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arXiv:1703.03818 · Physical Review A

Why a hole is like a beam splitter: A general diffraction theory for multimode quantum states of light

Zhihao Xiao, R. Nicholas Lanning, Mi Zhang, Irina V. Novikova, Eugeniy E. Mikhailov, Jonathan P. Dowling

Published Aug 14, 2017• 12 citations
arXivHTMLPDF

Abstract

Within the second-quantization framework, we develop a formalism for describing a spatially multimode optical field diffracted through a spatial mask and show that this process can be described as an effective interaction between various spatial modes. We demonstrate a method to calculate the quantum state in the diffracted optical field for any given quantum state in the incident field. We also give several additional examples of how the theory works, for various quantum input states, which may be easily tested in the laboratory, including two single-mode squeezed vacuums, single- and two-photon inputs, where we show that the diffraction process produces a two-mode squeezed vacuum, number-path entanglement, and a Hong-Ou-Mandel-like effect analogous to that of a beam splitter.

DOI
10.1103/physreva.96.023829

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Within the second-quantization framework, we develop a formalism for describing a spatially multimode optical field diffracted through a spatial mask and show that this process can be described as an effective interaction between various spatial modes. We demonstrate a method to calculate the quantum state in the diffracted optical field for any given quantum state in the incident field. We also give several additional examples of how the theory works, for various quantum input states, which may be easily tested in the laboratory, including two single-mode squeezed vacuums, single- and two-photon inputs, where we show that the diffraction process produces a two-mode squeezed vacuum, number-path entanglement, and a Hong-Ou-Mandel-like effect analogous to that of a beam splitter.
We consider multimode two-photon interference at a beam splitter by photons created by spontaneous parametric down-conversion. The resulting interference pattern is shown to depend upon the transverse spatial symmetry of the pump beam. In an experiment, we employ the first-order Hermite-Gaussian modes in order to show that, by manipulating the pump beam, one can control the resulting two-photon interference behavior. We expect these results to play an important role in the engineering of quantum states of light for use in quantum information processing and quantum imaging.
We theoretically demonstrate that a photonic crystal microcavity coupled to waveguides works as an optical beam splitter. Numerical simulations illustrate that spatially separated two output modes have a relative phase close to $\ensuremath{\pi}/2$ and tunable transmission coefficients. The general transformation of this beam splitter gives a single-qubit rotation operator about the xxx axis. Using such beam splitters, we show single-photon interference via the Mach-Zehnder interferometer and two-photon entanglement via Hong-Ou-Mandel effect. This system may have great applications in constructing interferometers and integrated circuits for optical quantum information processing with solid-state materials.
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Two-photon interference is a fundamental quantum optics effect with numerous applications in quantum information science. Here, we study two-photon interference in multiple transverse-spatial modes along a single beam-path. Besides implementing the analog of the Hong-Ou-Mandel interference using a two-dimensional spatial-mode splitter, we extend the scheme to observe coalescence and anticoalescence in different three- and four-dimensional spatial-mode multiports. The operation within spatial modes, along a single beam path, lifts the requirement for interferometric stability and opens up new pathways of implementing linear optical networks for complex quantum information tasks.
The second-order coherence of optical fields at output ports of a beam splitter is investigated. The properties of the second-order coherence are not only strongly dependent on the input states but also quite involved with the nature of the beam splitter. When parameters of both the input states and the beam splitter are chosen appropriately, the fields at the output ports will exhibit quantum behaviour. Between the two output modes, there stands mode–mode correlation which can even be anticorrelation. Some nonclassical properties, such as the photon antibunching effect of the input fields, may be destroyed by the beam splitter.
The accurate theory describing interaction of two coherent light beams with ultrasonic wave where one of the light beam is incident at positive Bragg angle while the other at negative was developed by Leroy and Blomme in 1984 . The assumed geometry in this theory resembles an interferometer wherein ultrasonic beam plays a role of the output beam splitter of the light. The output light intensity is modulated with ultrasonic frequency as a result of interference between the light in diffraction orders in which the frequency is changed due to the Doppler effects with those diffracted with unchanged frequency. Temporal modulation of the light intensities in diffraction orders, that was foreseen by Leroy and Blomme, was confirmed experimentally and the results will be presented. In addition, the Leroy-Blomme theory has been extended to include the effect of the initial phase of the ultrasonic wave on the phase of modulation of light intensity in diffraction orders. According to the best knowledge of the author, Leroy and Blomme theory has not been verified experimentally, although many years have passed. This theory could be applied to the quantum optics phenomenon when pairs of entangled photons are incident on ultrasonic wave instead of two coherent light beams. The geometry of interaction of two coherent light beams with ultrasonic wave in Leroy-Blomme theory is the same as for Hong-Ou-Mandel interferometer wherein a beam splitter is replaced by the ultrasonic wave. Similarities and differences between Leroy-Blomme and Hong-Ou-Mandel interferometers will be presented.