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Optics Express

Trajectory-engineered geometric phase in two-level photonic systems

Jing Liang, Huijie Pei, Yanwen Hu, Zhen Li

Published Oct 16, 2026• 0 citations
Publisher

Abstract

Although the geometric phase (GP) in two-level systems has been extensively studied, realizing tailored photonic-state evolution trajectories and experimentally resolving the corresponding phase responses within a controllable platform remain challenging. Here, we establish a synthetic two-level photonic system using rotated electro-optic crystals, in which two orthogonal polarization components serve as a pseudospin basis whose evolution is governed by a tunable synthetic magnetic field. Independent control of the field direction and magnitude enables tailored trajectories on the Poincaré sphere, while an explicit intensity-phase mapping provides a non-interferometric route to phase retrieval. We experimentally observe continuous GP evolution, abrupt π -phase jumps associated with antipodal crossings, and path-dependent GP accumulation along composite trajectories. These results provide a controllable electro-optic route to engineering and observing trajectory-dependent GP in synthetic photonic systems.

DOI
10.1364/oe.616556

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Although the geometric phase (GP) in two-level systems has been extensively studied, realizing tailored photonic-state evolution trajectories and experimentally resolving the corresponding phase responses within a controllable platform remain challenging. Here, we establish a synthetic two-level photonic system using rotated electro-optic crystals, in which two orthogonal polarization components serve as a pseudospin basis whose evolution is governed by a tunable synthetic magnetic field. Independent control of the field direction and magnitude enables tailored trajectories on the Poincaré sphere, while an explicit intensity-phase mapping provides a non-interferometric route to phase retrieval. We experimentally observe continuous GP evolution, abrupt π -phase jumps associated with antipodal crossings, and path-dependent GP accumulation along composite trajectories. These results provide a controllable electro-optic route to engineering and observing trajectory-dependent GP in synthetic photonic systems.
We present both theoretical and experimental demonstrations of geometric phase (GP) transitions in a synthetic two-level photonic system. We demonstrate the use of this nontrivial phenomenon to achieve an unprecedented birefringence change detection, with a precision up to Δn∼10−11. This result represents a significant improvement of 3 orders of magnitude over conventional techniques (typically limited to the level of Δn∼10−8). In this system, two orthogonal polarizations of light constitute the effective two-level system, which is driven by a birefringence-sensitive synthetic magnetic field. A given photonic spin is able to precess around the “magnetic” field and exhibits a topological transition in its evolution trajectory in parameter space. We establish a direct relationship between the GP transition and birefringence change, which is the basis of our detection technique. This approach significantly improves detection sensitivity and stability as compared with previous detection techniques. Moreover, the presented results benefit many fundamental and applied research fields such as structured light shaping and photonic switching.
The geometric phase, originating from the cyclic evolution of a state, such as polarization on the Poincaré sphere, is typically measured through interferometric approaches that often include unwanted contributions from the dynamic phase. Here, we present a non-interferometric technique based on quantum correlation of pair photons to measure the geometric phase of a classical beam. The transfer of geometric phase of the classical pump beam arising from the cyclic evolution of its polarization state on the Poincaré sphere onto the polarization-entangled pair photons generated via spontaneous parametric down-conversion in a Sagnac interferometer enables easy control over the quantum state. Characterization of the generated quantum states reveals that the geometric phase of the pump beam controls the coincidence counts, entanglement visibility, Bell's parameter, quantum state tomography, and fidelity in close agreement with theoretical predictions. We observe sinusoidal modulation of the Bell's parameter and state fidelity with changes in the geometric phase, resulting in transitions between orthogonal Bell states and Bell-like maximally entangled states. Our results establish the geometric phase of the classical pump as a tunable parameter for quantum state control, offering a compact, passive platform for phase manipulation in quantum photonic systems, enabling geometric phase-based quantum gates, and compensating unwanted phase acquired by the quantum state on propagation.
We propose a geometric hybrid Poincaré sphere (GHPS) as a unified geometrical framework for describing structured photon states with independently controllable spin angular momentum (SAM) and orbital angular momentum (OAM). Unlike the conventional higher-order Poincaré sphere, in which the SAM and OAM are intrinsically coupled through fixed basis states, the GHPS is constructed by defining its poles as direct products of arbitrary orthogonal bases on the Poincaré sphere (PS) and orbital Poincaré sphere (OPS) and by superposing these pole states. Using numerical simulations, we analyze representative GHPS states and show that the GHPS spherical coordinates govern the amplitude ratio and relative phase between the pole bases. This framework enables spatially inhomogeneous polarization distributions and intensity patterns, including nonseparable structures in which polarization and intensity are intrinsically intertwined, and provides a systematic state-space description for the coherent geometrical control of advanced structured light fields.
Photonic orbital angular momentum (OAM) offers a promising platform for high-dimensional quantum information processing. While geometric phase (GP) is the crucial tool in enabling intrinsically fault-tolerant quantum computation, the measurement of GP using linear optics remains relatively unexplored in the OAM state space. Here, we propose an experimental scheme to detect GP shifts resulting from the cyclic evolution of OAM qutrit states. Distinguished with the conventional evolution along cyclic path on the Poincaré sphere (PS), the nontrivial evolution in our theoretical scheme is along a cyclic path residing within the SU(3)/U(2) parameter space. By employing a combination of X-gates, dove prisms, and double cylindrical lenses, we achieve the cyclic evolution and analyse the resultant GP through our designed Sagnac interferometer. Our theoretical study may find potential in high-dimensional quantum computation using twisted photons and in exploring the geometric structure of such optical systems.
We investigate a class of cyclic evolutions for %the cyclic evolution of driven two-level quantum systems (effective spin-1/2) with a particular focus on the geometric characteristics of the driving and their specific imprints on the quantum dynamics. By introducing the concept of geometric field curvature for any field trajectory in the parameter space we are able to unveil underlying patterns in the overall quantum behavior: the knowledge of the field curvature provides a non-standard and fresh access to the interrelation between field and spin trajectories, and the corresponding quantum phases acquired in non-adiabatic cyclic evolutions. In this context, we single out setups in which the driving field curvature can be employed to demonstrate a pure geometric control of the quantum phases. Furthermore, the driving field curvature can be naturally exploited to introduce the geometrical torque and derive a general expression for the total quantum phase acquired in a cycle. Remarkably, such relation allows to access the mechanisms controlling the changeover of the quantum phase across a topological transition and to disentangle the role of the spin and field topological windings. As for implementations, we discuss a series of physical systems and platforms to demonstrate how the geometric control of the quantum phases can be realized for pendular field drivings. This includes setups based on superconducting islands coupled to a Josephson junction and inversion asymmetric nanochannels with suitably tailored geometric shapes.
We report the experimental observation of the non-local geometric phase in Hanbury Brown-Twiss polarized intensity interferometry.The experiment involves two independent, polarized, incoherent sources, illuminating two polarized detectors.Varying the relative polarization angle between the detectors introduces a geometric phase equal to half the solid angle on the Poincaré sphere traced out by a pair of single photons.Local measurements at either detector do not reveal the effect of the geometric phase, which appears only in the coincidence counts between the two detectors, showing a genuinely non-local effect.We show experimentally that coincidence rates of photon arrival times at separated detectors can be controlled by the two-photon geometric phase.This effect can be used for manipulating and controlling photonic entanglement.