Extended phase-matching conditions for improved entanglement generation

Entanglement is the cornerstone of quantum information technology. Its wide-ranging applications include tests of the foundations of quantum mechanics, implementations of quantum algorithms, cryptographic communication procedures, measurement-accuracy enhancements, etc. Arguably, the most important source of entanglement is spontaneous parametric down-conversion ~SPDC! in nonlinear optical crystals. In the down-conversion process, a photon from an intense pump beam is absorbed in a nonlinear crystal and two different photons ~conventionally called signal and idler! are created. Energy conservation dictates that the frequencies of the signal and idler photons sum to that of the pump photon. Frequency entanglement arises from quantum superposition of the various ways in which this energy constraint is fulfilled by signal and idler photons belonging to a certain frequency interval. This interval is governed by phase matching within the nonlinear crystal, a condition that is dependent on the crystal’s length and the refractive indices along its principal optical axes @1#. In this paper we show how to obtain new kinds of frequency entanglement by generalizing the conventionally used phase-matching condition. The resulting biphoton states, which were previoudly obtained by Erdmann et al. @14#, can be characterized using two interferometric setups based on the Hong-Ou-Mandel and on the Mach-Zehnder interferometers, as described below. As will be shown, the extended conditions can be fulfilled with available crystals. They allow the use of long nonlinear crystals that improve the generation efficiency. Furthermore, they allow the use of short-duration pump pulses that increase the experimental signal-idler generation rate compared to that of continuouswave ~cw! pumping. These two advantages are not readily available with the conventional phase-matching condition: on one hand the use of long type II phase-matched crystals narrows the spectrum over which the down-converted photons are entangled, and, on the other hand, the use of pulsed pumping reduces the entanglement because the indistinguishability of the down-converted photons is adversely affected @2–4#. The paper is organized as follows. In Sec. II we introduce our notation and give a brief derivation of the conventional phase-matching condition. In Sec. III we derive the extended phase-matching conditions and analyze their role in obtaining frequency entanglement. Section IV shows how one can