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

Optical trapping (OT) uses optical forces to trap particles at intensity extrema, and optical binding (OB) can assemble multiple particles via light scattering. The long-standing question is whether OB can assemble many particles into macroscopic optical matter. The authors argue this is impossible in typical situations once the particle number is modest (N ≳ 10): clusters become intrinsically unstable under optical forces alone and require additional dissipative forces (e.g., viscosity) to overcome instability. The key framework is non-Hermitian physics. Because light-matter systems are open with incoming light and radiative loss, the force matrices governing stability are non-Hermitian. Unlike commonly studied non-Hermitian models with complex diagonal terms, here the matrices are real and asymmetric. Such systems exhibit exceptional points (EPs): below EPs all eigenvalues are real; beyond EPs, eigenvalues coalesce and split into complex-conjugate pairs, leading to exponentially growing or decaying modes. The paper shows that as N increases, EPs are inevitably crossed, producing complex modes and instability irrespective of particle details or illumination, while damping can suppress the run-away dynamics.

Foundational work established OT and OB and their widespread applications (e.g., Ashkin et al.; Burns et al.). Numerous studies explored optical binding in various configurations, including near interfaces and with different particle types and shapes. Nonconservative forces in optical systems have been measured and analyzed, revealing departures from conservative force fields. In parallel, non-Hermitian and PT-symmetric physics highlighted the roles of gain/loss and exceptional points in wave systems. Prior optical manipulation studies typically assumed effective conservative behavior or did not fully link instability to non-Hermitian EPs. This work connects OB stability to non-Hermitian matrix properties of the force operator, emphasizing the distinct case of real but asymmetric matrices and showing how EPs and mode coalescence govern stability as particle number grows.

Stability analysis uses Newtonian dynamics linearized about equilibrium for N identical spherical particles acted upon solely by optical forces: m d^2(ΔX)/dt^2 = F(ΔX) ≈ K · ΔX, where K_ij = ∂F_i/∂(ΔX_j) is the 3N×3N force matrix at equilibrium. A system is conservative iff ∂F_i/∂x_j = ∂F_j/∂x_i everywhere; in general OB systems this symmetry is broken, rendering K non-Hermitian (real and asymmetric). Stability is governed by the eigenvalues of K: writing ΔX as a sum over eigenvectors with time dependence exp(i Ω_i t), modes are unstable if any eigenvalue K_i = −m Ω_i^2 is positive (purely imaginary Ω_i) or complex (complex Ω_i), the latter associated with EPs. Single-particle OT is analyzed by block-diagonalizing the 3×3 force matrix into a 2×2 real matrix K_OT = S + A with S symmetric (conservative trap stiffnesses) and A antisymmetric (nonconservative torque). Parameters a (average stiffness), b (half-level spacing of S), and g (nonconservative torque strength) determine eigenvalues: K_{1,2} = a ± sqrt(b^2 − g^2) for |g| < |b| (real) and a ± i sqrt(g^2 − b^2) for |g| > |b| (complex), with an EP at |g| = |b|. Electromagnetic fields for particle clusters are computed using the generalized multi-particle Mie theory by expanding fields in vector spherical wavefunctions, applying translation-addition theorems and boundary conditions to solve for expansion coefficients. Strongly focused Gaussian beams are modeled via the vector Debye integral linked to a geometrical optics description of the lens. Optical forces are evaluated using the time-averaged Maxwell stress tensor integrated over particle surfaces. Equilibria (zero-force configurations) are found by dynamic simulations that propagate particle motion with a fictitious damping until convergence. At equilibrium, K is computed by finite differences of forces with respect to particle coordinates. EPs are identified when K becomes defective upon parameter variation (e.g., particle size, polarization). To probe generic behaviors with matrix size, random real asymmetric matrices are constructed by decomposing K into S = (K + K^T)/2 and A = (K − K^T)/2, diagonalizing S via a similarity transform, and generating random S′ (diagonal with entries uniformly in [−1, 0]) and A′ (antisymmetric with entries uniformly in [−1, 1]). Composite matrices M = S_random + γ_asym A_random are used to determine γ_threshold (via bisection) needed to produce the first complex pair versus minimum level spacing δ. Ensembles of 10,000 matrices are generated for sizes 10–100 (step 2), with both uniform and Gaussian distributions considered.

• Single-particle OT exhibits EPs controlled by the competition between stiffness anisotropy (b) and nonconservative torque (g). Crossing |g| = |b| produces a complex-conjugate pair leading to spiral-in/spiral-out dynamics. In a focused Gaussian-beam trap, varying polarization from linear to circular forces a symmetry crossover that guarantees an EP at an intermediate ellipticity; in the reported example the EP occurs at ζ = 2/15 of the polarization parameter.
• For OB, small clusters already show asymmetric, non-Hermitian force matrices even when certain mirror symmetries exist. Example force matrices for a 3-sphere chain and a triangular trimer reveal significant asymmetries; symmetry reduction increases asymmetry ratios of off-diagonal elements (e.g., 4:15 vs 12:15), making EPs more likely.
• In a six-particle symmetric cluster, scanning particle radius reveals an EP where two real eigenmodes coalesce and become complex; after the EP, the real parts merge while the imaginary parts split into equal and opposite values, indicating unstable/stable spirals.
• Prevalence with size: The percentage of complex modes (CMs) increases monotonically with particle number N across diverse 2D and 3D geometries and illuminations. For clusters bound by non-standing waves, the first complex pair appears at N < 10; most modes are complex for N > 50; all studied clusters with N > 70 possess CMs. Systems driven by non-standing waves (which directly impart nonconservative forces) develop CMs more rapidly than those driven by standing waves.
• Spectral crowding drives EPs: As N increases, the eigenvalue bandwidth of the conservative part S remains bounded (set by finite laser intensity), but the number of modes grows (∝ N), so average level spacing shrinks toward zero. This enhances the probability that small asymmetries coalesce modes at EPs, making CMs inevitable in the large-N limit.
• Random-matrix corroboration: In ensembles of large real asymmetric matrices, the minimum asymmetry γ_threshold needed to induce the first complex pair strongly correlates with minimum level spacing δ (correlations ≈ 0.75 for uniform and ≈ 0.65 for Gaussian distributions). As matrix size grows and δ → 0, γ_threshold → 0, meaning even tiny asymmetry suffices to generate CMs.
• Stability phase diagrams: Phase maps for triangular and square lattices versus particle size and refractive index show that: • A small green domain (true optical binding by light alone) exists at small N and weak scattering, but it shrinks rapidly with N. • An orange domain (opto-hydrodynamic binding) appears where equilibria exist but are unstable due to CMs; sufficient damping stabilizes these clusters, and the required damping increases with N. • A gray domain (no equilibrium) expands with N, indicating optical crystallization cannot be achieved even with damping in those regions, as zero-force configurations do not exist.
• Physical interpretation: Nonconservative optical forces continually pump energy into particle degrees of freedom; beyond EPs this yields run-away modes that “melt” optically bound structures. Ambient fluid damping can remove injected energy and restore stability, explaining observations of large clusters in liquids.

The work addresses whether optical binding can create macroscopic optical matter. The results show that the force matrices governing optical manipulation are intrinsically non-Hermitian (real and asymmetric) in open light–matter systems. As particle number increases, spectral crowding reduces level spacings in the conservative sector, so even small nonconservative asymmetries inevitably drive the system through exceptional points. Beyond EPs, modes acquire complex eigenvalues, producing unstable spirals that destabilize the cluster under optical forces alone. Therefore, large clusters cannot be stably assembled purely by optical forces under typical conditions. This reframes prior experimental observations: large optically bound clusters seen in fluids rely on viscous dissipation to extract injected energy, effectively stabilizing complex modes. The findings underscore EPs and non-Hermiticity as central to the mechanics of optical trapping and binding, point to the importance of system symmetries, and clarify conditions under which optical crystallization is impossible (no-equilibrium regions) or requires dissipation (opto-hydrodynamic binding).

Optical manipulation of many-particle systems is fundamentally governed by non-Hermitian physics. Real but asymmetric force matrices naturally lead to exceptional points; as clusters grow, spectral crowding guarantees EP encounters and the emergence of complex modes, rendering optical forces alone insufficient to stabilize large clusters. Damping can restore stability by removing energy injected by light, explaining observations in liquids and motivating the term opto-hydrodynamic binding. Perfectly symmetry-protected configurations can avoid complex instabilities but are experimentally fragile. Beyond optics, the analysis generalizes to other large non-Hermitian systems (e.g., acoustics, mechanics), especially where waves exchange energy with trapped objects. Future research directions include: systematic exploration of symmetry-protected designs and their robustness; quantitative studies of damping pathways (hydrodynamic, material) needed to stabilize complex modes; extensions to non-spherical particles with internal degrees of freedom; experimental probing of EP signatures in many-particle optical and acoustic traps; and leveraging or mitigating non-Hermitian effects in large-scale particle manipulation.

Stability without damping can persist only in special, perfectly symmetry-protected configurations (e.g., infinite 1D periodic chains with exact particle identity and perfectly balanced counter-propagating waves). Such protection is fragile: any symmetry breaking or disorder reintroduces nonconservative couplings and complex instabilities. The main analysis considers spherical, identical particles and focuses on optical forces; while hydrodynamic effects are discussed qualitatively and via supplementary simulations, comprehensive fluid-structure interactions and thermal fluctuations are not the core focus. Reported stability thresholds and phase maps pertain to the studied geometries and illumination schemes; different configurations may shift boundaries but not the fundamental trend that complex modes proliferate with system size.