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arXiv:1409.5696 · Physical Review Letters

Induced Coherence, Vacuum Fields, and Complementarity in Biphoton Generation

Axel Heuer, R. W. Menzel, Peter W. Milonni

Published Feb 4, 2015• 50 citations
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Abstract

It is well established that spontaneous parametric down-conversion with induced coherence across two coupled interferometers results in high-visibility single-photon interference. We describe experiments in which additional photon channels are introduced such that "which-path" information is made possible and the fringe visibility in single-photon interference is reduced in accordance with basic notions of complementarity. However, these additional pathways result in nearly perfect visibility when photons are counted in coincidence. A simplified theoretical model accounts for these observations and attributes them directly to the vacuum fields at the different crystals.

DOI
10.1103/physrevlett.114.053601

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It is well established that spontaneous parametric down-conversion with induced coherence across two coupled interferometers results in high-visibility single-photon interference. We describe experiments in which additional photon channels are introduced such that "which-path" information is made possible and the fringe visibility in single-photon interference is reduced in accordance with basic notions of complementarity. However, these additional pathways result in nearly perfect visibility when photons are counted in coincidence. A simplified theoretical model accounts for these observations and attributes them directly to the vacuum fields at the different crystals.
Coherence can be induced or stimulated in parametric down-conversion using two or three crystals when, for example, the idler modes of the crystals are aligned. Previous experiments with induced coherence [Phys. Rev. Lett. 114, 053601 (2015)] focused on which-path information and the role of vacuum fields in realizing complementarity via reduced visibility in single-photon interference. Here we describe experiments comparing induced and stimulated coherence. Different single-photon interference experiments were performed by blocking one of the pump beams in a three-crystal setup. Each counted photon is emitted from one of two crystals and which-way information may or not be available, depending on the setup. Distinctly different results are obtained in the induced and stimulated cases, especially when a variable transmission filter is inserted between the crystals. A simplified theoretical model accounts for all the experimental results and is also used to address the question of whether the phases of the signal and idler fields in parametric down-conversion are correlated.
Using two crystals for spontaneous parametric down-conversion in a parallel setup, we observe two-photon interference with high visibility. The high visibility is consistent with complementarity and the absence of which-path information. The observations are explained as the effects of entanglement or equivalently in terms of interfering probability amplitudes and also by the calculation of a second-order field correlation function in the Heisenberg picture. The latter approach brings out explicitly the role of the vacuum fields in the down-conversion at the crystals and in the photon coincidence counting. For comparison, we show that the Hong–Ou–Mandel dip can be explained by the same approach in which the role of the vacuum signal and idler fields, as opposed to entanglement involving vacuum states, is emphasized. We discuss the fundamental limitations of a theory in which these vacuum fields are treated as classical, stochastic fields.
Single photon first order interferences of spatially separated regions from the cone structure of spontaneous parametric down conversion allow for analyzing the role of the mode function in quantum optics. In earlier experiments the role of the vacuum fields could be demonstrated in induced coherence experiments as the source of complementarity \cite{Heu14}. Here the spatial coherence properties of these vacuum fields are measured and as the physical reason for complementarity in single photon quantum optics demonstrated.
The visibilities of second-order (one-photon) and fourth-order (two-photon) interference have been observed in a Young's double-slit experiment using light generated by spontaneous parametric down-conversion and a photon-counting intensified charge-coupled device camera. Coherence and entanglement underlie one- and two-photon interference, respectively. As the effective source size is increased, coherence is diminished while entanglement is enhanced, so that the visibility of one-photon interference decreases while that of two-photon interference increases. This experiment demonstrates the complementarity between one- and two-photon interference (coherence and entanglement) in the spatial domain.
Entangled-photon pairs (biphotons) generated by spontaneous optical parametric down-conversion exhibit a number of properties that are analogous to those of ordinary photons generated by incoherent sources. The spatial pump-field distribution and the two-particle wave function in the biphoton case play the respective roles of the source intensity distribution and the second-order coherence function in the incoherent case. The van Cittert--Zernike theorem, which is applicable for incoherent optical sources emitting independent photons, has a counterpart for biphotons. Likewise, the partial-coherence theory of image formation has an analogous counterpart for biphoton beams transporting spatial information. However, an underlying duality, rather than analogy, accompanies the mathematical similarity between incoherent and biphoton emissions if the comparison is made between the photon count rate in the incoherent case and the biphoton count rate in the entangled-photon case. The smaller the size of an incoherent source, the more separable is the coherence function and the more coherent is the field, and therefore the higher the visibility of ordinary interference fringes. In contrast, the narrower the size of a biphoton pump source, the more separable is the wave function and the less entangled is the field, and therefore the lower the visibility of biphoton interference fringes. This duality is similar to the complementarity between single and two-photon interference exhibited for biphotons.
In earlier experiments the role of the vacuum fields could be demonstrated as the source of complementarity with respect to the temporal properties (Heuer et al., Phys. Rev. Lett. 114:053601, 2015). Single photon first order interferences of spatially separated regions from the cone structure of spontaneous parametric down conversion allow for analyzing the role of the mode function in quantum optics regarding the complementarity principle. Here the spatial coherence properties of these vacuum fields are demonstrated as the physical reason for complementarity in these single photon quantum optical experiments. These results are directly connected to the mode picture in classical optics. The properties of the involved vacuum fields selected via the measurement process are the physical background of the complementarity principle in quantum optics.