Calibration and energy measurement of optically levitated nanoparticle sensors

Mass measurement under medium vacuum in optically levitated nanoparticles based on Maxwell speed distribution law

As one of the directions of optical levitation technology, the mass measurement of micro-nano particles has always been a research hotspot in extremely weak mechanical measurements. When nanoscale particles are trapped in an optical trap, parameters such as density, diameter, and shape are unknown. Here we propose what we believe to be a new method to measure mass by fitting particle motion information to the Maxwell speed distribution law, with an accuracy better than 7% at 10 mbar. This method has the characteristics of requiring no external driving force, no precise natural frequency, no prior information such as density, and non-destructive testing within the medium vacuum range. With the increasing iterations, the uncertainty of mass measurement is reduced, and the accuracy of mass measurement of levitated particles is verified under multiple air pressures. It provides what we believe is a new method for the future non-destructive testing of nanoscale particles, and provides an apparently new way for the sensing measurement and metrology application fields of levitation dynamics systems.

Nanoscale feedback control of six degrees of freedom of a near-sphere

Manipulating the rotational as well as the translational degrees of freedom of rigid bodies has been a crucial ingredient in diverse areas, from optically controlled micro-robots, navigation, and precision measurements at macroscale to artificial and biological Brownian motors at nanoscale. Here, we demonstrate feedback cooling of all the angular motions of a near-spherical neutral nanoparticle with all the translational motions feedback-cooled to near the ground state. The occupation numbers of the three translational motions are 6 ± 1, 6 ± 1, and 0.69 ± 0.18. A tight, anisotropic optical confinement allows us to clearly observe three angular oscillations and to identify the ratio of two radii to the longest radius with a precision of 0.08 %. We develop a thermometry for three angular oscillations and realize feedback cooling of them to temperatures of lower than 0.03 K by electrically controlling the electric dipole moment of the nanoparticle.

Simulations and theory of power spectral density functions for time-dependent and anharmonic Langevin oscillators

Simulations and theory are presented for the power spectral density functions (PSDs) of particles in time-dependent and anharmonic potentials, including the effects of a thermal environment leading to damping and fluctuating forces. We investigate three one-dimensional perturbations to the harmonic oscillator of which two are time-dependent changes in the natural frequency of the oscillator, while the other is a time-independent extension of the quadratic potential to include a quartic term. We investigate the effect of these perturbations on two PSDs of the motion that are used in experiments on trapped nano-oscillators. We also derive and numerically test the PSDs for the motion of a spherical nanoparticle in a Paul trap. We found that the simple harmonic Langevin oscillator's PSDs are good approximations for the x and y coordinates' PSDs for small values of the parameter q of the Mathieu equation, but the difference can be more than a factor of two as "q" increases. We also numerically showed that the presence of a permanent electric dipole on the nanosphere can significantly affect the PSDs in the x and y coordinates.

Scalable all-optical cold damping of levitated nanoparticles

Motional control of levitated nanoparticles relies on either autonomous feedback via a cavity or measurement-based feedback via external forces. Recent demonstrations of the measurement-based ground-state cooling of a single nanoparticle employ linear velocity feedback, also called cold damping, and require the use of electrostatic forces on charged particles via external electrodes. Here we introduce an all-optical cold damping scheme based on the spatial modulation of trap position, which has the advantage of being scalable to multiple particles. The scheme relies on programmable optical tweezers to provide full independent control over the trap frequency and position of each tweezer. We show that the technique cools the centre-of-mass motion of particles along one axis down to 17 mK at a pressure of 2 × 10^-6 mbar and demonstrate its scalability by simultaneously cooling the motion of two particles. Our work paves the way towards studying quantum interactions between particles; achieving three-dimensional quantum control of particle motion without cavity-based cooling, electrodes or charged particles; and probing multipartite entanglement in levitated optomechanical systems.

Giant Diffusion of Nanomechanical Rotors in a Tilted Washboard Potential

We present an experimental realization of a biased optical periodic potential in the low friction limit. The noise-induced bistability between locked (torsional) and running (spinning) states in the rotational motion of a nanodumbbell is driven by an elliptically polarized light beam tilting the angular potential. By varying the gas pressure around the point of maximum intermittency, the rotational effective diffusion coefficient increases by more than 3 orders of magnitude over free-space diffusion. These experimental results are in agreement with a simple two-state model that is derived from the Langevin equation through using timescale separation. Our work provides a new experimental platform to study the weak thermal noise limit for diffusion in this system.

Position Measurement of a Levitated Nanoparticle via Interference with Its Mirror Image

Interferometric methods for detecting the motion of a levitated nanoparticle provide a route to the quantum ground state, but such methods are currently limited by mode mismatch between the reference beam and the dipolar field scattered by the particle. Here we demonstrate a self-interference method to detect the particle's motion that solves this problem. A Paul trap confines a charged dielectric nanoparticle in high vacuum, and a mirror retro-reflects the scattered light. We measure the particle's motion with a sensitivity of 1.7×10^{-12}  m/sqrt[Hz], corresponding to a detection efficiency of 2.1%, with a numerical aperture of 0.18. As an application of this method, we cool the particle, via feedback, to temperatures below those achieved in the same setup using a standard position measurement.

Precision Calibration of the Duffing Oscillator with Phase Control

The Duffing oscillator is a nonlinear extension of the ubiquitous harmonic oscillator and as such plays an outstanding role in science and technology. Experimentally, the system parameters are determined by a measurement of its response to an external excitation. When changing the amplitude or frequency of the external excitation, a sudden jump in the response function reveals the nonlinear dynamics prominently. However, this bistability leaves part of the full response function unobserved, which limits the precise measurement of the system parameters. Here, we exploit the often unknown fact that the response of a Duffing oscillator with nonlinear damping is a unique function of its phase. By actively stabilizing the oscillator's phase we map out the full response function. This phase control allows us to precisely determine the system parameters. Our results are particularly important for characterizing nanoscale resonators, where nonlinear effects are observed readily and which hold great promise for next generation of ultrasensitive force and mass measurements. We demonstrate our approach experimentally with an optically levitated particle in high vacuum.

Levitodynamics: Levitation and control of microscopic objects in vacuum

[Figure: see text].

Kovacs Memory Effect with an Optically Levitated Nanoparticle

The understanding of the dynamics of nonequilibrium cooling and heating processes at the nanoscale is still an open problem. These processes can follow surprising relaxation paths due to, e.g., memory effects, which significantly alter the expected equilibration routes. The Kovacs effect can take place when a thermalization process is suddenly interrupted by a change of the bath temperature, leading to a nonmonotonic evolution of the energy of the system. Here, we demonstrate that the Kovacs effect can be observed in the thermalization of the center of mass motion of a levitated nanoparticle. The temperature is controlled during the experiment through an external source of white Gaussian noise that mimics an effective thermal bath at a temperature that can be changed faster than any relaxation time of the system. We describe our experiments in terms of the dynamics of a Brownian particle in a harmonic trap without any fitting parameter, suggesting that the Kovacs effect can appear in a large variety of systems.

Sub-Kelvin Feedback Cooling and Heating Dynamics of an Optically Levitated Librator

Rotational optomechanics strives to gain quantum control over mechanical rotors by harnessing the interaction of light and matter. We optically trap a dielectric nanodumbbell in a linearly polarized laser field, where the dumbbell represents a nanomechanical librator. Using measurement-based parametric feedback control in high vacuum, we cool this librator from room temperature to 240 mK and investigate its heating dynamics when released from feedback. We exclude collisions with residual gas molecules as well as classical laser noise as sources of heating. Our findings indicate that we observe the torque fluctuations arising from the zero-point fluctuations of the electromagnetic field.

Quantum control of a nanoparticle optically levitated in cryogenic free space

Tests of quantum mechanics on a macroscopic scale require extreme control over mechanical motion and its decoherence^1-3. Quantum control of mechanical motion has been achieved by engineering the radiation-pressure coupling between a micromechanical oscillator and the electromagnetic field in a resonator^4-7. Furthermore, measurement-based feedback control relying on cavity-enhanced detection schemes has been used to cool micromechanical oscillators to their quantum ground states⁸. In contrast to mechanically tethered systems, optically levitated nanoparticles are particularly promising candidates for matter-wave experiments with massive objects^9,10, since their trapping potential is fully controllable. Here we optically levitate a femtogram (10^-15 grams) dielectric particle in cryogenic free space, which suppresses thermal effects sufficiently to make the measurement backaction the dominant decoherence mechanism. With an efficient quantum measurement, we exert quantum control over the dynamics of the particle. We cool its centre-of-mass motion by measurement-based feedback to an average occupancy of 0.65 motional quanta, corresponding to a state purity of 0.43. The absence of an optical resonator and its bandwidth limitations holds promise to transfer the full quantum control available for electromagnetic fields to a mechanical system. Together with the fact that the optical trapping potential is highly controllable, our experimental platform offers a route to investigating quantum mechanics at macroscopic scales¹¹.

Escape dynamics of active particles in multistable potentials

Rare transitions between long-lived metastable states underlie a great variety of physical, chemical and biological processes. Our quantitative understanding of reactive mechanisms has been driven forward by the insights of transition state theory and in particular by Kramers' dynamical framework. Its predictions, however, do not apply to systems that feature non-conservative forces or correlated noise histories. An important class of such systems are active particles, prominent in both biology and nanotechnology. Here, we study the active escape dynamics of a silica nanoparticle trapped in a bistable potential. We introduce activity by applying an engineered stochastic force that emulates self-propulsion. Our experiments, supported by a theoretical analysis, reveal the existence of an optimal correlation time that maximises the transition rate. We discuss the origins of this active turnover, reminiscent of the much celebrated Kramers turnover. Our work establishes a versatile experimental platform to study single particle dynamics in non-equilibrium settings.

Using the transient trajectories of an optically levitated nanoparticle to characterize a stochastic Duffing oscillator

We propose a novel methodology to estimate parameters characterizing a weakly nonlinear Duffing oscillator represented by an optically levitating nanoparticle. The method is based on averaging recorded trajectories with defined initial positions in the phase space of nanoparticle position and momentum and allows us to study the transient dynamics of the nonlinear system. This technique provides us with the parameters of a levitated nanoparticle such as eigenfrequency, damping, coefficient of nonlinearity and effective temperature directly from the recorded transient particle motion without any need for external driving or modification of an experimental system. Comparison of this innovative approach with a commonly used method based on fitting the power spectrum density profile shows that the proposed complementary method is applicable even at lower pressures where the nonlinearity starts to play a significant role and thus the power spectrum density method predicts steady state parameters. The technique is applicable also at low temperatures and extendable to recent quantum experiments. The proposed method is applied on experimental data and its validity for one-dimensional and three-dimensional motion of a levitated nanoparticle is verified by extensive numerical simulations.

Robust Optical-Levitation-Based Metrology of Nanoparticle's Position and Mass

Light has shown an incredible capability in precision measurement based on optomechanic interaction in high vacuum by isolating environment noises. However, there are still obstructions, such as displacement and mass estimation error, highly hampering the improvement of absolute accuracy at the nanoscale. Here, we present a nonlinearity based metrology to precisely measure the position and mass of a nanoparticle with optical levitation under 10^{-5}  mbar. By precisely controlling the oscillation amplitude of the levitated nanoparticle at the nonlinear regime for high accuracy calibration, we realized a feasible sub-picometer-level position measurement with an uncertainty of 1.0% without the prior information of mass, which can be further applied to weigh the femtogram-level mass with an uncertainty of 2.2%. It will also pave the way to construct a fine-calibrated optomechanic platform in high vacuum for high sensitivity and accuracy measurement in force and acceleration at the nanoscale and the study in quantum superposition at the mesoscopic scale.

Optomechanics with levitated particles

Optomechanics is concerned with the use of light to control mechanical objects. As a field, it has been hugely successful in the production of precise and novel sensors, the development of low-dissipation nanomechanical devices, and the manipulation of quantum signals. Micro- and nano-particles levitated in optical fields act as nanoscale oscillators, making them excellent low-dissipation optomechanical objects, with minimal thermal contact to the environment when operating in vacuum. Levitated optomechanics is seen as the most promising route for studying high-mass quantum physics, with the promise of creating macroscopically separated superposition states at masses of 10⁶ amu and above. Optical feedback, both using active monitoring or the passive interaction with an optical cavity, can be used to cool the centre-of-mass of levitated nanoparticles well below 1 mK, paving the way to operation in the quantum regime. In addition, trapped mesoscopic particles are the paradigmatic system for studying nanoscale stochastic processes, and have already demonstrated their utility in state-of-the-art force sensing.

Motional Sideband Asymmetry of a Nanoparticle Optically Levitated in Free Space

The hallmark of quantum physics is Planck's constant h, whose finite value entails the quantization that gave the theory its name. The finite value of h gives rise to inevitable zero-point fluctuations even at vanishing temperature. The zero-point fluctuation of mechanical motion becomes smaller with growing mass of an object, making it challenging to observe at macroscopic scales. Here, we transition a dielectric particle with a diameter of 136 nm from the classical realm to the regime where its zero-point motion emerges as a sizable contribution to its energy. To this end, we optically trap the particle at ambient temperature in ultrahigh vacuum and apply active feedback cooling to its center-of-mass motion. We measure an asymmetry between the Stokes and anti-Stokes sidebands of photons scattered by the levitated particle, which is a signature of the particle's quantum ground state of motion.

Resolved-Sideband Cooling of a Levitated Nanoparticle in the Presence of Laser Phase Noise

We investigate the influence of laser phase noise heating on resolved sideband cooling in the context of cooling the center-of-mass motion of a levitated nanoparticle in a high-finesse cavity. Although phase noise heating is not a fundamental physical constraint, the regime where it becomes the main limitation in Levitodynamics has so far been unexplored and hence embodies from this point forward the main obstacle in reaching the motional ground state of levitated mesoscopic objects with resolved sideband cooling. We reach minimal center-of-mass temperatures comparable to T_{min}=10  mK at a pressure of p=3×10^{-7}  mbar, solely limited by phase noise. Finally we present possible strategies towards motional ground state cooling in the presence of phase noise.

Accurate Mass Measurement of a Levitated Nanomechanical Resonator for Precision Force-Sensing

Nanomechanical resonators are widely operated as force and mass sensors with sensitivities in the zepto-Newton (10^-21) and yocto-gram (10^-24) regime, respectively. Their accuracy, however, is usually undermined by high uncertainties in the effective mass of the system, whose estimation is a nontrivial task. This critical issue can be addressed in levitodynamics, where the nanoresonator typically consists of a single silica nanoparticle of well-defined mass. Yet, current methods assess the mass of the levitated nanoparticles with uncertainties up to a few tens of percent, therefore preventing to achieve unprecedented sensing performances. Here, we present a novel measurement protocol that uses the electric field from a surrounding plate capacitor to directly drive a charged optically levitated particle in moderate vacuum. The developed technique estimates the mass within a statistical error below 1% and a systematic error of ∼2%, and paves the way toward more reliable sensing and metrology applications of levitodynamics systems.

Cold Damping of an Optically Levitated Nanoparticle to Microkelvin Temperatures

We implement a cold-damping scheme to cool one mode of the center-of-mass motion of an optically levitated nanoparticle in ultrahigh vacuum (10^{-8}  mbar) from room temperature to a record-low temperature of 100  μK. The measured temperature dependence on the feedback gain and thermal decoherence rate is in excellent agreement with a parameter-free model. For the first time, we determine the imprecision-backaction product for a levitated optomechanical system and discuss the resulting implications for ground-state cooling of an optically levitated nanoparticle.

Cavity-Based 3D Cooling of a Levitated Nanoparticle via Coherent Scattering

We experimentally realize cavity cooling of all three translational degrees of motion of a levitated nanoparticle in vacuum. The particle is trapped by a cavity-independent optical tweezer and coherently scatters tweezer light into the blue detuned cavity mode. For vacuum pressures around 10^{-5}  mbar, minimal temperatures along the cavity axis in the millikelvin regime are observed. Simultaneously, the center-of-mass (c.m.) motion along the other two spatial directions is cooled to minimal temperatures of a few hundred millikelvin. Measuring temperatures and damping rates as the pressure is varied, we find that the cooling efficiencies depend on the particle position within the intracavity standing wave. This data and the behavior of the c.m. temperatures as functions of cavity detuning and tweezer power are consistent with a theoretical analysis of the experiment. Experimental limits and opportunities of our approach are outlined.

Sensing Static Forces with Free-Falling Nanoparticles

Miniaturized mechanical sensors rely on resonant operation schemes, unsuited to detect static forces. We demonstrate a nanomechanical sensor for static forces based on an optically trapped nanoparticle in vacuum. Our technique relies on an off-resonant interaction of the particle with a weak static force, and a resonant readout of the displacement caused by this interaction. We demonstrate a sensitivity of 10 aN to static gravitational and electric forces. Our work provides a tool for the closer investigation of short-range forces, and marks an important step towards the realization of matter-wave interferometry with macroscopic objects.

Levitated Nanoparticles for Microscopic Thermodynamics-A Review

Levitated Nanoparticles have received much attention for their potential to perform quantum mechanical experiments even at room temperature. However, even in the regime where the particle dynamics are purely classical, there is a lot of interesting physics that can be explored. Here we review the application of levitated nanoparticles as a new experimental platform to explore stochastic thermodynamics in small systems.