Talking with a ghost: semi-virtual coupled levitated oscillators

Mesoscopic particles levitated by optical, electrical or magnetic fields act as mechanical oscillators with a range of surprising properties, such as tuneable oscillation frequencies, access to rotational motion, and remarkable quality factors. Coupled levitated particles display rich dynamics and non-reciprocal interactions, with applications in sensing and the exploration of non-equilibrium and quantum physics. In this work, we present a single levitated particle displaying coupled-oscillator dynamics by generating an interaction with a virtual or ``ghost’’ particle. This ghost levitated particle is simulated on an analogue computer, and its properties can thus be dynamically varied. Our work represents a new angle on measurement-based bath engineering and physical simulation and, in the future, could lead to the generation of novel cooling mechanisms and complex physical simulation.

💡 Research Summary

In this work the authors demonstrate a novel “semi‑virtual” coupled‑oscillator platform that merges a real levitated microsphere with a simulated “ghost” particle generated on an analogue computer. The real particle is a 5 µm silica sphere, positively charged to (6 ± 1) × 10³ e, trapped in a linear Paul trap at a pressure of ~2 × 10⁻² mbar. Its one‑dimensional centre‑of‑mass motion follows a Langevin equation with damping Γ_r, resonance ω_r and Brownian noise ξ_r. Position is tracked by an event‑based camera (EBC) and streamed via an FPGA as a continuous analog voltage to the analogue computer.

The analogue computer, built from integrators, summers, multipliers and potentiometers, solves in real time the Langevin equation of a driven harmonic oscillator representing the ghost particle. Its parameters—effective mass M_g, damping Γ_g, resonance ω_g and noise amplitude ξ_g—are independently tunable through four potentiometers (controlling M_g ω_g², M_g Γ_g, 1/M_g) and an external white‑noise source. By varying the noise voltage from 0.1 V to 3 V the effective temperature T_g of the ghost can be swept over a wide range; by adjusting M_g the accessible resonance frequencies span 2 Hz–160 Hz, and Γ_g can be set between 0.8 Hz and 27.5 Hz, matching the real particle’s quality factor.

Bidirectional coupling is realized by feeding back the measured real‑particle position to the analogue computer (producing a force F_r = k_g (q_r − q_g)/M_g) and by converting the ghost’s simulated position into a voltage that drives an electrode near the real particle (force F_g = k_r (q_g − q_r)/M_r). The coupling constants k_r and k_g, as well as the processing delay τ (≪ one oscillation period), are adjustable, allowing the system to emulate a linear spring interaction F_i = k_i(q_r − q_g).

When uncoupled, the power spectral densities (PSDs) of the real and ghost particles each follow the expected Lorentzian form, with the ghost’s PSD confirming the tunability of frequency, damping, and effective temperature. Upon activation of the coupling, the two oscillators hybridize into in‑phase (i) and out‑of‑phase (o) normal modes. In a representative configuration where ω_r ≈ ω_g ≈ 123 Hz and Γ_r ≈ Γ_g ≈ 1 Hz, the coupled system exhibits peaks at 123 Hz (the original mode) and at 142.5 Hz (the split out‑of‑phase mode). Coherence |S_rg|²/(S_rr S_gg) reaches unity at both frequencies, while the relative phase jumps by π, confirming the formation of distinct collective modes.

The authors compare the measured spectra with analytical expressions derived from the coupled Langevin equations (including the delay τ) and find excellent agreement. The depth of the dip between the two peaks is explained by the unequal noise terms ξ_r/M_r and ξ_g/M_g, which are fully captured by the model.

Key contributions of the paper are: (1) demonstration that an analogue computer can serve as a real‑time, continuously tunable bath for a levitated particle, providing dynamic control over mass, damping, resonance and temperature; (2) implementation of a hardware‑in‑the‑loop architecture that creates a genuine bidirectional interaction between a physical system and its simulated counterpart, opening the door to engineered non‑reciprocal or non‑linear couplings; (3) concrete realization of measurement‑based bath engineering, where the ghost particle acts as a controllable synthetic environment that could be used for novel cooling schemes, non‑equilibrium studies, or as a building block for larger arrays of hybrid real‑virtual oscillators. The approach promises extensions to multi‑particle levitated arrays, quantum‑level coupling, and exploration of exotic phenomena such as gravity‑induced entanglement, making it a versatile platform for both applied sensing and fundamental physics investigations.