Structural reconfiguration of interacting multi-particle systems through parametric pumping

Abstract

Processes from crystallization to protein folding to micro-robot self-assembly rely on achieving specific configurations of microscopic objects with short-ranged interactions. However, the small scales and large configuration spaces of such multi-body systems render targeted control challenging. Inspired by optical pumping manipulation of quantum states, we develop a method using parametric pumping to selectively excite and destroy undesired structures to populate the targeted one. This method does not rely on free energy considerations and therefore works for systems with non-conservative and even non-reciprocal interactions, which we demonstrate with an acoustically levitated five-particle system in the Rayleigh limit. With results from experiments and simulations on three additional systems ranging up to hundreds of particles, we show the generality of this method, offering a new path for non-invasive manipulation of strongly interacting multi-particle systems.

Fig. 1. Pump-quench control of multi-particle systems.

a Selective excitation of state A in a system of trapped atoms, creating a dark state B. The vertical axis indicates the potential energy of the system, E_pot. When state A is excited by light with appropriate energy, it will be pumped into an excited state (orange arrow), from which it can randomly decay into one of the base states (black arrows). State B cannot be activated by incident light; thus, pumping concentrates atoms in state B. b Cycling between pumping and quenching to create an absorbing 'quiet' state, which does not rely on the presence of well-defined potential energy levels and therefore is not limited to conservative systems. The vertical axis indicates the kinetic energy, E_kin, where E_kin = 0 refers to a base state of the system. By activating a multi-particle system’s vibration mode with parametric pumping, the system will oscillate to the point of transitioning into an unstable state (red arrow). When the pumping is turned off, the system in the unstable state is quenched by damping and will randomly decay back into one of the base states. By cycling between pumping and quenching, the time-averaged outcome is the creation of an absorbing state B, similar to the diagram in (a). c–f Application of pump-quench cycling to control the configuration of, c an acoustically levitated cluster of five particles, d a simulated cluster of a rod and two spheres, e a simulated cluster of 13 spheres, and f the size of an acoustically levitated granular raft.

Fig. 2. Acoustic parametric pumping.

a Diagram of experimental setup. b Tunable acoustic interactions between particles. The right plot has a higher acoustic power than the left. The thickness of the red lines connecting the particle centers indicates the strength of the effective spring constant and the background shows the streaming flow velocity field magnitude v_st, from finite-element method simulation. c Sound-induced net force between two spheres, for acoustic energy densities E₁ < E₂ < E₃. d Piezo input voltage with modulating frequency f_am and modulation depth ε_am. e Pumping state diagram for the in-plane vibration mode of the 2-sphere `molecule', showing the mode’s growth rate ${\gamma }_2^{*}$ as function of f_am and ε_am. Gray isolines indicate ${\gamma }_2^{*}=1$, where pumping balances dissipation. Maximum modulation depth in the experiments at two sound frequencies is shown by red and pink lines. f Video snapshots of a 2-sphere 'molecule' pumped with parameters marked by the pink circle in (e). g Center-to-center distance r between two spheres when pumped. Dashed vertical lines refer to the snapshots in (f).

Fig. 3. Reversible reconfiguration of a levitated 5-sphere cluster with non-reciprocal interactions.

a Snapshot of five spheres in the unstable state (left) and in their two stable states, pentagon and cross (right). b, c Simulation of steady streaming flow velocity field magnitude v_st around the five spheres when one sphere is displaced by 0.3D (original position indicated by gray dashed circle). Red arrows show the net force felt by the sphere at the other end of the dashed pink lines. d Overlay of pumping state diagrams for the cross state, ${\gamma }_c^{*}$ (blue) and the pentagon state, ${\gamma }_p^{*}$ (red). The maximum modulation depth in the experiment is indicated by the black line. Pumping parameter combinations marked by the blue cross and the red pentagon are chosen to activate one or the other state. e, f Transitioning between states. Plots show total kinetic energy when the pentagon, or the cross state, is repeatedly pumped (time intervals shaded light red or blue) and quenched (white), with snapshots of the evolving configurations. g Reversible switching between cross and pentagon states by altering the pumping parameters on the fly. h Cumulative probability of successful transitioning as a function of cycles needed. Solid lines: experiments; dashed lines: Markov model. Error bars: standard deviation.

Fig. 4. Extension to systems with more complicated configuration space.

a Pumping state diagram for a simulated 3-state system comprising a rod and two spheres. Inset: color code for the particle configurations corresponding to the three states and their activation. At certain spots in the state diagram, only one state is activated (red, blue, and green) or one state is stable (pink, cyan, and gold), or all three states are activated (black). b Probability of not reaching the targeted state decreases exponentially. Protocol I (square symbols) applies pump-quench cycles alternating between two sets of parameters marked by crosses with different colors in a. Protocol II (circle symbols) applies a single-step process using parameters marked by the circles in a. Open symbols: simulation data. Color of the circle symbol denotes the targeted structure. Lines: Markov model. c Pumping state diagram for a simulated cluster of 13 spheres with Lennard-Jones interactions at T = 0. In this three-dimensional cluster all spheres are in direct contact with neighbors. Red line shows the boundary above which pumping is possible (γ^* = 1) for the icosahedral sphere arrangement; gray lines are the boundaries for other structures. Red cross: the specific pumping parameter combination used. d Probability of not reaching the targeted icosahedral configuration (sketched in red) decreases exponentially. Examples of other, non-targeted configurations are sketched in gray.

Fig. 5. Controlling the size of an acoustically levitated granular raft.

a A levitated raft being pumped to undergo shape oscillations to shed particles, before being quenched and stabilized with fewer particles. The yellow dashed circle has the same size in each image and serves as a visual guide. b Evolution under repeated pump-quench cycling of the number of particles in a raft for different pump frequencies. Images to the right of the plot show the initial and final raft configuration, demonstrating the ability to control the raft size.