[Paper Review] Experimental Test of Hyper-Complex Quantum Theories

Lorenzo M. Procopio, Lee A. Rozema+6 more|arXiv (Cornell University)|Feb 4, 2016

Quantum Mechanics and Applications3 citations

TL;DR

This experiment tests hyper-complex quantum theories by probing phase non-commutativity using a Sagnac interferometer with a broadband negative-index metamaterial and a liquid crystal phase shifter. The results show phase commutativity within 0.03°, placing the tightest experimental bounds to date on non-commutativity in hyper-complex quantum theories, supporting standard complex quantum mechanics.

ABSTRACT

In standard quantum mechanics, complex numbers are used to describe the wavefunction. Although complex numbers have proven sufficient to predict the results of existing experiments, there is no apparent theoretical reason to choose them over real numbers or generalizations of complex numbers, i.e. hyper-complex numbers. Experiments performed to date have proven that real numbers are insufficient, but whether or not hyper-complex numbers are required remains an open question. Quantum theories based on hyper-complex numbers are one example of a post-quantum theory, which must be put on a firm experimental foundation. Here we experimentally probe hyper-complex quantum theories, by studying one of their deviations from complex quantum theory: the non-commutativity of phases. We do so by passing single photons through a Sagnac interferometer containing two physically different phases, having refractive indices of opposite sign. By showing that the phases commute with high precision, we place limits on a particular prediction of hyper-complex quantum theories.

Motivation & Objective

• To test predictions of hyper-complex quantum theories, particularly quaternionic quantum mechanics (QQM), which predict non-commuting phases.
• To determine whether complex quantum mechanics (CQM) remains sufficient or if hyper-complex extensions are required by experiment.
• To experimentally probe the non-commutativity of phases—a key deviation of QQM from CQM—using single photons and engineered optical media.
• To develop and integrate a broadband, low-loss negative-index metamaterial for use in single-photon experiments at visible wavelengths.
• To set the tightest experimental bounds to date on phase non-commutativity in post-quantum theories.

Proposed method

• A Sagnac interferometer is used to apply two distinct phases to single photons: one via a liquid crystal (positive refractive index) and one via a fishnet-type negative-index metamaterial (NIM) with a refractive index of -0.4 at 790 nm.
• The NIM is fabricated from silver and dielectric layers, designed to exhibit broadband negative permeability and effective refractive index across 650–850 nm, with a zero-crossing at ~750 nm.
• Phase shifts are measured by monitoring interference fringes in both clockwise and counter-clockwise paths of the interferometer, with the NIM inserted in one arm.
• The NIM is mounted on a translation stage and precisely aligned to maximize transmission in both interferometer paths, with a 0.44° tilt to suppress back-reflection coupling to detectors.
• The effective refractive index is extracted from measured phase shifts using the Fresnel equations, validated by numerical simulations.
• The experiment compares the net phase accumulated when the NIM and liquid crystal are applied in opposite orders, testing for non-commutativity.

Experimental results

Research questions

• RQ1 Does the phase accumulated by a photon depend on the order of application when passing through a negative-index metamaterial and a conventional phase shifter?
• RQ2 Can experimental evidence for non-commuting phases be observed, as predicted by quaternionic quantum mechanics?
• RQ3 To what extent do current experimental results constrain the validity of hyper-complex quantum theories compared to standard complex quantum mechanics?
• RQ4 What is the achievable precision in measuring phase non-commutativity using single photons and engineered metamaterials?
• RQ5 Can broadband negative-index metamaterials be effectively integrated into quantum optical setups for high-precision tests of foundational physics?

Key findings

• The experiment observes no significant difference in the net phase when the NIM and liquid crystal are applied in reverse order, demonstrating phase commutativity within 0.03°.
• The measured phase non-commutativity is bounded to less than 0.03°, representing a one-order-of-magnitude improvement over previous experiments.
• The negative-index metamaterial exhibits a refractive index of -0.4 at 790 nm, with a broad transmission window and high transmission (15% at 790 nm), suitable for single-photon experiments.
• Numerical simulations and experimental measurements of the NIM’s refractive index show excellent agreement, with a smooth transition from positive to negative index around 750 nm.
• The time evolution of phase fronts inside the NIM confirms backward wave propagation, with the Poynting vector and wave vector oriented antiparallel, a hallmark of negative-index behavior.
• The integration of the NIM into a single-photon Sagnac interferometer demonstrates the feasibility of using broadband metamaterials in precision quantum optical measurements.