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New Journal of Physics

Holography-based quantum projector in a state space of linear photon momentum

Mark T. Gruneisen, James P Black, Raymond C. Dymale, Kurt E. Stoltenberg

Published Feb 18, 2015• 4 citations
Publisher

Abstract

A transmission volume hologram is evaluated as a quantum projector operating on linear momentum states of individual photons in a four-dimensional state space. A state space is defined by four momentum vectors that are either efficiently diffracted or transmitted by the hologram. The influence of the volume hologram on the complex amplitudes of the basis states is modeled using coupled-wave theory. Measurement probabilities obtained with a second hologram introduced as a state-space analyzer are compared to measurement probabilities associated with a projection operator in quantum mechanics. A compact sequential hologram configuration for projecting superposition states onto all of the basis states is demonstrated for individual photons prepared in all 20 states of the five mutually unbiased bases of the four-dimensional state space. The complex amplitudes associated with superpositions of the basis states are imparted to individual photons via computer-generated holography and a liquid-crystal spatial light modulator. The coupled-wave theory analysis indicates that relative phase relationships in transmitted superposition states can be preserved to within 1/28th of an optical cycle and measured detection probabilities compare favorably to those associated with a quantum projector in a basis of discrete orthonormal states. Small departures of the measured statistics from theoretical expectations are quantified and attributed to imperfections in the hologram and experimental setup.

DOI
10.1088/1367-2630/17/2/023059

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A transmission volume hologram is evaluated as a quantum projector operating on linear momentum states of individual photons in a four-dimensional state space. A state space is defined by four momentum vectors that are either efficiently diffracted or transmitted by the hologram. The influence of the volume hologram on the complex amplitudes of the basis states is modeled using coupled-wave theory. Measurement probabilities obtained with a second hologram introduced as a state-space analyzer are compared to measurement probabilities associated with a projection operator in quantum mechanics. A compact sequential hologram configuration for projecting superposition states onto all of the basis states is demonstrated for individual photons prepared in all 20 states of the five mutually unbiased bases of the four-dimensional state space. The complex amplitudes associated with superpositions of the basis states are imparted to individual photons via computer-generated holography and a liquid-crystal spatial light modulator. The coupled-wave theory analysis indicates that relative phase relationships in transmitted superposition states can be preserved to within 1/28th of an optical cycle and measured detection probabilities compare favorably to those associated with a quantum projector in a basis of discrete orthonormal states. Small departures of the measured statistics from theoretical expectations are quantified and attributed to imperfections in the hologram and experimental setup.
Transmission volume holograms are evaluated as quantum projectors operating on spatial modes of the photon in mutually unbiased bases (MUBs). With applications to free-space quantum key distribution (QKD) in mind, state spaces based on rectilinear and azimuthal phase modes (i.e. photon linear and orbital angular momenta) are considered. Rectilinear phase modulation is shown to result in both greater mode densities for a receiving aperture in the far field and better efficiency and cross-talk characteristics when volume holograms are used as de-multiplexing elements. Twoand four-dimensional state spaces are defined via rectilinear phase modes and the complex optical fields of the MUBs are calculated and generated with a spatial light modulator in order to record and subsequently illuminate transmission volume holograms. Using holograms prepared in lithium niobate, diffraction efficiencies are measured for the 36 permutations associated with projecting the six MUB states of a two-dimensional state space onto the same six MUB states. Quantum measurements associated with cascaded projection operations in a four-dimensional state space are performed using photo-thermo-refractive glass holograms. Experimental results show approximate agreement with the inner-product relationships that describe quantum projection probabilities.
There has been considerable interest recently in the generation of azimuthal phase functions associated with photon orbital angular momentum (OAM) for high-dimensional quantum key distribution. The generation of secure quantum keys requires not only this pure phase basis but also additional bases comprised of orthonormal superposition states formed from the pure states. These bases are also known as mutually unbiased bases (MUBs) and include quantum states whose wave functions are modulated in both phase and amplitude. Although modulo 2pi optical path control with high-resolution spatial light modulators (SLMs) is well suited to creating the azimuthal phases associated with the pure states, it does not introduce the amplitude modulation associated with the MUB superposition states. Using computer-generated holography (CGH) with the Leith-Upatnieks approach to hologram recording, however, both phase and amplitude modulation can be achieved. We present a description of the OAM states of a three-dimensional MUB system and analyze the construction of these states via CGH with a phase-modulating SLM. The effects of phase holography artifacts on quantum-state generation are quantified and a prescription for avoiding these artifacts by preconditioning the hologram function is presented. Practical effects associated with spatially isolating the first-order diffracted field are also quantified, and a demonstration utilizing a liquid-crystal SLM is presented.
The scalar attributes of the optical field are considered as an approach to enhancing bit rates in quantum key distribution over free-space optical links. The implications of spatial-mode encoding to system aperture requirements are considered for rectilinear and azimuthal bases of phase modulation. Diffractive-optics-based approaches to generating and discriminating photons based on spatial mode characteristics are quantified through analysis and experimentation. Spatial modes based on phase modulation are generated via modulo-2π phase modulation with a liquid-crystal spatial light modulator (LC SLM). The phase and amplitude modulation associated with superpositions of phase modulated fields are achieved via computer-generated holography with single LC SLM. Transmission volume holograms are evaluated as a technique for de-multiplexing phase modulated modes and cascaded volume holograms are considered as an approach to performing projective measurements on optical fields.
We report the results of two-photon quantum holography where spatial information stored in phase holograms is retrieved by measuring quantum spatial correlations between two images formed by spatially entangled twin photons with a Schmidt number of 450 in the two-dimensional transverse space. In our experiments, the entire flux of spontaneous down-converted photons illuminates the phase holograms and the photons of signal-idler pairs transmitted by the holograms are detected separately in the far field on two electron-multiplying charge coupled device cameras.
The concept of holography has lured philosophers of science for decades, and is becoming more and more popular on several fronts of science, e. g. in the physics of black holes. In this paper we try to understand things as if the visible universe were a reading of a low-dimensional hologram generated in hyperspace. We performed the whole process of creating and reading the hologram of a point in virtual space by using computer simulations. We claim that the fuzzieness in quantum mechanics, in statistical physics and thermodynamics is due to the fact that we do not see the real image of the object, but a holographic projection of it. We found that the projection of a point particle is a de Broglie-type wave. This indicates that holography could be the origin of the wave nature of a particle. We have also noted that one cannot stabilize the noise (or fuzzieness) in terms of the integration grid-points of the hologram, it means that one needs to give the grid-points a physical significance. So we further claim that the space is quantized, which supports the basic assumption of quantum gravity. Our study in the paper, although it is more qualitative, yet gives a smoking gun hint of a holographic basis of physical reality.