Bogoliubov Hamiltonians and one parameter groups of Bogoliubov transformations

TL;DR

This paper studies the implementation of one-parameter groups of Bogoliubov transformations on bosonic Fock space, introducing conditions for their associated Hamiltonians to be well-defined and establishing their unitary evolution.

Contribution

It proves that under general conditions, Bogoliubov transformations can be implemented by a one-parameter unitary group with a well-defined Hamiltonian, introducing two types of such Hamiltonians.

Findings

Existence of a unitary implementation for Bogoliubov transformations

Introduction of two types of Bogoliubov Hamiltonians

Conditions ensuring well-defined Hamiltonians

Abstract

On the bosonic Fock space, a family of Bogoliubov transformations corresponding to a strongly continuous one-parameter group of symplectic maps R(t) is considered. Under suitable assumptions on the generator A of this group, which guarantee that the induced representations of CCR are unitarily equivalent for all time t, it is known that the unitary operator U_{nat}(t) which implement this transformation gives a prjective unitary representation of R(t). Under rather general assumptions on the generator A, we prove that the corresponding Bogoliubov transformations can be implemented by a one-parameter group U(t) of unitary operators. The generator of U(t) will be called a Bogoliubov Hamiltonian. We will introduce two kinds of Bogoliubov Hamiltonians (type I and II) and give conditions so that they are well defined.