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Bi-fractional transforms in phase space

S. Agyo

Published Jan 1, 2016• 0 citations

Abstract

The displacement operator is related to the displaced parity operator through a two dimensional \nFourier transform. Both operators are important operators in phase space \nand the trace of both with respect to the density operator gives the Wigner functions \n(displaced parity operator) and Weyl functions (displacement operator). The generalisation \nof the parity-displacement operator relationship considered here is called \nthe bi-fractional displacement operator, O(α, β; θα, θβ). Additionally, the bi-fractional \ndisplacement operators lead to the novel concept of bi-fractional coherent states. \nThe generalisation from Fourier transform to fractional Fourier transform can be \napplied to other phase space functions. The case of the Wigner-Weyl function is considered \nand a generalisation is given, which is called the bi-fractional Wigner functions, \nH(α, β; θα, θβ). Furthermore, the Q−function and P−function are also generalised to \ngive the bi-fractional Q−functions and bi-fractional P−functions respectively. The \ngeneralisation is likewise applied to the Moyal star product and Berezin formalism for \nproducts of non-commutating operators. These are called the bi-fractional Moyal star \nproduct and bi-fractional Berezin formalism. \nFinally, analysis, applications and implications of these bi-fractional transforms \nto the Heisenberg uncertainty principle, photon statistics and future applications are \ndiscussed.

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CRC Press eBooks•Jun 1, 2023•0 citations

The displacement operator is related to the displaced parity operator through a two dimensional \nFourier transform. Both operators are important operators in phase space \nand the trace of both with respect to the density operator gives the Wigner functions \n(displaced parity operator) and Weyl functions (displacement operator). The generalisation \nof the parity-displacement operator relationship considered here is called \nthe bi-fractional displacement operator, O(α, β; θα, θβ). Additionally, the bi-fractional \ndisplacement operators lead to the novel concept of bi-fractional coherent states. \nThe generalisation from Fourier transform to fractional Fourier transform can be \napplied to other phase space functions. The case of the Wigner-Weyl function is considered \nand a generalisation is given, which is called the bi-fractional Wigner functions, \nH(α, β; θα, θβ). Furthermore, the Q−function and P−function are also generalised to \ngive the bi-fractional Q−functions and bi-fractional P−functions respectively. The \ngeneralisation is likewise applied to the Moyal star product and Berezin formalism for \nproducts of non-commutating operators. These are called the bi-fractional Moyal star \nproduct and bi-fractional Berezin formalism. \nFinally, analysis, applications and implications of these bi-fractional transforms \nto the Heisenberg uncertainty principle, photon statistics and future applications are \ndiscussed.

The Heisenberg-Weyl group

HW(d)

related to a

d

-dimensional Hilbert space

H(d)

, is enlarged into the Heisenberg-Weyl-parity group

HWP(d)

that incorporates parity transformations. It consists of

2d^3

elements, of which

d^3

elements belong to the

HW(d)

subgroup, and extra

d^3

elements which are related through a Fourier transform with the former ones. It is shown that

HWP(d)

is a generalised version of the dihedral group. The properties of operators that combine displacements and parity, are discussed.

HWP(d)

is shown to be a solvable group, and commutators of its elements perform displacement and parity transformations of quantum states, along loops in the discrete phase space.

2d^2

coherent states related to the

HWP(d)

group are introduced, which consist of

d^2

coherent states related to the

HW(d)

subgroup, and extra

d^2

coherent states which are related through a Fourier transform with the former ones. In noisy cases, expansion of an arbitrary state in terms of the

2d^2

coherent states with Bargmann coefficients, is advantageous in comparison to expansion in terms of the

d^2

coherent states related to

HW(d)

. One of the consequences of the

HWP(d)

group, is a natural unification of the Wigner and Weyl functions. The properties of the unified Wigner-Weyl function are discussed.

Bifractional transformations which lead to quantities that interpolate between other known quantities are considered. They do not form a group, and groupoids are used to describe their mathematical structure. Bifractional coherent states and bifractional Wigner functions are also defined. The properties of the bifractional coherent states are studied. The bifractional Wigner functions are used in generalizations of the Moyal star formalism. A generalized Berezin formalism in this context is also studied.

The Heisenberg-Weyl group H W ( d ) related to a d -dimensional Hilbert space H ( d ) , is enlarged into the Heisenberg-Weyl-parity group H W P ( d ) that incorporates parity transformations. It consists of 2 d 3 elements, of which d 3 elements belong to the H W ( d ) subgroup, and extra d 3 elements which are related through a Fourier transform with the former ones. It is shown that H W P ( d ) is a generalised version of the dihedral group. The properties of operators that combine displacements and parity, are discussed. H W P ( d ) is shown to be a solvable group, and commutators of its elements perform displacement and parity transformations of quantum states, along loops in the discrete phase space. 2 d 2 coherent states related to the H W P ( d ) group are introduced, which consist of d 2 coherent states related to the H W ( d ) subgroup, and extra d 2 coherent states which are related through a Fourier transform with the former ones. In noisy cases, expansion of an arbitrary state in terms of the 2 d 2 coherent states with Bargmann coefficients, is advantageous in comparison to expansion in terms of the d 2 coherent states related to H W ( d ) . One of the consequences of the H W P ( d ) group, is a natural unification of the Wigner and Weyl functions. The properties of the unified Wigner–Weyl function are discussed.

The coupled fractional Wigner-Ville distribution is a more general version of the fractional Wigner-Ville distribution. Main properties including boundedness, Moyal's formula and inversion formula are studied in detail for the transformation. Additionally, the relation of the coupled fractional Wigner-Ville distribution with the two-dimensional Fourier transform is studied. We also present the relationship between the coupled fractional Wigner-Ville distribution with the two-dimensional Wigner-Ville distribution. We show how the properties and relations allow us to derive several versions of the uncertainty inequalities related to the coupled fractional Wigner-Ville distribution.

The language of statistical moments and covariance matrices we introduced in the previous chapter offers a compact and efficient formalism to deal with Gaussian states. A much more general approach may be taken to describe any quantum state in a setting which is reminiscent of classical phase space. This is the framework of quantum characteristic functions and quasi-probability distributions, which goes back to seminal work by Wigner on quantum corrections to classical statistical mechanics, and bloomed in the sixties with the rise of theoretical quantum optics and the emergence of a general unifying picture. Conceptually, the phase space description of quantum states hinges on the completeness of the set of displacement operators, which we shall prove in the form of the Fourier–Weyl relation between density matrices and characteristic functions. This relationship will constitute the bridge between phase space and Hilbert space descriptions, which will be useful in several applications in the context of quantum information. It will be convenient to handle most of the proofs and mathematical arguments concerning characteristic functions and quasi-probability distributions on a single mode of the system. Because displacement operators of multimode systems are just tensor products of local displacement operators, the extension of the formalism to systems with many degrees of freedom will be straightforward. Nevertheless, we shall always take care of explicitly linking the single-mode formulae which will appear in this chapter to the general multimode description adopted in the previous one.