Nonreciprocal wave-mediated interactions power a classical time crystal

An acoustic standing wave acts as a lattice of evenly spaced potential energy wells for sub-wavelength-scale objects. Trapped particles interact with each other by exchanging waves that they scatter from the standing wave. Unless the particles have identical scattering properties, their wave-mediated interactions are nonreciprocal. Pairs of particles can use this nonreciprocity to harvest energy from the wave to sustain steady-state oscillations despite viscous drag and the absence of periodic driving. We show in theory and experiment that a minimal system composed of two acoustically levitated particles can access four distinct dynamical states, two of which are emergently active steady states. Under some circumstances, these emergently active steady states break spatiotemporal symmetry and therefore constitute a classical time crystal.

💡 Research Summary

In this work the authors demonstrate that a pair of sub‑wavelength particles levitated in an acoustic standing‑wave field can exhibit self‑sustained oscillations without any external periodic driving, and that under certain conditions these oscillations constitute a continuous classical time crystal. The acoustic levitator (based on the TinyLev2 design) operates at 40 kHz, creating a one‑dimensional lattice of pressure nodes spaced by 4.3 mm. Millimetre‑scale expanded polystyrene (EPS) beads are trapped at the nodes by the time‑averaged Gor’kov force, which provides a harmonic restoring potential whose stiffness depends on the bead’s material parameters (density, compressibility).

When two beads have different scattering properties—principally because their radii differ—the waves they scatter back into the standing field are not reciprocal. The authors extend the classic Kӧnig expression for wave‑mediated forces to include higher‑order multipole contributions (quadrupole, octupole). These contributions generate a non‑reciprocal correction term χ_{ij} that changes sign upon exchanging the particles. The resulting inter‑particle force decays as 1/r² along the axis and is given by Eq. (4)–(5) in the manuscript.

Combining the Gor’kov restoring force, viscous Stokes drag, and the non‑reciprocal Kӧnig interaction yields a set of coupled dimensionless equations of motion (Eq. 6). Linearising around the equilibrium (ζ₁=ζ₂=0) leads to a Jacobian whose eigenvalues λ determine the stability of the symmetric (in‑phase) and antisymmetric (out‑of‑phase) normal modes. The authors introduce three stability functions Λ(n) (n = 1, 3, 5) that depend on the dimensionless size parameters x_j = k a_j. The zeros of Λ(1) and Λ(5) mark the boundaries where the symmetric and antisymmetric modes, respectively, become linearly unstable (Re λ ≥ 0). The zero of Λ(3) separates regions where the symmetric mode has higher frequency from those where the antisymmetric mode dominates.

In the parameter space of (a₁, a₂) the authors identify four dynamical regimes: (i) a passive fixed point where both modes are damped; (ii) an “active symmetric” regime where the in‑phase mode grows and settles into a steady oscillation (an active oscillator); (iii) an “active antisymmetric” regime where the out‑of‑phase mode grows and, after nonlinear saturation, forms a stable limit cycle; (iv) a region where both modes are active, leading to more complex dynamics. Crucially, the active antisymmetric regime breaks both parity and time‑parity symmetry without any external drive, satisfying the definition of a continuous time crystal: a spontaneously broken continuous time‑translation symmetry manifested as a linearly stable limit cycle with an emergent frequency.

Experimentally, the authors vary the bead radii to explore different points in the (a₁, a₂) plane. High‑speed video (170 fps) records the three‑dimensional trajectories, which are projected onto the levitator axis and analysed by principal‑component analysis to separate the common and relative motions. Power spectral densities S(f) reveal distinct peaks for the symmetric mode (always near the natural frequency Ω₀ ≈ 66 Hz) and, for larger beads, a second peak corresponding to the antisymmetric mode. In the “small‑bead” case (k a₁≈0.8, k a₂≈1.0) only the symmetric peak is observed, confirming the system resides in the active‑symmetric regime. In the “large‑bead” case (k a₁≈1.07, k a₂≈1.21) a clear antisymmetric peak appears at 66.787 Hz, with a fitted Lorentzian linewidth corresponding to a coherence time τ≈100 s—orders of magnitude longer than the viscous relaxation time (≈0.16 s). The oscillations persist for hours, indicating that the true coherence time of the nonequilibrium steady state is likely much larger.

The authors also discuss how higher‑order damping mechanisms (beyond simple Stokes drag) can stabilize the growing modes, allowing them to settle into the observed limit cycles. They verify that the active region shrinks to zero when the beads are identical (a₁ = a₂), because χ_{ij}=0 and the interaction becomes reciprocal, eliminating the source of energy influx.

Overall, the paper provides (1) a clear theoretical framework that links non‑reciprocal wave‑mediated forces to active matter behaviour, (2) experimental validation that a minimal two‑particle system can exhibit self‑sustained oscillations, and (3) the first demonstration of a continuous classical time crystal realized without any external periodic forcing. The work opens avenues for engineering active metamaterials and time‑crystalline phases in acoustic, optical, or other wave‑based platforms by exploiting controlled heterogeneity and non‑reciprocal interactions.