Nonlinear stochastic and quantum motion from Coulomb forces

communications physics Article A Nature Portfolio journal https://doi.org/10.1038/s42005-025-02106-0 Nonlinear stochastic and quantum motion from Coulomb forces Check for updates Luca Ornigotti ,DarrenW.Moore & Radim Filip Controllable nonlinear quantum interactions are a much sought after target for modern quantum technologies. They are typically difficult and costly to engineer for bespoke purposes. However controllable nonlinearities may have always been in reach via the natural and fundamental forces between quantum particles. The Coulomb interaction between charged particles is the simplest example. We show that after eliminating the harmonic part of the Coulomb force by an auxiliary linear force, the remaining reciprocal nonlinear part results in a directly observable non-reciprocal nonlinear effect: increase of the signal-to-noise ratio (SNR) of the coherent displacement of one particle, driven by the position noise, or uncertainty in quantum regime, in another particle. This essential evidence of nonlinear forces is present across large ranges of trap frequency and mass scales, as well as visible in both stochastic and quantum regimes. The control of quantum systems has proceeded apace, and many experimental settings possess precise control over linear quantum systems1,2,asin quantum optics3,4and quantum optomechanics5,andthemostsimple nonlinear systems such as the Jaynes–Cummings model, exemplified in cavity-QED6or trapped ion systems7. To construct a large-scale nonlinear system from such simple nonlinear systems is a current challenge, and therefore, many experiments are pushing into exciting nonlinear regimes where standard modes of analysis fail, often catastrophically. Such nonlinear regimes are, as might be expected, also the site of yet uncovered quantum phenomena and absolutely required for the most advanced quantum technologies to come to fruition8,9. Similar statements can be made regarding recently developed platforms operating entirely in the classical regime and near the classical-quantum boundary, such as levitated nanoobjects10–21. Such systems are on the road from being inherently stochastic in a high temperature environment to gradually reaching semiclassical and quantum domains. The resulting classical-to-quantum transition with macroscopic objects will likely open access to unexplored physics once they enter nonlinear regimes, such as entanglement-by-heating22, and with the possibility of applications in topics such as quantum sensing for parameter estimation23,24 or of gravity25, quantum thermodynamics26,27 and even quantum computing28,29. In fact, the most prospective systems work in a regime derived from the natural forces between the particles. Perhaps the mostwidelyexploited is the direct reciprocal interaction by the Coulomb force between a pair of charged particles. With state-of-the-art technology, such interactions can be controllably realised on large ranges of mass and frequency scales, from charged levitated nano-particles30–32 all the way down to trapped individual ions33 and even recently between pairs of electrons34.Whenconfined by effective harmonic traps, the linearisation of the force results in coupled harmonic oscillations, with each directional mode decoupled from the others35.We promote an approach to nonlinear effects which show themselves in the intermodal coupling achievable by going beyond the harmonic approximation. Such effects have already been observed in the rotating wave approximation for trapped ion systems36–38. However, the steps beyond this approximation in macroscopic mechanical systems have not been exploited yet. On the other hand, in the context of microscopic single-ion heat engines, cubic interactions between different radial and axial modes have been engineered by tailoring the trap geometry39,40. As the first step, we investigate directly observable nonlinear effects arising from fundamental forces beyond the harmonic approximation and outside any rotating wave approximations, in both classical stochastic and quantum mechanical regimes. In essence, the nonlinear effects are made manifest non-reciprocally through the noise or uncertainty stimulated properties of the system, in our case, a coherent displacement of one particle driven via the noise or uncertainty in the other. By definition, this can only occur via nonlinear interaction of the particles. Even in an approximation where the intermodal effects along different directions are suppressed, this effect can be predicted by expanding the Coulomb interaction only to third order, the first nontrivial nonlinear term, and focusing on a single nonlinear interaction along the straight line between the particles. The interaction is compound, maintaining the reciprocity of the Coulomb force, but does not prevent the observation of non-reciprocal effects in classical and quantum regimes. In what follows, we will make these propositions more precise by introducing the simplest model, presenting testable results, and then demonstrating how such effects can be understood in terms of a suitable approximation. The detection of nonlinear motion, quantified by a noise/ uncertainty-driven increase in the signal-to-noise ratio (SNR) of the Department of Optics, Palacký University Olomouc, 17. listopadu 1192/12, Olomouc, 77900, Czech Republic. e-mail: [email protected] Communications Physics | (2025) 8:195 1 1234567890():,; 1234567890():,; momentum displacement, demonstrates the activity of the nonlinear part of the Coulomb interaction. These effects are shown to be observable across a range of experimentally relevant parameters. This forms a proof-ofprinciple step towards experimental verification in the stochastic and, later, the quantum regime. We expect that the observation of such effects will open further directions for further investigations beyond Gaussian entanglement30,31. Results Nonlinear motion from the Coulomb force A pair of equally charged particles of masses m i ,i∈{1, 2}, confined to a three-dimensional harmonic trap and interacting via the Coulomb potential, can be arranged along the z-axis by having a much greater trap frequency in that direction15,20,21. The axes can then be chosen so that the equilibrium points are dðiÞ¼00zi;0  . Even beyond the harmonic approximation, where the motional axes are not decoupled, the interactions depend upon the distances between the two particles along the respective axes. If motion along the xand yaxes is sufficiently cooled so that fluctuations along these axes are small, then the interactions between xor y,and zare suppressed. The Hamiltonian of the system can then be written as H¼1 2X i p2 i miþmiω2 iðzizi;0Þ2  þκ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðz1z2dÞ2 q;ð1Þ where ω i are the trap frequencies, κ=q 1 q 2 /4πϵ 0 is the coupling strength of the reciprocal interaction calculated via the charge q i and the electric permittivity in vacuum ϵ 0 ,andd=z 1,0 −z 2,0 is the initial distance between the particles’equilibrium positions. The setup is illustrated in Fig. 1a. In the harmonic approximation, expanding around the distance between the particles in each direction to second order, results in an extra displacement term in the zdirection for both particles. The motion of the chargedparticles is then described by harmonically coupled oscillators with modified frequencies. The coupling induced between the particles is intramodal, such that the modes along different axes do not talk to each other. The third-order term introduces nonlinear and intermodal interactions. To isolate them, a compensating force eliminating the lower-order contributions is necessary. The constant force can be compensated via linear tilt through an electrostatic force41, whereas the linear force can be compensated by parametric control, such as feedback42–44. In the regime of optimal compensation, assumed throughout the manuscript, the still reciprocal interaction Hamiltonian can be approximated as H3κ d4ðz1z2Þ3¼κ d4z3 1z3 2þ3ðz1z2 2z2 1z2Þ  :ð2Þ A non-optimal compensation still results in the non-reciprocal nonlinear effect, albeit reduced in visibility, see Supplementary Material (SM) Note 3. Due to the nonlinearity of the Coulomb force, a reciprocal cubic interaction emerges along the z-axis. This is a minimal model of the nonlinear Coulomb interaction between two particles. While similar nonlinear interactions can be constructed for ions directly via the trap geometry for hybrid radial and axial modes39,40, here the nonlinearity emerges directly from the Coulombic interaction between the particles. Both the trap and the compensation Fig. 1 | Noise and uncertainty-induced momentum displacement via a nonlinear Coulomb interaction. a(i), (ii) Illustration of the principle in classical and quantum regimes. Two harmonically confined particles (blue, yellow) experience a nonlinear Coulomb interaction (red spring) along a single axis, together with a compensating force (green). In the classical regime, the stochastic particle 1 is prepared in oscillator equilibrium states via dissipation to the thermal environment at temperature T 1 , while particle 2 is initially cooled to T0¼10 mK in a room temperature environment T 1 =T 2 = 300 K. In the quantum regime, the particles are prepared in ground states, and the only fluctuations arise from the quantum uncertainties. A weak linear damping with a rate Γ, acting only on particle 2, is present in order to ensure the stability of the effect. The trap frequency modulation, and mass disproportion of particle 1, as shown in the table, are used to generate unidirectional flow of fluctuations to particle 2, thus avoiding backactions. b(i), (ii) Time evolution of mean momentum hpz2i, normalised to the initial standard deviation (top) and signal-to-noise ratio SNRpz2(bottom). For large noise (full circles), the SNR ¼1=ffiffiffi 2 pis quickly reached by all regimes, but tuning frequency (blue) allows for better noise control. At lower noise (empty circles), the parametric symmetry is the only regime reaching the SNR bound (grey). In the quantum regime (b(ii)), the ground state fluctuations (empty circles) are equally harnessed by all regimes, whereas an initial uncertainty amplification, by free-fall (full circle), allows the SNR bound to be reached by all regimes. Symmetric (grey) and tuning frequency (blue) further experience the faster uncertainty growth (SNR drop), not visible for tuning mass (orange), which is also the best regime here. https://doi.org/10.1038/s42005-025-02106-0 Article Communications Physics | (2025) 8:195 2 operate in the harmonic approximation, and therefore cannot contribute to the nonlinear effects outlined below. Differently than for optical cubic potentials45, the interaction in Eq. (2) combines competing cubic nonlinearities. That is, the cubic single particle potential z3 1;z3 2,andthe cubic interparticle potential z1z2 2;z2 1z2. Their vying nature may limit the direct observation of the interparticle nonlinear noise or uncertaintyinduced phenomena. Quantisation can be accomplished by promoting the canonical variables to operators satisfying the commutation relations [z i ,p i ]=i. Classical noise-induced momentum. Nonlinear interactions such as z2 1z2allow for the possibility to coherently displace the momentum of one particle via increases in the initial position noise of the other. Heuristically, the complete reciprocal interaction term of Eq. (2) indicates that for initially uncorrelated states the mean momentum displacement in one mode is rapidly driven by the noise in the other i.e. hpz2ihpz2;0i3κðhz2 1;0i2hz2;0ihz1;0iþhz2 2;0iÞt=d4. The reciprocal nature of the interaction Hamiltonian of Eq. (2) suggests that a separate asymmetrical manipulation performed solely on one of the two particles can enhance the interparticle non-reciprocal noise-induced nonlinear effect observed on the other. For parametrically symmetric interactions, m 1 =m 2 and ω 1 =ω 2 , the minimal asymmetrical manipulation is accomplished by unbalancing the initial distribution of thermal noise. That is, prepare the initial thermal state at temperature T= 300 K of particle 2 at an effective temperature of T0¼10 mK by cooling, while particle 1 is prepared in an oscillator thermal-equilibrium state at room temperature T 1 = 300 K. These are zero-mean Gaussian states with variances in position σ2 z¼kBT=mωand momentum σ2 p¼mkBT, where Tis the effective temperature of mode zand k B is the Boltzmann constant. During the dynamics, the modes are immersed in thermal environments with T 1 ,T 2 = 300 K. This initial thermal distribution imbalance between T 1 and T0minimises the thermal fluctuations hp2 z2;0i, which otherwise obscure the noise-induced effect, while simultaneously minimising any unwanted back-action effects on the particle whose noise drives the noise-induced effect. While the technical details of this preparation depend strongly on the chosen platform, we suggest a proof-of-principle state preparation scheme in SM Note 4. In Fig. 2a we show the noise-induced motion for this parametrically symmetric case (grey). The classical simulations are carried out with the parameters ω i =50kHz,m 1 =m 2 =8×10 −17 kg, κ=2.3×10 −24 Nm 2,and d=3μm, inspired by optical levitation15,20,21,32 and enriched with magnetic levitation16,17 platforms in mind as they operate with a wider trapping frequency range. No displacement occurs in z 1 , however, the noise grows rapidly. The mean momentum of the second particle, however, experiences a positive sharp increase away from zero as well as an increase in noise. This displacement is not critically outperformed by noise, as seen in the SNR in Fig. 1b(i) (empty circles). This is a sign of a direct noise-induced coherent effect. Importantly, the SNR grows with increasing σz1;0(full circles) and can saturate the maximum of 1=ffiffiffi 2 pat the cost of large noise in the initial position of z 1 . This effect can be explained by examining the Langevin equations of motion for the third-order Coulomb term. These are given by m2 € z2ðtÞþm2Γ_ z2ðtÞ3κ d4z2 1ðtÞþz2 2ðtÞ  m2ω2 2þ6κ d4z1ðtÞ  z2ðtÞ þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ΓkBT2 pξ2ðtÞ; m1 € z1ðtÞþm1Γ_ z1ðtÞ3κ d4z2ðtÞ2þz2 1ðtÞ  m1ω2 16κ d4z2ðtÞ  z1ðtÞ þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ΓkBT1 pξ1ðtÞ; ð3Þ where Γis the drag coefficient whose value and phenomenological origin depends upon implementation, and ξ 1 ,ξ 2 are independent zero-mean Gaussian white noises with hξiðt0Þξiðt00Þi¼ δðt0t00Þ. In what follows, we focus on the underdamped regime, with Γ=10 −4Hz,asintheoverdamped regime, the nonlinear effect in momentum pz2is not visible (see SM Note 1). For parametric symmetry, it is useful to discuss the dynamics using the mean value approximation, by reducing the two-body interaction into a one-body problem by virtue of the effective potentials ~ Vðz2Þ¼3κhz1i2z2=d4þ ~ ω2z2 2κz3 2=d4and ~ Vðz1Þ¼3κhz2i2z1=d4þ~ ω1z2 1þκz3 1=d4.Inthis framework, the unwanted back-action of z 2 on z 1 is understood as the mean displacement 〈z 2 〉which (i) generates a drift in z 1 ,asvisiblefromthefirst term in ~ Vðz1Þ, and (ii) modifiesthefrequencyoftheharmonicconfinement of z 1 via ~ ω1¼m1ω2 1=23κhz2i=d4. The imbalance in the initial noise properties minimises both back-action contributions at short transients t<20μs. That is, low temperature T0makes the noise-induced shift generated by the cubic potential in ~ Vðz2Þnegligible, keeping the position below the critical value of 〈z 2 〉≈1μm (calculated for the parameters used in numerical simulation) after which the effective frequency ~ ω1becomes negative and the harmonic confinementbecomesaninvertedquadratic potential, leading to unstable diverging trajectories. However, the noise in z 1 is still substantially increasing in time, and can, in general, complicate both predictions and applications of nonlinearities. For large initial noise, as visible in Fig. 2a (grey), the higher-order nonlinear terms of the Coulomb interaction make for a positive back-action z 2 on z 1 . It results in a decrease of the noise of z 1 , even below that of its initial thermal state. When the reciprocity in the nonlinear effect is further broken by either tuning the mass (at fixed frequency) or frequency (at fixed mass) of particle 1, the fluctuations in z 1 are modified and the properties of the coherent motion are altered. The initial thermal state of z 1 is determined by its dynamical variables m 1 ,ω 1 , and local environmental temperature T 1 via the Fig. 2 | Analysis of the time evolution of variables undergoing nonlinear motion. Analysis of the time evolution of position z 1 and momentum pz2at different initial fluctuations of z 1 . The shaded area represents the standard deviation around the mean evolution (solid). All quantities are normalised to the standard deviation of their respective initial states. Symmetry breaking by mass tuning (orange) and frequency tuning (blue) allows to control divergence in pz2in both mean and standard deviation. In the classical regime (a), the mass tuning visibly performs better than the other strategies as it produces larger momentum drift hpz2i. For the quantum regime (b), the symmetric (grey) and frequency tuned (blue) outperform the mass tuned with the same metric. It confirms the result presented in Fig. 1b(i), (ii). https://doi.org/10.1038/s42005-025-02106-0 Article Communications Physics | (2025) 8:195 3 standard deviation σz1;0¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kBT=m1ω1 p. Changing the mass or frequency to pursue the symmetry-breaking techniques results in a modification of the initial fluctuations. We therefore fix the noise properties to σz1;0¼30 nm to observe the effect arising from the nonlinear interaction with constant initial noise across different parameter regimes. That is, z 1 is not prepared in an equilibrium state of its local oscillator but rather in an out-of-equilibrium thermally squeezed state. Trapping a massive particle m 1 ≫m 2 minimises its kinetic term p2 z1=2m1, and as a result, the position does not move away from its initial mean condition z1  z1;0  . That is, the back-action is negligible for short transients, as is visible. Differently than parametric symmetry, the additional imbalance of the mass reshapes the noise evolution of z 1 , keeping it close to its initial distribution σz1;0for low initial fluctuations. However, as visible in Fig. 2a (orange), large initial noise promotes a positive back-action loop over time, decreasing the fluctuations of z 1 below the initial thermal state. It is similar to the parametric symmetry (grey) back-action effect, however, its effect on particle 2 showcases a larger displacement, although with larger fluctuations (orange shaded area). The resulting SNR in Fig. 1b(i), while increasing even at low fluctuations (empty circles), only saturates the bound for larger initial noise σz1;0(full circles). The short transient of p 2 evolves as pz2ðt0Þpz2;0þ3κRt0 0ds0z2 1ðs0Þ=d4m2ω2 2z2;0t0 in the limit of zero damping Γ= 0. For mass tuning, it leads to the following moments hp2ðtÞi  3κtσ2 z1;0 d4;SNR ¼hp2i σp21 ffiffiffi 2 p:ð4Þ Alternatively, trapping particle 1 in a stiffer harmonic potential ω 1 ≫ω 2 confines its noise dynamics to that of a harmonic oscillator for short transients. Its evolution is described by coherent oscillations, approximately described by z1z1;0cosðω1tÞunder the assumption of vanishing initial velocity _ z1;0¼0. As visible in Fig. 2a (blue), this oscillatory evolution dominates over the back-action for times of a few tenths of microseconds. This added imbalance of frequency negatively impacts the dynamics, generating a lower momentum drift hpz2i,andalowerSNR output Fig. 1b(i) (blue) at small initial noise (empty circles), but it too saturates the 1=ffiffiffi 2 pbound at large initial noise σz1;0(full circles). For tuning frequencies, the moments of momentum p 2 approach hpz2ðtÞi  3κ 4d4ω1 2θþsinð2θÞ ½ σ2 z1;0;SNR 1 ffiffiffi 2 p;ð5Þ where θ=ω 1 t. In the low noise limit, the added symmetry breaking lowers the SNR relative to the symmetric case (see Fig. 1b(i), empty circle). To reach it, an extra cost of increasing the initial noise σz1;0is required. Specifically, for tuning stiffness, the same SNR is reached at a smaller displacement (see Fig. 2a, blue), while for tuning mass, the SNR saturation is obtained with a much larger displacement (orange), making it a favourable strategy. This is further highlighted in Fig. 3a, where a target SNR ¼1=ffiffiffi 2 pis fixed, and the output momentum displacement hpz2i(top) and standard deviation σpz2 (bottom) are plotted against input noise cost σz1;0. It shows that for initial noise below σz1;0≲100 nm, the parametric symmetry (grey) harnesses the noise of z 1 through the Coulomb interaction more efficiently. However, for initial noise input beyond σz1;0≳100 nm, tuning frequency (blue) generates the same SNR with smaller output noise and momentum drift, while tuning mass (orange) reaches it with larger momentum drift. Fig. 3shows that the best strategy, for large initial noise, is to break the symmetry by tuning the mass to reach the target signal-to-noise with the largest momentum drift. Quantum uncertainty-induced momentum. As we decrease the initial noise to the ground state extension, that is σz1;0¼0:01 nm, the nonlinear effect in the stochastic framework described by Eq. (3) vanishes (see SM Note 2). Operating in the quantum regime of Eq. (3), using pure initial states, an analogue of the previous noise-induced phenomena comes directly from the quantum fluctuations in z 1 . As expected, it is sufficient to produce momentum displacement on z 2 , as visible in Fig. 2b. In this section, the numerical simulations are performed directly on Eq. (3), see SM Note 2, using the same parameters outlined in the previous Fig. 3 | Noise/Uncertainty-induced momentum under noise confinement. a The output displacement hpz2i(i), and standard deviation σpz2(ii) at the target signal-to-noise ratio SNRpz2¼1=ffiffiffi 2 pare plotted for the stochastic classical dynamics against the input noise σz1;0. At low initial input noise, the parametric symmetry (grey) always reaches the target. At large input noise, all regimes reach the target, but breaking the symmetry via tuning frequency (blue) provides the least noise output, and thus even smaller momentum displacement. Breaking symmetry by tuning mass (orange) is useful only between noise input 60 ≲σz1;0≲100 nm. bThe output displacement hpz2i(i), and standard deviation σpz2(ii) at the target SNR for different initial uncertainty σz1;0. All regimes reach the target, but at different times. For parametric symmetry (grey) and tuning frequency (blue), the target is reached at t≈1μs, while for tuning mass (orange), the target is reached at larger times t≈2μs. Regardless of the required time, when the target is reached, all regimes produce the same displacement and standard deviation output, making the parametric symmetry (grey) the preferred strategy to reach the target with the minimum noise cost. Note, breaking symmetry requires extra squeezing to reach the same initial noise input. The dashed filled circles record the value of momentum displacement and standard deviation when the target SNR is not reached. https://doi.org/10.1038/s42005-025-02106-0 Article Communications Physics | (2025) 8:195 4 section. All quantities are rescaled by the standard deviation of the initial state. For parametric symmetry, the quantum fluctuations of z 1 induce a small drift in the momentum, further enhanced by the quantum fluctuation of z 2 through the cubic nonlinearity, reaching hpz2i3κðσ2 z1;0þσ2 z2;0Þt=d4. The unwanted back-action force is too small to induce instability, therefore, the noise of z 1 does not increase larger than its reference state. It drives the momentum hpz2iwith an increasing SNR (Fig. 3panel b(i), empty circles) that does not reach the maximum of 1=ffiffiffi 2 p,asfortimest>2μs, the cubic nonlinearity dominates both dynamics and the back-action strongly drives z 1 to instability. It enhances the noise of p 2 beyond the initial ground state, thus resulting in a drop of the SNR. The conservative symmetry-breaking strategies, introduced for the stochastic dynamics, result in a qualitatively similar time evolution (see Fig. 2b, blue and orange). The initial standard deviation σz1;0¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi _=ð2m1ω1Þ pis calculated using the same parameters of the stochastic system. For parametric symmetry, i.e. m 1 =8×10 −17 kg and ω 1 = 50 kHz, the initial standard deviation results in σz1;0¼0:01 nm. For tuning frequency, i.e. ω 1 = 2500 kHz, and tuning mass m 1 =8×10 −16 kg, the initial standard deviations assume different values. Respectively σz1;0¼0:001 nm, and σz1;0¼0:003 nm. To observe only the effects of the nonlinear interaction given by the dynamics, under the same initial noise conditions, the initial state of z 1 is squeezed by a factor ξ¼ logðσtrg ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m1ω1=_ pÞ=2 to reach the unified target standard deviation of σ trg = 0.01 nm. That is, the position variance is amplified σz1;0¼ ξffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi _=ð2m1ω1Þ pby ξ, while the momentum variance is attenuated by the inverse amount σpz1;0¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi _m1ω1=2 p=ξ. When the initial ground state of z 1 is further squeezed in momentum, it realises a larger drift reaching a SNR ¼1=ffiffiffi 2 pat short transients (Fig. 1, panel b(ii)). Notice that parametric symmetry (grey) and tuning frequency (blue) are subjected to instability for times larger than t≈1μs, resulting in a drop of the SNR, while tuning mass (orange) is not yet affected by it. For tuning mass m 1 ≫m 2 , the short transient of the moments of p 2 approach hpz2i3κtσ2 z1;0 d4; SNRpz21 ffiffiffi 2 p1þm2 2ω4 2d8σ2 z2;0 18κ2σ4 z1;0þ d8σ2 pz2;0 18κ2σ4 z1;0t2 "# 1 2 :ð6Þ For short transients, the ground state momentum noise σ2 pz2;0prevents the SNR from reaching the 1=ffiffiffi 2 pbound. At larger times it becomes negligible, leaving the position noise σ2 z2;0as the dominant limiting term of the evolution. For fixed m 2 ,ω 2 ,anamplification of position noise σ2 z1;0¼ξσ2 z1;0 by ξallows to reach a SNR ¼1ffiffiffi 2 pas visible in Fig. 1b(ii), full orange circle. That is, the initial state is further squeezed in momentum. Squeezing the position noise σ2 z2;0can, in principle, improve upon the SNR of Eq. (6). However, the complementary amplification of momentum noise σ2 pz2;0 increases the back-action to particle 1 at larger times, thus leading to the divergence quicker. For tuning frequency ω 1 ≫ω 2 ,thenoiseofz 1 is confined in a stiffer harmonic bound, and its dynamics is described as z1z1;0cosðθÞ,with θ=ω 1 t.Inthisregime,thedynamicsevolvessimilarlytothatoftheparametric symmetry as visible in Fig. 1b(ii), blue and grey circles. Its momenta evolve in short transients according to hpz2i3κσ2 z1;0 4d4ω1 2θþsinð2θÞ ½ ; SNRpz21 ffiffiffi 2 p1þ 8ω2 2d8σ2 pz2;0þ8ω2 1d8t2σ2 z2;0  9κ2σ4 z1;02θþsinð2θÞ ½ 2 2 43 5 1 2 : ð7Þ For an initial ground state σ2 z1;0¼_=ð2m1ω1Þ, the momentum and position noise of the initial state of particle 2, i.e. σ2 z2;0;σ2 pz2;0hinders the uncertainty-induced effect from reaching the SNR ¼1=ffiffiffi 2 pbound at short transients. For a longer time, the instability from the cubic potential, not accounted for in Eq. (7), dominates the dynamics, resulting in a drop of the SNR. Similar to tuning mass, to reach the signal-to-noise bound for short transients, the amplification of position noise σ2 z1;0is required, as visible in Fig. 1b(ii), full blue circles. Moreover, squeezing the position noise σz2;0 results in faster divergence, similar to the case of the tuning mass. The hidden cost of parametric symmetry breaking lies in the preparation of the initial state, which requires squeezing of the ground state. That is, to have comparable noise in position of σz1;0¼0:01 nm after tuning mass and frequency, the ground state must be squeezed, by ξ=1.15and ξ= 1.96, respectively. For the large noise case, a further amplification by freefall is then used to reach the position noise in all regimes of σz1;0¼0:08 nm. In Fig. 3b, the target SNR ¼1=ffiffiffi 2 pis reached equally by all regimes for initial input noise σz1;0≳0:06 nm, but at different times. Ultimately, the parametric symmetry emerges as the best regime, as it does not require any extra costs, i.e. the initial squeezing to σz1;0¼0:01nm,asisthecaseofthe mass and frequency tuning regimes. Conclusion Experiments involving the interaction of trapped charged particles typically operate in the harmonic approximation30,31,35, where the motion decouples into oscillations along each coordinate axis. This can only result in Gaussian effects: single particle and multiparticle squeezing, and potentially, entanglement in the quantum regime. Here we have shown a proof-of-principle method to go beyond this approximation, showing that the inherently nonlinear effect of noise-induced coherent motion can be seen and explained using the first nontrivial nonlinear term in an expansion of the reciprocal Coulomb interaction between two charged particles. This is observed in Fig. 1b(i) by the non-reciprocal effect: improvement in SNR of one particle due to increased position thermal noise in another nonlinearly coupled particle. This effect persists into the quantum regime, where now the quantum uncertainty rather than classical noise drives the dynamics of the test particle to the same SNR bound of classical dynamics, at smaller momentum displacement. It is visible in Fig. 1b(ii). This method provides the first step to analysing one of the most basic two-particle nonlinear effects, as visible in Fig. 3. Noise control is an important basic tool for control of technologies that are inherently stochastic. Advantageously, this can be done autonomously through naturally occurring forces, lowering the engineering requirements to make such effects visible across a broad range of parameters and platforms. It should be noted that nano-particles possess a natural advantage compared to ions as their mass-charge ratio favours observation of the short-time nonlinear effects. That is, their high charge and high mass range capabilities allow for a stronger interaction tunability and slower divergence, respectively. To date, the most in-depth studies of motional nonlinear systems are of single-mode nonlinearities, possibly linked to other linear systems. Increasingly detailed studies of experimentally accessible two-particle nonlinear interactions will likely unveil many unexpected effects. Breaching further into genuinely multipartite systems combines the complexity of the many possible configurations (chains, triangles, lattices, or clusters)46–48 with that of a nonlinear interaction distributed across multiple particles, likely hiding exciting nonlinear effects. This then opens the possibility to further exploit such nonlinearity in naturally occurring interactions. Code availability Code available upon reasonable request. Received: 20 September 2024; Accepted: 22 April 2025; References 1. Weedbrook, C. et al. Gaussian quantum information. Rev. Mod. Phys. 84, 621–669 (2012). https://doi.org/10.1038/s42005-025-02106-0 Article Communications Physics | (2025) 8:195 5 2. Adesso, G., Ragy, S. & Lee, A. R. Continuous variable quantum information: Gaussian states and beyond. Open Syst. Inf. Dyn. 21, 1440001 (2014). 3. Cerf, N. J., Leuchs, G. & Polzik, E. S. Quantum Information with Continuous Variables of Atoms and Light.https://www. worldscientific.com/worldscibooks/10.1142/p489 (Imperial College Press, 2007). 4. Asavanant, W. & Furusawa, A. Optical Quantum Computers: A Route to Practical Continuous Variable Quantum Information Processing. https://pubs.aip.org/books/book/50/Optical-QuantumComputersA-Route-to-Practical (AIP Publishing LLC, 2022). 5. Bowen, W. P. & Milburn, G. J. Quantum Optomechanics.https://www. taylorfrancis.com/books/quantum-optomechanics-warwick-bowen/ 10.1201/b19379 (CRC Press, 2015). 6. Blais, A., Grimsmo, A. L., Girvin, S. M. & Wallraff, A. Circuit quantum electrodynamics. Rev. Mod. Phys. 93, 025005 (2021). 7. Monroe, C. et al. Programmable quantum simulations of spin systems with trapped ions. Rev. Mod. Phys. 93, 025001 (2021). 8. Calcluth, C., Reichel, N., Ferraro, A. & Ferrini, G. Sufficient condition for universal quantum computation using bosonic circuits. PRX Quantum 5, 020337 (2024). 9. Asavanant, W. & Furusawa, A. Multipartite continuous-variable optical quantum entanglement: generation and application. Phys. Rev. A 109, 040101 (2024). 10. Timberlake, C., Gasbarri, G., Vinante, A., Setter, A. & Ulbricht, H. Acceleration sensing with magnetically levitated oscillators above a superconductor. Appl. Phys. Lett. 115, 224101 (2019). 11. Vinante, A. et al. Ultralow mechanical damping with Meissner-levitated ferromagnetic microparticles. Phys. Rev. Appl. 13, 064027 (2020). 12. Conangla, G. P., Rica, R. A. & Quidant, R. Extending vacuum trapping to absorbing objects with hybrid Paul-optical traps. Nano Lett. 20, 6018–6023 (2020). 13. Svak, V. et al. Stochastic dynamics of optically bound matter levitated in vacuum. Optica 8, 220–229 (2021). 14. Tebbenjohanns, F., Mattana, M. L., Rossi, M., Frimmer, M. & Novotny, L. Quantum control of a nanoparticle optically levitated in cryogenic free space. Nature 595, 378–382 (2021). 15. Rieser, J. etal. Tunable light-induceddipole-dipole interaction between optically levitated nanoparticles. Science 377,987–990 (2022). 16. Hofer, J. et al. High-qmagnetic levitation and control of superconducting microspheres at millikelvin temperatures. Phys. Rev. Lett. 131, 043603 (2023). 17. Gutierrez Latorre, M., Higgins, G., Paradkar, A., Bauch, T. & Wieczorek, W. Superconducting microsphere magnetically levitated in an anharmonic potential with integrated magnetic readout. Phys. Rev. Appl. 19, 054047 (2023). 18. Brown, C. D. et al. Superfluid helium drops levitated in high vacuum. Phys. Rev. Lett. 130, 216001 (2023). 19. Bykov, D. S., Dania, L., Goschin, F. & Northup, T. E. 3d sympathetic cooling and detection of levitated nanoparticles. Optica 10, 438–442 (2023). 20. Brzobohatý, O. et al. Synchronization of spin-driven limit cycle oscillators optically levitated in vacuum. Nat. Commun. 14, 5441 (2023). 21. Reisenbauer, M. et al. Non-Hermitian dynamics and non-reciprocity of optically coupled nanoparticles. Nat. Phys. 20, 1629–1635 (2024). 22. Laha, P., Moore, D. W. & Filip, R. Entanglement growth via splitting of a few thermal quanta. Phys. Rev. Lett. 132, 210201 (2024). 23. Ivanov, P. A. Quantum parameter estimation of nonlinear coupling in a trilinear Hamiltonian with trapped ions. Phys. Rev. A 105, 032617 (2022). 24. Mahmoudi, P., Ritboon, A. & Filip, R. Optimal quantum sensing of the nonlinear bosonic interactions using Fock states. Phys. Rev. Res. 6, 043215 (2024). 25. Carlesso, M., Bassi, A., Paternostro, M. & Ulbricht, H. Testing the gravitational field generated by a quantum superposition. N. J. Phys. 21, 093052 (2019). 26. Martinez, E. A. Dynamics and thermodynamics of linear quantum open systems. Phys. Rev. Lett. 110, 130406 (2013). 27. Ray,K.J.,Boyd,A.B.,Guarnieri,G.&Crutchfield, J. P. Thermodynamic uncertainty theorem. Phys. Rev. E 108, 054126 (2023). 28. Moore, D. W. Quantum hypergraph states in continuous variables. Phys. Rev. A 100, 062301 (2019). 29. Vandré’, L., Jing, B., Xiang, Y., Gühne, O. & He, Q. Graphical calculus for non-Gaussian quantum states. http://arxiv.org/abs/2409.07300 (2024). 30. Rudolph, H., Delič, U., Aspelmeyer, M., Hornberger, K. & Stickler, B. A. Force-gradient sensing and entanglement via feedback cooling of interacting nanoparticles. Phys. Rev. Lett. 129, 193602 (2022). 31. Penny, T. W., Pontin, A. & Barker, P. F. Sympathetic cooling and squeezing of two colevitated nanoparticles. Phys. Rev. Res. 5, 013070 (2023). 32. Deplano, Q., Pontin, A., Ranfagni, A., Marino, F. & Marin, F. Coulomb coupling between two nanospheres trapped in a bichromatic optical tweezer. Optica 11, 1773–1777 (2024). 33. Nguyen, C.-H., Tseng, K.-W., Maslennikov, G., Gan, H. C. J. & Matsukevich, D. Experimental SWAP test of infinite dimensional quantum states. arXiv:2103.10219 [physics, physics:quant-ph].http:// arxiv.org/abs/2103.10219 (2021). 34. Beysengulov, N. R. et al. Coulomb interaction-driven entanglement of electrons on helium. PRX Quantum 5, 030324 (2024). 35. Deng, X.-L., Porras, D. & Cirac, J. I. Quantum phases of interacting phonons in ion traps. Phys. Rev. A 77, 033403 (2008). 36. Ding, S., Maslennikov, G., Hablützel, R., Loh, H. & Matsukevich, D. Quantum parametric oscillator with trapped ions. Phys. Rev. Lett. 119, 150404 (2017). 37. Ding, S., Maslennikov, G., Hablützel, R. & Matsukevich, D. Quantum simulation with a trilinear Hamiltonian. Phys. Rev. Lett. 121, 130502 (2018). 38. Maslennikov, G. et al. Quantum absorption refrigerator with trapped ions. Nat. Commun. 10, 202 (2019). 39. Abah, O. et al. Single-ion heat engine at maximum power. Phys. Rev. Lett. 109, 203006 (2012). 40. Roßnagel, J. et al. A single-atom heat engine. Science 352, 325–329 (2016). 41. Ciampini, M. A. et al. Experimental nonequilibrium memory erasure beyond Landauer’s bound. http://arxiv.org/abs/2107.04429 (2021). 42. Setter, A., Toroš, M., Ralph, J. F. & Ulbricht, H. Real-time Kalman filter: cooling of an optically levitated nanoparticle. Phys. Rev. A 97, 033822–(2018). 43. Conangla, G. P. et al. Optimal feedback cooling of a charged levitated nanoparticle with adaptive control. Phys. Rev. Lett. 122, 223602– (2019). 44. Magrini, L. et al. Real-time optimal quantum control of mechanical motion at room temperature. Nature 595, 373–377 (2021). 45. Rakhubovsky, A. A. & Filip, R. Stroboscopichigh-order nonlinearityfor quantum optomechanics. npj Quantum Inf. 7,1–7 (2021). 46. Lu, Y. et al. Implementing Arbitrary Ising Models with a Trapped-Ion Quantum Processor. Phys. Rev. Lett. 134, 050602 (2025). 47. Frattini, N. E. et al. 3-wave mixing Josephson dipole element. Appl. Phys. Lett. 110, 222603 (2017). 48. Bykov, D. S., Meusburger, M., Dania, L. & Northup, T. E. Hybrid electro-optical trap for experiments with levitated particles in vacuum. Rev. Sci. Instrum. 93, 073201 (2022). Acknowledgements We acknowledge the project GA23-06224S of the Czech Science Foundation, EU and MEYS Czech Republic No. CZ.02.01.01/00/22_008/0004649 (QUEENTEC). R.F. also acknowledges funding from the MEYS of the Czech Republic (Grant Agreement 8C22001). Project SPARQL has received funding from the European Union’s Horizon 2020 Research and Innovation Programme under Grant Agreement no. 731473 and 101017733 (QuantERA). https://doi.org/10.1038/s42005-025-02106-0 Article Communications Physics | (2025) 8:195 6 Author contributions L.O. performed the numerical simulations and analytical solutions to the classical dynamics, and D.M. performed the quantum mechanical calculations. Both received theory inputs from R.F. R.F. conceived and supervised the project. All authors contributed to the analysis of the results and composition of the article. Competing interests The authors declare no competing interests. Additional information Supplementary information The online version contains supplementary material available at https://doi.org/10.1038/s42005-025-02106-0. Correspondence and requests for materials should be addressed to Darren W. Moore. Peer review information Communications Physics thanks Anton Zasedatelev and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Reprints and permissions information is available at http://www.nature.com/reprints Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. © The Author(s) 2025 https://doi.org/10.1038/s42005-025-02106-0 Article Communications Physics | (2025) 8:195 7