Tunable light-induced dipole-dipole interaction between optically levitated nanoparticles

Quantum computing predicts particle trajectories in optical tweezers

A recent study demonstrated advancements in quantum computing by applying it to address a non-Hermitian optical manipulation problem. The emergence of exceptional points and the dynamics of optically trapped single or multiple particles were simulated using a quantum computing approach.

Levitation and controlled MHz rotation of a nanofabricated rod by a high-NA metalens

An optically levitated nanoparticle in a vacuum provides an ideal platform for ultra-precision measurements and fundamental physics studies because of the exceptionally high-quality factor and rich motion modes, which can be engineered by manipulating the optical field and the geometry of the nanoparticle. Nanofabrication technology with the ability to create arbitrary nanostructure arrays offers a precise way of engineering the optical field and the geometry of the nanoparticle. Here, for the first time, we optically levitate and rotate a nanofabricated nanorod via a nanofabricated a-Si metalens which strongly focuses a 1550 nm laser beam with a numerical aperture of 0.953. By manipulating the laser beam's polarization, the levitated nanorod's translation frequencies can be tuned, and the spin rotation mode can be switched on and off. Then, we showed the control of rotational frequency by changing the laser beam's intensity and polarization as well as the air pressure. Finally, a MHz spin rotation frequency of the nanorod is achieved in the experiment. This is the first demonstration of controlled optical spin in a metalens-based compact optical levitation system. Our research holds promise for realizing scalable on-chip integrated optical levitation systems.

Optical trapping of mesoscale particles and atoms in hollow-core optical fibers: principle and applications

Hollow-core fiber (HCF) is a special optical waveguide type that can guide light in the air or liquid core surrounded by properly designed cladding structures. The guiding modes of the fiber can generate sufficient optical gradient forces to balance the gravity of the particles or confine the atom clouds, forming a stable optical trap in the hollow core. The levitated objects can be propelled over the fiber length along the beam axis through an imbalance of the optical scattering forces or by forming an optical lattice by the counter-propagating beams. The ability to overcome the diffraction of the laser beam in HCF can significantly increase the range of the optical manipulation compared with standard free-space optical tweezers, opening up vast ranges of applications that require long-distance optical control. Since the first demonstration of optical trapping in HCF, hollow-core-fiber-based optical trap (HCF-OT) has become an essential branch of optical tweezer that draws intense research interests. Fast progress on the fundamental principle and applied aspects of HCF-OT has been visible over the past two decades. In recent years, significant milestones in reducing the propagation loss of HCF have been achieved, making HCF an attractive topic in the field of optics and photonics. This further promotes the research and applications of HCF-OT. This review starts from the mechanism of light guidance of HCF, mainly focusing on the issues related to the optical trap in the hollow core. The basic principles and key features of HCF-OT, from optical levitation to manipulation and the detection of macroscopic particles and atoms, are summarized in detail. The key applications of HCF-OT, the challenges and future directions of the technique are also discussed.

Deviation of a system of nonreciprocally coupled harmonic oscillators from a conservative system

Discrete systems of coupled linear mechanical oscillators with nonreciprocal interaction are a model for a variety of physical systems. In general, the presence of nonreciprocal interactions renders their dynamics nonconservative, but under certain conditions it remains conservative. In this paper we show which thermodynamic properties induced by nonreciprocity can be observed in conservative systems and which are specific to nonconservative systems. To this end, we formulate a criterion for identifying conservative systems and construct a measure to quantify the deviation from conservativity.

Quantum collective motion of macroscopic mechanical oscillators

Collective phenomena arise from interactions within complex systems, leading to behaviors absent in individual components. Observing quantum collective phenomena with macroscopic mechanical oscillators has been impeded by the stringent requirement that oscillators be identical. We demonstrate the quantum regime for collective motion of N = 6 mechanical oscillators, a hexamer, in a superconducting circuit optomechanical platform. By increasing the optomechanical couplings, the system transitions from individual to collective motion, characterized by a [Formula: see text] enhancement of cavity-collective mode coupling, akin to superradiance of atomic ensembles. Using sideband cooling, we prepare the collective mode in the quantum ground state and measure its quantum sideband asymmetry, with zero-point motion distributed across distant oscillators. This regime of optomechanics opens avenues for studying multipartite entanglement, with potential advances in quantum metrology.

Quantum Optical Binding of Nanoscale Particles

Optical binding refers to the light-induced interaction between two or more objects illuminated by laser fields. The high tunability of the strength, sign, and reciprocity of this interaction renders it highly attractive for controlling nanoscale mechanical motion. Here, we discuss the quantum theory of optical binding and identify unique signatures of this interaction in the quantum regime. We show that these signatures are observable in near-future experiments with levitated nanoparticles. In addition, we prove the impossibility of entanglement induced by far-field optical binding in free space and identify strategies to circumvent this no-go theorem.

Continuous Space-Time Crystal State Driven by Nonreciprocal Optical Forces

We show that the continuous time crystal state can arise in an ensemble of linear oscillators from nonconservative coupling via optical radiation pressure forces. This new mechanism comprehensively explains observations of the time crystal state in an array of nanowires illuminated with light [T. Liu et al., Nat. Phys. 19, 986 (2023).NPAHAX1745-247310.1038/s41567-023-02023-5]. Being fundamentally different from regimes of nonlinear synchronization, it has relevance to a wide range of interacting many-body systems, including in the realms of chemistry, biology, weather, and nanoscale matter.

Vacuum levitation and motion control on chip

By isolating from the environment and precisely controlling mesoscopic objects, levitation in vacuum has evolved into a versatile technique that has already benefited diverse scientific directions, from force sensing and thermodynamics to materials science and chemistry. It also holds great promise for advancing the study of quantum mechanics in the unexplored macroscopic regime. However, most current levitation platforms are complex and bulky. Recent efforts in miniaturization of vacuum levitation set-ups have comprised electrostatic and optical traps, but robustness is still a concern for integration into confined settings, such as cryostats or portable devices. Here we show levitation and motion control in high vacuum of a silica nanoparticle at the surface of a hybrid optical-electrostatic chip. By combining fibre-based optical trapping and sensitive position detection with cold damping through planar electrodes, we cool the particle motion to a few hundred phonons. We envisage that our fully integrated platform is the starting point for on-chip devices combining integrated photonics and nanophotonics with precisely engineered electric potentials, enhancing control over the particle motion towards complex state preparation and read-out.

One-way optomechanical interaction between nanoparticles

Within a closed system, physical interactions are reciprocal. However, the effective interaction between two entities of an open system may not obey reciprocity. Here, we describe a non-reciprocal interaction between nanoparticles which is one-way, almost insensitive to the interparticle distance, and scalable to many particles. The interaction we propose is based on the non-conservative optical forces between two nanoparticles with highly directional scattering patterns. However, we elucidate that scattering patterns can in general be very misleading about the interparticle forces. We introduce zeroth- and first-order non-reciprocity factors to precisely quantify the merits of any optomechanical interaction between nanoparticles. Our proposed one-way interaction could constitute an important step in the realization of mesoscopic heat pumps and refrigerators, the study of non-equilibrium systems, and the simulation of non-Hermitian quantum models.

Non-Hermitian dynamics and non-reciprocity of optically coupled nanoparticles

Non-Hermitian dynamics, as observed in photonic, atomic, electrical and optomechanical platforms, holds great potential for sensing applications and signal processing. Recently, fully tuneable non-reciprocal optical interaction has been demonstrated between levitated nanoparticles. Here we use this tunability to investigate the collective non-Hermitian dynamics of two non-reciprocally and nonlinearly interacting nanoparticles. We observe parity-time symmetry breaking and, for sufficiently strong coupling, a collective mechanical lasing transition in which the particles move along stable limit cycles. This work opens up a research avenue of non-equilibrium multi-particle collective effects, tailored by the dynamic control of individual sites in a tweezer array.

Tunable on-chip optical traps for levitating particles based on single-layer metasurface

Optically levitated multiple nanoparticles have emerged as a platform for studying complex fundamental physics such as non-equilibrium phenomena, quantum entanglement, and light-matter interaction, which could be applied for sensing weak forces and torques with high sensitivity and accuracy. An optical trapping landscape of increased complexity is needed to engineer the interaction between levitated particles beyond the single harmonic trap. However, existing platforms based on spatial light modulators for studying interactions between levitated particles suffered from low efficiency, instability at focal points, the complexity of optical systems, and the scalability for sensing applications. Here, we experimentally demonstrated that a metasurface which forms two diffraction-limited focal points with a high numerical aperture (∼0.9) and high efficiency (31 %) can generate tunable optical potential wells without any intensity fluctuations. A bistable potential and double potential wells were observed in the experiment by varying the focal points' distance, and two nanoparticles were levitated in double potential wells for hours, which could be used for investigating the levitated particles' nonlinear dynamics, thermal dynamics and optical binding. This would pave the way for scaling the number of levitated optomechanical devices or realizing paralleled levitated sensors.

Acoustic manipulation of multi-body structures and dynamics

Sound can exert forces on objects of any material and shape. This has made the contactless manipulation of objects by intense ultrasound a fascinating area of research with wide-ranging applications. While much is understood for acoustic forcing of individual objects, sound-mediated interactions among multiple objects at close range gives rise to a rich set of structures and dynamics that are less explored and have been emerging as a frontier for research. We introduce the basic mechanisms giving rise to sound-mediated interactions among rigid as well as deformable particles, focusing on the regime where the particles' size and spacing are much smaller than the sound wavelength. The interplay of secondary acoustic scattering, Bjerknes forces, and micro-streaming is discussed and the role of particle shape is highlighted. Furthermore, we present recent advances in characterizing non-conservative and non-pairwise additive contributions to the particle interactions, along with instabilities and active fluctuations. These excitations emerge at sufficiently strong sound energy density and can act as an effective temperature in otherwise athermal systems.

Exponentially Enhanced Non-Hermitian Cooling

Certain non-Hermitian systems exhibit the skin effect, whereby the wave functions become exponentially localized at one edge of the system. Such exponential amplification of wavefunction has received significant attention due to its potential applications in, e.g., classical and quantum sensing. However, the opposite edge of the system, featured by exponentially suppressed wave functions, remains largely unexplored. Leveraging this phenomenon, we introduce a non-Hermitian cooling mechanism, which is fundamentally distinct from traditional refrigeration or laser cooling techniques. Notably, non-Hermiticity will not amplify thermal excitations, but rather redistribute them. Hence, thermal excitations can be cooled down at one edge of the system, and the cooling effect can be exponentially enhanced by the number of auxiliary modes, albeit with a lower bound that depends on the dissipative interaction with the environment. Non-Hermitian cooling does not rely on intricate properties such as exceptional points or nontrivial topology, and it can apply to a wide range of excitations.

Synchronization of optically self-assembled nanorotors

Self-assembly of nanoparticles by means of interparticle optical forces provides a compelling approach toward contact-free organization and manipulation of nanoscale entities. However, exploration of the rotational degrees of freedom in this process has remained limited, primarily because of the predominant focus on spherical nanoparticles, for which individual particle orientation cannot be determined. Here, we show that gold nanorods, which self-assemble in water under the influence of circularly polarized light, exhibit synchronized rotational motion at kilohertz frequencies. The synchronization is caused by strong optical interactions and occurs despite the presence of thermal diffusion. Our findings elucidate the intricate dynamics arising from the transfer of photon spin angular momentum to optically bound matter and hold promise for advancing the emerging field of light-driven nanomachinery.

Non-Hermitian non-equipartition theory for trapped particles

The equipartition theorem is an elegant cornerstone theory of thermal and statistical physics. However, it fails to address some contemporary problems, such as those associated with optical and acoustic trapping, due to the non-Hermitian nature of the external wave-induced force. We use stochastic calculus to solve the Langevin equation and thereby analytically generalize the equipartition theorem to a theory that we denote the non-Hermitian non-equipartition theory. We use the non-Hermitian non-equipartition theory to calculate the relevant statistics, which reveal that the averaged kinetic and potential energies are no longer equal to kBT/2 and are not equipartitioned. As examples, we apply non-Hermitian non-equipartition theory to derive the connection between the non-Hermitian trapping force and particle statistics, whereby measurement of the latter can determine the former. Furthermore, we apply a non-Hermitian force to convert a saddle potential into a stable potential, leading to a different type of stable state.

Cavity-mediated long-range interactions in levitated optomechanics

The ability to engineer cavity-mediated interactions has emerged as a powerful tool for the generation of non-local correlations and the investigation of non-equilibrium phenomena in many-body systems. Levitated optomechanical systems have recently entered the multiparticle regime, which promises the use of arrays of strongly coupled massive oscillators to explore complex interacting systems and sensing. Here we demonstrate programmable cavity-mediated interactions between nanoparticles in vacuum by combining advances in multiparticle optical levitation and cavity-based quantum control. The interaction is mediated by photons scattered by spatially separated particles in a cavity, resulting in strong coupling that is long-range in nature. We investigate the scaling of the interaction strength with cavity detuning and interparticle separation and demonstrate the tunability of interactions between different mechanical modes. Our work will enable the exploration of many-body effects in nanoparticle arrays with programmable cavity-mediated interactions, generating entanglement of motion, and the use of interacting particle arrays for optomechanical sensing.

Simulation of optomechanical interaction of levitated nanoparticle with photonic crystal micro cavity

We propose and analyze theoretically a promising design of an optical trap for vacuum levitation of nanoparticles based on a one-dimensional (1D) silicon photonic crystal cavity (PhC). The considered cavity has a quadratically modulated width of the silicon wave guiding structure, leading to a calculated cavity quality factor of 8 × 105. An effective mode volume of approximately 0.16 μm3 having the optical field strongly confined outside the silicon structure enables optical confinement on nanoparticle in all three dimensions. The optical forces and particle-cavity optomechanical coupling are comprehensively analyzed for two sizes of silica nanoparticles (100 nm and 150 nm in diameter) and various mode detunings. The value of trapping stiffnesses in the microcavity is predicted to be 5 order of magnitudes higher than that reached for optimized optical tweezers, moreover the linear single photon coupling rate can reach MHz level which is 6 order magnitude larger than previously reported values for common bulk cavities. The theoretical results support optimistic prospects towards a compact chip for optical levitation in vacuum and cooling of translational mechanical degrees of motion for the silica nanoparticle of a diameter of 100 nm.

Macroscopic Quantum Superpositions via Dynamics in a Wide Double-Well Potential

We present an experimental proposal for the rapid preparation of the center of mass of a levitated particle in a macroscopic quantum state, that is a state delocalized over a length scale much larger than its zero-point motion and that has no classical analog. This state is prepared by letting the particle evolve in a static double-well potential after a sudden switchoff of the harmonic trap, following initial center-of-mass cooling to a sufficiently pure quantum state. We provide a thorough analysis of the noise and decoherence that is relevant to current experiments with levitated nano- and microparticles. In this context, we highlight the possibility of using two particles, one evolving in each potential well, to mitigate the impact of collective sources of noise and decoherence. The generality and scalability of our proposal make it suitable for implementation with a wide range of systems, including single atoms, ions, and Bose-Einstein condensates. Our results have the potential to enable the generation of macroscopic quantum states at unprecedented scales of length and mass, thereby paving the way for experimental exploration of the gravitational field generated by a source mass in a delocalized quantum state.

Near-Field GHz Rotation and Sensing with an Optically Levitated Nanodumbbell

A levitated nonspherical nanoparticle in a vacuum is ideal for studying quantum rotations and is an ultrasensitive torque detector for probing fundamental particle-surface interactions. Here, we optically levitate a silica nanodumbbell in a vacuum at 430 nm away from a sapphire surface and drive it to rotate at GHz frequencies. The relative linear speed between the tip of the nanodumbbell and the surface reaches 1.4 km s-1 at a submicrometer separation. The rotating nanodumbbell near the surface demonstrates a torque sensitivity of (5.0 ± 1.1) × 10-26 N m Hz-1/2 at room temperature. Moreover, we probed the near-field laser intensity distribution beyond the optical diffraction limit with a nanodumbbell levitated near a nanograting. Our numerical simulations show that the system can measure the Casimir torque and will improve the detection limit of non-Newtonian gravity by several orders of magnitude.

Trapping Microparticles in a Structured Dark Focus

We experimentally demonstrate stable trapping and controlled manipulation of silica microspheres in a structured optical beam consisting of a dark focus surrounded by light in all directions-the dark focus tweezer. Results from power spectrum and potential analysis demonstrate the nonharmonicity of the trapping potential landscape, which is reconstructed from experimental data in agreement to Lorentz-Mie numerical simulations. Applications of the dark tweezer in levitated optomechanics and biophysics are discussed.

Synchronization of spin-driven limit cycle oscillators optically levitated in vacuum

We explore, experimentally and theoretically, the emergence of coherent coupled oscillations and synchronization between a pair of non-Hermitian, stochastic, opto-mechanical oscillators, levitated in vacuum. Each oscillator consists of a polystyrene microsphere trapped in a circularly polarized, counter-propagating Gaussian laser beam. Non-conservative, azimuthal forces, deriving from inhomogeneous optical spin, push the micro-particles out of thermodynamic equilibrium. For modest optical powers each particle shows a tendency towards orbital circulation. Initially, their stochastic motion is weakly correlated. As the power is increased, the tendency towards orbital circulation strengthens and the motion of the particles becomes highly correlated. Eventually, centripetal forces overcome optical gradient forces and the oscillators undergo a collective Hopf bifurcation. For laser powers exceeding this threshold, a pair of limit cycles appear, which synchronize due to weak optical and hydrodynamic interactions. In principle, arrays of such Non-Hermitian elements can be arranged, paving the way for opto-mechanical topological materials or, possibly, classical time crystals. In addition, the preparation of synchronized states in levitated optomechanics could lead to new and robust sensors or alternative routes to the entanglement of macroscopic objects.

Levitated Optomechanics with Meta-Atoms

We propose to introduce additional control in levitated optomechanics by trapping a meta-atom, i.e., a subwavelength and high-permittivity dielectric particle supporting Mie resonances. In particular, we theoretically demonstrate that optical levitation and center-of-mass ground-state cooling of silicon nanoparticles in vacuum is not only experimentally feasible but it offers enhanced performance over widely used silica particles in terms of trap frequency, trap depth, and optomechanical coupling rates. Moreover, we show that, by adjusting the detuning of the trapping laser with respect to the particle's resonance, the sign of the polarizability becomes negative, enabling levitation in the minimum of laser intensity, e.g., at the nodes of a standing wave. The latter opens the door to trapping nanoparticles in the optical near-field combining red and blue-detuned frequencies, in analogy to two-level atoms, which is of interest for generating strong coupling to photonic nanostructures and short-distance force sensing.

Nonreciprocal forces enable cold-to-hot heat transfer between nanoparticles

We study the heat transfer between two nanoparticles held at different temperatures that interact through nonreciprocal forces, by combining molecular dynamics simulations with stochastic thermodynamics. Our simulations reveal that it is possible to construct nano refrigerators that generate a net heat transfer from a cold to a hot reservoir at the expense of power exerted by the nonreciprocal forces. Applying concepts from stochastic thermodynamics to a minimal underdamped Langevin model, we derive exact analytical expressions predictions for the fluctuations of work, heat, and efficiency, which reproduce thermodynamic quantities extracted from the molecular dynamics simulations. The theory only involves a single unknown parameter, namely an effective friction coefficient, which we estimate fitting the results of the molecular dynamics simulation to our theoretical predictions. Using this framework, we also establish design principles which identify the minimal amount of entropy production that is needed to achieve a certain amount of uncertainty in the power fluctuations of our nano refrigerator. Taken together, our results shed light on how the direction and fluctuations of heat flows in natural and artificial nano machines can be accurately quantified and controlled by using nonreciprocal forces.

Interaction between an Optically Levitated Nanoparticle and Its Thermal Image: Internal Thermometry via Displacement Sensing

We propose and theoretically analyze an experiment where displacement sensing of an optically levitated nanoparticle in front of a surface can be used to measure the induced dipole-dipole interaction between the nanoparticle and its thermal image. This is achieved by using a surface that is transparent to the trapping light but reflective to infrared radiation, with a reflectivity that can be time modulated. This dipole-dipole interaction relies on the thermal radiation emitted by a silica nanoparticle having sufficient temporal coherence to correlate the reflected radiation with the thermal fluctuations of the dipole. The resulting force is orders of magnitude stronger than the thermal gradient force, and it strongly depends on the internal temperature of the nanoparticle for a particle-to-surface distance greater than two micrometers. We argue that it is experimentally feasible to use displacement sensing of a levitated nanoparticle in front of a surface as an internal thermometer in ultrahigh vacuum. Experimental access to the internal physics of a levitated nanoparticle in vacuum is crucial to understanding the limitations that decoherence poses to current efforts devoted to preparing a nanoparticle in a macroscopic quantum superposition state.

Highly sensitive gyroscope based on a levitated nanodiamond

A gyroscope is one of the core components of an inertial navigation system. Both the high sensitivity and miniaturization are important for the applications of the gyroscope. We consider a nitrogen-vacancy (NV) center in a nanodiamond, which is levitated either by an optical tweezer or an ion trap. Based on the Sagnac effect, we propose a scheme to measure the angular velocity with ultra-high sensitivity through the matter-wave interferometry of the nanodiamond. Both the decay of the motion of the center of mass of the nanodiamond and the dephasing of the NV centers are included when we estimate the sensitivity of the proposed gyroscope. We also calculate the visibility of the Ramsey fringes, which can be used for estimating the limitation of gyroscope sensitivity. It is found that the sensitivity ∼6.86×10^-7 r a d/s/H z can be achieved in an ion trap. As the working area of the gyroscope is extremely small (∼0.01~μm²), it could be made on-chip in the future.

Optimal Cooling of Multiple Levitated Particles through Far-Field Wavefront Shaping

Light forces can be harnessed to levitate mesoscopic objects and cool them down toward their motional quantum ground state. Roadblocks on the way to scale up levitation from a single to multiple particles in close proximity are the requirements to constantly monitor the particles' positions as well as to engineer light fields that react fast and appropriately to their movements. Here, we present an approach that solves both problems at once. By exploiting the information stored in a time-dependent scattering matrix, we introduce a formalism enabling the identification of spatially modulated wavefronts, which simultaneously cool down multiple objects of arbitrary shapes. An experimental implementation is suggested based on stroboscopic scattering-matrix measurements and time-adaptive injections of modulated light fields.

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.

Stroboscopic thermally-driven mechanical motion

Unstable nonlinear systems can produce a large displacement driven by a small thermal initial noise. Such inherently nonlinear phenomena are stimulating in stochastic physics, thermodynamics, and in the future even in quantum physics. In one-dimensional mechanical instabilities, recently made available in optical levitation, the rapidly increasing noise accompanying the unstable motion reduces a displacement signal already in its detection. It limits the signal-to-noise ratio for upcoming experiments, thus constraining the observation of such essential nonlinear phenomena and their further exploitation. An extension to a two-dimensional unstable dynamics helps to separate the desired displacement from the noisy nonlinear driver to two independent variables. It overcomes the limitation upon observability, thus enabling further exploitation. However, the nonlinear driver remains unstable and rapidly gets noisy. It calls for a challenging high-order potential to confine the driver dynamics and rectify the noise. Instead, we propose and analyse a feasible stroboscopically-cooled driver that provides the desired detectable motion with sufficiently high signal-to-noise ratio. Fast and deep cooling, together with a rapid change of the driver stiffness, are required to reach it. However, they have recently become available in levitating optomechanics. Therefore, our analysis finally opens the road to experimental investigation of thermally-driven motion in nonlinear systems, its thermodynamical analysis, and future quantum extensions.

Force-Gradient Sensing and Entanglement via Feedback Cooling of Interacting Nanoparticles

We show theoretically that feedback cooling of two levitated, interacting nanoparticles enables differential sensing of forces and the observation of stationary entanglement. The feedback drives the two particles into a stationary, nonthermal state which is susceptible to inhomogeneous force fields and which exhibits entanglement for sufficiently strong interparticle couplings. We predict that force-gradient sensing at the zepto-Newton per micron range is feasible and that entanglement due to the Coulomb interaction between charged particles can be realistically observed in state-of-the-art setups.

Two nanoparticles dancing as a pair

Lasers induce and control interactions between two nanoparticles.