Quantum theory of electrically levitated nanoparticle-ion systems: Motional dynamics and sympathetic cooling

TL;DR

This work develops a complete quantum framework for a nanoparticle co-trapped with ions in a dual-frequency Paul trap, deriving the classical secular dynamics, the quantum Hamiltonian, and a master equation that captures the Coulomb coupling and the dominant dissipation channels. It analyzes sympathetic cooling of the nanoparticle by a Doppler-cooled ion, providing analytic steady-state occupations and cooling rates, and shows sub-kelvin temperatures are reachable with current setups, even without feedback, while micromotion can limit performance. Extending to an $N$-ion ensemble reveals a linear enhancement of the cooling rate with $N$ and a $1/N$ scaling of the nanoparticle occupation, with the center-of-mass mode primarily responsible for cooling; this establishes a scalable but challenging path toward ground-state cooling. The results offer a practical theoretical toolbox to explore ion-assisted preparation of non-Gaussian motional states and suggest future routes (e.g., cavity-mediated coupling or parametric modulation) to further close the gap to quantum-ground-state levitation.

Abstract

We develop the theory describing the quantum coupled dynamics of the center-of-mass motion of a nanoparticle and an ensemble of ions co-trapped in a dual-frequency linear Paul trap. We first derive analytical expressions for the motional frequencies and classical trajectories of both nanoparticle and ions. We then derive a quantum master equation for the ion-nanoparticle system and quantify the sympathetic cooling of the nanoparticle motion enabled by its Coulomb coupling to a continuously Doppler-cooled ion. We predict that motional cooling down to sub-kelvin temperatures is achievable in state-of-the-art experiments even in the absence of motional feedback and in the presence of micromotion. We then extend our analysis to an ensemble of $N$ ions, predicting a linear increase of the cooling rate as a function of $N$ and motional cooling of the nanoparticle down to tenths of millikelvin in current experimental platforms. Our work establishes the theoretical toolbox needed to explore the ion-assisted preparation of non-Gaussian motional states of levitated nanoparticles.

Figures (6)

• Figure 1: Scheme of the system under study. A charged dielectric nanoparticle is co-trapped with a single or multiple ions in a two-tone Paul trap. The center-of-mass motion of the nanoparticle is coupled to the ion motion by Coulomb interaction. The ion motion can be continuously Doppler-cooled using laser beams.
• Figure 2: Coupling rate between ion and nanoparticle center-of-mass motion along the $x-$ (Eq. \ref{['gr']}, magenta), $y-$ (Eq. \ref{['gr']}, blue), and $z-$axis (Eq. \ref{['gz']}, orange) and equilibrium distance between the ion and nanoparticle (Eq. \ref{['separation']}, dashed), versus nanoparticle charge $Q_p$ (panel a) and nanoparticle mass $M_p$ (panel b), for the parameters of Table \ref{['tab:parameters']}. The equilibrium position corresponds to nanoparticle and ion lying along the $z-$axis. The open dots mark the points beyond which the coupled system becomes unstable, see main text for details.
• Figure 3: Steady state phonon number of nanoparticle motion along the $z-$(panel a) and $x-$(panel b) axes versus total damping rate, $\gamma_p = \gamma^{\rm fb}+\gamma^{\rm gas}$, for the parameters of Table \ref{['tab:parameters']}. The solid and dotted black curves correspond to the trap-displacement noise values of Ref. Bykov_2025 and to the case of no trap displacement noise $\Gamma_j^{\rm td}=0$, respectively. The horizontal and slanted gray dashed lines show our approximate expressions for the occupation in the sympathetic cooling regime and the feedback cooling regime, Eq. \ref{['gammaDOP']} and $\langle\hat{n}_{jp}\rangle_{\rm ss}\approx \Gamma_{jp}/\gamma^{\rm fb}$ respectively (see main text for details). The inset in (a) shows the steady state phonon number as a function of the trap-displacement noise in the system, $\Gamma^{\rm td}_z$, for $\gamma_p = 2 \pi \times 44.5 ~\mathrm{nHz}$ (blue) and $\gamma_p = 2 \pi \times 1 ~\mathrm{Hz}$ (green).
• Figure 4: (a) Potential and kinetic energy of nanoparticle motion along the $x-$axis as a function of time in the long-time limit. The kinetic energy is multiplied by a factor 50 for visibility. The inset shows the fast oscillations of the micromotion at frequency $\omega_f$. (b) Time-averaged purity (Eq. \ref{['averagepurity']}) and effective occupation (Eq. \ref{['neff']}) of the nanoparticle motion along the $x-$axis in the long-time limit as a function of nanoparticle damping $\gamma_p=\gamma^{\rm gas}+\gamma^{\rm fb}$. Triangles and solid lines indicate the values with and without micromotion. Parameters are taken from Table \ref{['tab:parameters']}.
• Figure 5: (a) Optimal ion and nanoparticle equilibrium positions for sympathetic cooling, for $N=5$ and $N=6$ ions, respectively. (b) Steady state motional occupation of the nanoparticle along the $z-$axis, $\expval{\hat{n}_{zp}}_{\rm ss}$, versus ion number $N$, for the best-performing equilibrium configuration for each $N$. Parameters are taken from Table \ref{['tab:parameters']}, with trap-displacement noise $\Gamma^{\rm td}_z=0$. The exact solution (black) is compared to an independent-ion model (green, see text for details) and to a model where only the center-of-mass mode of the chain couples to the nanoparticle motion (red, Eq. \ref{['gammaDOP2']}). (c) Normal mode frequencies of the $N-$ion system (green circles) and coupling rate between nanoparticle and normal mode $\alpha=1$ (black). The dashed gray curves show a $\sqrt{N}$ scaling. The vertical dotted lines in panels (b-c) mark the transition between one-sided and symmetric equilibrium configurations.