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

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.