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Journal of Optics B Quantum and Semiclassical Optics

Nonunitary squeezing and biorthogonal scalar products in polarization optics

Y. Ben-Aryeh

Published Nov 4, 2005• 7 citations
Publisher

Abstract

The polarization vectors of monochromatic electromagnetic fields can be transformed by multiplying them by 2 ? 2 matrices representing, respectively, retardation plates, rotators and partial polarizers. The present description of partial polarizers is related to a two-dimensional Lorentz algebra SU(1,1) leading to nonunitary squeezing as the present SU(1,1) algebra includes non-Hermitian operators in addition to the Hermitian ones. The present analysis exploits the use of similarity transformations which are more general than the unitary transformations used usually in quantum mechanics. By performing a similarity transformation on a set of Hermitian physical operators the transformed operators are Hermitian with respect to new scalar product defined by the use of a new metric. It is shown that such transformations can be implemented experimentally by the use of polarization optics and the metric obtained in such transformations is analysed.

DOI
10.1088/1464-4266/7/12/004

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The polarization vectors of monochromatic electromagnetic fields can be transformed by multiplying them by 2 ? 2 matrices representing, respectively, retardation plates, rotators and partial polarizers. The present description of partial polarizers is related to a two-dimensional Lorentz algebra SU(1,1) leading to nonunitary squeezing as the present SU(1,1) algebra includes non-Hermitian operators in addition to the Hermitian ones. The present analysis exploits the use of similarity transformations which are more general than the unitary transformations used usually in quantum mechanics. By performing a similarity transformation on a set of Hermitian physical operators the transformed operators are Hermitian with respect to new scalar product defined by the use of a new metric. It is shown that such transformations can be implemented experimentally by the use of polarization optics and the metric obtained in such transformations is analysed.
This paper constitutes an application of the polarization optics in the problem of quantum measurement. The non-Hermitian operators of the nonorthogonal multilayer optical polarizers represent observables in the sense of the generalized quantum theory of measurement. The intimate spectral structure of these polarizers can be disclosed in the frame of skew-angular vector bases and biorthonormal vector systems. We show that these polarizers correspond to skew projectors; their operators are "generated" by skew projectors in the sense of the spectral theorem of linear operators theory. Thus the common feature of all the polarizers (Hermitian and non-Hermitian) is that their "nuclei" are (orthogonal or skew) projectors--the generating projectors.
Despite acute interest in the dynamics of non-Hermitian systems, there is a lack of consensus in the mathematical formulation of non-Hermitian quantum mechanics in the community. Different methodologies are used in the literature to study non-Hermitian dynamics. This ranges from consistent frameworks like biorthogonal quantum mechanics and metric approach characterized by modified inner products, to normalization by time-dependent norms inspired by open quantum systems. In this work, we systematically explore the similarities and differences among these various methods. Utilizing illustrative models with exact solutions, we demonstrate that these methods produce not only quantitatively different results but also distinct physical interpretations. For dissipative systems where non-Hermiticity arises as an approximation, we find that simply dividing by the norm in the PT -broken regime closely aligns with the full master equation solutions. In contrast, for quantum systems where non-Hermiticity can be engineered exactly, incorporating metric dynamics is crucial for the probabilistic interpretation of quantum mechanics, necessitating the generalizations of similarity transformations and unitarity to non-Hermitian systems. This study lays the groundwork for further exploration of non-Hermitian Hamiltonians, potentially leveraging generalized transformations for novel physical phenomena.
The normally ordered Hilbert space image of the general (complex) linear similarity transformation in phase space is obtained in a coherent state representation. Although preserving the commutator of a and a + , these quantum mechanical images of classical transformations are generally not unitary. Remarkably, although the kets and bras produced by the non-unitary similarity transformations are not Hermitian conjugates, squeezed state analogues produced using the similarity transformation still satisfy an overcompleteness relation. The results are extended to two-mode Fock space and simple examples of the utility of the similarity transformation are presented. The evaluation of the normally ordered operators is greatly facilitated by the use of integration within ordered products.
In recent years, non-Hermitian quantum physics has gained a great deal of popularity in the quantum optics and condensed matter communities in order to model quantum systems with varying symmetries. In this paper, we identify a non-standard inner product that implies bosonic commutator relations for local electric and magnetic field observables and leads to a natural local biorthogonal description of the quantised electromagnetic field. When comparing this description with an alternative local Hermitian description, in which the states of local photonic particles, i.e., of so-called bosons localised in position (blips), are orthogonal under the conventional Hermitian inner product, we find that there is an equivalence between the two approaches. Careful consideration needs to be given to the physical interpretation of the different descriptions. Whether a Hermitian or a non-Hermitian approach is more suitable depends on the circumstances that we want to model.
Continuing the line of development of Paper I [J. Math. Phys. 15, 1295 (1974)], we enlarge the concept of canonical transformations in quantum mechanics in two directions: first, by allowing the definition of a canonical transformation to be made through the preservation of an so(2,1) algebra, rather than the usual Heisenberg algebra, and providing the bridge between the classical and quantum mechanical descriptions, and, second, through the complexification of the transformation group. In this paper we study specifically the transformations which can be interpreted as the radial part of n-dimensional complex linear transformations in Paper I. We show that we can build Hilbert spaces of analytic functions with a scalar product defined through integration over half the complex plane of a variable which has the meaning of a complex radius. A unitary mapping to the ordinary Hilbert space Lrn−12(0,∞) is provided with a kernel involving a Bessel function. Special cases of this are shown to be the Barut-Girardello, one-dimensional Bargmann and Hankel transforms. The transform kernels provide a series of representations of a subsemigroup of SL(2,C) and allow the construction of coherent states for the harmonic oscillator with an extra centrifugal force. We present a hyperdifferential operator realization of these transforms which yields new Baker-Campbell-Hausdorff and special function relations.