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Journal of the Optical Society of America B

Multipartite continuous-variable entanglement in nondegenerate optical parametric amplification system

Chao Zhao, Wei Han Tan, Jiang Rong Xu, Fan Ge

Published Apr 12, 2011• 6 citations
Publisher

Abstract

A nondegenerate optical parametric amplification (NOPA) system with N modes can be resolved into combinations of N degenerate parametric amplification (DOPA j, j=1–N) systems. Based on the quantum fluctuations, we extend the method of V1 criterion for bipartite entanglement to multipartite entanglement. The entanglement characteristics for N=2–5 are investigated, the variance V1 decreases as a function of interaction time t, the great N brings a very small variance V1.

DOI
10.1364/josab.28.001067

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A nondegenerate optical parametric amplification (NOPA) system with N modes can be resolved into combinations of N degenerate parametric amplification (DOPA j, j=1–N) systems. Based on the quantum fluctuations, we extend the method of V1 criterion for bipartite entanglement to multipartite entanglement. The entanglement characteristics for N=2–5 are investigated, the variance V1 decreases as a function of interaction time t, the great N brings a very small variance V1.
We experimentally demonstrate that the quantum entanglement between amplitude and phase quadratures of optical modes produced from a nondegenerate optical parametric amplifier (NOPA) can be enhanced and manipulated phase sensitively by means of another NOPA. When both NOPAs operate at deamplification, the entanglement degree is increased at the cavity resonance of the second NOPA. When the first NOPA operates at deamplification and the second one at amplification, the spectral features of the correlation variances are significantly changed. The experimental results are in good agreement with the theoretical expectation.
The multipartite continuous-variable (CV) entanglement characteristic of a multi-mode phased-locked period pumping non-degenerate optical parametric amplification system was investigated for the case when finite bandwidth is considered. A continuously modulated pumping with a sinusoidal envelope can produce an improvement in the minimum values of the multipartite CV entanglement. The minimum variances can be described approximately by the linear theory formula 1/(1+μ). We should take a proper bandwidth Δω in order to obtain a long-term entanglement. The larger values of pump parameter μ and particle number N result in long-term entanglement.
In this paper, we first find out the analytic solution of the time-dependent Fokker-Planck equation of the non-degenerate optical parametric amplification (NOPA) system under the consideration of the dispersion, the loss and the pump depletion effects. Then, through the numerical calculation, we obtain the squeezing characteristic of the degenerate optical parametric amplification (DOPA) system with dispersion. the research indicates: the dispersion effect stems from the nonlinear susceptibility change from χ″ toχ″/{1+σ2/}/+2, with the increasing of the dispersion effect, the general feature of the squeezing curves beeps unchanged, and the curves contract toward left. The maximum squeezing approaches to the linear theory 1/(1+μ). Finally, we obtain the entanglement characteristic of the NOPA system with dispersion. We find out when σ is given, with the increasing of pump parameter μ, he corresponding phase makes a large change. The nonlinear susceptibility changes many times. When the polarity is positive, the system obtains the gain, when the polarity is negative, the system suffers the loss, but the gain is mainly dissipated by the loss, so the net gain is small, the squeezing is also small. The minimum variance V1 reduces gradually, and the whole curve moves to the right, approaches to the linear theory 1/(1+μ).
Usually the quantum fluctuation characteristic of a non-degenerate optical parametric amplifier is analysed under the assumption of monochromatic pumping. However, in experiments, a driving beam with finite bandwidth is used to obtain the non-degenerate signal and idler beam amplifications. On account of that, we derive an analytical solution for the non-degenerate optical parametric amplification system with finite bandwidth laser pumping, and evaluate the associated quantum fluctuation. Finally, the application of the V1 criterion to bipartite entanglement is discussed.
Quantum entanglement shared by more than two partites is the essential bases for developing quantum communication networks and quantum computation. The multipartite entanglement of continuous variables can be generated by distributing or combining squeezed states of light using linear optics. We experimentally obtained a tripartite entangled state of bright optical field by means of a nondegenerate optical parametric amplifier (NOPA) and optical beam-splitters. The NOPA is operated in the state of parametric deamplification by locking the relative phase between the pump laser and the injected seed wave to (2n+1)π, so the generated Einstein-Podolsky-Rosen (EPR) beam for continuous variables is an entangled state of light with the quantum correlation of phase quadratures and the quantum anticorrelation of amplitude quadratures [1]. The output EPR beam consists of two frequency-degenerate optical modes with horizontal and vertical polarizations which are distributed to three parties with two linear beam-splitters. We have theoretically and experimentally demonstrated the three parties are in a tripartite entangled state with three-mode “position” (amplitude quadrature) correlation and relative “momentum” (phase quadrature) correlation [2]-[4]. The obtained GHZ-like state is a fully inseparable tripartite continuous-variables state violating the three inequalities used for the criterion of the inseparability [5][6]. The controlled dense coding quantum communication is experimentally accomplished by exploiting the tripartite entanglement. The three entangled modes are sent to a sender (Alice), a receiver (Bob) and a controller (Claire), respectively. The information transmission capacity of the quantum channel between Alice and Bob is controlled by Claire. The channel capacity completed under Claire’s help is always larger than that without her help.