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Journal of Physics B Atomic Molecular and Optical Physics

Entanglement in a parametric converter

Su‐Yong Lee, Shahid Qamar, Hai-Woong Lee, Muhammad Suhail Zubairy

Published Jun 30, 2008• 12 citations
Publisher

Abstract

In this paper, we consider a parametric converter as a source of entangled radiation. We examine recently derived conditions (Hillery and Zubairy 2006 Phys. Rev. Lett. 96 050503, Duan et al 2000 Phys. Rev. Lett. 84 2722) for determining when the two output modes in a parametric converter are entangled. We show that for different initial field states, the two criteria give different conditions that ensure that the output states are entangled. We also present an input–output calculation for the entanglement of the output field.

DOI
10.1088/0953-4075/41/14/145504

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In this paper, we consider a parametric converter as a source of entangled radiation. We examine recently derived conditions (Hillery and Zubairy 2006 Phys. Rev. Lett. 96 050503, Duan et al 2000 Phys. Rev. Lett. 84 2722) for determining when the two output modes in a parametric converter are entangled. We show that for different initial field states, the two criteria give different conditions that ensure that the output states are entangled. We also present an input–output calculation for the entanglement of the output field.
We examine the implications of several recently derived conditions [Hillery and Zubairy, Phys. Rev. Lett. 96, 050503 (2006)] for determining when a two-mode state is entangled. We first find examples of non-Gaussian states that satisfy these conditions. We then apply the entanglement conditions to the study of several linear devices, the beam splitter, the parametric amplifier, and the linear phase-insensitive amplifier. For the first two, we find conditions on the input states that guarantee that the output states are entangled. For the linear amplifier, we determine in the limit of high and no gain, when an entangled input leads to an entangled output. Finally, we show how application of two two-mode entanglement conditions to a three-mode state can serve as a test of genuine three-mode entanglement.
Abstract Generating and controlling continuous-variable entanglement is a central requirement for quantum information processing protocols. In the present contribution, we investigate the bipartite Gaussian entanglement of the two-mode Gaussian state produced by a parametric converter, with the two cavity modes initially prepared as arbitrary, non-classical single-mode Gaussian states of tunable purity and non-classicality. The entanglement is witnessed using the Simon's separability criterion, which is necessary and sufficient for Gaussian states, along with the Duan's inseparability criterion. We solve its dynamics as functions of the coherent pump-field phase, the cavity decay rates, and the non-classicality and purity of the initial modes. The entanglement evolves as a damped oscillation and emerges only once there is a difference between the non-classicalities of the two input states, becoming stronger as this difference grows. Likewise, the phase of the pump-field controls both the strength and time duration of entanglement. Higher input purity moderately strengthens and prolongs the correlations, whereas the cavity decay rates are the main source of decoherence. Finally, we compare the two criteria and find that the Simon's condition shows nearly continuous entanglement, whereas Duan's criterion responds only where the correlations are strong, leaving transient windows in which the state is separable under Duan yet still inseparable under Simon. These theoretical results suggest that, the non-classicality of the input field and the phase of the pump field can serve as effective control parameters for achieving tunable bipartite Gaussian entanglement, offering guidance for future practical realizations in cavity-based systems.
In this paper, we use a criterion provided by E. Shchukin and W. Vogel [Phys. Rev. Lett. 95, 230502 (2005)] for determining when a bipartite quantum state is entangled. We first show that the entanglement criterion is necessary and sufficient condition for the partial transposition of the bipartite quantum states. Furthermore, previously known entanglement criteria are proved to be special cases of this criterion. We then apply the entanglement criterion to determine entangled property of several non-classical two-mode states. Finally, we provide the conclusion about this criterion.
Recently quantum beat lasers have been considered as a source of entangled radiation [S. Qamar, F. Ghafoor, M. Hillery, and M. S. Zubairy, Phys. Rev. A 77, 062308 (2008)]. We investigate and quantify the entanglement of this system when the initial cavity modes are prepared in a Gaussian two-mode state, one being a nonclassical state and the other a thermal state. It is investigated how the output entanglement varies with the nonclassicality of the input Gaussian state, thermal noise, and the strength of the driving field.