Approaching the Tsirelson bound with a Sagnac source of polarization-entangled photons

High-fidelity polarization-entangled photons are a powerful resource for quantum communication, distributing entanglement and quantum teleportation. The Bell-CHSH inequality S\leq2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>S</mml:mi><mml:mo>≤</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math> is violated by bipartite entanglement and only maximally entangled states can achieve S=2\sqrt{2} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>S</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn><mml:msqrt><mml:mn>2</mml:mn></mml:msqrt></mml:mrow></mml:math> , the Tsirelson bound. Spontaneous parametric down-conversion sources can produce entangled photons with correlations close to the Tsirelson bound. Sagnac configurations offer intrinsic stability, compact footprint and high collection efficiency, however, there is often a trade off between source brightness and entanglement visibility. Here, we present a Sagnac polarization-entangled source with 2\sqrt{2}-S=(5.65\pm0.57\times10^{-3}) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mn>2</mml:mn><mml:msqrt><mml:mn>2</mml:mn></mml:msqrt><mml:mo>−</mml:mo><mml:mi>S</mml:mi><mml:mo>=</mml:mo><mml:mo stretchy="false" form="prefix">(</mml:mo><mml:mn>5.65</mml:mn><mml:mo>±</mml:mo><mml:mn>0.57</mml:mn><mml:mo>×</mml:mo><mml:msup><mml:mn>10</mml:mn><mml:mrow><mml:mo>−</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false" form="postfix">)</mml:mo></mml:mrow></mml:math> , on-par with the highest S <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>S</mml:mi></mml:math> parameters recorded, while generating and detecting (4660\pm70)pairs/s/mW <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mo stretchy="false" form="prefix">(</mml:mo><mml:mn>4660</mml:mn><mml:mo>±</mml:mo><mml:mn>70</mml:mn><mml:mo stretchy="false" form="postfix">)</mml:mo><mml:mi>p</mml:mi><mml:mi>a</mml:mi><mml:mi>i</mml:mi><mml:mi>r</mml:mi><mml:mi>s</mml:mi><mml:mi>/</mml:mi><mml:mi>s</mml:mi><mml:mi>/</mml:mi><mml:mi>m</mml:mi><mml:mi>W</mml:mi></mml:mrow></mml:math> , which is a substantially higher brightness than previously reported for Sagnac sources and around two orders of magnitude brighter than for traditional cone sources with the highest S <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>S</mml:mi></mml:math> parameters. Our source records 0.9953\pm0.0003 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mn>0.9953</mml:mn><mml:mo>±</mml:mo><mml:mn>0.0003</mml:mn></mml:mrow></mml:math> concurrence and 0.99743\pm0.00014 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mn>0.99743</mml:mn><mml:mo>±</mml:mo><mml:mn>0.00014</mml:mn></mml:mrow></mml:math> fidelity to an ideal Bell state. By studying systematic errors in Sagnac sources, we identify that the precision of the collection focal point inside the crystal plays the largest role in reducing the S <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>S</mml:mi></mml:math> parameter in our experiment. We provide a pathway that could enable the highest S <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>S</mml:mi></mml:math> parameter recorded with a Sagnac source to-date while maintaining very high brightness.