Nonlinear stochastic and quantum motion from Coulomb forces

TL;DR

The paper investigates nonlinear motion arising from Coulomb forces between two trapped charged particles, going beyond the harmonic approximation. By compensating the linear part of the interaction, a cubic interparticle term $H_3 \approx \frac{\kappa}{d^4}(z_1 - z_2)^3$ yields a non-reciprocal transfer of fluctuations, enabling a noise- or uncertainty-driven momentum displacement of one particle conditioned on the other's state. The authors analyze both classical stochastic and quantum regimes, deriving SNR benchmarks and showing that the non-reciprocal effect persists across a broad range of trap frequencies, masses, and damping, with quantum fluctuations producing analogous momentum shifts. This work provides a proof-of-principle pathway to harness natural Coulomb nonlinearities for quantum control and sensing, suggesting platforms such as levitated nano-objects or trapped ions where the effect can be observed without relying on rotating-wave or other approximations.

Abstract

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.

Figures (3)

• Figure 1: Noise and uncertainty-induced momentum displacement via a nonlinear Coulomb interaction.a (i), a (ii) Illustration of the principle idea in classical and quantum regime. 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 thermal environment at temperature $T_1$, while particle $2$ is initially cooled to $T_2'=10$ 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 $\Gamma$, 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 showed in the table, are used to generate unidirectional flow of fluctuations to particle $2$, thus avoiding back-actions. b (i), b (ii) Time evolution of mean momentum $\langle p_{z_2} \rangle$, normalised to the initial standard deviation (top) and signal-to-noise ratio $\mathrm{SNR}_{p_{z_2}}$ (bottom). For large noise (full circles), the $\mathrm{SNR}=1/\sqrt{2}$ is 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 $\mathrm{SNR}$ 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 freefall, (full circle) allows the $\mathrm{SNR}$ bound to be reached by all regimes. Symmetric (grey) and Tuning Frequency (blue) further experience the faster uncertainty growth ($\mathrm{SNR}$ drop), not visible for tuning mass (orange), which is also the best regime here.
• Figure 2: Analysis of time evolution of variables undergoing nonlinear motion. Analysis of time evolution of position $z_1$ and momentum $p_{z_2}$ at 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 $p_{z_2}$ in 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 $\langle p_{z_2} \rangle$. 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. \ref{['fig0']} ($b$ (i),$b$ (ii)).
• Figure 3: Noise/Uncertainty induced momentum under noise confinement. (a): The output displacement $\langle p_{z_2} \rangle$ (i), and standard deviation $\sigma_{p_{z_2}}$ (ii) at the target signal-to-noise ratio $\mathrm{SNR}_{p_{z_2}}=1/\sqrt{2}$ are plotted for the stochastic classical dynamics against the input noise $\sigma_{z_1,0}$. 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 $60 \lesssim \sigma_{z_1,0} \lesssim 100$ nm. (b): The output displacement $\langle p_{z_2} \rangle$ (i), and standard deviation $\sigma_{p_{z_2}}$ (ii) at the target $\mathrm{SNR}$ for different initial uncertainty $\sigma_{z_1,0}$. All regimes reach the target, but at different times. For parametric symmetry (grey) and tuning frequency (blue) the target is reached at $t\approx 1\, \mu$s, while for tuning mass (orange) the target is reached at larger times $t \approx 2\, \mu$s. 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 $\mathrm{SNR}$ is not reached .