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arXiv:0908.0381 · Journal of Optics B Quantum and Semiclassical Optics

Single-mode quantum properties of the codirectional Kerr nonlinear coupler: frequency mismatch and exact solution

Faisal A A El-Orany, M Sebawe Abdalla, Jan Peřina

Published Sep 18, 2004• 19 citations
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Abstract

In this paper, we investigate the single-mode quantum properties of the codirectional Kerr nonlinear coupler when the frequency mismatch is involved and a condition for an exact solution of equations of motion is fulfilled. Particularly, we investigate quadrature and principal squeezing, Wigner function, quadrature distribution, phase distribution and phase variance. We show that the quadrature squeezing and the phase variance can exhibit collapse–revival and collapse–revival–subrevival phenomena, respectively, based on the values of the detuning parameter. Furthermore, we analytically demonstrate that the system can generate cat states, in particular, Yurke–Stoler states.

DOI
10.1088/1464-4266/6/11/006

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In this paper, we investigate the single-mode quantum properties of the codirectional Kerr nonlinear coupler when the frequency mismatch is involved and a condition for an exact solution of equations of motion is fulfilled. Particularly, we investigate quadrature and principal squeezing, Wigner function, quadrature distribution, phase distribution and phase variance. We show that the quadrature squeezing and the phase variance can exhibit collapse–revival and collapse–revival–subrevival phenomena, respectively, based on the values of the detuning parameter. Furthermore, we analytically demonstrate that the system can generate cat states, in particular, Yurke–Stoler states.
In this paper, we investigate the single mode quantum properties of the codirectional Kerr nonlinear coupler when the frequency mismatch is involved and a condition for an exact solution of equations of motion is fulfilled. Particularly, we investigate quadrature and principal squeezing, Wigner function, quadrature distribution, phase distribution and phase variance. We show that the quadrature squeezing and the phase variance can exhibit collapse-revival and collapse-revivalsubrevival phenomena, respectively, based on the values of the detuning parameter. Furthermore, we analytically demonstrate that the system can generate cat states, in particular, Yurke-Stoler states. PACS numbers: 42.50Dv,42.60.Gd
Quantum statistical properties of interacting optical fields in Kerr nonlinear couplers may be regarded as one of the most fundamental problems yet to be solved in quantum optics. As an analytical solution to this system is not always possible, a semi-analytic or direct numerical solution must be employed in order to obtain a more complete description of their evolution. This study investigates the quantum properties of four-mode couplers composed of two and four Kerr nonlinear waveguides, both with and without cavity set up via phase space representation. In these systems, the electromagnetic fields are described by their Hamiltonians, while the time-evolution of the systems are described by the Von-Neumann equation. Following standard techniques, the master equations for the density matrices of the systems are obtained and further converted to the corresponding classical Fokker- Planck equations using both positive P and Wigner representations. The corresponding set of Langevin Stochastic equations is then obtained from the Fokker-Planck equations by means of Ito rules. Finally, the systems are integrated numerically and averaged over many trajectories to get the relevant information. The effects of self-action nonlinearity, cross-action coupling and the initial energy of the coherent light on the evolution of the field quadrature variances are investigated. Furthermore, the influences of multimode interaction on the dynamics of the squeezed states are discussed. In this research, we show that the system may produce squeezed states in both quadrature variances for all calculated parameters. The multimode two channel model can provide better squeezing in contrast to the conventional device. The amplitude oscillation of maximal squeezing may be amplified by manipulation of the initial state of the coherent light in the first mode, and the range may be extended through multimode interaction. When the model is contained in a cavity, the system is able to generate better amplitude oscillations and squeezing range with leaf-revival-collapse-like behaviour; with the whole oscillation able to evolute below the standard quantum limit Regular photon transition and self trapping effects of the interacting modes in the four channels model are simulated, and squeezing is predicted for various initial intensities of the coherent field and nonlinear coupling parameter g. The cavity setup caused the system to undergo fluctuation amplification and the potential for different types of squeezing to be generated is observed. Depending on how we choose the interaction parameter values, the system may generate better squeezing in comparison to the cavity-less model.
Quantum statistical properties of interacting optical fields in Kerr nonlinear coupler can be regarded as one of the most fundamental problems in quantum optics. As the analytical solution to these systems is not always possible, a semi-analytic or direct numerical solution has to be employed to obtain more complete description of their evolution. This paper investigates the quantum properties of a four-mode coupler composed of two Kerr nonlinear waveguides via phase-space representation. In this system, the electromagnetic field is described by its Hamiltonian and the time evolution of the system is described by Von-Neumann equation. Following the standard techniques, the master equation for the density matrix of the system is obtained and converted to the corresponding classical Fokker-Planck equation using both positive-P and Wigner representations. The corresponding set of Langevin Stochastic equations are then obtained from the Fokker-Planck equation using Ito rules. Finally, the system is integrated numerically and averaged over many trajectories to get the relevant information. The dynamics of the mean fields, the effects of self-action nonlinearity, cross-action coupling and initial energy of the coherent light on the evolution of the field quadrature variances are investigated. Further, the influences of multimode interaction on the dynamics of the squeezed states are discussed. We show that the system can produce squeezed states in both quadrature variances for all calculated parameters. In addition, the multimode interaction extends the range of squeezing at the expense of its amplitude.
The properties of squeezing, in relation to the different interaction frequencies of the input modes in three-mode Kerr nonlinear coupler consisting of two linearly-coupled Kerr waveguides, were explored. Utilizing the exact representation of a set of noisy-stochastic equations, an investigation was made of the first-order properties of squeezing for two- and three-mode interactions with frequency mismatch. It was observed that enhanced squeezing could be obtained under certain conditions of interactions.
The authors examine both codirectional and contradirectional nonlinear couplers operating by second harmonic generation with particular attention to the effect of linear and nonlinear phase mismatches under the assumption that the second harmonic mode is stimulated by a strong coherent field. Quantum statistical properties are described by a photocount distribution, its factorial moments, quadrature and integrated intensity variances and the uncertainty product, with particular attention to generation and transmission of nonclassical light. Incident beams can be coherent or squeezed with mixed external noise. It is shown that, in general, the nonlinear phase mismatch, which effectively makes the interaction weaker, may reduce nonclassical effects and their transmission to signal linear waveguide. However, as a consequence of the phase mismatches some initial regimes of the interaction can be repeated, which leads to new interesting nonclassical effects.