Continuous-variable Einstein-Podolsky-Rosen paradox with traveling-wave second-harmonic generation

The early days of quantum mechanics saw many debates about the applicability of classical concepts, such as position and momentum. Two of the main protagonists were Einstein and Bohr, with Einstein, Podolsky, and Rosen (EPR) publishing a famous paper in 1935 [1], setting out what became known as the Einstein-Podolsky-Rosen paradox, and claiming that quantum mechanics was incomplete. Bohr replied with a paper arguing that EPR did not establish this incompleteness, but rather showed the inapplicability of classical descriptions in the quantum domain [2]. Although Bell suggested a set of inequalities that could be violated by quantum mechanics, but not by local hidden variable theories [3], a direct and feasible demonstration of the EPR paradox with continuous variables was first suggested by Reid [4]. The proposal was for an optical demonstration of the paradox via quadrature phase amplitudes, using nondegenerate parametric amplification (also known as the OPA), and was closely related to the original version, which considered position and momentum. The essential step in the EPR argument is to introduce correlated (entangled) states of at least two particles which persist even when the particles become spatially separated. According to EPR, depending on which property of one group of particles that we choose to measure, we can predict with some certainty the values of physical quantities of the other group of particles. If these properties are represented by noncommuting operators (such as position and momentum, or quadrature amplitudes), we may seemingly violate the Heisenberg uncertainty principle. The EPR conclusion was therefore that the description of physical reality given by quantum mechanics is not complete. Quantum optics was one of the first areas of physics which allowed for simple investigations of some of these fundamental mysteries and paradoxes of quantum mechanics [5]. Among the simplest possible quantum optical systems which can exhibit nonclassical behaviour are traveling-wave second-harmonic generation (SHG) and parametric downconversion. The process of traveling-wave downconversion has been of special interest because it allows for many experiments concerned with the fundamentals of quantum mechanics, among these violations of Bell’s inequalities [6–8] and the closely related topic of preparation of EinsteinPodolsky-Rosen states [4,9]. Intracavity second-harmonic generation with transverse degrees of freedom has also been suggested as a source of EPR correlations [10]. What we will demonstrate in this article is that the experimentally simple system of traveling-wave SHG is also a good candidate for a demonstration of the EPR paradox.