Coherence Lab · Field Explainer · TC-01

Kicked once a beat, it keeps a beat of two

Flip a chain of spins every period, a little short each time. Does it settle into the rhythm you gave it, or hold one of its own that your sloppy kicks can't break?

drum · every kick crystal · every other kick period 0
—crystal's beat
× drive frequency
—impostor's beat
× drive frequency
—crystal order Φ
last 100 periods
—pendulum memory
coupled / loose
DATA · every number on this page is measured live by the simulations below, in your browser, from seeded starts. Nothing is typed in. The engine is checked against an independent numpy model before it ships.
00

A crystal in time

An ordinary crystal is matter that picks its own spacing. The laws of physics are the same at every point in space, yet the atoms settle into a lattice that repeats every few ångströms. In 2012 Frank Wilczek asked whether matter could do the same in time [Wilczek 2012]. For a system left alone in equilibrium, the answer turned out to be no [Watanabe & Oshikawa 2015].

For a system that is driven, the answer is yes. Kick a chain of spins once every period T. A discrete time crystal answers with a rhythm of 2T, a beat the drive never played, and it keeps that beat when the kicks are imperfect [Else, Bauer & Nayak 2016; Khemani et al. 2016; Yao et al. 2017]. Labs have seen it in trapped ions, in nitrogen-vacancy centres in diamond, and on Google's superconducting processor [Zhang et al. 2017; Choi et al. 2017; Mi et al. 2022].

The sloppiness is the whole test. A perfect half-turn flips any spin, so any spin comes back after two kicks. That is an echo, not a crystal. Make every kick fall short by a fraction ε and a lone spin drifts off the beat. In a crystal the spins hold each other in place. Each spin's energy depends on its neighbours, so an almost-flip keeps landing as a flip, and the response stays locked at exactly half the drive frequency.

01

The chain and its impostor

Ten quantum spins, simulated exactly: all 1024 complex amplitudes, every period. Both chains get identical kicks and identical local fields. The only difference is that the impostor's spin–spin bonds are cut.

Crystal · bonds on

measuredL=10 · exact
spins · arrow = Bloch vector of each sitefree
⟨Z⟩ per site · last periods →
beat m(n) = Σ z₀·⟨Z⟩ / L, one stem per period

Impostor · bonds cut

measuredL=10 · exact
same fields, same kicks, J = 0free
⟨Z⟩ per site · last periods →
beat m(n), one stem per period
spectrum of m(n) over the last 128 periods · frequency in units of the drivemeasured
crystalimpostor ½ · the beat a crystal must hold (guide) each lone spin's exact frequency (guide, not data)
0.10
1.5
HOW TO READ IT · at ε = 0 both chains flip perfectly, and the impostor looks just as good. Slide ε to 0.1. Within this 128-period window the crystal's spectral peak sits on the dashed ½ line. A window that short cannot tell ½ from 0.499 on its own; the stronger test is that the crystal's order stays flat over thousands of periods (the tests check 4,000), which a near miss could not do. The impostor's peaks move off ½ to near the slate ticks, each tick the exact frequency of one lone spin. In the space-time strips the crystal's stripes keep their pattern, while each impostor spin drifts in and out of step at its own rate, so its rows go mottled.
02

The pendulum shelf

The classical version needs no quantum mechanics at all, just noise, friction and springs [Yao, Nayak, Balents & Zaletel 2020].

Shake a pendulum's pivot up and down at twice its natural frequency and it starts to swing at half the shaking frequency. That is parametric resonance, the way a child pumps a swing. The pendulum can lock to the shaking in two timings, half a swing apart, and random jostling (heat) occasionally knocks a lone pendulum from one timing into the other. Join the pendulums with springs and a single one can't switch alone: it would fight its neighbours. Twenty-four noisy pendula then keep one shared beat. That is the mechanism behind a classical time crystal. Calling it a phase of matter takes more: you have to know how the beat's lifetime scales with chain length, coupling and noise. That is what Yao and colleagues studied, and what this shelf of 24 does not.

Coupled by springs

measuredN=24 · noisy
beat timing per pendulum · drive periods →memory —

Loose · no springs

measuredN=24 · same noise
beat timing per pendulum · drive periods →memory —
on the founding beatslipped to the other timing faded = small swing
0.050
1.50
1.6
MEMORY · how well each pendulum's timing now agrees with its timing 101 drive periods earlier, averaged over the last 50 such pairs (+1 = every pendulum kept its timing, 0 = no relation). A pendulum that is barely swinging counts as 0, and the lag is odd so that a pendulum that merely sat still cannot score. Turn the noise up and the coupled row loses its memory too. The crystal melts.
03

Sloppier kicks, until the beat breaks

The phase rig. Each point is a fresh run: 8 spins, 200 periods, 6 disorder draws. Φ is the period-two order over the second half of the run: +1 means the beat held perfectly and 0 means it was lost.

order Φ vs pulse error εrunning
crystal · bonds + disorder no disorder · every bond and field at its average impostor · bonds cut impostor, exact single-spin formula (guide) your runs · L=10 chain above, after 200 periods
CHECK · the measured impostor dots must sit on the dashed curve, which comes from a closed-form single-spin rotation and not from the simulation. If they ever drift off it, the engine is wrong. Near ε = 0 that curve swings well below zero: a lone spin with a tiny error drifts slowly in and out of step, and the late window catches it out of step. That is a slow beat, not a lock. The crystal's dots stay well above zero long after the impostor's have collapsed. Disorder matters most for small errors. The clean chain, with every bond and field set to the disorder's average, loses about half its order at the first nudge while the disordered crystal barely moves. Around ε ≈ 0.2 the two curves meet and cross. That crossing is an observed result for 8 spins. Its cause is not established here, so don't read it as a general law. In long chains a driven chain without disorder generally heats up and loses the beat. Disorder is one way to stop that, and it is the route this model takes [Khemani et al. 2016]. A fast drive that delays the heating for a very long time is another, and disorder-free prethermal crystals have been observed too [Kyprianidis et al. 2021].
04

Where this stops

BOUNDARY
  • Ten spins is not a material. At this size the crystal melts gradually as ε grows. The sharp phase transition belongs to long chains, and this page does not show it. The rig uses 8 spins and the live chain uses 10.
  • The quantum model is the idealised one. The energy is spin–spin bonds plus random fields along one axis, the same family as Khemani et al. 2016 and the Google experiment. It has no bath and no decoherence. Real devices fight both, and this page does not model them.
  • Pendulum units are dimensionless: natural frequency 1, damping 0.1, drive depth 0.3, noise as a temperature. In one dimension the classical crystal's lifetime is finite but grows quickly with spring stiffness [Yao et al. 2020]. Here it simply outlasts the window you watch.
  • Twenty-four pendula is not a phase either. A fair worry is that springs just average the noise away: a perfectly rigid chain feels 1/24 of it. Measured, that is not what happens. A single pendulum given 1/24 of the noise keeps perfect memory where the chain of 24 loses some, so the chain slips locally, a few pendula at a time, and the springs suppress those local slips. Whether that adds up to a true phase needs chains of growing length [Yao et al. 2020].
  • Scar tissue. The first pendulum measurement counted every sign flip of a pendulum's phase as a slip. Noise jitter near the switching line counted too, so coupled chains looked worse than loose ones. Comparing against 100 periods earlier fixed that, and an outside review then found two more holes: a pendulum that had stopped swinging scored perfect memory, and the +100 button recorded at the wrong point of the drive cycle. Now small swings count as zero, the lag is odd, and every record is taken at the same drive phase. Memory still compares two moments, so two slips that cancel between them go unseen.
sources

F. Wilczek, Quantum time crystals, PRL 109, 160401 (2012) · H. Watanabe & M. Oshikawa, PRL 114, 251603 (2015) · D. Else, B. Bauer & C. Nayak, PRL 117, 090402 (2016) · V. Khemani, A. Lazarides, R. Moessner & S. Sondhi, PRL 116, 250401 (2016) · N. Yao, A. Potter, I.-D. Potirniche & A. Vishwanath, PRL 118, 030401 (2017) · J. Zhang et al., Nature 543, 217 (2017) · S. Choi et al., Nature 543, 221 (2017) · X. Mi et al., Nature 601, 531 (2022) · A. Kyprianidis et al., Observation of a prethermal discrete time crystal, Science 372, 1192 (2021) · N. Yao, C. Nayak, L. Balents & M. Zaletel, Classical discrete time crystals, Nature Physics 16, 438 (2020).