A detailed analysis of and a general decomposition theorem for in general unbounded symplectic transformations on an arbitrary complex pre-Hilbert space (one–boson test function space) are given. The structure of strongly continuous symplectic groups on such spaces is determined. The connection between quadratic Hamiltonians, Bogoliubov transformations, and symplectic transformations is discussed in the Fock representation, and their relevance for squeezing operations in quantum optics is pointed out. The results for this rather general class of transformations are proved in a self-contained fashion.
A detailed analysis of and a general decomposition theorem for in general unbounded symplectic transformations on an arbitrary complex pre-Hilbert space (one–boson test function space) are given. The structure of strongly continuous symplectic groups on such spaces is determined. The connection between quadratic Hamiltonians, Bogoliubov transformations, and symplectic transformations is discussed in the Fock representation, and their relevance for squeezing operations in quantum optics is pointed out. The results for this rather general class of transformations are proved in a self-contained fashion.
For the case of infinitely many photon (Boson) modes we investigate the unitary implementability of a class of symplectic one-parameter groups (more exactly, of the associated groups of Bogoliubov automorphisms on the CCR algebra) in the Fock representation and in representations of the CCR algebra, which are symplectically related and inequivalent to Fock. Furthermore, the existence of the associated (squeezing) quadratic Hamiltonians is discussed. Finally, applications in the theory of quantization in QED and in the Luttinger model are pointed out.
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.
The features of the paper are these: 1) we discuss {\bf especially} quadratic (alias bilinear) Bose-Hamiltonians, the related Bogoliubov transformations and {\bf especially} quasi-free-like (alias coherent or Fock-like) states 2) we discuss {\bf any} quadratic Bose-Hamiltonians and Bogoliubov transformations, whether diagonalizable or not, whether proper or improper, and {\bf arbitrary} quasi-free-like states, whether regular or non-regular they are 3) we associate notions and terms of the CCRs theory {CCRs = Canonical Commutation Relations} with notions and terms of the indefinite inner product spaces theory. Then, we apply the corresponding `bilingual dictionary' so as to construct invariant states of some of the quadratic Hamiltonians.
This thesis is concerned with the properties of dynamical systems with Hamiltonians which are general quadratic combinations of classical or quantum canonical variables or of Boson or Fermion annihilation and creation operators.Chapter I, consisting of six sections, deals mainly with sets of canonical forms for quadratic Hamiltonians under the action of the group of linear canonical transformations.By viewing the group of C.C.R.Bogoliubov transformations as Sp(2tf, i?) , a full set of canonical Boson Hamiltonians, labelled by the invariants for symplectic conjugacy classes, is obtained.Since the conjugacy class of a real orthogonal matrix is determined by its eigenvalues, there is only one type of canonical form for quadratic Fermion Hamiltonians.Unlike the Fermion case, indefinite Boson Hamiltonians can not in general be reduced to a sum of quasi-particle number operators.However, it is shown how spectral theory in Krein space rigorizes the full reduction of a strictly positive quadratic Boson Hamiltonian for which the single particle space is infinite dimensional.In the case of infinite degrees of freedom, there are various existence criteria for a unitary operator which implements a given Bogoliubov transformation and the equivalence of these is discussed in Section 6.Cahpter II, consisting of seven sections, applies the techniques of Chapter I to algebraic quantization.The Boson commutation relations , among formal mode operators obtained in heuristic quantization, are shown to be incompatible with the canonical commutation relations among conjugate variables unless the motion is stable.Segal quantization of any real
Gaussian unitaries are specified by a second order polynomial in the bosonic operators, that is, by a quadratic polynomial and a linear term. From the Hamiltonian other equivalent representations of the Gaussian unitaries are obtained, such as Bogoliubov and real symplectic transformations. The paper develops an alternative representation, called complex symplectic transformation, which is more compact and is comprehensive of both Bogoliubov and real symplectic transformations. Moreover, it has other advantages. One of the main results of the theory, not available in the literature, is that the final displacement is not simply given by the linear part of the Hamiltonian, but depends also on the quadratic part. In particular, it is shown that by combining squeezing and rotation, it is possible to achieve a final displacement with an arbitrary amount.
Squeezing processes are commonly described in terms of quadratic Hamiltonians, which generate unitary implementations of Bogoliubov transformations of the quantized electromagnetic field. Here the behaviour of the quasifree, the classical, and the coherent photon states under general squeezing Bogoliubov transformations is investigated. It is found that there is a great variety of mixed classical states, which remain classical under the squeezing operation, whereas each pure classical state becomes non-classical. Especially, some classical, microscopic first order coherent states remain classical and coherent of first order under one-mode squeezing. This contrasts squeezing of macroscopic coherent states.