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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Free-space quantum key distribution with spatial modes of the optical field

Mark T. Gruneisen, Raymond C. Dymale, Kurt E. Stoltenberg

Published Oct 6, 2011• 2 citations
Publisher

Abstract

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.

DOI
10.1117/12.898926

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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.
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.
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.
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.
In the diffraction-limited near-field propagation regime, free-space optical quantum key distribution (QKD) systems can employ multiple spatial modes to improve their key rate. This improvement can be effected by means of high-dimensional QKD or by spatial-mode multiplexing of independent QKD channels, with the latter, in general, offering higher key rates. Here, we theoretically analyze spatial-mode-multiplexed, decoy-state BB84 whose transmitter mode set is either a collection of phase-tilted, flat-top focused beams (FBs) or the Laguerre-Gaussian (LG) modes. Although for vacuum propagation the FBs suffer a QKD rate penalty relative to the LG modes, their potential ease of implementation make them an attractive alternative. Moreover, in the presence of turbulence, the FB modes may outperform the LG modes.
A phase-only hologram applies a modal transformation to an optical transverse spatial mode via phase encoding and intensity masking. Accurate control of the optical field crucially depends on the method employed to encode the hologram. In this Letter, we present a method to encode the amplitude and the phase of an optical field into a phase-only hologram, which allows the exact control of spatial transverse modes. Any intensity masking method modulates the amplitude and alters the phase of the optical field. Our method consists in correcting for this unwanted phase alteration by modifying the phase encryption accordingly. We experimentally verify the accuracy of our method by applying it to the generation and detection of transverse spatial modes in mutually unbiased bases of dimension two and three.
Spatial modes of light provide a high-dimensional space that can be used to encode both classical and quantum information. Current approaches for dynamically generating and measuring these modes are slow, due to the need to reconfigure a high-resolution phase mask such as a spatial light modulator or digital micromirror device. The process of updating the spatial mode of light can be greatly accelerated by multiplexing a set of static phase masks with a fast, image-preserving optical switch, such as an acousto-optic modulator (AOM). We experimentally realize this approach, using a double-pass AOM to generate one of five orbital angular momentum states with a switching rate of up to 500 kHz. We then apply this system to perform fast quantum state tomography of spatial modes of light in a 2-dimensional Hilbert space by projecting the unknown state onto six spatial modes comprising three mutually unbiased bases. We are able to reconstruct arbitrary states in under 1 ms with an average fidelity of 96.9%.