Nonreciprocal wave-mediated interactions power a classical time crystal

Abstract

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.

Figures (8)

• Figure 1: (a) Experimental realization of a steady-state time crystal composed of two millimeter-scale spheres of expanded polystyrene levitated in air by an acoustic standing wave at 40kHz. Images captured at 170framess reveal sustained oscillations without periodic driving and despite dissipation due to viscous drag. Curly braces denote twice the 16 predicted period of the 61 antisymmetric mode. (b) Model for the forces acting on spheres localized at the nodes of an acoustic standing wave, including restoring forces, $\boldsymbol{F}_j$, and nonreciprocal interparticle forces, $\boldsymbol{F}_{ij}$ and $\boldsymbol{F}_{ji}$. Dashed springs represent the possibility of extending the system to more than two particles.
• Figure 2: (a) Dynamical states for a pair of levitated spheres of radii $a_1$ and $a_2$. Dashed curve: Rayleigh limit: $ka_1, ka_2 \leq 1$. Solid curves: roots of the stability functions: $\Lambda(n) = 0$. Activity surpasses dissipation in the (yellow) region bounded by $\Lambda(1) = 0$ and $\Lambda(5) = 0$. Circles denote experimental conditions from Fig. \ref{['fig:experimental']}(a) and Fig. \ref{['fig:powerspectrum']}. Shading represents the frequency of the antisymmetric mode relative to the natural frequency. (b) Frequency and (c) growth rate for the symmetric (blue) and antisymmetric (red) modes for pairs with $a_1 = \qty{1}{\mm}$ ($ka_1 = 0.732$). Activity powers spontaneous emergence of oscillations in the (yellow) region where $\mathrm{Re}\left\{ \lambda \right\} \ge 0$. The steady-state symmetric mode is an active oscillator. The steady-state antisymmetric mode is a time crystal.
• Figure 3: Power spectral density OSF of the symmetric (blue) and antisymmetric (red) modes of two acoustically-coupled EPS beads in an acoustic levitator. Solid curves are computed from 20000.0-frame video sequences acquired at 200frames. Shaded bands indicate the predicted frequency ranges for the normal modes from Eq. \ref{['eq:eigenvalues']}. Results for a single-particle trajectory (gray) represent the measurement's noise floor. (a) Small particles ($ka_1 = 0.8(2)$ and $ka_2 = 1.02(2)$) oscillate in the symmetric mode at the predicted common-mode frequency, $\Omega_0$. (b) Larger particles ($ka_1 = 1.07(4)$ and $ka_2 = 1.21(3)$) break spatiotemporal symmetry with antisymmetric oscillations at a higher frequency. Inset: Expanded view of the antisymmetric mode's spectral peak (red points) compared with a Lorentzian response (solid red curve) and the measured noise floor (gray points).
• Figure 4: (a) The fixed point interaction force between two particles of radii $a_1$ (blue curve) and $a_2$ (orange curve) as a function of the particle size ratio, $a_2/a_1$. When the particle size ratio is small, the interaction force between the particles is repulsive and approximately reciprocal. A large particle size ratio results in an attractive interparticle force that is nonreciprocal: the force on particle 2 due to particle 1 is much stronger than the force on particle 1 due to particle 2. The yellow region corresponds to the active region in Fig. 2 in the main text, within which nonreciprocal interactions replace the energy lost to dissipation. (b) The Frobenius norm of the difference between the Jacobian for the levitated two-particle system and the parity-transformed Jacobian as a function of the particle size ratio. Parity is approximately conserved when the interaction force is repulsive. When the interaction force is attractive, parity is strongly broken. (c) The Frobenius norm of the difference between the Jacobian for the levitated two-particle system and the $PT$-transformed Jacobian as a function of the particle size ratio. Parity-time symmetry is broken within the (yellow) active range. The horizontal dotted line represents a noise floor below which all norm differences are indistinguishable from zero. (d) The norm difference between all unique pairs of the Jacobian's eigenvectors, $v_i$. An exceptional point occurs when the norm difference is zero, as marked by the vertical dotted line. The exceptional point occurs within the active range.
• Figure 5: Numerical solutions and Poincaré maps of Eq. \ref{['eq:neom']} for two-particle systems with $a_1 = \qty{1}{\mm}$. (a) $a_2 = \qty{1.5}{\mm}$: In the absence of noise ($s = 0$), the system's oscillations ring down to a stable fixed point ($\dot{\zeta}_1 = \dot{\zeta}_2 = 0$). (b) $a_2 = \qty{2.12}{\mm}$, $s = 0$: Analytically predicted conditions for a true time crystal. The trajectory starts from rest and evolves toward a steady-state limit cycle corresponding to the true continuous time crystal. (c) $a_2 = \qty{2.12}{\mm}$, $s = e-2$: A small amount of additive force noise does not prevent the system from evolving toward the steady-state limit cycle, confirming that the time crystal is linearly stable.