut-off, and discuss the resulting improvement of exponential localization of ground state eigenvectors. For the semi-relativistic model we also discuss the increase of the renormalized electron mass and determine the linear leading order term in the asymptotics of the self-energy, as the ultra-violet cut-off goes to infinity.

The Mass Shell in the Semi-Relativistic Pauli–Fierz Model

We consider the semi-relativistic Pauli-Fierz model for a single free electron interacting with the quantized radiation field. Employing a variant of Pizzo's iterative analytic perturbation theory we construct a sequence of ground state eigenprojections of infra-red cutoff, dressing transformed fiber Hamiltonians and prove its convergence, as the cutoff goes to zero. Its limit is the ground state eigenprojection of a certain Hamiltonian unitarily equivalent to a renormalized fiber Hamiltonian acting in a coherent state representation space. The ground state energy is an exactly two-fold degenerate eigenvalue of the renormalized Hamiltonian, while it is not an eigenvalue of the original fiber Hamiltonian unless the total momentum is zero. These results hold true, for total momenta inside a ball about zero of arbitrary radius p>0, provided that the coupling constant is sufficiently small depending on p and the ultra-violet cutoff. Along the way we prove twice continuous differentiability and strict convexity of the ground state energy as a function of the total momentum inside that ball.

Minimization of the energy of the nonrelativistic one-electron Pauli-Fierz model over quasifree states

In this article the existence of a minimizer for the energy for the nonrelativistic one-electron Pauli-Fierz model within the class of quasifree states is established. To this end it is shown that the minimum of the energy on quasifree states coincides with the minimum of the energy on pure quasifree states, where existence and uniqueness of a minimizer holds. Infrared and ultraviolet cutoffs are assumed, along with sufficiently small coupling constant and momentum of the dressed electron. A perturbative expression of the minimum of the energy on quasifree states for a small momentum of the dressed electron and small coupling constant is given. We also express the Lagrange equation for the minimizer in terms of the generalized one particle density matrix of the pure quasifree state.

On the ground state energy of the translation invariant Pauli-Fierz model. II.

In this note, we determine the ground state energy of the translation invariant Pauli-Fierz model to subleading order $O(\alpha^3)$ with respect to powers of the finestructure constant $\alpha$, and prove rigorous error bounds of order $O(\alpha^{4})$. A main objective of our argument is its brevity.

The no-binding regime of the Pauli-Fierz model

The Pauli-Fierz model H(α) in nonrelativistic quantum electrodynamics is considered. The external potential V is sufficiently shallow and the dipole approximation is assumed. It is proven that there exist constants 0 &amp;lt; α− &amp;lt; α+ such that H(α) has no ground state for |α| &amp;lt; α−, which complements an earlier result stating that there is a ground state for |α| &amp;gt; α+. We develop a suitable extension of the Birman-Schwinger argument. Moreover, for any given δ &amp;gt; 0 examples of potentials V are provided such that α+ − α− &amp;lt; δ.

REGULARITY OF BOUND STATES

We study regularity of bound states pertaining to embedded eigenvalues of a self-adjoint operator H, with respect to an auxiliary operator A that is conjugate to H in the sense of Mourre. We work within the framework of singular Mourre theory which enables us to deal with confined massless Pauli–Fierz models, our primary example, and many-body AC-Stark Hamiltonians. In the simpler context of regular Mourre theory, our results boil down to an improvement of results obtained recently in [8, 9].

Spectral Analysis of an Effective Hamiltonian in Nonrelativistic Quantum Electrodynamics

We investigate spectral properties of an effective Hamiltonian which is obtained as a scaling limit of the Pauli–Fierz model in nonrelativistic quantum electrodynamics. The Lamb shift of a hydrogen-like atom is derived as the lowest order approximation (in the fine structure constant) of an energy level shift of the effective Hamiltonian.

Quantitative Estimates on the Binding Energy for Hydrogen in Non-Relativistic QED

In this paper, we determine the exact expression for the hydrogen binding energy in the Pauli–Fierz model up to the order α 5 log α −1, where α denotes the fine structure constant, and prove rigorous bounds on the remainder term of the order o(α 5 log α −1). As a consequence, we prove that the binding energy is not a real analytic function of α, and verify the existence of logarithmic corrections to the expansion of the ground state energy in powers of α, as conjectured in the recent literature.

GIBBS MEASURES WITH DOUBLE STOCHASTIC INTEGRALS ON A PATH SPACE

We investigate Gibbs measures relative to Brownian motion in the case when the interaction energy is given by a double stochastic integral. In the case when the double stochastic integral is originating from the Pauli–Fierz model in nonrelativistic quantum electrodynamics, we prove the existence of its infinite volume limit.