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Canonical Transformations

Bryan J. Dalton, John Jeffers, Stephen Mark Barnett

Published Nov 13, 2014• 0 citations
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Abstract

Abstract This chapter focuses on linear canonical unitary transformations of mode annihilation and creation operators, both for bosons and for fermions. The generators of such transformations (which form a Lie algebra) are introduced for the specific case of two-mode systems, and the effects of these transformations on two-mode Fock states are evaluated. Applications of linear canonical transformations to two-mode interference associated with beam splitters and the distinction between boson and fermion cases are presented, followed by a treatment of pair creation processes such as squeezing in one- and two-mode bosonic systems. Treatments of thermofields – used to map mixed thermal states of single-mode systems onto pure states of two-mode systems, which are related to the two-mode vacuum state by a linear transformation – and the Bogoliubov transformations – used in describing excitations in Bose–Einstein condensates – round off the chapter.

DOI
10.1093/acprof:oso/9780199562749.003.0006

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Abstract This chapter focuses on linear canonical unitary transformations of mode annihilation and creation operators, both for bosons and for fermions. The generators of such transformations (which form a Lie algebra) are introduced for the specific case of two-mode systems, and the effects of these transformations on two-mode Fock states are evaluated. Applications of linear canonical transformations to two-mode interference associated with beam splitters and the distinction between boson and fermion cases are presented, followed by a treatment of pair creation processes such as squeezing in one- and two-mode bosonic systems. Treatments of thermofields – used to map mixed thermal states of single-mode systems onto pure states of two-mode systems, which are related to the two-mode vacuum state by a linear transformation – and the Bogoliubov transformations – used in describing excitations in Bose–Einstein condensates – round off the chapter.
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
Canonical transformations are defined and discussed along with the coherent and the ultracoherent vectors. It is shown that the single-mode and the n -mode squeezing operators are elements of the group of canonical transformations. An application of canonical transformations is made, in the context of open quantum systems, by studying the effect of squeezing of the bath on the decoherence properties of the system. Two cases are analysed. In the first case, the bath consists of a massless bosonic field with the bath reference states being the squeezed vacuum states and squeezed thermal states while in the second case a system consisting of a harmonic oscillator interacting with a bath of harmonic oscillators is analysed with the bath being initially in a squeezed thermal state.
We study in this series the group theoretical structure of Fermion many-body systems arising from the canonical anticommutation relation of the annihilation-creation operators. Owing to the canonical anticommutation relation, a Fermion system with N single particle states has at least six Lie algebras of Fermion operators, a U(N), an SO(2N), an SO(2N+1), an SO(2N+2) and two U(N+1) Lie algebras. There are also two Clifford algebras of 2N and 2N+1 dimensions. The Fermion space is shown to belong to the spinor representations of the SO(2N), SO(2N+1) and SO(2N+2) groups. The canonical transformations generated by the U(N), SO(2N) and SO(2N+1) Lie algebras are characterized as the transformations to induce the linear U(N), SO(2N) and SO(2N+1) transformations for the Clifford algebras. The independent (quasi-) particle type wave functions of three kinds including the Hartree-Fock and Hartree-Bogoliubov wave functions are constructed by means of the canonical transformations and their relationship is studied. We derive three exact generator coordinate representations for state vectors in which the generator coordinates are the U(N), SO(2N) or SO(2N+1) group and the generating functions are the independent (quasi-) particle type wave functions. We characterize the structures of state vectors in the generator coordinate representations and study the relationship of the three representations.
Abstract This chapter sets out the second-quantisation treatment for systems of identical bosons or fermions. Key features such as the symmetrisation principle and super-selection rules are presented. Orthogonality and completeness requirements for mode functions are set out, and Fock states are defined in terms of mode occupancy, leading to the definitions of mode annihilation and creation operators and the determination of their commutation (bosons) or anticommutation (fermions) rules. Important two-mode states such as binomial states and relative-phase eigenstates are considered. Field creation and annihilation operators are defined and related to multi-particle position eigenstates. The Hamiltonian is expressed in terms of both mode operators and field operators, state dynamics being treated via Liouville–von Neumann, master or Matsubara equations for the density operator. Normal ordering is introduced and applied to expressions for the vacuum state projector. Multi-particle position probabilities are considered and related to normally ordered quantum correlation functions.
Classical linear transformations in coordinate-momentum phase space are mapped to unitary quantum-mechanical operators in Hilbert space to produce a generalization of squeeze operators. The classical to quantum transition is manifestly apparent in the derivation. The unitary operators are evaluated in the coherent-state representation using the ``integration within ordered products'' technique in the one- and two-mode cases. Application of these operators results in a new generalization of coherent states.