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arXiv:quant-ph/9911114 · Journal of Optics B Quantum and Semiclassical Optics

Ladder operator formalisms and generally deformed oscillator algebraic structures of quantum states in Fock space

Xiaoguang Wang

Published Jul 31, 2000• 3 citations
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Abstract

We show that various kinds of one-photon quantum states studied in the field of quantum optics admit ladder operator formalisms and have the generally deformed oscillator (GDO) algebraic structure. The two-photon case is also considered. We obtain the ladder operator formalisms of two general states defined in the even/odd Fock space. The two-photon states may also have a GDO algebraic structure. Some interesting examples of one- and two-photon quantum states are given.

DOI
10.1088/1464-4266/2/4/312

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We show that various kinds of one-photon quantum states studied in the field of quantum optics admit ladder operator formalisms and have the generally deformed oscillator (GDO) algebraic structure. The two-photon case is also considered. We obtain the ladder operator formalisms of two general states defined in the even/odd Fock space. The two-photon states may also have a GDO algebraic structure. Some interesting examples of one- and two-photon quantum states are given.
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The generally deformed oscillator (GDO) and its multiphoton realization as well as the coherent and squeezed vacuum states are studied. We discuss, in particular, the GDO depending on a complex parameter q (therefore we call it q-GDO) together with the finite dimensional cyclic representations. As a realistic physical system of GDO the isospectral oscillator system is studied and it is found that its coherent and squeezed vacuum states are closely related to those of the oscillator. It is pointed out that starting from the q-GDO with q root of unity one can define the hermitian phase operators in quantum optics consistently and algebraically. The new creation and annihilation operators of the Pegg-Barnett type phase operator theory are defined by using the cyclic representations and these operators degenerate to those of the ordinary oscillator in the classical limit q → 1.
The ladder operator formalism of a general quantum state for su(1,1) Lie algebra is obtained. The state bears the generally deformed oscillator algebraic structure. It is found that the Peremolov's coherent state is a su(1,1) nonlinear coherent state. The expansion and the exponential form of the nonlinear coherent state are given. We obtain the matrix elements of the su(1,1) displacement operator in terms of the hypergeometric functions and the expansions of the displaced number states and Laguerre polynomial states are followed. Finally some interesting su(1,1) optical systems are discussed.
We construct the deformed ladder operators in the presence of a minimal length to study the one- and two-mode squeezed harmonic oscillator. The generalized Hamiltonian of the system is expressed in terms of a deformed su(1,1)su(1,1)su(1,1) algebra. The realizations of this algebra allow us to convert the purely quantum mechanical problem of the model into a differential equation. By means of the Nikiforov–Uvarov method, the energy eigenvalues are obtained and the corresponding wave functions, in the momentum space, are expressed in terms of hypergeometric functions. Our study shows that the domain of existence of the energy levels is extended and this extension is due to the deformation parameter.
The ladder operator formalism of a general quantum state for su(1, 1) Lie algebra is obtained. The state bears the generally deformed oscillator algebraic structure. It is found that the Perelomov's coherent state is a su(1, 1) nonlinear coherent state. The expansion and the exponential form of the nonlinear coherent state are given. We obtain the matrix elements of the su(1, 1) displacement operator in terms of the hypergeometric functions and the expansions of the displaced number states and Laguerre polynomial states are followed. Finally some interesting su(1, 1) optical systems are discussed.
A description of the Fock space of a parabose oscillator of order p based on commutation relations bilinear in creation and annihilation operators rather than trilinear ones is developed. A new statement of the well-known Green’s ansatz is introduced to help understand the significance of the bilinear commutation relations. It is also applied to show how all parabose oscillators of even order (respectively, odd order) can be obtained as irreducible constituents of the Fock space associated with a Green’s ansatz of two (three) terms. Analogs of the form in which the parabose Green’s ansatz is presented are provided for one parafermi oscillator and for systems of many parabose and parafermi oscillators of the same order. Methods used in the paper allow various new results for q-deformed parabose oscillators to be derived.