Cavity-mediated long-range interactions in levitated optomechanics

Exploring quantum physics at macroscopic scales presents significant challenges, primarily the need for either large-scale delocalization of a single object or entanglement of multiple objects, alongside the difficulty of ground-state cooling massive objects. Levitodynamics, which focuses on controlling the motion of massive oscillators in vacuum, has shown progress in multi-particle systems, demonstrating cooling and short-range coupling between levitated nanoparticles. However, free-space entanglement is hindered by insufficient entangling rates compared to decoherence rates. This necessitates the use of optical cavities to mediate coupling between particles. This paper introduces a method to engineer programmable cavity-mediated interactions between multiple spatially separated particles in vacuum. Programmability is achieved using acousto-optic deflectors (AODs) to create tweezer arrays for trapping particles, allowing precise control over optical frequencies, cavity detuning, mechanical frequencies, and particle positions. This control is crucial for tuning interaction strength and selecting which particles and modes couple. Most experimental systems studying many-body physics use localized short-range interactions or a common cavity mode, limiting connectivity. This new approach in levitodynamics enables generating quantum correlations and entanglement, exploring complex phases from interacting particles, and employing multiparticle quantum resources for optomechanical sensing.

The authors review existing literature on macroscopic quantum physics, highlighting the challenges of ground-state cooling and entanglement of massive objects. They discuss the advancements in levitodynamics, specifically mentioning previous work on cooling and short-range coupling of levitated nanoparticles. The limitations of free-space entanglement due to decoherence are addressed, emphasizing the need for cavity-mediated interactions. The paper also contrasts its approach with existing many-body physics systems, noting the limitations of localized short-range interactions and common-mode cavity-mediated interactions in terms of connectivity and tunability. The authors cite relevant works on superconducting qubits, cold atoms, and other optomechanical systems to establish the novelty and significance of their approach.

The experiment uses near-spherical SiO₂ nanoparticles (150 nm diameter) levitated in vacuum (~10⁻⁴ mbar) using optical tweezers (NA = 0.75) at 1,550 nm. The particles are placed in an optical cavity (linewidth κ/2π = 600 kHz) with mirrors separated by 9.6 mm. Two tweezers with identical optical frequencies are generated using two AODs. The cavity resonance is detuned by Δ from the tweezers. Light scattered by the particles, carrying information about their center-of-mass motion, leaks through the cavity mirror and is combined with a local oscillator for balanced heterodyne detection. Particle positions and separation are controlled by the AOD RF inputs. The interparticle separation (typically 6 μm) is large enough to suppress short-range Coulomb and free-space optical binding interactions. Optical power and mechanical frequencies are controlled by the AOD RF amplitudes. To engineer interactions, the mechanical frequencies of the two particles are brought close together using linear ramps of optical powers. Coherent scattering, where light scattered by each nanoparticle populates the cavity mode, results in optomechanical coupling (gᵢ) and effective cavity-mediated interactions (Gμ) between nanoparticles. For tweezers polarized along the cavity axis, there is no particle-particle coupling. The coupled system has two normal modes (λ₁ and λ₂), with the minimal splitting at the avoided crossing being 2|Gyy|. Spectrograms of particle modes during power sweeps are analyzed for different cavity detunings to extract mode splittings. The distance dependence is investigated by keeping one particle stationary and scanning the position of the second particle along the cavity standing wave. The tunability of interacting modes is explored by keeping the interparticle separation fixed and moving the particle pair along the standing wave to vary the relative interaction strengths of transverse (y) and longitudinal (z) modes. The authors also detail the theoretical modelling of coherent scattering with two particles, including the derivation of the effective cavity-mediated couplings and the method for extracting mode splittings from spectrograms.

The experiments demonstrate cavity-mediated long-range interactions between levitated nanoparticles. The interaction strength scales with cavity detuning as predicted by theory. The measured mode splittings are in excellent agreement with theoretical estimates. The interaction exhibits a periodic dependence on interparticle distance, confirming cavity mediation. The authors demonstrate tunability of interactions between different mechanical modes (y and z) by changing the particle positions within the cavity standing wave. The maximum interaction strength achieved is Gzz/Ωz = 0.238 ± 0.005 for the longitudinal z modes, exceeding values reported in free-space experiments despite larger separations. Short-range interactions (Coulomb and direct optical) are shown to be negligible compared to the cavity-mediated interaction.

The results confirm the successful engineering and characterization of programmable cavity-mediated long-range interactions between levitated nanoparticles. The high degree of control over experimental parameters allows for detailed investigation of the interaction's dependence on cavity detuning, interparticle distance, and mechanical mode. The observation of tunable interactions between different modes opens up possibilities for controlling the dynamics of coupled systems. The strong coupling achieved, despite the relatively large interparticle separations, highlights the advantages of the cavity-mediated approach compared to free-space interactions. The findings are significant for advancing macroscopic quantum physics and developing ultra-precise optomechanical sensors.

This work establishes a new platform for optomechanics by combining cavity-based coherent scattering and multiparticle levitation. The high degree of control enables engineering and investigating cavity-mediated long-range interactions between mechanical oscillators. Future research could focus on scaling up to more particles, achieving the quantum regime by improving trap frequencies and pressure, and employing cavities with smaller mode volumes or narrower linewidths to further enhance interaction strength for motional entanglement. The programmable cavity-mediated interactions offer a powerful resource for exploring the boundaries of quantum physics with massive interacting mechanical systems and building ultra-precise sensors.

The study is currently limited to two nanoparticles. Scaling up to larger arrays of nanoparticles could be challenging. The current setup might face limitations in achieving extremely low temperatures necessary for observing quantum effects. Further advancements in controlling and reducing noise sources would enhance the precision and sensitivity of the measurements.