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arXiv:quant-ph/0612104 · Physical Review Letters

Anisotropically and High Entanglement of Biphoton States Generated in Spontaneous Parametric Down-Conversion

Mikhail Vladimirovich Fedorov, Maxim A. Efremov, P. A. Volkov, Ekaterina V. Moreva, Stanislav S. Straupe, S. P. Kulik

Published Aug 6, 2007• 98 citations
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Abstract

We show both theoretically and experimentally that biphoton wave packets generated via spontaneous parametric down-conversion can be strongly anisotropic and highly entangled. The conditions under which these effects exist are found and discussed.
DOI
10.1103/physrevlett.99.063901

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We show that the wave packet of a biphoton generated via spontaneous parametric down-conversion is strongly anisotropic. Its anisotropic features manifest themselves very clearly in comparison with measurements performed in two different schemes: When the detector scanning plane is perpendicular or parallel to the plane containing the crystal optical axis and the laser axis. The first of these two schemes is traditional whereas the second one gives rise to such unexpected results, such as anomalously strong narrowing of the biphoton wave packet measured in the coincidence scheme and very high degree of entanglement. The results are predicted theoretically and confirmed experimentally.
Wave packet structure of biphoton states arising in the process of spontaneous parametric down-conversion (SPDC) is studied theoretically and experimentally, and its relation with entanglement is discussed. In theory, a new expression is found for the biphoton wave function, in which anisotropy of the crystal refractive index is properly taken into account. Under the conditions of the experiment we report, the anisotropy effects are shown to be very strong. A scheme, optimal for observing extremely narrow coincidence angular photon distributions is found to be that in which the plane containing both the laser axis and the optical axis of a crystal coincides with the observation plane. The opposite, commonly investigated, case is that when these two planes are orthogonal to each other. We show that in the optimal geometry the coincidence angular distribution can be almost 20 times narrower than in the traditional scheme of measurements, and about 10 times narrower than the pump. Narrowing of coincidence distributions in the optimal geometry is accompanied by broadening the single-particle distributions, and this broadening is stronger pronounced in the optimal than in the commonly used geometry. The ratio R of widths of the single-particle and coincidence distributions in the optimal geometry appears to be as large as about 100, and we interpret this as an appropriate evaluation of the degree of entanglement accumulated in biphoton states. The results described are obtained theoretically and perfectly confirmed experimentally.
We investigate theoretically the strong anisotropy in the wave packet distribution of biphoton pairs generated via spontaneous parametric down-conversion (SPDC) process with biaxial crystals. From the angle relationship in biaxial crystals, we deduce the wave function of type I SPDC by considering the longitudinal detuning under paraxial approximation. Taking BiB3O6 (BIBO) as an example, we numerically simulate the single-particle and coincidence distributions in two observation planes under the optimum phase-matching scheme. In contrast to uniaxial crystals, a stronger anisotropy effect and a higher entanglement of the photon pair states are obtained, which are attributed to the complex structure of biaxial crystals. These results are important for the generation of highly entangled biphoton pairs, especially for multi-photon pairs generation.
We investigate spectral and temporal features and entanglement of biphoton wave packets formed in spontaneous parametric down-conversion with a pulsed laser pump. The degree of entanglement is characterized by the experimentally measurable parameter RRR defined as the ratio of the coincidence and single-particle spectral widths. In the frequency representation, this parameter is found as a function of the pump-pulse duration $\ensuremath{\tau}$. The function $R(\ensuremath{\tau})$ is shown to have a minimum and even in the minimum, at rather natural conditions, the degree of entanglement is found to be very high (Rmin=73)({R}_{\mathrm{min}}=73)(Rmin​=73). The Schmidt number KKK is found analytically for both short and long pump pulses and interpolated for arbitrary pulse durations. All functional dependences of RRR and KKK are found to be identical and numerical coefficients are found to be rather close. Two-time temporal wave function of a biphoton state is investigated in detail, and a rather significant difference between the cases of short and long pump pulses is found to occur. In the case of long pulses, the temporal parameter Rt{R}_{t}Rt​ (defined as the ratio of durations of the single-particle and coincidence signals) is shown to be very close to the Schmidt number KKK.