Classical Many-Body Time Crystals

Discrete time crystals are a many-body state of matter where the extensive system’s dynamics are slower than the forces acting on it. Nowadays, there is a growing debate regarding the specific properties required to demonstrate such a many-body state, alongside several experimental realizations. In this work, we provide a simple and pedagogical framework by which to obtain many-body time crystals using parametrically coupled resonators. In our analysis, we use classical period-doubling bifurcation theory and present a clear distinction between single-mode time-translation symmetry breaking and a situation where an extensive number of degrees of freedom undergo the transition. We experimentally demonstrate this paradigm using coupled mechanical oscillators, thus providing a clear route for time crystal realizations in real materials.

Figure 1

(a) Stability diagram of a single parametrically driven resonator mode with eigenfrequency ${\omega }_0$ as a function of parametric pumping frequency and depth, ${\mathrm{\Omega }}_p$ and $\lambda$, respectively [1, 48]; cf. Eq. (1). Green tongue shapes indicate regions where the linear resonator becomes unstable. (b) Beyond the instability threshold in the first lobe, ${\mathrm{\Omega }}_p\sim 2{\omega }_0$, the nonlinear resonator undergoes a period-doubling bifurcation and oscillates with frequency ${\mathrm{\Omega }}_p/2$. Because of the period doubling, there exist two possible phase states with equal amplitude but opposite phase. In some frequency ranges, there is an additional zero-displacement stable solution. (c) Enlargement of the stability diagram around the first instability lobe for $N$ coupled resonators. Here, $N$ nondegenerate normal modes (marked by different color and symbols) generically arise; cf. Eq. (1). The red area indicates the region of many-body TTSB. (d) Inside the many-body TTSB, each of the normal modes reside in one of the phase states. The resulting multistate configuration depends on the coupling coefficients ${\beta }_{ij}$, the nonlinearities, and noise fluctuations.

Figure 2

Strongly coupled oscillators: (a) Schematic setup representing two parametrically driven strings coupled via an additional mechanical connection. (b) Calculated normal-mode stability diagram of the symmetric (I) and antisymmetric (II) eigenmodes of the coupled system. (c) Measured amplitude and (d) phase of strings 1 and 2 for the up sweep (orange and brown) and down sweep (light and dark blue) where both oscillators are parametrically driven at frequency $2\omega$. The fixed phase relations in regions I and II are signatures of the corresponding normal mode symmetries. (e)–(f) Simulated steady-state solutions of oscillators 1 and 2 in the rotating frame phase space ($u$, $v$) calculated from the slow-flow equations, cf. Eq. (2) as a function of $\omega$. The thick (thin) tubes are stable (unstable) solutions and white spheres indicate the positions of bifurcations. The stable branches corresponding to the experiment are highlighted in matching colors for up- and down sweeps. Arrows indicate the sweep directions. In agreement with the experiment, fixed symmetric and antisymmetric phase relations appear in the normal modes I and II, respectively.

Figure 3

Weakly coupled oscillators: (a) Schematic setup representing two parametrically driven strings weakly coupled via the driving plate. (b) Normal mode stability diagram. (c) Amplitude and (d) phase of strings 1 and 2 for the up sweep (orange and brown) and down sweep (light and dark blue) where both oscillators are parametrically driven at frequency $2\omega$. Both symmetric and antisymmetric stable phase relations are observed depending on the system path in parameter space. This is the signature of a many-body time crystal. (e) and (f) The simulated steady-state solutions of oscillators 1 and 2 in the rotating-frame phase space ($u$, $v$) calculated from the slow-flow equations, cf. Eq. (2) as a function of $\omega$. The thin tubes are unstable solutions and all other colored tubes represent stable solutions. The white spheres in these plots denote bifurcations. In agreement with the experiment, either symmetric or antisymmetric phase relations appear depending on the sweep direction (indicated by arrows).