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arXiv:quant-ph/0410081 · Physical Review A

Effects of mode coupling on the generation of quadrature Einstein-Podolsky-Rosen entanglement in a type-II optical parametric oscillator below threshold

Julien Laurat, T. Coudreau, G. Keller, Nicolas Treps, Claude Fabre

Published Feb 18, 2005• 58 citations
arXivHTMLPDF

Abstract

We study theoretically and experimentally the quantum properties of a type-II frequency-degenerate optical parametric oscillator below threshold with a quarter-wave plate inserted inside the cavity which induces a linear coupling between the orthogonally polarized signal and idler fields. This original device provides a good insight into general properties of two-mode Gaussian states, illustrated in terms of the covariance matrix. We report on the experimental generation of a two-mode squeezed vacuum on nonorthogonal quadratures depending on the plate angle. After a simple operation, the entanglement is maximized and put into standard form, i.e., quantum correlations and anticorrelations on orthogonal quadratures. A half sum of squeezed variances as low as $0.33\ifmmode\pm\else\textpm\fi{}0.02$, well below the unit limit for inseparability, is obtained and the entanglement measured by the entropy of formation.

DOI
10.1103/physreva.71.022313

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We study theoretically and experimentally the quantum properties of a type-II frequency-degenerate optical parametric oscillator below threshold with a quarter-wave plate inserted inside the cavity which induces a linear coupling between the orthogonally polarized signal and idler fields. This original device provides a good insight into general properties of two-mode Gaussian states, illustrated in terms of the covariance matrix. We report on the experimental generation of a two-mode squeezed vacuum on nonorthogonal quadratures depending on the plate angle. After a simple operation, the entanglement is maximized and put into standard form, i.e., quantum correlations and anticorrelations on orthogonal quadratures. A half sum of squeezed variances as low as 0.33\ifmmode±\else\textpm\fi0.020.33\ifmmode\pm\else\textpm\fi{}0.020.33\ifmmode±\else\textpm\fi0.02, well below the unit limit for inseparability, is obtained and the entanglement measured by the entropy of formation.
We report on the experimental demonstration of strong quadrature Einstein-Podolsky-Rosen entanglement and squeezing at very low noise sideband frequencies produced by a single type-II, self-phase-locked, frequency degenerate optical parametric oscillator below threshold. The generated two-mode squeezed vacuum state is preserved for noise frequencies as low as 50kHz50\phantom{\rule{0.3em}{0ex}}\mathrm{kHz}50kHz. Designing simple setups able to generate nonclassical states of light in the kHz regime is a key challenge for high sensitivity detection of ultraweak physical effects such as gravitational wave or small beam displacement.
We study the far-field spatial distribution of the quantum fluctuations in the transverse profile of the output light beam generated by a type-II optical parametric oscillator below threshold, including the effects of transverse walk-off. We study how quadrature field correlations depend on the polarization. We find spatial Einstein-Podolsky-Rosen entanglement in quadrature-polarization components. For the far-field points not affected by walk-off there is almost complete noise suppression in the proper quadratures difference of any orthogonal polarization components. We show the entanglement of the state of symmetric, intense, or macroscopic, spatial light modes. We also investigate nonclassical polarization properties in terms of the Stokes operators. We find perfect correlations in all Stokes parameters measured in opposite far-field points in the direction orthogonal to the walk-off, while locally the field is unpolarized and we find no polarization squeezing.
The optical parametric oscillator is one of the most well-characterized devices in nonlinear optics, with many useful applications, especially as a frequency converter and as a wide-band amplifier. Novel discoveries made with these devices in the quantum domain include demonstrations of large amounts of both single-mode and two-mode squeezing, and, in the two-mode case, significant quantum intensity correlations between the two modes and quadrature correlations, which provided the first experimental demonstration of the original Einstein–Podolsky–Rosen (EPR) paradox. In this chapter, a systematic theory is developed of quantum fluctuations in the degenerate parametric oscillator. The quantum statistical effects produced by the degenerate parametric amplifier and oscillator have been intensely studied. As we have seen, the amount of transient squeezing produced in an ideal lossless parametric amplifier was calculated using asymptotic methods by Crouch and Braunstein. It was found to scale inversely with the square root of the initial pump photon number. When the system is put into a cavity, and driving and dissipation are introduced, we find that the steady state of a driven degenerate parametric oscillator bifurcates. There is a single steady state at low driving strengths, but, as the driving field increases, a threshold value is reached and the single state splits and forms two stable states. Below the threshold, a linearized analysis of squeezing in the quantum quadratures is possible; while above this threshold, or critical point, the tunneling time between the two states due to quantum fluctuations can be calculated.
For measurements describable by a discrete set of possibilities, numerous demonstrations of the violation of Bell's inequalities have been reported in optical physics based on observations of polarization correlations for photon pairs. By contrast for observables with a continuous spectrum, there appears to be no general inequalities corresponding to those of Bell and no experimental realization of the original Einstein-Podolsky-Rosen paradox. Following Reid and Drummond,1 we are attempting to demonstrate the EPR paradox in the context of the measurement of continuous quantum variables with a subthreshold optical parametric oscillator operated in a frequency degenerate but polarization nondegenerate mode. In this case, the EPR paradox is formulated in terms of two sets of quadrature phase amplitudes for the spatially separated signal and idler fields. As the parametric gain is increased toward threshold, the quadrature amplitudes of the signal and idler beams become increasingly faithful quantum copies of one another. Our measurements of quadrature amplitudes are carried out using independent balanced homodyne detectors at two sites. When operated for the detection of squeezing at one site, noise reductions of 4 dB relative to the vacuum state limit are observed, which is most encouraging since noise reductions >3 dB in the difference photocurrent from the two sites are sufficient for inference at a distance of a signal observable from a measurement of the corresponding idler variable.
The bright Einsten-Podolsky-Rosen (EPR) beams with the quantum correlations between the quadrature-phase amplitudes of the spatially separated signal and idler beams have been experimentally generated from a cw nondegenerate optical parametric amplifier injected by seed waves with degenerate frequency but orthogonal polarization. The correlation degree of $0.853\ifmmode\pm\else\textpm\fi{}0.004$ between the quadrature-phase amplitudes of the output entangled beams is directly inferred from the measured quadrature-phase squeezing of the output vacuum squeezed-state light field formed from superposition of the original signal and idler modes. Our theoretical calculation and experimental measurements provide a reliable method to confirm the quadrature phase-squeezing and EPR correlation of bright light field.