The displacement operator is related to the displaced parity operator through a two dimensional
Fourier transform. Both operators are important operators in phase space
and the trace of both with respect to the density operator gives the Wigner functions
(displaced parity operator) and Weyl functions (displacement operator). The generalisation
of the parity-displacement operator relationship considered here is called
the bi-fractional displacement operator, O(α, β; θα, θβ). Additionally, the bi-fractional
displacement operators lead to the novel concept of bi-fractional coherent states.
The generalisation from Fourier transform to fractional Fourier transform can be
applied to other phase space functions. The case of the Wigner-Weyl function is considered
and a generalisation is given, which is called the bi-fractional Wigner functions,
H(α, β; θα, θβ). Furthermore, the Q−function and P−function are also generalised to
give the bi-fractional Q−functions and bi-fractional P−functions respectively. The
generalisation is likewise applied to the Moyal star product and Berezin formalism for
products of non-commutating operators. These are called the bi-fractional Moyal star
product and bi-fractional Berezin formalism.
Finally, analysis, applications and implications of these bi-fractional transforms
to the Heisenberg uncertainty principle, photon statistics and future applications are
discussed.
| Identifer | oai:union.ndltd.org:BRADFORD/oai:bradscholars.brad.ac.uk:10454/14522 |
| Date | January 2016 |
| Creators | Agyo, Sanfo D. |
| Contributors | Vourdas, Apostolos |
| Publisher | University of Bradford, Faculty of Engineering and Informatics, School of Electrical Engineering and Computer Science |
| Source Sets | Bradford Scholars |
| Language | English |
| Detected Language | English |
| Type | Thesis, doctoral, PhD |
| Rights | <a rel="license" href="http://creativecommons.org/licenses/by-nc-nd/3.0/"><img alt="Creative Commons License" style="border-width:0" src="http://i.creativecommons.org/l/by-nc-nd/3.0/88x31.png" /></a><br />The University of Bradford theses are licenced under a <a rel="license" href="http://creativecommons.org/licenses/by-nc-nd/3.0/">Creative Commons Licence</a>. |
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