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Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences

Quantum algebras and parity-dependent spectra

C V Sukumar, Andrew Hodges

Published Jul 10, 2007• 12 citations
Publisher

Abstract

We study the structure of a quantum algebra in which a parity-violating term modifies the standard commutation relation between the creation and annihilation operators of the simple harmonic oscillator. We discuss several useful applications of the modified algebra. We show that the Bernoulli and Euler numbers arise naturally in a special case. We also show a connection with Gaussian and non-Gaussian squeezed states of the simple harmonic oscillator. Such states have been considered in quantum optics. The combinatorial theory of Bernoulli and Euler numbers is developed and used to calculate matrix elements for squeezed states.

DOI
10.1098/rspa.2007.0003

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We study the structure of a quantum algebra in which a parity-violating term modifies the standard commutation relation between the creation and annihilation operators of the simple harmonic oscillator. We discuss several useful applications of the modified algebra. We show that the Bernoulli and Euler numbers arise naturally in a special case. We also show a connection with Gaussian and non-Gaussian squeezed states of the simple harmonic oscillator. Such states have been considered in quantum optics. The combinatorial theory of Bernoulli and Euler numbers is developed and used to calculate matrix elements for squeezed states.
This work addresses the study of the oscillator algebra, defined by four parameters ppp, qqq, ααα, and ννν. The time-independent Schrödinger equation for the induced deformed harmonic oscillator is solved; explicit analytic expressions of the energy spectrum are given. Deformed states are built and discussed with respect to the criteria of coherent state construction. Various commutators involving annihilation and creation operators and their combinatorics are computed and analyzed. Finally, the correlation functions of matrix elements of main normal and antinormal forms, pertinent for quantum optics analysis, are computed.
A parity-dependent squeezing operator is introduced which imposes different SU(1,1) rotations on the even and odd subspaces of the harmonic oscillator Hilbert space. This operator is used to define parity-dependent squeezed states which exhibit highly non-classical properties such as strong antibunching, quadrature squeezing, strong oscillations in the photon-number distribution, etc. In contrast to the usual squeezed states whose Q and Wigner functions are simply Gaussians, the parity-dependent squeezed states have much more complicated Q and Wigner functions that exhibit an interesting interference in phase space. The generation of these states by parity-dependent quadratic Hamiltonians is also discussed.
We introduce the inverse of the harmonic-oscillator creation and annihilation operators by their actions on the number states. We then show that three of the two-photon annihilation operators, a^ $^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}\mathrm{\ensuremath{-}}1}$a^, a^a^ $^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}\mathrm{\ensuremath{-}}1}$, and a^ 2^{2}2, possess eigenstates that are analogous to the often-used coherent and squeezed states. A family of the eigenstates of the operator a^ $^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}\mathrm{\ensuremath{-}}1}$a^ is the customary squeezed vacuum S(\ensuremath{\sigma})\ensuremath{\Vert}0〉 while another family of the eigenstates of a^a^ $^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}\mathrm{\ensuremath{-}}1}$ is the state S(\ensuremath{\sigma})\ensuremath{\Vert}n=1〉. We find some interesting properties related to squeezing, bunching, and antibunching exhibited by these states. An alternative method of summing some series by using these states is also considered. It is hoped that these states will find applications in quantum optics and quantum mechanics in general.