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Chinese Physics

The solution of the time-dependent Fokker–Planck equation of non-degenerate optical parametric amplification and its application to the optimum realization of EPR paradox

Chao Zhao, TAN WEI-HAN

Published Feb 26, 2007• 7 citations
Publisher

Abstract

In this paper, the solution of the time-dependent Fokker–Planck equation of non-degenerate optical parametric amplification is used to deduce the condition demonstrating the Einstein–Podolsky–Rosen (EPR) paradox. The analytics and numerical calculation show the influence of pump depletion on the error in the measurement of continuous variables. The optimum realization of EPR paradox can be achieved by adjusting the parameter of squeezing. This result is of practical importance when the realistic experimental conditions are taken into consideration.

DOI
10.1088/1009-1963/16/3/015

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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).

Acta Physica Sinica•Jan 1, 2010•0 citations
In this paper, the solution of the time-dependent Fokker–Planck equation of non-degenerate optical parametric amplification is used to deduce the condition demonstrating the Einstein–Podolsky–Rosen (EPR) paradox. The analytics and numerical calculation show the influence of pump depletion on the error in the measurement of continuous variables. The optimum realization of EPR paradox can be achieved by adjusting the parameter of squeezing. This result is of practical importance when the realistic experimental conditions are taken into consideration.
The solution of the time-dependent periodic pumping non-degenerate optical parametric amplifier (NOPA) is derived when the pump depletion is considered both above and below thresholds. Based on this solution, the quantum fluctuation calculated shows that a high entanglement and a good squeezing degree of the parametric light beams are achieved near and above thresholds. We adopt two kinds of pump fields: (i) a continuously modulated pump with a sinusoidal envelope; (ii) a sequence of laser pulses with Gaussian envelopes. We analyse the time evolution of continuous variable entanglement by analytical and numerical calculations, and show that the periodic driven pumping also improves the degree of entanglement. The squeezing and Einstein–Podolsky–Rosen (EPR) entanglement by using the two pumping driven functions are investigated from below to above the threshold regions, the tendencies are nearly the same, and the Gaussian driven function is superior to that of the sine driven function, when the maximum squeezing and the minimum variance of quantum fluctuation are considered. In the meantime, the generalization of frequency degenerate OPA to frequency non-degenerated OPA problem is investigated.
Over the past three decades, numerous demonstrations of the violation of Bell’s inequalities have been reported in optical experiments based upon observations of polarization correlations for photon pairs. Without exception, these measurements as well as those in other systems have involved discrete variables for which the Bell inequalities are applicable. By contrast, an experiment to demonstrate the original proposal by Einstein, Podolsky, and Rosen (EPR)1 for a system of observables with a continuous spectrum has not previously been realized. From many possible theoretical avenues, we have followed the suggestion of Reid and Drummond2 to achieve a demonstration of the EPR paradox for continuous variables by employing a subthreshold optical parametric oscillator (OPO) operated in a frequency-degenerate but polarization-nondegenerate mode.
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).