Melting Coulomb clusters through nonreciprocity-enhanced parametric pumping

Abstract

Complex systems out of equilibrium often experience intermittent oscillations between quiescent and highly dynamic states. The type of intermittency depends on how energy is pumped into the system, and how it is dissipated. While intermittency is usually driven by stochastic noise or external forcing, energy can also be sourced from field-mediated interactions between particles, which are often nonreciprocal and effectively violate Newton's 3rd law. Here we demonstrate how nonreciprocal interactions produce intermittency in clusters of charged micron-sized particles confined in a plasma sheath. Through three-dimensional particle tracking, we observe that vertical oscillations, induced by fluctuations of the plasma environment, can be parametrically coupled to the horizontal modes. Experiments and simulations show that nonreciprocal interactions strongly amplify this parametric coupling, creating a positive feedback loop that drives explosive growth of both the horizontal and vertical modes. This mechanism triggers abrupt melting transitions from an ordered cluster to an ergodic gas-like state, and leads to intermittent switching between states over long time scales. Overall, our work identifies nonreciprocal interactions as a key mechanism through which strongly coupled finite systems transform interaction-mediated activity into dynamical nonequilibrium states.

Figures (10)

• Figure 1: Sample preparation and imaging of dusty plasma clusters. ($\mathbf{a}$) Schematic of the plasma chamber and the laser tomographic imaging. Micron-sized dust particles are dropped into the argon plasma discharge in the chamber. The high speed camera is synchronized with a scanning laser sheet which illuminates particles at different height. ($\mathbf{b}$) A stack of 40 frames of images are captured within one cycle of laser scan, where particles at different $z$ positions are illuminated in different frames. ($\mathbf{c}$) The charged dust particles are levitated above the electrode and confined within the center of the electrode due to the electric field gradient. The ions stream towards the electrode and form the ion wakes below the particles which make the interactions between particles effectively nonreciprocal. The down-streaming ions also drift in the azimuthal direction due to the magnetic field, which induces the cluster rotation. Note that the ion drift velocity is much smaller than the streaming velocity.
• Figure 2: Intermittent melting of Coulomb clusters composed of 2, 3, 7, and 18 particles. ($\mathbf{a}$-$\mathbf{d}$) 3D trajectories of the quiescent crystalline state of the clusters. The trajectories are plotted in the $xy$-plane over 0.25 s. The size of each panel is 4 mm $\times$ 4 mm. The $z$-coordinate of each particle is represented by the color of the traces, where $\Delta z=0$ corresponds to the center of mass averaged over the whole time series. ($\mathbf{e}$-$\mathbf{h}$) 3D trajectories of the melted state of the clusters. ($\mathbf{i}$-$\mathbf{l}$) Total kinetic energy normalized by its time average over 60 s for each cluster. The blue (red) dashed line highlights the quiescent (melted) state shown above.
• Figure 3: Rotation and oscillation modes of the 3-particle cluster in experiments. (a) Reduced momentum of inertia $\bar{I}$, and reduced angular momentum $\bar{L}$. Both $\bar{I}$ and $\bar{L}$ increase when the breathing oscillation mode is excited. (b) Orientation angle of the cluster. The cyan line is a linear fit between $t=4.5$ s to $t=9.5$ s. (c-e) Evolution of the $(w_1,w_2,w_3)$ variables. The insets show the corresponding displacements of the particles during the oscillation. The red data in (e) shows the deviation of $\theta$ from the linear fit in (b).
• Figure 4: PCA mode analysis of the 3-particle cluster experiment in the rotating reference frame. ($\mathbf{a}$) The $x$-coordinate of each particle (red, blue, black) in the laboratory frame oscillates due to rotation. The rotation angle of each particle varies linearly until the breathing mode is excited. The 5 s region between the vertical cyan dashed lines denote the quiescent state where the angular velocity of the rotating frame is computed. In the rotating frame ($x'$), the positions are fixed prior to the breathing mode excitation. ($\mathbf{b},\mathbf{c}$) Oscillations modes from PCA in the horizontal (1-6) and vertical (7-9) directions. Modes 1 and 2 are center of mass translations. Modes 3 and 4 are the two degenerate vibrating modes ($w_1$ and $w_2$). Mode 5 is the breathing oscillation, and mode 6 is the zero-frequency rotational mode. Mode 7 is the $z$ center of mass oscillation, and modes 8-9 are asymmetric vertical oscillations. The length of the arrows denote relative displacements of the particles. ($\mathbf{d}$) Fourier spectrum of the PCA modes. Modes 1 and 2 have split peaks due to the Coriolis force. The vertical fluctuations are dominated by mode 7, whose frequency is approximately twice the breathing mode 5.
• Figure 5: Simultaneous growth of the horizontal breathing mode and vertical center of mass mode drives melting in both experiment ($\text{a}$-$\text{c}$) and simulation with $\tilde{q}=0.4, \sigma=0.1$. ($\text{d}$-$\text{f}$). ($\text{a}$,$\text{d}$) Cluster radius, $R$, and the $z$ center of mass, $z_{\text{com}}$, of the 3-particle cluster. The blue shaded regions indicate $w_3<0$. ($\text{b}$,$\text{e}$) $R$ and $z_{\text{com}}$ of the 3-particle cluster near the onset of melting. The breathing mode amplitude increases until the cluster melts and ${w}_3$ begins to switch sign. The amplitude of the $z$ center of mass oscillation grows with the breathing mode. ($\text{c}$,$\text{f}$) Fourier spectrum of ${R}$ and ${z}_{\text{com}}$. The two dashed lines mark the peak frequencies of $f_b=7.42$ s$^{-1}$ ($7.28$ s$^{-1}$), and $f_z=14.57$ s$^{-1}$ ($14.95$ s$^{-1}$) in experiment (simulation).