Non-Hermitian physics for optical manipulation uncovers inherent instability of large clusters

Optical trapping (OT) uses optical forces to trap particles at intensity extrema, while optical binding (OB) involves trapping multiple particles through light scattering. Decades of research have not definitively answered whether OB can create macroscopic "optical matter." This paper investigates the inherent stability limits of OB in assembling large particle clusters using the framework of non-Hermitian physics. Unlike conservative systems described by Hermitian matrices with real eigenvalues, OT and OB involve open systems with energy exchange between light and particles. This results in real but asymmetric (non-Hermitian) force matrices that yield complex eigenvalues beyond exceptional points (EPs). The research hypothesizes that the presence of EPs, leading to complex eigenvalues, renders large optically bound clusters inherently unstable unless additional stabilizing forces (like viscous damping) are present. This challenges the conventional understanding of optical trapping and binding, suggesting that optical forces alone are insufficient for large-scale manipulation.

Existing literature extensively documents optical trapping and binding phenomena. Early work demonstrated the feasibility of trapping and binding using optical forces, but the scalability of these techniques to form macroscopic structures remained largely unexplored. Previous studies explored the optical forces acting on particle clusters, modelling them often as conservative systems. However, the non-conservative nature of optical forces due to radiative losses and incoming light has been acknowledged in some studies, yet its impact on the stability of large clusters wasn't fully understood. The role of exceptional points in non-Hermitian systems has also gained attention, but its application to the stability of optically bound clusters was a novel area of exploration in this research.

The study analyzes the stability of N identical spherical particles subject to optical forces, using the equation of motion d²ΔX/mdt² = F(ΔX) ≈ K⋅ΔX, where m is the particle mass, t is time, ΔX represents displacement from equilibrium, F represents optical force, and K is the 3N x 3N force matrix at equilibrium. The force matrix K, governing stability, is evaluated using the generalized multi-particle Mie theory and Maxwell stress tensor. The stability is assessed based on the eigenvalues of K; complex eigenvalues indicate instability. A single-particle OT is first analyzed, modeling its force matrix as K_OT = S + A, where S is the conservative symmetric part, and A is the nonconservative antisymmetric part. Eigenvalues of K_OT are derived, revealing EPs at which eigenvalues transition from real to complex, indicating instability. This framework is extended to OB, where the force matrices for various particle cluster geometries (chains, triangles, larger clusters) are numerically evaluated. The presence of EPs and the percentage of complex modes are determined for different cluster sizes and geometries. Random matrix theory is employed to examine the likelihood of complex eigenvalues in large asymmetric matrices.

The core finding is that instability stemming from complex eigenvalues is prevalent in optically bound clusters with approximately 10 or more particles. This instability is linked to exceptional points (EPs) in the non-Hermitian force matrix, occurring when the nonconservative components overcome the conservative components. The research demonstrates that for larger clusters, the probability of encountering an EP and exhibiting complex eigenvalues increases dramatically. This increase is not solely due to the strength of the non-conservative force, but also because of the decreasing level spacing between eigenvalues of the conservative component of the force matrix as the number of particles increases. Analysis using random matrices supports this observation; large asymmetric matrices tend to have complex eigenvalues regardless of the magnitude of asymmetry, especially when the level spacing is small. The study reveals a phase diagram showing the transition from stable optically bound clusters to unstable clusters needing damping for stability, and ultimately to states where no equilibrium is attainable. This indicates that optical forces alone become insufficient for binding in large clusters. The fraction of the phase space exhibiting stable equilibrium diminishes rapidly with increasing particle number, further supporting the inherent instability.

The results directly address the central question of whether OB can create macroscopic structures solely via optical forces. The findings demonstrate that this is not feasible for typical scenarios. The instability unveiled is a fundamental consequence of the non-Hermitian nature of optical forces in open systems, and is not due merely to a simple imbalance between conservative and nonconservative forces. The transition to instability is shown to be intricately linked to the decreasing separation between eigenvalues, a consequence of the increasing density of states as the number of particles increases. The research highlights the crucial role of damping, especially in fluid environments, in stabilizing these clusters. This suggests that experimentally observed large optically bound clusters are stabilized by viscous damping, effectively transforming the system from purely optical binding to opto-hydrodynamic binding. This necessitates a refined understanding of optical binding, particularly in regimes with strong scattering and prominent nonconservative forces. This impacts the interpretation of existing literature that focuses on stable optically bound clusters. The theory expands our understanding of non-Hermitian physics beyond the typical cases of symmetric matrices with complex diagonal terms.

This study demonstrates that optical forces alone are insufficient to bind large particle clusters due to the inherent instability stemming from the non-Hermitian nature of optical forces. The emergence of complex eigenvalues beyond exceptional points, coupled with diminishing level spacing in large systems, leads to this instability. This necessitates considering damping mechanisms, such as viscous forces in fluids, for stabilizing larger optically bound clusters. Future research could explore the implications of this work for other non-Hermitian systems, like acoustic trapping, and investigate ways to mitigate the instability for large-scale optical manipulation.

The study primarily focuses on idealized spherical particles. The inclusion of non-spherical particles and their rotational degrees of freedom could potentially alter the stability characteristics. The model assumes homogeneous particle properties and uniform incident illumination; variations in these parameters could affect the observed instability. The numerical simulations, while computationally demanding, are limited to a specific size of particle clusters due to limitations in computing power. The generalizability of the findings to different experimental settings and types of damping needs further investigation.