^{1}Research Center for Advanced Computational Sensing and Intelligent Processing, Zhejiang Lab, Hangzhou 310000, China

^{2}State Key Laboratory of Extreme Photonics and Instrumentation, College of Optical Science and Engineering, Zhejiang University, Hangzhou 310027, China

^{3}Research Center for Frontier Fundamental Studies, Zhejiang Lab, Hangzhou 310000, China

Arrays of optically levitated nanoparticles with fully tunable light-induced dipole–dipole interactions have emerged as a platform for fundamental research and sensing applications. However, previous experiments utilized two optical traps with identical polarization, leading to an interference effect. Here, we demonstrate light-induced dipole–dipole interactions using two orthogonally polarized optical traps. Furthermore, we achieve control over the strength and polarity of the optical coupling by adjusting the polarization and propose a method to simultaneously and stably measure conservative and non-conservative coupling rates. Our results provide a new scheme for exploring entanglement and topological phases in arrays of levitated nanoparticles.

1. INTRODUCTION

Pioneering works in optical levitation systems, which employ tightly focused lasers to trap and manipulate microparticles in a vacuum, date back to the 1970s [1–3]. Over the subsequent decades, due to their ability to avoid mechanical dissipation and effectively isolate environmental thermal noise, optical levitation systems have garnered widespread attention and attained full development in the fields of basic scientific research [4–8] and sensing applications [9–13].

Cooling the center of mass (CoM) motion of optically trapped particles is essential for their stable levitation in a high vacuum, thus becoming a primary research focus. Recent years have seen significant advancements in this area. Motional ground-state cooling of a single silica nanoparticle has been successfully achieved through cavity cooling [14] and measurement-based feedback [15,16]. Additionally, cavity cooling technology has been extended to synchronously cooling multiple degrees of freedom of the optically levitated nanoparticle [17,18].

Following the advancements in cooling technology for single nanoparticles, attention is increasingly shifting towards arrays of optically levitated nanoparticles, which offer a much broader spectrum of physical degrees of freedom [19–23]. Optical trap arrays can be experimentally realized using acousto-optic deflectors (AODs) and spatial light modulators (SLMs). Unlike the optical trap array generated by AODs, each trap in the optical trap array produced by SLM maintains the same frequency. This uniformity makes it particularly suitable for the investigation of light-induced dipole–dipole interactions among particles. In 2022, Rieser et al. demonstrated the use of this approach to study fully tunable and nonreciprocal optical interactions between two silica nanoparticles through the coherent control of light-induced dipole–dipole interactions [23]. Combined with the rapidly developing ground-state cooling technology mentioned above, their work provides a powerful tool for studies in quantum simulations involving mechanical degrees of freedom, enhanced quantum sensing, collective effects, and the study of phonon transport and thermalization.

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Although considerable effort has been made in studying light-induced dipole–dipole interactions between optically levitated nanoparticles, such studies to date are based on two distinct optical traps with identical polarization [22,23]. This configuration imposes an interference effect not considered in previous studies. For example, at the position of particle 1, the trapping field of particle 1 interferes with the tail of the trapping field for particle 2. Consequently, the dipole-dipole interaction model is unable to accurately predict the measured coupling rate. In this study, we demonstrate light-induced dipole–dipole interactions between silica nanoparticles levitated by two distinct optical traps with orthogonal polarization. Our theoretical model indicates that orthogonal polarization cancels the interference effect attributed to the tails of the trapping fields. Unlike previous studies, adjusting the polarization not only adjusts the strength of the optical coupling but also its polarity. Furthermore, we propose a method to simultaneously measure conservative and non-conservative coupling rates by assessing the polarization dependence of normal-mode frequency splitting after introducing a suitable power difference in the trapping beams. This approach successfully mitigates resonance caused by non-conservative interaction. Our work expands the toolbox of optical binding based on dipole–dipole interactions, potentially enabling further studies of complex many-body physics, such as macroscopic quantum entanglement, quantum correlations, and topological phases [24,25].

2. THEORY

A. Optical Coupling Coefficients

In our experiment, the radius of levitated nanoparticles (${r}_{s}\sim 100\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$) was significantly smaller than the wavelength of the trapping beams and their beam waists ($\lambda =1064\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$, ${w}_{0}\sim 650\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$, ${r}_{s}/\lambda \sim 0.1$). Consequently, the isotropic nanoparticles optically levitated at position ${\mathit{r}}_{i}$ can be treated as light-induced dipoles characterized by dipole moments ${\mathit{p}}_{i}={\alpha}_{i}\mathit{E}({\mathit{r}}_{i})$. The real part of particle polarizability, ${\alpha}_{i}$, is defined as $3{\u03f5}_{0}{V}_{i}(\u03f5-1)/(\u03f5+2)$, where ${V}_{i}$ represents the particle volume, ${\u03f5}_{0}$ denotes the vacuum permittivity, and $\u03f5\sim 2.1$ is the particle relative permittivity. Figure 1(a) illustrates the dipole model of two nanoparticles trapped in two distinct optical traps with orthogonal polarization. Both traps are located in the same $x\u2013y$ plane, perpendicular to the direction of propagation of the two trapping beams along the $z$-axis. The red arrows represent the dipole moments, aligned with the polarization of their respective trapping beams. The angle ${\theta}_{i}$ quantifies the orientation of the dipole moment ${\mathit{p}}_{i}$ with respect to the $y$-axis, adopting the convention that angles measured clockwise from the $y$-axis are positive and those angles measured counterclockwise are negative. The $x$-axis is parallel, and the $y$-axis is perpendicular to the line connecting the two particles. ${\theta}_{1}\in [0,\pi ]$ and ${\theta}_{2}\in [-\pi /2,\pi /2]$, satisfying ${\theta}_{1}-{\theta}_{2}=\pi /2$.

Figure 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 ${\theta}_{i}$ determines the orientation of the dipole moment ${\mathit{p}}_{i}$ 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. ${\theta}_{1}\in [0,\pi ]$ and ${\theta}_{2}\in [-\pi /2,\pi /2]$, satisfying ${\theta}_{1}-{\theta}_{2}=\pi /2$. $O$ indicates the origin of the coordinate system used for expressing the particle positions. (b) The distribution of the amplitudes of the trapping field $|{\mathit{E}}^{\mathrm{T}}|$, and the field emitted by a $y$-polarized particle $|\overline{\mathit{G}}\alpha {\mathit{E}}_{\text{origin}}^{\mathrm{T}}|$ 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 ($\mathrm{NA}=0.8$) and the focal length ($f=2\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{mm}$) of the microscope objective, the waist of the beam incident on the objective (${w}_{0}=2.1\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{mm}$), and the radius and relative permittivity of the ${\mathrm{SiO}}_{2}$ particle (${r}_{s}=100\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$, $\u03f5=2.1$).

In experiment, the trapping field exhibits an Airy function shape at the focus, evident from the black curve in Fig. 1(b), which represents the normalized amplitude of the trapping field calculated under experimental conditions [26]. This phenomenon is ascribed to the diffraction of light on the circular aperture of the objective lens [27]. Further, we simulated the dipole radiation field emitted by a $y$-polarized nanoparticle located at the origin for distances $|r|\ge \lambda $. The red curve in Fig. 1(b) presents the amplitude distribution of the dipole radiation field along the $x$-axis, normalized against the peak amplitude of the trapping field. Clearly, the amplitude of the Airy tail of the trapping field surpasses that of the dipole radiation field from the polarized nanoparticle positioned at the coordinate origin. Therefore, the total electric field ${\mathit{E}}_{i}({\mathit{r}}_{i})$ at the position of particle $i$ comprises three components [28], $${\mathit{E}}_{i}({\mathit{r}}_{i})={\mathit{E}}_{i}^{\mathrm{T}}({\mathit{r}}_{i})+{\mathit{E}}_{j}^{\mathrm{T}}({\mathit{r}}_{i})+\overline{\mathit{G}}({\mathit{r}}_{i}-{\mathit{r}}_{j}){\alpha}_{j}{\mathit{E}}_{j}({\mathit{r}}_{j}).$$Here, ${\mathit{E}}_{i}^{\mathrm{T}}({\mathit{r}}_{i})$ and ${\mathit{E}}_{j}^{\mathrm{T}}({\mathit{r}}_{i})$ represent the local trapping field of particle $i$ and the tail of the trapping field for particle $j$, respectively. The dipole field propagator $\overline{\mathit{G}}$, also known as the dyadic Green’s tensor, is expressed in the form [23,28] $$\overline{\mathit{G}}(r)=\frac{{e}^{ikr}}{4\pi {\u03f5}_{0}{r}^{3}}[(-{k}^{2}{r}^{2}-3ikr+3)\frac{\mathit{r}\otimes \mathit{r}}{{r}^{2}}+({k}^{2}{r}^{2}+ikr-1)\mathbb{I}].$$Here, $k=2\pi /\lambda $ is the wave number of the light with a wavelength $\lambda $, $\mathbb{I}$ represents the identity matrix, $\otimes $ denotes an outer product, and $r=|\mathit{r}|$ is the modulus of the displacement vector $\mathit{r}$ pointing from source to observation point. Once the total electric component of the optical field is determined by the Eq. (1), the time-averaged optical gradient force acting on particle $i$ can be expressed as [23,28–30] $${\mathit{F}}_{i}^{\mathrm{grad}}({\mathit{r}}_{i})\sim \frac{{\alpha}_{i}}{4}{\nabla}_{i}[{\mathit{E}}_{i}^{\mathrm{T}*}({\mathit{r}}_{i})\xb7{\mathit{E}}_{i}^{\mathrm{T}}({\mathit{r}}_{i})]+{\nabla}_{i}\Re \{\frac{{\alpha}_{i}}{2}{\mathit{E}}_{i}^{\mathrm{T}*}({\mathit{r}}_{i})\xb7[{\mathit{E}}_{j}^{\mathrm{T}}({\mathit{r}}_{i})+\overline{\mathit{G}}({\mathit{r}}_{i}-{\mathit{r}}_{j}){\alpha}_{j}{\mathit{E}}_{j}^{\mathrm{T}}({\mathit{r}}_{j})]\}.$$Here, $\Re \{X\}$ represents the real part of $X$, and ${\nabla}_{i}$ is the nabla operator acting on the coordinate ${\mathit{r}}_{i}$. Note that the ratio of the imaginary part of the polarizability to the real part is approximately 0.01. Therefore, the contribution of the imaginary part to the net optical force was omitted [28]. Additionally, in the derivation of Eq. (3), higher-order terms [$\sim \mathcal{O}({V}^{3},{V}^{4})$], such as ${\mathit{E}}_{j}^{\mathrm{T}*}({\mathit{r}}_{i})\xb7{\mathit{E}}_{j}^{\mathrm{T}}({\mathit{r}}_{i})$, ${\mathit{E}}_{j}^{\mathrm{T}*}({\mathit{r}}_{i})\xb7\overline{\mathit{G}}({\mathit{r}}_{i}-{\mathit{r}}_{j}){\alpha}_{j}{\mathit{E}}_{j}^{\mathrm{T}}({\mathit{r}}_{j})$, and $\overline{{\mathit{G}}^{*}}({\mathit{r}}_{i}-{\mathit{r}}_{j}){\alpha}_{j}^{*}{\mathit{E}}_{j}^{\mathrm{T}*}({\mathit{r}}_{j})\xb7\overline{\mathit{G}}({\mathit{r}}_{i}-{\mathit{r}}_{j}){\alpha}_{j}{\mathit{E}}_{j}^{\mathrm{T}}({\mathit{r}}_{j})$, were omitted. Similarly, the dipole field radiated by particle $j$ is approximated to be proportional to the local trapping field ${\mathit{E}}_{j}^{\mathrm{T}}({\mathit{r}}_{j})$. The first term of Eq. (3) signifies the well-recognized gradient force of the local trapping beam, whereas the second term specifies the force exerted on particle $i$ by the interference terms between the two trapping beams as well as between the trapping beam for particle $i$ and the dipole field of particle $j$. In contrast to previous studies [23], our optical traps feature orthogonal polarizations, leading to ${\mathit{E}}_{i}^{\mathrm{T}*}({\mathit{r}}_{i})\xb7{\mathit{E}}_{j}^{\mathrm{T}}({\mathit{r}}_{i})\equiv 0$, thereby effectively suppressing the impact of the tail of the trapping fields. Consequently, the optical binding force exerted on particle $i$ can be expressed as [23] $${\mathit{F}}_{i}^{\mathrm{bind}}({\mathit{r}}_{i})=\frac{{\alpha}_{i}{\alpha}_{j}}{2}{\nabla}_{i}\Re \{{\mathit{E}}_{i}^{\mathrm{T}*}({\mathit{r}}_{i})\xb7\overline{\mathit{G}}({\mathit{r}}_{i}-{\mathit{r}}_{j}){\mathit{E}}_{j}^{\mathrm{T}}({\mathit{r}}_{j})\}.$$

If the particles are deeply trapped in their respective Gaussian beam traps, the electric field in Eq. (4) can be treated locally as Gaussian beams traveling along the $z$-direction as [31] $${\mathit{E}}_{i}^{\mathrm{T}}({\mathit{r}}_{i})\sim \sqrt{\frac{4{P}_{i}}{cn{\u03f5}_{0}\pi {w}_{0}^{2}}}\mathrm{exp}[-i(k-\frac{1}{{z}_{R}}){z}_{i}+{\phi}_{i}],$$where ${P}_{i}$ is the beam power, ${w}_{0}$ is the radius of the beam waist, $c$ is the speed of light in a vacuum, $n$ is the refractive index of the environment, ${z}_{R}$ is the Rayleigh range, and ${\phi}_{i}$ represents the local phase in the focal plane. Note that the beam waist radius of the focus created by the tight focusing of the 1064 nm laser is approximately 650 nm, with a Rayleigh distance of around 1.25 μm, which is significantly larger than the oscillation amplitude of the particles. Consequently, in Eq. (5), the first-order approximation of the Gouy phase shift, $\mathrm{arctan}({z}_{i}/{z}_{R})\sim {z}_{i}/{z}_{R}$, is adopted. The reference point for measuring the axial coordinates ${z}_{i}$ and ${z}_{j}$ is the common focal plane of the two trapping beams. For the two nanoparticles trapped in two orthogonally polarized optical traps, as illustrated in Fig. 1(a), the trapping fields are characterized as follows: $${\mathit{E}}_{1}^{\mathrm{T}}({\mathit{r}}_{1})=({\mathit{E}}_{1}^{\mathrm{T}}({\mathit{r}}_{1})\mathrm{sin}\text{\hspace{0.17em}}{\theta}_{1},{\mathit{E}}_{1}^{\mathrm{T}}({\mathit{r}}_{1})\mathrm{cos}\text{\hspace{0.17em}}{\theta}_{1},0),$$$${\mathit{E}}_{2}^{\mathrm{T}}({\mathit{r}}_{2})=(-{\mathit{E}}_{2}^{\mathrm{T}}({\mathit{r}}_{2})\mathrm{cos}\text{\hspace{0.17em}}{\theta}_{1},{\mathit{E}}_{2}^{\mathrm{T}}({\mathit{r}}_{2})\mathrm{sin}\text{\hspace{0.17em}}{\theta}_{1},0).$$Here, the condition ${\theta}_{1}-{\theta}_{2}=\pi /2$ is used.

By substituting Eqs. (2), (5), (6), and (7) into Eq. (4) and expanding the optical binding forces around the equilibrium positions of the two particles, while retaining only the first-order term of ${z}_{1}-{z}_{2}$, we obtained the optical binding force along the $z$-axis as $${\mathit{F}}_{iz}^{\mathrm{bind}}\sim \frac{{G}_{0}\text{\hspace{0.17em}}\mathrm{sin}(2{\theta}_{1})}{k{r}_{0}}[\mathrm{cos}(k{r}_{0})\mathrm{cos}(\mathrm{\Delta}{\phi}_{0})\pm \mathrm{sin}(k{r}_{0})\mathrm{sin}(\mathrm{\Delta}{\phi}_{0})](\mp {z}_{1}\pm {z}_{2}){\widehat{e}}_{z},$$with constant ${G}_{0}={k}^{3}{\alpha}_{1}{\alpha}_{2}\sqrt{{P}_{1}{P}_{2}}{(k-1/{z}_{R})}^{2}/(4cn{\pi}^{2}{\u03f5}_{0}^{2}{w}_{0}^{2})$. ${r}_{0}$ denotes the trap separation, $\mathrm{\Delta}{\phi}_{0}={\phi}_{1}-{\phi}_{2}$ represents the initial phase difference in the focal plane, and ${\widehat{e}}_{z}$ is the unity vector along the $z$-axis. The top and bottom of the signs ($\pm ,\mp $) apply to particles 1 and 2, respectively. In subsequent experimental studies, we investigated the dipole–dipole interactions between nanoparticles when ${r}_{0}>3\lambda /2$. Within this range, the dipole approximation and the far-field approximation ($k{r}_{0}\gg 1$) hold true. Therefore, in the derivation of Eq. (8), only the first-order terms [${(k{r}_{0})}^{-1}$] were retained. The coupling coefficient between particles is defined as ${k}_{ij}={k}_{c}\pm {k}_{\mathrm{n}\mathrm{c}}$, which signifies a non-reciprocal interaction. The conservative and non-conservative coupling coefficients, represented by ${k}_{c}$ and ${k}_{\mathrm{nc}}$, respectively, are determined as follows: $${k}_{c}={G}_{0}\text{\hspace{0.17em}}\mathrm{sin}(2{\theta}_{1})\mathrm{cos}(k{r}_{0})\mathrm{cos}(\mathrm{\Delta}{\phi}_{0})/(k{r}_{0}),\phantom{\rule{0ex}{0ex}}{k}_{\mathrm{nc}}={G}_{0}\text{\hspace{0.17em}}\mathrm{sin}(2{\theta}_{1})\mathrm{sin}(k{r}_{0})\mathrm{sin}(\mathrm{\Delta}{\phi}_{0})/(k{r}_{0}).$$

The trap separation, ${r}_{0}$, and the phase difference, $\mathrm{\Delta}{\phi}_{0}$, enable the tuning of dipole–dipole interactions between purely conservative and non-conservative. Unlike previous studies [23], adjusting the polarization in our experiment not only controls the strength of these interactions but also allows switching between positive (attractive) and negative (repulsive) nature. At ${\theta}_{1}=0$ or $\pi /2$, the dipole–dipole interactions are closed, whereas at ${\theta}_{1}=\pi /4$ and $3\pi /4$, their amplitudes are maximized, albeit with opposite polarity. This ability to toggle polarity represents a unique feature of our scheme using two orthogonally polarized optical traps.

B. Eigenfrequencies of the Coupled System

Assuming the particles have trapping frequencies ${\mathrm{\Omega}}_{\mathrm{1,2}}$ and share the same mass $m$, the system comprising two optically trapped and coupled particles is described by the linear dynamics equations in the frequency domain [23], $$-m{\omega}^{2}{\tilde{z}}_{1}+im\omega \gamma {\tilde{z}}_{1}=-(m{\mathrm{\Omega}}_{1}^{2}+{k}_{c}+{k}_{\mathrm{nc}}){\tilde{z}}_{1}+({k}_{c}+{k}_{\mathrm{nc}}){\tilde{z}}_{2},\phantom{\rule{0ex}{0ex}}-m{\omega}^{2}{\tilde{z}}_{2}+im\omega \gamma {\tilde{z}}_{2}=-(m{\mathrm{\Omega}}_{2}^{2}+{k}_{c}-{k}_{\mathrm{nc}}){\tilde{z}}_{2}+({k}_{c}-{k}_{\mathrm{nc}}){\tilde{z}}_{1}.$$Here, ${\tilde{z}}_{\mathrm{1,2}}$ are the spectral representations of particle displacement coordinates. $\gamma $ is the mechanical linewidth, also known as the damping rate. ${k}_{c}$ and ${k}_{\mathrm{nc}}$ are the conservative and non-conservative coupling coefficients, respectively. For simplicity, Eq. (10) does not include the fluctuating forces acting on the nanoparticles due to Brownian motion and other noise sources. Upon solving Eq. (10), the eigenfrequencies of the normal modes of the coupled system are derived as [23] $${\mathrm{\Omega}}_{\pm}=\sqrt{{\mathrm{\Omega}}^{2}+{k}_{c}/m\mp \sqrt{{\mathrm{\Omega}}^{4}{\eta}^{2}+2{\mathrm{\Omega}}^{2}\eta {k}_{\mathrm{nc}}/m+{({k}_{c}/m)}^{2}}},$$where ${\mathrm{\Omega}}_{1}=\mathrm{\Omega}\sqrt{1+\eta}$ and ${\mathrm{\Omega}}_{2}=\mathrm{\Omega}\sqrt{1-\eta}$ with $\eta $ signifying the power difference. $\mathrm{\Omega}$ is the eigenfrequency of the particles at $\eta =0$ in the absence of interparticle interactions.

We define the conservative and non-conservative coupling rates as $g={k}_{c}/2m\mathrm{\Omega}$ and $\overline{g}={k}_{\mathrm{nc}}/2m\mathrm{\Omega}$, respectively. The conservative coupling rate originates from the reciprocal interaction between particles and represents the coupling form of energy conservation in optical binding. In contrast, the non-conservative coupling rate arises from the radiation pressure of the scattered field, which continuously pumps energy into the system. Consequently, the non-conservative coupling rate cannot be derived from the system Hamiltonian, and the energy of the coupled system is not conserved. By setting $\eta =0$, the conservative coupling rate $g$ can be obtained through normal-mode frequency splitting measurements [23] $$g=({\mathrm{\Omega}}_{-}-{\mathrm{\Omega}}_{+})/2.$$To ascertain the non-conservative coupling rate, we introduce the intermediate parameter $S(\eta )$, $$S(\eta )={({\mathrm{\Omega}}_{-}^{2}-{\mathrm{\Omega}}_{+}^{2})}^{2}/4={\eta}^{2}{\mathrm{\Omega}}^{4}+4\eta {\mathrm{\Omega}}^{3}\overline{g}+4{\mathrm{\Omega}}^{2}{g}^{2}.$$Here, the frequencies of the normal modes ${\mathrm{\Omega}}_{\pm}$ can be experimentally obtained from the $z$-mode signal of one of the particles, while the parameters $\eta $ and $\mathrm{\Omega}$ can be determined by independently measuring the frequencies of the $z$-modes of the two particles after turning off optical and electrostatic coupling. By adjusting the optical power of the two optical traps so that the frequencies of the two particles are identical, one can obtain the value of $\mathrm{\Omega}$, where $\mathrm{\Omega}={\mathrm{\Omega}}_{1}={\mathrm{\Omega}}_{2}$. When $\eta $ is not zero, its value can be determined using the definition of the parameter $\mathrm{\Omega}$ after measuring the frequencies ${\mathrm{\Omega}}_{1}$ and ${\mathrm{\Omega}}_{2}$ of the two particles, respectively.

If ${\overline{g}}^{2}>{g}^{2}$, the modes become degenerate within the range $[{\eta}_{1},{\eta}_{2}]$, where ${\eta}_{\mathrm{1,2}}=2(-\overline{g}\mp \sqrt{{\overline{g}}^{2}-{g}^{2}})/\mathrm{\Omega}$. By selecting $\eta \notin [{\eta}_{1},{\eta}_{2}]$, mode degeneracy is effectively prevented. Moreover, the constant ${G}_{0}\propto \sqrt{{P}_{1}{P}_{2}}$ implies that coupling rates $g,\overline{g}\propto \sqrt{1-{\eta}^{2}}$, demonstrating that a minor $\eta $ has a negligible impact on the coupling rates. By combining with Eqs. (9) and (12), the conservative and non-conservative coupling rates can be experimentally quantified through the measurement of the polarization dependence of the parameter $S$ for minor $\eta $.

Note that we present the results obtained only for the motion along the optical axis ($z$-axis) in this paper. However, standard optical coupling exists in all three directions of particle motion. The optical coupling forces along the $x$- and $y$-axes have the same magnitude as those along the $z$-axis. Nonetheless, the ratio of the coupling rates along the $z$- and $x$- ($y$-) directions depends on the inverse ratio of the mechanical frequencies ($\sim 3$) [23].

3. EXPERIMENTAL SETUP

Figure 2 shows our experimental setup for investigating light-induced interactions between two nanoparticles trapped using two orthogonally polarized optical traps. We employed an infrared laser source characterized by low intensity noise ($\lambda =1064\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$, ALS-IR-1064-5-I-SF, Azur Light Systems) as both the trapping beam and the reference beam for detecting the $z$-mode of the coupled particles. Note that the $x$- and $z$-modes mentioned in the following text refer to the oscillation modes of the coupled particles along the $x$- and $z$-axes, respectively. The trapping beam was expanded to a diameter of 4.2 mm in order to fill the aperture of the spatial light modulator (SLM, HOLOEYE, PLUTO-2-NIR-149, $1920\times 1080$ pixels) and the microscope objective (OBJ, CFI60 TU Plan Epi ELWD $100\times $, Nikon Corp., $\mathrm{NA}=0.8$, $\mathrm{WD}=4.5\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{mm}$). The SLM can only modulate the incident beam with polarization along its long display axis. Consequently, to create two beams with orthogonal polarization, we encoded a grating phase onto the SLM along its horizontal direction ($x$-axis) and adjusted the incident polarization to 45° relative to the horizontal plane using a half-wave plate (HWP2). This configuration allows the modulated and unmodulated light to form two optical traps with orthogonal polarization in the focal plane of the OBJ that corresponds to the Fourier plane of the trapping optics. By rotating HWP2, the power distribution between the two traps could be controlled. The phase difference and spacing of the traps were independently regulated by the computer-generated hologram (CGH), which was imaged onto the trapping optics via a 1:1 telescope set in a $4f$ configuration, consisting of lenses L1 and L2, each with a focal length of 300 mm. Subsequently, the SLM-modulated beam was tightly focused by the OBJ, in conjunction with the unmodulated zeroth-order beam, to establish two distinct optical traps featuring orthogonal polarization. The beam waists of the two traps were about 650 nm, as estimated through tight focusing simulation [26]. The total power used in front of the vacuum chamber was about 1.5 W, resulting in a frequency of approximately $2\pi \times 65\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{kHz}$ for the $z$-modes of the two particles without optical coupling trapped in two nearly perfectly balanced traps. We maintained a stable pressure of $p\sim 1.5\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{mbar}$ in the vacuum chamber, at which the particles are in thermal equilibrium with the environment and the net charge of the particles can be efficiently controlled.

Figure 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 ${r}_{s}\sim 100\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$ trapped in the two traps at a distance ${r}_{0}\sim 2.65\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{\mu m}$ 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.

A bare electrode, connected to a high voltage DC source (HV, ${V}_{\mathrm{HV}}\sim \pm 800\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{V}$), was employed to control the net charges of the particles via the process of corona discharge. Adjacent to the beam focus, a pair of electrodes mounted on a 3D piezo stage (AG-LS25V6, Newport) was aligned along the $x$-axis. These electrodes were connected to an amplified signal from a function generator (FG) to drive the particles at a frequency of ${\omega}_{\mathrm{dr}}$. For stability, the driving frequency was maintained at around 10 kHz away from the resonance frequency of the $x$-mode (around $2\pi \times 180\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{kHz}$), i.e., ${\omega}_{\mathrm{dr}}\sim {\mathrm{\Omega}}_{x}\pm 10\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{kHz}$. The $x$-mode signal was recorded by a dual-channel lock-in amplifier (UHFLI, Zurich Instruments), which was synchronized with the function generator. This setup enables the demodulation of the oscillation amplitude ${A}_{\mathrm{LI}}$ and phase ${\varphi}_{\mathrm{LI}}$ within a narrow frequency range centered around ${\omega}_{\mathrm{dr}}$, facilitating the determination and control of the net charge on the particles.

After tight focusing, the trapping beams were collimated using an aspheric collection lens (CL, AL1512-C, Thorlabs, $\mathrm{NA}=0.55$) positioned on a 3D piezo stage. In our experiment, the two optical traps had orthogonal polarization, allowing for separating the signals of the two particles using a combination of an HWP and a polarization beam splitter (PBS). Specifically, our experiment utilized two electronically controlled half-wave plates (EHWPs), each consisting of a half-wave plate and a commercially available piezoelectric-driven rotating stage (AG-PR100, Newport). The first EHWP was positioned in front of the vacuum chamber to rotate the polarization of the traps, while the second was placed behind the chamber to function with PBS2 in separating the signals of different particles. The two EHWPs were synchronized to start rotating by an external trigger signal. Although the step length of each piezoelectric rotating stage may vary slightly, the experimental error resulting from this is negligible for our study focusing on mode splitting, as the movements of the two particles along the $z$-axis consist of eigen oscillation modes with identical frequencies but differing amplitudes and phases. The intensity of these separated beams was then reduced to a suitable level using a neutral density filter (NDF) and subsequently focused onto two homebuilt quadrant photodetectors (QPDs). A reference beam was directed onto the corresponding reference photodiode to suppress common mode noise for $z$-mode signals. The common-mode rejection ratios of our QPDs were approximately 45 dB at 65 kHz for $z$-mode signals and 47 dB at 180 kHz for $x$-mode signals, respectively. The signals from QPDs were then fed into the dual-channel lock-in amplifier. The $x$-mode signal was used to determine the net charge of the particles, while the $z$-mode signal was used to investigate the interactions between the coupled particles.

A beam of 532 nm light was focused onto the particles along the $y$-axis for imaging through the microscope objective. For simplicity, the optical path along the $y$-axis was not illustrated in Fig. 2. The imaging optical path in the $y\u2013z$ plane was depicted as the green path. Additionally, two bandpass filters (BPFs) were employed to eliminate unwanted trapping beams. The inset presents an image of two nanoparticles with radius ${r}_{s}\sim 100\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$ trapped in the two traps at a distance ${r}_{0}\sim 2.65\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{\mu m}$ along the $x$-axis. Note that the actual experimental separation of the particles was determined by comparing the images of the nanoparticles with those of the calibration target (R1L3S2P, Thorlabs) captured through the same imaging optical path.

4. RESULTS AND DISCUSSION

A. Neutralizing Trapped Nanoparticles

Optically levitated nanoparticles usually carry an initial net charge of tens of elementary charges. The coupling rate induced by electrostatic interaction was given by [23] $${g}_{e}=-\frac{{q}_{1}{q}_{2}}{8\pi {\u03f5}_{0}m{\mathrm{\Omega}}^{\prime}{r}_{0}^{3}},$$where ${\mathrm{\Omega}}^{\prime}=\sqrt{{\mathrm{\Omega}}^{2}-\frac{{q}_{1}{q}_{2}}{4\pi {\u03f5}_{0}m{r}_{0}^{3}}}$. The electrostatic coupling rate was approximately proportional to ${r}_{0}^{-3}$, increasing more rapidly with decreasing interparticle distance than the optical coupling rate, which was roughly proportional to ${r}_{0}^{-1}$. In our experiment, the electrostatic coupling rate, attributable to the initial net charge of the particles, was comparable in magnitude to the optical coupling rate of the coupled motion along the $z$-axis, with both rates in the tens of kHz range. To demonstrate this, we measured the electrostatic coupling rate ${g}_{e}$ as a function of the trap separation ${r}_{0}$ without optical coupling [${\theta}_{1}=0$, leading to $g=\overline{g}=0$; see Eq. (12) and related discussion], as shown in Fig. 3(a). The data were fitted using Eq. (14) with an experimentally determined $\mathrm{\Omega}=2\pi \times 65\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{kHz}$ (red curve). Note that only the data with the coupling rates greater than a half of mechanical linewidth, i.e., ${g}_{e}>\gamma /2$, were used for fitting, since normal-mode frequency splitting became indistinguishable when ${g}_{e}\le \gamma /2$. The parameter ${q}_{1}{q}_{2}/m$ was adjustable during the fitting process, and its resulting value was $2.42\times {10}^{-17}\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{C}}^{2}/\mathrm{kg}$. For nanoparticles with ${r}_{s}=100\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$ and $\rho =1850\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{kg}/{\mathrm{m}}^{3}$, the product of the charge numbers of the two particles was approximately 7325, verifying the statement that “optically levitated nanoparticles usually carry an initial net charge of tens of elementary charges.” Therefore, neutralizing their net charge was essential to accurately investigate light-induced dipole–dipole interactions between two nanoparticles.

Figure 3.(a) Electrostatic coupling rate ${g}_{e}$ as a function of trap separation ${r}_{0}$. Optical coupling was deactivated by setting the polarization of the trapping beam ${\theta}_{1}=0$. The avoided crossing was indiscernible for coupling rates smaller than a half of mechanical linewidth $\gamma /2$ (gray region). (b), (c) Simultaneous net charge neutralization process for the trapped nanoparticles 1 and 2, respectively. ${A}_{\mathrm{LI}}$ and ${\varphi}_{\mathrm{LI}}$ were the demodulated oscillation amplitude and phase within a narrow frequency range centered around ${\omega}_{\mathrm{dr}}$. Discrete steps of ${A}_{\mathrm{LI}}$ indicated the addition or removal of charges from the particle, while a 180° phase shift denoted a reversal in charge polarity. After time ${t}_{0}$, the amplitude ${A}_{\mathrm{LI}}$ for both particles dropped to zero, and the phase ${\varphi}_{\mathrm{LI}}$ became disordered, signifying the simultaneous neutralization of the charge of both particles.

We employed a high-voltage discharge method to neutralize the net charge of the two trapped nanoparticles simultaneously. The charge control system is depicted in Fig. 2. Figures 3(b) and 3(c) show the simultaneous net charge neutralization process for the trapped nanoparticles 1 and 2, respectively. The oscillation amplitude ${A}_{\mathrm{LI}}$ was proportional to the number of the elementary charges carried by the particle, and the phase ${\varphi}_{\mathrm{LI}}$ reflected the charge polarity. Specifically, 180° phase shift indicated charge polarity reversal. When a particle was neutralized, its phase became undefined, as shown by the disordered phase curve in Figs. 3(b) and 3(c). It is evident that the charges of both particles were simultaneously neutralized after time ${t}_{0}$.

B. Phase Difference $\mathbf{\Delta}{\mathit{\phi}}_{\mathbf{0}}$

To demonstrate the dependence of the optical coupling rates on the phase difference $\mathrm{\Delta}{\phi}_{0}$, we measured the frequency splitting of the normal-mode oscillations along the $z$-axis at a fixed separation of ${r}_{0}\sim 2.65\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{\mu m}$ and a polarization ${\theta}_{1}=3\pi /4$. The chosen separation distance gives $k{r}_{0}\sim 5\pi $, corresponding to a minimum of the factor $\mathrm{cos}(k{r}_{0})$ and a zero of the factor $\mathrm{sin}(k{r}_{0})$. Thus, the non-conservative coupling interaction should theoretically approach zero. The power distribution factor $\eta $ was set to zero, indicating that the conservative optical coupling rate $g$ was equal to half the normal-mode frequency splitting $({\mathrm{\Omega}}_{-}-{\mathrm{\Omega}}_{+})/2$. Figure 4(a) shows that the interactions at $\mathrm{\Delta}{\phi}_{0}=n\pi \text{\hspace{0.17em}\hspace{0.17em}}(n\in \mathbb{Z})$ were predominantly conservative, resulting in experimental data near these points aligning well with the theoretical model that considers only conservative interaction [conservative coefficient ${k}_{c}$ in Eq. (9), blue curve, with amplitude $g=27.5\pm 1.0\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{kHz}$]. For all other values of $\mathrm{\Delta}{\phi}_{0}$, the theoretical model failed to fully predict the experimental data due to residual non-conservative interaction. The non-conservative interaction continuously pumps energy into the system, thereby increasing the particle motional amplitude by an order of magnitude. Without an additional cooling mechanism, the particles can explore nonlinear terms in the Hamiltonian interaction due to the amplified motional amplitude, which affects the eigenfrequencies and modifies the normal-mode splitting. However, the concordance between our experimental data and the theoretical curve has significantly improved compared with Fig. 3(c) in Ref. [23]. In both cases, the trap separation ${r}_{0}$ was chosen to give $\mathrm{sin}(k{r}_{0})\sim 0$, thereby eliminating the non-conservative interaction. We attributed this improvement to the utilization of two orthogonally polarized optical traps, which significantly suppressed the interference effect, thereby enhancing the position stability of the particles.

Figure 4.(a) Half of the normal-mode frequency splitting, defined as $({\mathrm{\Omega}}_{-}-{\mathrm{\Omega}}_{+})/2$, versus the phase difference $\mathrm{\Delta}{\phi}_{0}$ at a trap separation of ${r}_{0}\sim 2.65\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{\mu m}$ and polarization ${\theta}_{1}=3\pi /4$. The power distribution factor $\eta $ was set to zero. The blue curve represented an ideal dependence, exclusively accounting for conservative interactions and being proportional to $\mathrm{cos}(\mathrm{\Delta}{\phi}_{0})$. The non-conservative force contributed to the total force for the values of $\mathrm{\Delta}{\phi}_{0}$ different from $n\pi \text{\hspace{0.17em}\hspace{0.17em}}(n\in \mathbb{Z})$ 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, $\mathrm{\Delta}{\phi}_{0}=\pi /3$) and resonance (orange data, $\mathrm{\Delta}{\phi}_{0}=\pi /2$).

Moreover, when tuning $\mathrm{\Delta}{\phi}_{0}$ such that the normal-mode frequency splitting approached zero, we observed phase locking and frequency degeneracy due to the non-reciprocal interactions being dominant. Figure 4(b) demonstrates that at a phase difference of $\mathrm{\Delta}{\phi}_{0}=\pi /3$, the power spectral density (PSD) of the $z$-mode displayed typical normal-mode frequency splitting. However, when the phase difference reached $\mathrm{\Delta}{\phi}_{0}=\pi /2$, the resonance amplified the motion of particles, as evidenced by the orange PSD data. During such resonance, mode degeneracy made it impossible to distinguish the modes, as illustrated in Fig. 4(a).

C. Trap Separation ${r}_{0}$

Dipole–dipole interactions originate from the interference between the trapping field and dipole scattering fields. The coupling rate exhibited oscillations with a period of about $\lambda $ as the trap separation ${r}_{0}$ changed, with its amplitude decreasing in accordance with ${r}_{0}^{-1}$. Likewise, the variation of the parameter $S$ displayed oscillations with a period of about $\lambda /2$, and its amplitude decreased as ${r}_{0}^{-2}$. These characteristics follow from the dependence of $S$ on the square of the coupling rate $g$, as expressed by Eq. (13). We measured the parameter $S$ across trap separations ranging from 1.75 to 4.24 μm, with $\mathrm{\Delta}{\phi}_{0}=0$ and ${\theta}_{1}=3\pi /4$, to maximize conservative interaction. Note that the actual interparticle distance was different from the trap separation owing to the radiation pressure force of the dipole radiation, which provided a constant displacement force along the $x$-axis. The results are shown in Fig. 5. The parameters $\eta $ and $\mathrm{\Omega}$ were set to 0.38 and $2\pi \times 65\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{kHz}$, respectively, at the initial point ${r}_{0}=4.24\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{\mu m}$.

Figure 5.Parameter $S$ varied with the trap separation ${r}_{0}$ at $\mathrm{\Delta}{\phi}_{0}=0$ and ${\theta}_{1}=3\pi /4$. The curve exhibited a periodicity $\sim \lambda /2$. The red points indicated the minima of the curve, where the conservative coupling rate $g\sim 0$. The power distribution factor $\eta $ of the two traps changed with the trap separation distance, resulting in the deviation of the parameter $S$ from the theoretical model as ${r}_{0}$ varied.

In our experiment, we adjusted the trap separation ${r}_{0}$ by moving one of the optical traps, resulting in a misalignment between the trapping beam and the microscope objective. This misalignment subsequently changed the power distribution factor $\eta $ and the oscillation frequency $\mathrm{\Omega}$ as ${r}_{0}$ varied, thereby modifying the trend of parameter $S$ in relation to ${r}_{0}$ and resulting in deviations from the theoretical model outlined in Eq. (13). However, the period of the curve still closely aligned with the expected $\lambda /2$. The minima of the curve, marked by red points, indicated a conservative coupling rate approaching zero. The relevant trap separations, ${r}_{0}\sim \mathrm{2.28,}\mathrm{2.88,}\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{3.50,}\text{\hspace{0.17em}\hspace{0.17em}}4.05\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{\mu m}$, aligned well with the values reported in Ref. [23]. In the future, one could monitor the output power of the two traps after the microscope objective and simultaneously apply feedback control on the total power and polarization direction of the laser illuminating on the SLM. This approach will ensure that $\eta $ and $\mathrm{\Omega}$ remain constant as the trap separation ${r}_{0}$ changes.

In addition, when ${r}_{0}<1.75\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{\mu m}$, the particles escaped. Because the escape occurred at a position where the conservative coupling rate approached zero, we speculated that the residual non-conservative force pumped energy into the system and amplified the motion of particles, thereby leading to their escape.

D. Polarization ${\mathit{\theta}}_{\mathbf{1}}$ and ${\mathit{\theta}}_{\mathbf{2}}$

Rotating the polarization of the trapping beams provided an easy-to-implement method to control dipole–dipole interactions. In our experiment, optical traps with orthogonal polarization consistently maintained ${\theta}_{1}-{\theta}_{2}=\pi /2$ throughout the rotation process. Figure 6(a) shows four distinct scenarios within the dipole scattering model for the two trapped particles, specifically at ${\theta}_{1}=0,\pi /4,\pi /2$, and $3\pi /4$. The unique spatial distribution of dipole radiation in the far field resulted in a reduced amplitude of dipole radiation along the $x$-axis, scaled by $\mathrm{sin}({\theta}_{1})$ and $\mathrm{cos}({\theta}_{1})$ for the two nanoparticles, respectively. Consequently, when ${\theta}_{1}$ equaled $0$ or $\pi /2$, the $x$-component of the dipole scattering field for one of the particles became zero, effectively closing the dipole–dipole interactions. At ${\theta}_{1}=\pi /4$ and $3\pi /4$, the amplitude of the product of the $x$-components of the dipole scattering fields reached its maximum, signifying the strongest dipole–dipole interactions. However, due to a phase shift of $\pi $ in ${\theta}_{1}$, although the amplitudes were identical, their polarities were opposite. These phenomena are clearly demonstrated in Figs. 6(b) and 6(c).

Figure 6.(a) Four special cases (${\theta}_{1}=0,\pi /4,\pi /2$, and $3\pi /4$) within the dipole radiation model for nanoparticles trapped in two orthogonally polarized optical traps, satisfying ${\theta}_{1}-{\theta}_{2}=\pi /2$. (b) At a trap separation ${r}_{0}\sim 2.65\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{\mu m}$, the polarization ${\theta}_{1}$ was changed while keeping the phase difference fixed at $\mathrm{\Delta}{\phi}_{0}=0$ (blue data) or $\mathrm{\Delta}{\phi}_{0}=\pi $ (orange data). For $\mathrm{\Delta}{\phi}_{0}=0$, the coupling rate $g$ was proportional to $-\mathrm{sin}(2{\theta}_{1})$. For $\mathrm{\Delta}{\phi}_{0}=\pi $, an additional factor of $-1$ enters into the expression for coupling rate $g$ due to the $\pi $-phase shift of $\mathrm{\Delta}{\phi}_{0}$. The power distribution factor $\eta $ was set to zero. (c) Dependence of the parameter ${S}^{\prime}$, defined as $(S-{\eta}^{2}{\mathrm{\Omega}}^{4})/4$, on the polarization ${\theta}_{1}$ for different phase differences $\mathrm{\Delta}{\phi}_{0}=\mathrm{0,}\text{\hspace{0.17em}\hspace{0.17em}}4\pi /\mathrm{9,}\text{\hspace{0.17em}\hspace{0.17em}}2\pi /3$. The other parameters were determined experimentally, including trap separation ${r}_{0}\sim 2.65\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{\mu m}$, power distribution factor $\eta =-0.2$, and oscillation frequency $\mathrm{\Omega}=2\pi \times 65.3\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{kHz}$.

Analysis of Fig. 4 revealed that at phase differences $\mathrm{\Delta}{\phi}_{0}=0$ and $\pi $, only conservative interaction was present. By setting the power distribution factor $\eta $ to zero, we obtained the conservative coupling rate $g$ through measurements of the normal-mode frequency splitting. The corresponding experimental results are illustrated in Fig. 6(b). At a trap separation ${r}_{0}\sim 2.65\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{\mu m}$, when $\mathrm{\Delta}{\phi}_{0}=0$, the coupling rate $g$ was proportional to $-\mathrm{sin}(2{\theta}_{1})$. For $\mathrm{\Delta}{\phi}_{0}=\pi $, an additional factor of $-1$ enters into the expression for coupling rate $g$ due to the $\pi $-phase shift of $\mathrm{\Delta}{\phi}_{0}$. This dependence of $g$ on the polarization parameter ${\theta}_{1}$ aligned perfectly with the predictions of our theoretical model. Compared to results from two traps with identical polarization, where the coupling rate $g$ was proportional to ${\mathrm{cos}}^{2}(\theta )$ [23], our orthogonally polarized traps enabled adjustment of the strength and polarity of dipole–dipole interactions through the rotation of the polarization. This capability made orthogonally polarized optical traps a significant advantage in experiments that require precise control of dipole–dipole interactions, especially where consistent trap separation and optical phase are essential. Experimentally, adjusting the polarization of the traps was straightforward, allowing for continuous adjustment while maintaining orthogonality simply by using a half-wave plate, as EHWP1 depicted in Fig. 2. Our work thereby expanded the toolkit for controlling light-induced dipole–dipole interactions.

Furthermore, we measured the non-conservative coupling rate and the amplitude of the conservative coupling rate under varying phase difference by fitting the polarization dependence of parameter ${S}^{\prime}$: $${S}^{\prime}({\theta}_{1})=[S(\eta )-{\eta}^{2}{\mathrm{\Omega}}^{4}]/4=\eta {\mathrm{\Omega}}^{3}{\overline{g}}_{m}\text{\hspace{0.17em}}\mathrm{sin}(2{\theta}_{1})+{\mathrm{\Omega}}^{2}{g}_{m}^{2}\text{\hspace{0.17em}}{\mathrm{sin}}^{2}(2{\theta}_{1}).$$Here, ${g}_{m}$ and ${\overline{g}}_{m}$ represent the conservative and non-conservative coupling rates at ${\theta}_{1}=\pi /4$, respectively, for a fixed trap separation ${r}_{0}$ and phase difference $\mathrm{\Delta}{\phi}_{0}$. To avoid phase locking and resonance, we introduced a power difference in the trapping beams. The parameters $\eta =-0.2$ and $\mathrm{\Omega}=2\pi \times 65.3\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{kHz}$ were determined experimentally by disabling the optical coupling at ${\theta}_{1}=0$ and measuring the trapping frequencies ${\mathrm{\Omega}}_{1}$ and ${\mathrm{\Omega}}_{2}$ of the two particles. Results for $\mathrm{\Delta}{\phi}_{0}=\mathrm{0,}\text{\hspace{0.17em}\hspace{0.17em}}4\pi /\mathrm{9,}\text{\hspace{0.17em}\hspace{0.17em}}2\pi /3$ are shown in Fig. 6(c). At $\mathrm{\Delta}{\phi}_{0}=0$, the amplitude of the conservative coupling rate, $|{g}_{m}|=25.9\pm 0.4\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{kHz}$, significantly exceeded the non-conservative coupling rate, ${\overline{g}}_{m}=0.6\pm 0.2\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{kHz}$, consistent with our measurement in Fig. 6(b), where $g=-28.4\pm 0.4\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{kHz}$. Moreover, ${S}^{\prime}$ exhibited axial symmetry around ${\theta}_{1}=\pi /2$ with a periodicity of $\pi /2$. At $\mathrm{\Delta}{\phi}_{0}=2\pi /3$, the non-conservative coupling rate, ${\overline{g}}_{m}=13.2\pm 0.4\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{kHz}$, was significantly higher than the nearly zero conservative coupling rate, in agreement with the zero crossing in Fig. 4(a). Additionally, ${S}^{\prime}$ showed central symmetry at about ${\theta}_{1}=\pi /2,{S}^{\prime}\sim 0$, with a periodicity of $\pi $. For other values of $\mathrm{\Delta}{\phi}_{0}$, such as $4\pi /9$, both coupling rates contributed, resulting in an asymmetric ${S}^{\prime}$ curve around ${\theta}_{1}=\pi /2$, as evidenced by the orange data points, where $|{g}_{m}|$ and ${\overline{g}}_{m}$ were $15.1\pm 0.5\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{kHz}$ and $9.3\pm 0.2\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{kHz}$, respectively. Note that the polarity of ${g}_{m}$ could be determined by the polarity of the normal-mode frequency splitting as described in Eq. (12). Due to the $\pi /2$-phase shift of ${\theta}_{1}$, the polarities of ${g}_{m}$ for $\mathrm{\Delta}{\phi}_{0}=0$ and $4\pi /9$ were opposite to those shown in Fig. 4(a), i.e., ${g}_{m}=-25.9\pm 0.4\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{kHz}$ and $-15.1\pm 0.5\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{kHz}$, respectively.

From Figs. 4(a) and 6(c), one could see that although Eq. (9) failed to accurately predict optical coupling rates when the phase difference $\mathrm{\Delta}{\phi}_{0}\ne 0$ or $\pi $, the dependency of the optical coupling rates on the polarization ${\theta}_{1}$ still held for a fixed $\mathrm{\Delta}{\phi}_{0}$. We attributed this to the residual interference effect, which was particularly sensitive to the phase difference of the trapping beams. This effect arose because the polarization state of our traps could not maintain perfect orthogonality, owing to the limited diffraction efficiency of our SLM. This finding underscored that adjusting the strength and polarity of the optical coupling rates by rotating the polarization was more reliable than making adjustments through the phase differences, highlighting the advantage of using orthogonally polarized optical traps in our study.

5. CONCLUSION

In summary, we have demonstrated the fully controllable light-induced dipole–dipole interactions between two silica nanoparticles levitated in distinct optical traps with orthogonal polarization. This orthogonal polarization not only effectively suppresses the interference effect arising from the tail of the trapping fields but also expands the toolbox for controlling dipole–dipole interactions, specifically enabling the adjustment of the coupling strength and polarity through polarization rotation. Furthermore, we introduced a power difference in the trapping beams and assessed the polarization dependence of parameter ${S}^{\prime}$ to simultaneously evaluate the conservative and non-conservative coupling rates. This method offers a broadly applicable and stable approach for measuring both types of coupling rates, effectively mitigating the impact of resonance instabilities typically induced by non-conservative interactions.

We note that our work can facilitate a number of experiments requiring two coupled nanoparticles. Quantum (or vacuum) friction is a prominent example of such a study [32]. Furthermore, combined with previously realized quantum state preparation, our work provides a platform for quantum simulation with mechanical degrees of freedom [33,34], enhanced quantum sensing [35], and entanglement with nanoscale objects [36,37].