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Annalen der Physik

Quantum Fluctuations of a Levitated Nanoparticle: An Analysis Within Nelson's Stochastic Mechanics Reformulation of Quantum Theory

Kai-Hendrik Henk, Wolfgang Paul

Published Feb 28, 2026• 0 citations
Publisher

Abstract

ABSTRACT Levitodynamics, i.e., the levitation of objects of mesoscopic size has made huge progress in the last decade allowing to address some questions of fundamental physics for the first time. It has become possible to cool a levitated particle of mesoscopic size down to its motional ground state and to observe its motion driven by quantum fluctuations. This defines a quantum system that is ideally suited for an analysis within Nelson's stochastic mechanics approach to quantum mechanics, which allows for a dynamic description of individual particle paths. In good approximation, a levitated nanoparticle is a quantum harmonic oscillator. The coherent states of such an oscillator are superpositions of classical paths and the stochastic ground state process. They are exact solutions not only of the Schrödinger equation but also of the quantum Hamilton equations of Nelson's stochastic mechanics, and they are amenable to path simulations. All model parameters are typically available from experiment. Analyzing the quantum motion of the levitated nanoparticle in terms of the coherent states of a quantum harmonic oscillator allows for a quantitative prediction of the experimentally measured power spectral density of its position fluctuations and predicts new experimentally accessible measures for the mean number of phonons present in the center of mass motion. It also explains the observed phase space trajectory of the nano‐particle during cooling.

DOI
10.1002/andp.202500279

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ABSTRACT Levitodynamics, i.e., the levitation of objects of mesoscopic size has made huge progress in the last decade allowing to address some questions of fundamental physics for the first time. It has become possible to cool a levitated particle of mesoscopic size down to its motional ground state and to observe its motion driven by quantum fluctuations. This defines a quantum system that is ideally suited for an analysis within Nelson's stochastic mechanics approach to quantum mechanics, which allows for a dynamic description of individual particle paths. In good approximation, a levitated nanoparticle is a quantum harmonic oscillator. The coherent states of such an oscillator are superpositions of classical paths and the stochastic ground state process. They are exact solutions not only of the Schrödinger equation but also of the quantum Hamilton equations of Nelson's stochastic mechanics, and they are amenable to path simulations. All model parameters are typically available from experiment. Analyzing the quantum motion of the levitated nanoparticle in terms of the coherent states of a quantum harmonic oscillator allows for a quantitative prediction of the experimentally measured power spectral density of its position fluctuations and predicts new experimentally accessible measures for the mean number of phonons present in the center of mass motion. It also explains the observed phase space trajectory of the nano‐particle during cooling.
Levitodynamics, i.e., the levitation of objects of mesoscopic size has made huge progress in the last decade, giving rise to new experimental opportunities for instance in materials science, but also allowing to address questions of fundamental physics for the first time. It has become possible to cool a levitated particle of mesoscopic size down to its motional ground state and to observe its motion driven by quantum fluctuations. Such an experiment is ideally suited for an analysis within the stochastic mechanics approach to quantum mechanics, which allows for a dynamic description of individual particle paths. We show that within our approach we reproduce the findings of a recent experiment [Magrini et al., Nature 595-373-377 (2021)]. The phase space dynamics reported there based on a quantum optimal control using Kalman filtering can be understood using the concept of coherent states.
State purity is a key resource for experimental tests of quantum mechanics with massive objects. Cavity cooling of a levitated nanoparticle via coherent scattering stands out as a strategy to prepare highly pure states of its roto-translational motion. In several cases, though, the performance of cooling by coherent scattering is limited by the classical fluctuations of the laser field. Here we present a quantum theory of coherent-scattering experiments which includes the effect of the laser amplitude and phase noise. We derive a master equation for the noise-averaged particle's dynamics, valid for both center-of-mass and rotational degrees of freedom, and quantify the cooling performance for different levels of noise in state-of-the-art experiments. We compute approximate analytical expressions for the laser-noise-induced heating rates and excess phonon number in both regimes of weak and strong optomechanical coupling. Moreover, our model can be applied to laser noise processes with generic power spectral densities and finite correlation times. This work offers a toolbox to quantify and optimize the state purity in coherent-scattering experiments, and opens new avenues towards the preparation of quantum states with levitated systems.
Manipulating the motion of macroscopic objects near their quantum mechanical uncertainties has been desired in diverse fields, including fundamental physics, sensing, and transducers. Despite progress in ground-state cooling of a levitated solid particle, realizing its nonclassical states has been elusive. Here, we demonstrate quantum squeezing of the motion of a single nanoparticle by rapidly varying its oscillation frequency. We reveal appreciable narrowing of the velocity variance to -4.9 ± 0.1 decibels of that of the ground state using free-expansion measurements. Our work shows that a levitated nanoparticle offers an ideal platform for studying nonclassical states of its motion and provides a route to developing applications in quantum sensing and exploring quantum mechanics at a macroscopic scale.
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 NNN ions, predicting a linear increase of the cooling rate as a function of NNN 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.
The center-of-mass motion of a single optically levitated nanoparticle resembles three uncoupled harmonic oscillators. We show how a suitable modulation of the optical trapping potential can give rise to a coupling between two of these oscillators, such that their dynamics are governed by a classical equation of motion that resembles the Schrödinger equation for a two-level system. Based on experimental data, we illustrate the dynamics of this parametrically coupled system both in the frequency and in the time domain. We discuss the limitations and differences of the mechanical analogue in comparison to a true quantum mechanical system.
A nanoparticle trapped and cooled Cooling massive particles to the quantum ground state allows fundamental tests of quantum mechanics to be made; it would provide an experimental probe of the boundary between the classical and quantum worlds. Delić et al. laser-cooled an optically trapped solid-state object (a ∼150-nanometer-diameter silic a nanoparticle) into its quantum ground state of motion starting from room temperature. Because the object is levitated using optical forces, the experimental configuration can be switched to free fall, thereby providing a test bed for several macroscopic quantum experiments. Science , this issue p. 892