Fig. 1: Overview of the experiment.

a Basic geometry and coordinate system. Two pairs of counter-propagating circularly polarized laser beams (red arrows) form two Gaussian standing waves. Two polystyrene particles are localized in the standing waves in the axial direction (z-axis), while in the lateral direction, they tend to orbit due to the azimuthal spin force at the ambient pressure. b Schematic describing the stochastic quasi-modes in the linear (sub-threshold) regime. Increasing the laser power leads to slightly larger radii of the lateral particles' trajectories and suppression of two quasi-modes QM₁, QM₃. c Experimentally observed biased stochastic motion, showing a tendency towards orbital rotation in the xy-plane illustrated using spatial probability densities (PDFs) in the particle displacement basis and in the quasi-mode basis. The picture depicts excitation of the breathing mode QM₂ for beam separation d = 8.6 μm, corresponding to a condition where the threshold power P_c for the center of mass mode QM₄ is larger than P_b for the breathing mode. For above threshold power (216 mW), the formation of fluctuating limit cycles is illustrated by PDFs with annular distributions. Left-hand/right-hand columns in quasi-mode basis (x_by_b and x_cy_c) show excited breathing (d = 8.6 μm)/CoM mode (d = 8.9 μm), respectively. d Particle motion calculated using stochastic simulation showing similar tendency towards orbital rotation, however, here the threshold power is (for the same pressure) an order of magnitude smaller than for experimental observation and thus the hydrodynamic interaction is relatively stronger than optical, favouring the CoM mode. The threshold power is affected mainly by imperfections and asymmetries in the experimental system. Note, elliptical shapes of PDF and different width of PDF for both particles.