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Retrieving Vortex-Driven Non-Hermiticity from Trajectory-Resolved Measurements in Optical Levitation

This paper experimentally demonstrates how to reconstruct non-Hermitian eigenmodes and force landscapes in optically levitated systems by analyzing trajectory-resolved measurements, revealing that non-Hermiticity arises from real force-field vortices in single-particle traps and virtual phase-space vortices in multi-particle assemblies.

Original authors: Jack Ng, Mian Wu, Xiao Li, Nan Li, Han Cai, Che Ting Chan, Huizhu Hu

Published 2026-08-24

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CC BY 4.0

Original authors: Jack Ng, Mian Wu, Xiao Li, Nan Li, Han Cai, Che Ting Chan, Huizhu Hu

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

For over a century, scientists have understood that light carries momentum, capable of pushing on matter with a force so faint it was once thought impossible to measure. The invention of the laser changed this, allowing researchers to use focused beams of light to hold tiny particles in mid-air, a technique known as optical levitation. In this suspended state, a particle is not merely sitting still; it is in a constant, energetic exchange with the light around it, absorbing and scattering photons in a way that keeps it trapped. While this system might seem like a simple balance of forces, it is actually an open system, meaning it constantly trades energy with its environment. This openness introduces a complex layer of physics where the usual rules of symmetry break down, creating a state where the particle's motion is influenced by forces that do not behave like the conservative pushes and pulls we see in everyday life. Understanding these subtle, non-conservative interactions is crucial because they govern how energy flows in many natural and engineered systems, from the microscopic machinery of cells to the behavior of advanced materials.

A team of researchers has now made a direct window into this hidden world by watching the exact paths of tiny glass beads suspended in a vacuum. By tracking the movement of these particles with extreme precision, the scientists were able to reconstruct the invisible landscape of forces acting upon them. They discovered that the light does not just push the particles; it creates a swirling, vortex-like force field that drives the particles in specific directions. This swirling motion is the physical signature of what physicists call non-Hermitian dynamics, a state where the system's behavior cannot be described by simple, symmetric rules. The team found that this effect arises in two distinct ways depending on how many particles are involved. When a single particle is trapped, the swirling force comes directly from the spin of the light beam itself, which carries a type of angular momentum that twists the particle. When multiple particles are trapped together, a different mechanism takes over: the light bounces back and forth between the particles, creating a "virtual" swirl in the space around them that drives their collective motion.

To see this, the researchers used a high-powered laser to trap a single silica sphere, about 250 nanometers in radius, inside a vacuum chamber. They carefully adjusted the polarization of the light, changing it from a straight line to a circle, which altered how the light's spin interacted with the particle. As they made this change, they observed the particle's motion shift from a simple back-and-forth vibration to a more complex, rotating behavior. By measuring the particle's position thousands of times per second, they mapped out the exact directions in which the particle preferred to move. They found that as the light's spin increased, the particle began to favor one direction of rotation over the other, effectively breaking the symmetry of time. In the beginning, the particle moved clockwise and counter-clockwise with equal likelihood, but as the non-conservative forces grew stronger, it began to spin almost exclusively in one direction, driven by the energy injected by the light. This directional bias is a hallmark of a system that is far from equilibrium, where the flow of energy creates a clear "arrow of time."

The researchers then expanded their study to a system of three particles held in a triangular formation. Here, the light did not need to be spinning to create the effect. Instead, the particles scattered light between one another, creating a complex web of interactions that generated a similar swirling force. This "virtual vortex" emerged purely from the way the particles communicated through the light field, pulling the entire group into a coordinated, non-symmetric dance. The team measured the frequencies at which these particles vibrated and the specific patterns of their movement, finding that the experimental data matched rigorous theoretical calculations with remarkable precision. They confirmed that the forces acting on the particles were not symmetric; the push in one direction was not simply the reverse of the push in the opposite direction. This lack of symmetry is the defining feature of the non-Hermitian state they were investigating.

The significance of this work lies in how it moves beyond simply measuring the frequencies of vibration to actually visualizing the shape of the forces themselves. In the past, scientists could only infer the nature of these complex systems by looking at their spectral signatures, much like trying to understand a machine by only listening to its hum. This new approach allows researchers to see the actual trajectory of the particle and reconstruct the full force landscape, revealing the non-orthogonal modes that define the system's behavior. The study demonstrates that these exotic physical properties are not just mathematical abstractions but are real, measurable features of the microscopic world. By resolving the motion of individual particles, the team has provided a general framework for understanding how energy flows in open, noisy systems. This clarity offers a new way to study the fundamental dynamics of levitated matter, bridging the gap between theoretical predictions and the tangible reality of particles suspended in light.

Technical Summary: Retrieving Vortex-Driven Non-Hermiticity from Trajectory-Resolved Measurements in Optical Levitation

Problem Statement
Non-Hermitian physics provides a framework for describing open and nonequilibrium systems, characterized by complex spectra and non-orthogonal eigenmodes. While optical levitation (optical tweezers) in vacuum is inherently an open, non-conservative system due to continuous energy and momentum exchange with light, previous experimental investigations have largely focused on spectral signatures, such as exceptional points (EPs) and complex eigenfrequencies. A critical gap remains in the direct experimental access to the eigenmodes themselves. Unlike Hermitian systems, non-Hermitian eigenmodes are generally non-orthogonal, a feature that significantly impacts physical interpretations, such as stiffness calibration and noise analysis. Furthermore, the microscopic origins of non-Hermiticity in these systems—specifically the role of optical vorticity in force fields—had not been directly reconstructed from trajectory data.

Methodology
The authors employ high-spatiotemporal-resolution trajectory-resolved measurements of silica nanoparticles in vacuum to reconstruct the full linear mechanical dynamics of levitated systems. The study utilizes two distinct experimental configurations:

Single-Particle Optical Trapping: A single silica nanosphere ( $r \approx 250.5$ nm) is trapped by a tightly focused, z-propagating Gaussian beam ( $\lambda = 1064$ nm) with tunable polarization (from linear to circular). The transverse motion is detected via a quadrant photodetector (QPD) using forward-scattered light.
Optical Binding: Three silica nanoparticles ( $r \approx 100$ nm) are levitated in an isosceles triangular configuration formed by a phase-modulated beam. The longitudinal ( $z$ -axis) motions are independently detected via three QPDs.

The core methodology involves reconstructing the force constant matrix ( $\mathbf{K}$ ), which acts as the Jacobian of the optical force. By measuring the power spectral density (PSD) of particle trajectories projected onto arbitrary directions, the authors identify eigenvector orientations (where spectral amplitude is maximized) and eigenfrequencies. For regimes beyond the EP, where eigenmodes become complex, phase differences between orthogonal trajectories are extracted using the Hilbert transform. These experimental data are used to reconstruct the full non-Hermitian matrix $\mathbf{K}$ via the relation $\mathbf{K} = \mathbf{V}\mathbf{\Lambda}\mathbf{V}^{-1}$ , where $\mathbf{V}$ contains the eigenvectors and $\mathbf{\Lambda}$ the eigenvalues. Theoretical predictions are derived rigorously using Mie scattering theory and the Maxwell stress tensor without classical electrodynamics approximations.

Key Contributions and Results
The paper successfully retrieves both eigenfrequencies and eigenmodes for non-Hermitian systems, demonstrating two distinct physical origins of non-Hermiticity:

Single-Particle Trapping (Real Vortices):
- By tuning the polarization from linear to circular, the system transitions from a Hermitian regime to a non-Hermitian one, culminating in an Exceptional Point (EP) at a polarization angle of $\theta \approx 31^\circ$ .
- The non-Hermiticity arises from a real vortex in the force field induced by the orbital angular momentum (OAM) of the incident light. This manifests as an antisymmetric component ( $g$ ) in the force constant matrix.
- Experimental results show the coalescence of real eigenfrequencies at the EP, followed by the emergence of complex-conjugate eigenfrequencies. The eigenmodes transition from non-orthogonal real vectors to parallel complex vectors, matching analytical predictions.
Optical Binding (Virtual Vortices):
- In a three-particle assembly, non-Hermiticity emerges from virtual vortices in the phase space induced by passive light scattering (multiple scattering) between particles, rather than the angular momentum of the incident beam.
- The system exhibits purely antisymmetric, non-reciprocal coupling between the $z$ -motions of the particles.
- The experiment successfully measures the coalescence of eigenvectors and the complex eigenfrequencies associated with this binding-induced non-Hermiticity.
Stochastic Signatures and Fluctuation Theorems:
- The study identifies characteristic directional probability currents in the particle trajectories. As non-Hermiticity increases, the system exhibits a strong bias toward counter-clockwise (CCW) or clockwise (CW) rotation, corresponding to the direction of positive work input.
- This directional asymmetry is shown to be consistent with the Crooks fluctuation theorem, linking the statistics of work performed on closed trajectories to entropy production. The ratio of forward to backward events serves as a proxy for the non-conservative work scale.

Significance and Claims
The paper claims to establish a general trajectory-based framework for reconstructing the dynamics of stochastic open systems. Its primary significance lies in demonstrating that non-Hermiticity in levitated systems is not merely a spectral feature but is fundamentally manifested in the geometry of particle trajectories and the structure of non-orthogonal eigenmodes.

By directly accessing the modal structure underlying non-Hermitian dynamics, the work moves beyond spectral analysis to provide a complete linear mechanical description of the system. The authors assert that their results, which show remarkable quantitative agreement with rigorous Mie theory and Maxwell stress tensor calculations, provide direct experimental evidence of two distinct microscopic origins of non-Hermiticity: real vortices driven by OAM and virtual vortices driven by multiple scattering. This approach offers a complementary paradigm to wave-based non-Hermitian platforms, enabling the study of non-conservative forces and non-orthogonal modes at the level of individual trajectories in noisy, open systems.

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