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Chinese Journal of Physics

Finite Dimensional Trio-Coherent States and Some Non-Classical Properties

A.‐S. F. Obada, M. M. A. Ahmed, Eied Mahmoud Khalil, Sellami Ali

Published Aug 1, 2008• 3 citations
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Abstract

Mathematics Department, Faculty of Science,Al-Azher University, Nassr City 11884, Cairo, Egypt(Received November 13, 2007)A new state of the electromagnetic field is introduced. It is a correlated three-modestate but in finite dimensions. Based on the resonant ion-cavity interaction, we propose ascheme to generate these states, revealing their connection with the nonlinear three-modeinteraction. A relation to the realization of the SU(2) algebra is sought. We also investigatesome of their non-classical properties.

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Mathematics Department, Faculty of Science,Al-Azher University, Nassr City 11884, Cairo, Egypt(Received November 13, 2007)A new state of the electromagnetic field is introduced. It is a correlated three-modestate but in finite dimensions. Based on the resonant ion-cavity interaction, we propose ascheme to generate these states, revealing their connection with the nonlinear three-modeinteraction. A relation to the realization of the SU(2) algebra is sought. We also investigatesome of their non-classical properties.
Using the nonlinear coherent states approach, a general formalism for the construction of various classes of nonlinear trio coherent states in the context of multi-mode quantum states has been proposed. In particular, taking into account the most popular nonlinearity function associated with f -deformed coherent states, i.e. the nonlinearity function of the centre-of-mass motion of a trapped ion, it is illustrated that the corresponding trio coherent state possesses some interesting nonclassical properties. To establish this observation, sub-Poissonian statistics, three-mode squeezing, the behaviour of Vogel's characteristic function and the existence of entanglement are investigated. Due to the intrinsic non-classicality nature of the considered three-mode states, their physical production may be of high interest in the quantum optics field. Thus, we have finally demonstrated how nonlinear motional trio coherent states can be generated in three-dimensional anisotropic traps, appropriately.
We introduce a novel class of higher-order, three-mode states called K-dimensional trio coherent states. We study their mathematical properties and prove that they form a complete set in a truncated Fock space. We also study their physical content by explicitly showing that they exhibit nonclassical features such as oscillatory number distribution, sub-poissonian statistics, Cauchy-Schwarz inequality violation and phase-space quantum interferences. Finally, we propose an experimental scheme to realize the state with K=2 in the quantized vibronic motion of a trapped ion.
In this contribution we study a superposition of two finite dimensional trio coherent states (FTCS). The state is regarded as a correlated three-mode state in finite dimensional bases. The framework of Pegg and Barnett formalism, and the phase distribution in addition to the Poissonian distribution are examined. It is shown that the eigenvalue of the difference of the photon number (the q-parameter) is responsible for the non-classical phenomenon. Furthermore, the quasi-probability distribution functions (the Wigner and Q-functions) are also discussed. In this case and for the Wigner function the non-classical behavior is only reported for the odd values of the q-parameter.