Talking with a ghost: semi-virtual coupled levitated oscillators

TL;DR

The paper presents a semi-virtual coupled-oscillator platform where a real levitated microsphere interacts with a ghost oscillator simulated on an analogue computer, forming a hardware-in-the-loop system. The authors characterize the ghost, implement a delayed, Coulomb-like coupling, and demonstrate tunable coupling between equal- and unequal-frequency oscillators, with quantitative agreement between experiment and theory in PSDs, coherence, and mode analysis. This approach enables dynamic bath engineering and physical analogue simulation beyond purely real systems and is scalable to multi-degree-of-freedom arrays. Potential applications include novel cooling strategies and exploration of non-reciprocal interactions and complex dynamics in levitated systems.

Abstract

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 hence its prperties can 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.

Figures (7)

• Figure 1: a Illustration of the experiment. A real charged microsphere is levitated in vacuum and tracked with an event-based camera (EBC) ren2022, producing a signal proportional to the real particle position $q_\mathrm{r}$ which is fed into an analogue computer. The motion of the ghost particle $q_\mathrm{g}$ is simulated on the same analogue computer, and used to produce a voltage on an electrode near the real particle. b Simplified diagram of the analogue computer and the coupling between the real and ghost particles. The analogue computer enables dynamic control over six quantities related to the ghost particle: 1 the initial position $q_0$, 2 a quantity related to the oscillation frequency $\omega_\mathrm{g}$, 3 a quantity related to the damping rate $\Gamma_\mathrm{g}$, 4 a quantity related to the mass of the oscillator $M_\mathrm{g}$, 5 the strength of the feedback force $F_\mathrm{r}$ from the real to the ghost particle, 6 the strength of the feedback force $F_\mathrm{g}$ from the ghost to the real particle.
• Figure 2: Characterization of the "ghost" virtual oscillator. a PSDs of the ghost particle's motion with different noise amplitude $\xi_\mathrm{g}$, effectively raising the ghost particle temperature $T_\mathrm{g}$. b PSDs illustrating variable oscillation frequencies for the ghost particle. Varying the effective mass $M_\mathrm{g}$ enables different ranges of accessible frequencies. c PSDs of the ghost particle's motion with different momentum damping rates $\Gamma_\mathrm{g}$. As described in the text, increasing $\Gamma_\mathrm{g}$ for fixed amplitude $\xi_\mathrm{g}$ effectively reduces the temperate $T_\mathrm{g}$. d Accessible oscillation frequency and damping rate range for the ghost particle, indicated by shaded regions, with the different colours representing different values of $M_\mathrm{g}$. Solid points represent example measured values and uncertainties obtained by fitting \ref{['equ:psd_formula_ghost']} to records of $q_\mathrm{g}(t)$ (length $150\,\unit{s}$, sampling rate $20\,\unit{kHz}$).
• Figure 3: Synthesizing a coupling between the real and ghost particles. a Position PSDs of the un-coupled real (red) and ghost (blue) particles, $\omega_\mathrm{r} = 2\pi\times (123\pm0.2)\,$Hz, $\omega_\mathrm{g} = 2\pi\times (123\pm0.2)\,$Hz, $\Gamma_\mathrm{r} = 2\pi\times (1.0\pm0.2)\,$Hz, $\Gamma_\mathrm{g} = 2\pi\times (1.2\pm0.2)\,$Hz. These parameters and their uncertainties are extracted by fitting \ref{['equ:psd_formula_real', 'equ:psd_formula_ghost']} to the data. b (left) After the semi-virtual oscillators are coupled both PSDs exhibit similar behavior with resonances at 123.0 Hz and 142.5 Hz. (right) PSDs of the in-phase (turquoise) and out-of-phase (orange) motion of the coupled semi-virtual particles, identifying two distinct collective modes. c Coherence and d phase between the real and ghost particle's motion, illustrating that the modes are coupled, and the nature of each mode, respectively. e Simulation of the data in b, showing good quantitative agreement. f Relationship between the coupling strength $g$ and the ratio of the in-phase and out-of-phase frequencies. The shaded region illustrates the values of $g$ accessible in our system, and the solid points the values we have measured in our experiment.
• Figure 4: Interaction between a real and ghost particle with different frequencies. a Position PSDs of the real particle with a fixed frequency (red) and a ghost particle with a variable frequency (blue), when they are uncoupled (left) and coupled (middle). The simulations (right) show good quantitative agreement. b Calculated (left) and simulated (right) coherence between the coupled real and ghost particle motion, with c-mode 1 and c-mode 2 marked by green and orange dotted lines respectively. c Variation in the coherence at the position of c-mode 1 (green) and c-mode 2 (orange) as a function of the mode separation $\Delta f$, compared to theory (dotted black line).
• Figure A5: Simulating the dynamics of a levitated microparticle on an analogue computer by solving its equations of motion. A description is given in the text.