Fig. 1. (a) Dipole model of the nanoparticles trapped in two distinct optical traps with orthogonal polarization. The red arrows signify the dipole moments, which are aligned with the polarization of the corresponding trapping beams. The angle θi determines the orientation of the dipole moment pi relative to the y-axis, following the convention that angles measured clockwise from the y-axis are considered positive, whereas those measured counterclockwise are deemed negative. The x-axis is parallel, and the y-axis is perpendicular to the line that connects the two particles. θ1∈[0,π] and θ2∈[−π/2,π/2], satisfying θ1−θ2=π/2. O indicates the origin of the coordinate system used for expressing the particle positions. (b) The distribution of the amplitudes of the trapping field |ET|, and the field emitted by a y-polarized particle |G¯αEoriginT| along the x-axis. Both of them are normalized by the peak amplitude of the trapping field. To clearly illustrate the amplitude of the dipole scattered field, we have separately plotted the red curve from the main figure in the inset. The parameters used in these calculations include the numerical aperture (NA=0.8) and the focal length (f=2  mm) of the microscope objective, the waist of the beam incident on the objective (w0=2.1  mm), and the radius and relative permittivity of the SiO2 particle (rs=100  nm, ϵ=2.1).

Fig. 2. Setup for generating two orthogonally polarized optical traps using a spatial light modulator. A beam of 532 nm light was focused onto the particles along the y-axis for imaging through the microscope objective, although this was not depicted in the figure for simplicity. The inset presents an image of two nanoparticles with radius rs∼100  nm trapped in the two traps at a distance r0∼2.65  μm along the x-axis. A bare electrode connected to a high voltage DC source (HV) was used to control the net charge of the particles. A pair of electrodes connected to the amplified signal from a function generator (FG) was used to determine the amount of net charge. A dual-channel lock-in amplifier was utilized to record the signal from the QPDs. ISO, optical isolator; HWP, half-wave plate; PBS, polarization beam splitter; BE, beam expander; M, mirror; SLM, phase-only spatial light modulator; L, lens; DM, dichroic mirror; BPF, bandpass filter; CCD, charge-coupled device; EHWP, electronically controlled half-wave plate; OBJ, microscope objective; CL, collection lens; NDF, neutral density filter; QPD, quadrant photodetector.

Fig. 3. (a) Electrostatic coupling rate ge as a function of trap separation r0. Optical coupling was deactivated by setting the polarization of the trapping beam θ1=0. The avoided crossing was indiscernible for coupling rates smaller than a half of mechanical linewidth γ/2 (gray region). (b), (c) Simultaneous net charge neutralization process for the trapped nanoparticles 1 and 2, respectively. ALI and ϕLI were the demodulated oscillation amplitude and phase within a narrow frequency range centered around ωdr. Discrete steps of ALI indicated the addition or removal of charges from the particle, while a 180° phase shift denoted a reversal in charge polarity. After time t0, the amplitude ALI for both particles dropped to zero, and the phase ϕLI became disordered, signifying the simultaneous neutralization of the charge of both particles.

Fig. 4. (a) Half of the normal-mode frequency splitting, defined as (Ω−−Ω+)/2, versus the phase difference Δφ0 at a trap separation of r0∼2.65  μm and polarization θ1=3π/4. The power distribution factor η was set to zero. The blue curve represented an ideal dependence, exclusively accounting for conservative interactions and being proportional to cos(Δφ0). The non-conservative force contributed to the total force for the values of Δφ0 different from nπ  (n∈Z) and was able to amplify the particle motion, thus modifying the normal-mode frequency splitting. (b) Power spectrum density of the z-mode of one of the coupled particles for normal-mode frequency splitting (blue data, Δφ0=π/3) and resonance (orange data, Δφ0=π/2).

Fig. 5. Parameter S varied with the trap separation r0 at Δφ0=0 and θ1=3π/4. The curve exhibited a periodicity ∼λ/2. The red points indicated the minima of the curve, where the conservative coupling rate g∼0. The power distribution factor η of the two traps changed with the trap separation distance, resulting in the deviation of the parameter S from the theoretical model as r0 varied.

Fig. 6. (a) Four special cases (θ1=0,π/4,π/2, and 3π/4) within the dipole radiation model for nanoparticles trapped in two orthogonally polarized optical traps, satisfying θ1−θ2=π/2. (b) At a trap separation r0∼2.65  μm, the polarization θ1 was changed while keeping the phase difference fixed at Δφ0=0 (blue data) or Δφ0=π (orange data). For Δφ0=0, the coupling rate g was proportional to −sin(2θ1). For Δφ0=π, an additional factor of −1 enters into the expression for coupling rate g due to the π-phase shift of Δφ0. The power distribution factor η was set to zero. (c) Dependence of the parameter S′, defined as (S−η2Ω4)/4, on the polarization θ1 for different phase differences Δφ0=0,  4π/9,  2π/3. The other parameters were determined experimentally, including trap separation r0∼2.65  μm, power distribution factor η=−0.2, and oscillation frequency Ω=2π×65.3  kHz.