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To test whether a driven many-body system forms a discrete time crystal, measure an order parameter at every drive period, look for a persistent response at a subharmonic frequency (usually half the drive frequency), and then test whether that frequency stays locked when the drive and interactions are perturbed. A half-frequency peak by itself is not enough: a credible result also needs rigidity, persistence, controls against ordinary period doubling, and finite-size and uncertainty analysis.
What a laboratory measurement must establish
A discrete time crystal (DTC) is a nonequilibrium phase of a periodically driven many-body system. The drive repeats after a Floquet period T, but the system’s collective order parameter repeats only after an integer multiple of that period. The simplest case is period doubling: the measured signal changes sign or alternates between two values on successive cycles, so its period is 2T and its frequency is f_drive/2.
The defining feature is not merely that an oscillator can be made to respond at half frequency. Nonlinear classical systems, heating transients, control errors, and finite observation windows can all produce a subharmonic-looking line. A DTC claim requires a combination of observations:
- Subharmonic response: an order parameter sampled stroboscopically contains a clear component at
f_drive/2(often written as normalized frequencyν = 1/2). - Frequency rigidity: that component remains locked to the same subharmonic when pulse angles, drive amplitude, interaction strength, disorder, or drive frequency are changed within the operating regime.
- Persistence: the oscillation survives for many cycles, with a lifetime extracted from a stated decay model or a sliding-window analysis.
- Many-body and control evidence: interactions, disorder, or a prethermal regime play the role expected for the platform, and controls rule out an externally forced period-doubling response.
- Scaling and error analysis: finite-size effects, shot noise, readout uncertainty, heating, and decoherence are quantified rather than hidden in an averaged trace.
No universal lifetime, peak height, or signal-to-noise threshold defines a time crystal. The relevant values depend on the drive frequency, interaction strength, disorder, system size, preparation, and detection noise, so every number must be reported with those conditions.
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Choose an order parameter and sample it synchronously
Spin magnetization
For trapped ions or other spin systems, use the magnetization after each complete Floquet cycle. If there are N spins, a common normalized observable is M(nT) = (1/N) Σᵢ ⟨σᶻᵢ(nT)⟩, where n is the cycle number. Site-resolved measurements are valuable because they reveal whether the response is collective or confined to a few sites. The ten-ion experiment reported in Nature in 2017 measured individual spin magnetizations and then analyzed their Fourier spectrum.
Polarization
In diamond spin ensembles, use the population difference between the two measured spin states as the order parameter. Choi and colleagues’ 2017 Nature experiment studied approximately one million dipolar spin impurities, recorded polarization over many drive periods, and followed the normalized ν = 1/2 component with sliding-window Fourier transforms.
Autocorrelation and energy observables
A prethermal experiment can supplement the stroboscopic order parameter with a magnetization autocorrelation, such as C(nT) = ⟨M(nT)M(0)⟩, or with energy-density dynamics. Kyprianidis and colleagues’ 2021 Science study compared these quantities as functions of drive frequency and initial energy density to identify the prethermal regime and its lifetime. Autocorrelation is especially useful when a single-site average obscures phase slips or when the mean magnetization is close to zero.
Sampling requirements
Measure at the same phase of the drive on every cycle. A stroboscopic record should contain one value per Floquet period, not a mixture of measurements taken at different points within a pulse sequence. Record the number of experimental shots, site-resolved values when possible, and the uncertainty of every cycle. Keep the unsmoothed trace so that transients, beats, phase slips, and abrupt loss of coherence remain visible.
Step-by-step protocol
1. Prepare the many-body state
Initialize the spins or other degrees of freedom in a state with a measurable order parameter. Document the preparation fidelity, initial energy or polarization, and any state-selection or cooling procedure. A DTC test is sensitive to heating and to the initial state, so preparation conditions are part of the measurement, not background information.
2. Apply a periodic Floquet sequence
Define one complete drive cycle and its period T. The sequence may include global rotations, interaction evolution, disorder, and refocusing or decoupling pulses. Calibrate the nominal pulse angle and drive amplitude before collecting the time trace. Repeat the same sequence for enough cycles to pass the initial transient and to observe the eventual decay. The trapped-ion Nature experiment repeated its Floquet evolution for 100 periods, then applied an analysis rotation and performed state detection.
3. Read out after every period
At t = nT, measure the chosen order parameter. For a spin experiment, record populations in the measurement basis and convert them to magnetization or polarization. Repeat the sequence for independent shots to estimate the mean and standard error at each cycle. If the instrument permits, retain individual sites rather than averaging immediately; spatial information can distinguish a synchronized collective mode from a local defect.
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4. Build and inspect the raw time trace
Plot the order parameter against cycle number before applying a filter or fit. A period-doubled signal should alternate in sign or amplitude from one cycle to the next. Mark the preparation transient and any intervals affected by missing shots, pulse faults, or detector saturation. An apparent clean oscillation that begins only after aggressive smoothing is not adequate evidence.
5. Transform the record into frequency space
For a record of L stroboscopic samples Mₙ = M(nT), calculate a discrete Fourier transform, for example F(k) = Σₙ₌₀ᴸ⁻¹ Mₙ exp(-2πikn/L). Convert the index to physical frequency using the sampling rate 1/T, or report normalized frequency ν = f/f_drive. A period-doubled response appears near ν = 1/2.
Report the peak height using a stated normalization, the peak width, the surrounding noise floor, and the signal-to-noise ratio. The finite record length sets the frequency resolution; a short record can make a broad transient look like a sharp line. Use the same windowing and normalization for every parameter scan. A rectangular window preserves the raw amplitude but produces strong sidelobes, while tapered windows reduce leakage at the cost of changing the amplitude calibration. Whichever choice you make, state it and apply it consistently.
6. Test rigidity under controlled perturbations
Repeat the measurement while varying one control at a time. Useful scans include:
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- drive amplitude and drive frequency;
- interaction strength or interaction range;
- disorder strength and disorder realization;
- initial energy density or preparation state; and
- system size.
In a rigid DTC regime, the subharmonic frequency remains at the same locked value while its amplitude, width, or lifetime changes. If the frequency drifts continuously with a control parameter, the response is more consistent with an ordinary driven oscillator or an accidental resonance. The trapped-ion work reported synchronized oscillations and enhanced Fourier subharmonic peaks as interactions increased, and its extended data included perturbation, disorder, and finite-size scans.
7. Quantify persistence and repeat the controls
Extract a lifetime from the decay of the subharmonic component, then repeat the analysis with alternative fit windows and uncertainty estimates. Include a no-interaction or no-disorder control when the apparatus allows it. Compare the driven sequence with a sequence in which the interaction term, synchronization mechanism, or relevant pulse is removed. A control that still produces the same half-frequency response indicates that the source may be the drive protocol rather than collective time-crystal order.
How to measure the subharmonic signal reliably
Peak height and variance
Compute the power or amplitude spectrum for each experimental realization or bootstrap resample, not only for one grand-average trace. The distribution of the ν = 1/2 peak height provides an uncertainty estimate and can reveal rare phase slips that disappear after averaging. The ten-ion Nature experiment used the subharmonic peak height and its variance as diagnostics.
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Phase and sign alternation
Fourier power discards phase information. Also inspect the complex Fourier component or fit the stroboscopic record to an alternating envelope, such as M(nT) ≈ A(n)(-1)ⁿ + B(n). A genuine period-doubled response should retain a coherent phase over the interval used for the lifetime estimate. Random sign flips can create a broad half-frequency feature without long-lived order.
Sliding-window spectra
For a signal whose coherence changes with time, calculate the Fourier transform in overlapping windows. Plot the normalized half-frequency peak versus the center of each window. Choose a window long enough to resolve the subharmonic but short enough to track decay, and report the window length, overlap, and normalization. The diamond experiment used moving windows to track a crystalline fraction and extract a lifetime.
Noise and uncertainty
Separate shot noise, detector noise, and run-to-run drift. Estimate the noise floor from neighboring frequencies or from a control sequence, but do not subtract a control spectrum without explaining the procedure. Propagate uncertainty from finite shots and from the fit. If the peak is comparable to the noise floor, report an upper limit or an inconclusive result instead of assigning a time-crystal lifetime.
Extracting the time-crystalline lifetime
Envelope fitting
One approach is to fit the magnitude of the subharmonic component or an alternating envelope to a decay model. A single exponential, A(t) = A₀ exp(-t/τ), is convenient but not automatically justified. Prethermal systems can show a long plateau followed by a rapid decay, and finite-size recurrences can produce nonmonotonic behavior. Compare the fit with a stretched exponential, a piecewise plateau-plus-decay model, or a model motivated by the platform, and state the fitting interval.
Sliding-window crystalline fraction
For long records, define a crystalline fraction from the normalized half-frequency peak in each moving window. The lifetime can then be the time at which that fraction crosses a stated threshold, or the characteristic decay time from a fit. Because the threshold is a convention, publish the threshold, window length, frequency band, and uncertainty rather than presenting one number as universal.
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Calculate the magnetization autocorrelation and compare decay in a symmetry-breaking regime with decay in a symmetry-preserving regime. A slowly decaying autocorrelation supports persistent order, but it should be interpreted with the Fourier and rigidity tests. Autocorrelation can remain nonzero because of a static offset, so subtract the appropriate mean or compare to a control before fitting.
Pulse-sequence and heating corrections
Interactions during pulses can produce apparent decay that is not the intrinsic lifetime of the ordered phase. The 2018 Physical Review Letters NMR DTC echo experiment specifically cautioned that pulse-sequence interactions complicate lifetime extraction. Record pulse durations, pulse amplitudes, refocusing details, heating indicators, and decoherence rates. If the signal decays when the pulse is lengthened but not when free-evolution time is changed, pulse-induced effects may dominate.
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What to report with every lifetime
- the definition of the order parameter and the sampling interval;
- the number of cycles and shots;
- the fit function, fit window, and weighting;
- the frequency band or threshold used for a sliding-window peak;
- the noise floor and confidence interval;
- drive frequency, interaction strength, disorder, system size, and initial state; and
- whether heating, pulse interactions, readout errors, or finite-size recurrences limit the estimate.
Controls that distinguish a DTC from ordinary period doubling
Detune the pulse angle
Change the nominal rotation away from the value that would produce an exact two-cycle response in a noninteracting model. A DTC’s subharmonic frequency should remain locked over a finite range, while its amplitude and lifetime vary. A response that follows the detuning continuously is not rigid.
Remove or weaken interactions
Turn off interactions or reduce them as far as the platform permits. Without the collective mechanism, a driven system may still show a short transient at half frequency, but it should lose the robust locking and long persistence associated with the interacting regime. Compare peak height, width, phase coherence, and lifetime rather than only the presence or absence of a line.
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Change disorder and symmetry-breaking conditions
Scan disorder strength and use several disorder realizations when disorder is part of the proposed mechanism. Also test a symmetry-preserving preparation or drive. For a prethermal DTC, compare regimes in which the relevant symmetry is broken and preserved; the magnetization autocorrelation should behave differently if the order is genuine.
Vary system size
A finite system cannot display an infinitely sharp thermodynamic transition. Expect a rounded crossover and possible recurrences. Repeat the analysis for several sizes and show how the peak, lifetime, and rigidity window change. The trapped-ion study reported finite-size scaling for chains of N = 10, 12, and 14 ions.
Check phase coherence, not only power
Average power spectra can retain a half-frequency peak even when different shots have random phases. Examine the phase relative to the drive and the shot-to-shot distribution. A synchronized response should show phase coherence over the stated analysis interval.
Examples from established laboratory platforms
| Platform and study | Measured quantity | Distinctive analysis | Important qualification |
|---|---|---|---|
| Ten trapped ions, Nature (2017) | Individual spin magnetizations in an interacting chain | Floquet evolution for 100 periods, analysis rotation, state detection, Fourier subharmonic peak and variance; perturbation, disorder, and finite-size scans | Finite systems produce a rounded crossover; reported sizes included 10, 12, and 14 ions |
| Approximately one million dipolar spin impurities in diamond, Nature (2017) | Polarization, defined by the population difference between spin states | Sliding-window Fourier transforms, normalized ν = 1/2 peak, and crystalline-fraction lifetime |
Lifetime depends on drive, interactions, disorder, finite size, and readout noise |
| Prethermal trapped-ion crystal, Science (2021) | Magnetization autocorrelation and energy-density dynamics | Comparison across drive frequency and initial energy density to identify a prethermal regime and its lifetime | Autocorrelation must be interpreted with controls for offsets, heating, and symmetry |
| Ordered dipolar crystal in NMR, Physical Review Letters (2018) | DTC echo and coherence during an NMR pulse sequence | Echo-based coherence measurements | Interactions during pulses can add apparent decay and complicate the intrinsic-lifetime estimate |
| Prethermal DTC sensing with dipolar-coupled 13C spins in diamond, Nature Physics (2026) | Time-crystal response used as a magnetic-field sensing signal | Linewidth set by the time-crystal lifetime | The demonstrated sensing range was 0.5–50 kHz; that range is platform-specific, not a universal DTC bandwidth |
Finite-size, drift, and statistical analysis
Finite-size effects
Small systems can look ordered for a limited time and can later recur because their spectrum is discrete. Treat the observation as a finite-size dynamical regime unless scaling supports a phase interpretation. Plot results for each system size separately before combining them, and state whether the reported lifetime ends because of decoherence, heating, or a recurrence.
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Interleave reference and perturbed runs rather than collecting all control data at the end. Monitor pulse amplitude, resonance frequency, temperature, laser or microwave power, and detector gain. Slow drift can mimic a changing lifetime or broaden the half-frequency peak.
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Bootstrap and hierarchical uncertainty
Use independent experimental runs as the resampling unit when technical noise is correlated across shots. If sites are not independent, do not treat every site as a separate experiment. Report confidence intervals for peak height, frequency, and lifetime, and show the raw number of runs used.
Frequency resolution and observation time
With L stroboscopic samples, the nominal Fourier-bin spacing is f_drive/L. A claim of frequency locking narrower than that resolution requires a model-based estimate and a stated uncertainty, not simply a visually narrow bin. Longer records improve resolution but can also include heating and drift, so analyze both early and late windows.
Troubleshooting common misleading results
A half-frequency peak appears for only a few cycles
Check the preparation transient, pulse calibration, and the control sequence. A short transient can be ordinary nonlinear period doubling. Extend the record, quantify the decay, and test rigidity before using the term time crystal.
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The peak shifts when the pulse angle changes slightly
This indicates a likely forced or resonant oscillator rather than a rigid DTC. Map the frequency continuously over the perturbation range and compare it with an interacting, disorder, or prethermal control.
The averaged trace is flat but the spectrum has a strong line
Inspect shot-to-shot phase. Random phases can cancel in the average while preserving power. Align phases only if the alignment procedure is physically justified and disclose it; otherwise report the phase incoherence.
The lifetime changes when the Fourier window changes
Vary the window length and overlap systematically. A strong dependence means the record may not support a single lifetime, or the system may have a plateau followed by a nonexponential decay. Report a range or a time-dependent crystalline fraction instead of one over-precise value.
The subharmonic peak survives without interactions
Investigate pulse imperfections, resonant driving, detector aliasing, and classical nonlinear response. A no-interaction signal is an essential warning that the observed period doubling is not necessarily collective.
Heating overwhelms the late-time signal
Measure energy density or another heating proxy alongside the order parameter. Restrict the lifetime claim to the interval in which the intended driven phase exists, and state that heating, rather than intrinsic loss of order, limits the observation.
A reporting checklist for a credible claim
- Define the Floquet period, drive frequency, pulse sequence, and sampling phase.
- Name the order parameter and give its normalization and readout basis.
- Show the raw stroboscopic trace with shot or run uncertainties.
- Show the Fourier spectrum, normalization, window function, frequency resolution, and noise floor.
- Identify the subharmonic peak and report its height, width, and signal-to-noise ratio.
- Demonstrate frequency locking while changing at least one relevant drive parameter.
- Show how amplitude and lifetime, rather than frequency, respond to those perturbations.
- Include interaction, disorder, symmetry, or no-drive controls appropriate to the platform.
- Measure persistence with a stated fit or sliding-window definition and include confidence intervals.
- Repeat across system sizes and discuss finite-size rounding and recurrences.
- Quantify heating, decoherence, pulse-induced interactions, detector noise, and missing data.
- Use cautious language if any of these tests is absent: call the result a subharmonic response or period-doubled transient, not a confirmed time crystal.
What the measurement can and cannot prove
A synchronized half-frequency oscillation that survives perturbations, persists for a quantified interval, and disappears or weakens under appropriate controls is strong evidence for discrete time-crystalline behavior in the measured regime. It does not establish an infinite-system phase by itself. The conclusion is bounded by the platform, system size, drive interval, disorder realization, and lifetime over which the tests were performed.
The most defensible laboratory result therefore presents the time trace, spectrum, rigidity map, lifetime analysis, controls, and scaling together. The 2017 trapped-ion report described its result as an experimental observation of a discrete time crystal in an interacting spin chain; subsequent prethermal, NMR, and sensing experiments illustrate why the same observables must always be accompanied by platform-specific caveats.
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