Measure T1 with a population-recovery experiment and T2 with a calibrated Hahn-echo experiment. For an optically readable defect such as the negatively charged nitrogen-vacancy (NV−) center in diamond, identify the microwave transition with optically detected magnetic resonance (ODMR), calibrate pulse durations with Rabi oscillations, then record fluorescence while varying the appropriate delay. T1 comes from recovery after a resonant π pulse and dark wait; Hahn-echo T2 comes from decay of spin-echo contrast in a π/2–τ/2–π–τ/2–π/2 sequence.
The numbers are properties of the defect and its environment, not fixed constants. Temperature, magnetic-field magnitude and direction, defect and nitrogen concentration, nearby spins, nanomagnetization and the measurement sequence can all change the result.
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T1 and T2 measure different physical processes
T1 is the longitudinal or spin-lattice relaxation time. It describes how quickly a prepared spin population returns toward thermal equilibrium. In an NV− experiment, a microwave π pulse moves population from the optically prepared ms = 0 state to ms = +1 or ms = −1; T1 is obtained by watching that population difference disappear during a dark delay.
T2 is the transverse spin-spin relaxation time, or phase-coherence lifetime. It describes the loss of a phase relationship between spin states. A Hahn echo refocuses static or slowly varying frequency offsets, so the resulting value is specifically Hahn-echo T2. It is not interchangeable with free-induction T2*, which is measured without the refocusing π pulse and is usually shorter when inhomogeneous broadening is present.
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| Property | T1 experiment | Hahn-echo T2 experiment |
|---|---|---|
| Physical quantity | Population recovery along the quantization axis | Transverse phase-coherence decay |
| Core sequence | Optical initialization → resonant π pulse → dark delay τ → optical readout | π/2 → τ/2 → π → τ/2 → π/2 → optical readout |
| Measured curve | Population or fluorescence difference versus τ | Normalized echo contrast versus total evolution time τ |
| Main sensitivities | Spin-lattice channels, temperature, magnetic field, cross-relaxation and nearby spins | Magnetic and electric-frequency noise, fluctuating magnetization and pulse errors |
| Typical use | Relaxometry and noise or cross-relaxation spectroscopy | Coherent control and sensing under refocusing |
Prepare and calibrate the experiment
Identify the transition with ODMR
First record an ODMR spectrum while monitoring spin-dependent fluorescence. Select the resonance associated with the defect orientation you intend to address. For an NV center, the bias field should be controlled in both magnitude and direction so that the selected transition is separated from other orientations and nearby resonances. Record the resonance frequency used in the pulse sequence; it can shift with field, temperature and local strain.
Calibrate microwave pulses with Rabi oscillations
Drive the selected transition while varying microwave pulse duration and record the resulting fluorescence oscillations. A π pulse is the duration that produces population inversion, and a π/2 pulse produces an equal superposition. Use the calibrated durations at the actual microwave amplitude, carrier frequency and delivery geometry used for the relaxation measurement. Recalibrate after changing attenuation, frequency, bias field, sample position or the microwave path.
Assemble the optical and microwave hardware
A documented NV measurement combines confocal fluorescence collection, 532 nm excitation, a microwave source, a programmable pulse generator and a microwave delivery structure such as a 70 μm copper wire or an integrated stripline. Use a controlled bias magnetic field to select and characterize the ODMR transition. Synchronize optical and microwave timing through the pulse generator, and use a detection gate that captures the spin-dependent part of the fluorescence rather than the entire decay curve.
- Optical excitation and collection path with stable focus and photon-counting or equivalent fluorescence detection.
- Microwave source tuned to the selected ODMR transition.
- Programmable timing hardware capable of nanosecond-scale pulse and delay control appropriate to the defect.
- Bias magnet and field-control method that preserve the chosen orientation and resonance.
- Ability to average repeated shots and to interleave reference sequences.
No single laser power, microwave power, pulse width or fitting bound applies to every spin-defect platform. Calibrate those parameters for the defect, sample geometry and hardware instead of transferring settings from another experiment.
How to measure T1 step by step
1. Initialize the spin optically
Start each shot with a green initialization pulse. One published NV protocol uses a 1 ms green-light pulse to optically pump NV− centers into ms = 0. Treat 1 ms as that protocol’s condition, not a universal requirement; adjust the duration until the initial fluorescence and ODMR contrast are stable for your sample.
2. Invert the selected transition
Apply the resonant microwave π pulse calibrated from Rabi oscillations. The pulse transfers population from ms = 0 to the selected ms = +1 or ms = −1 state. Keep the laser off during the subsequent delay so that the optical field does not repump or dephase the spin.
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3. Sweep the dark delay
Choose a series of dark delays τ that cover the expected decay from a short fraction of T1 to several times T1. Use approximately logarithmic spacing when the lifetime is not known, then add densely spaced points around the visible roll-off. Keep the optical and microwave timing identical for every delay except the value of τ.
4. Read out the population
Turn on the optical readout pulse and record the early fluorescence window, where the spin-dependent contrast is largest. Store the photon count or calibrated fluorescence value for every shot. Repeat enough times that shot noise is smaller than the change expected across the relaxation curve.
5. Interleave a no-π reference
Run a matched sequence in which the resonant π pulse is omitted. Interleave reference and measurement shots rather than collecting them in separate long blocks. Subtract the paired traces or otherwise normalize them to remove common changes in laser power, collection efficiency, focus and detector response. This common-mode subtraction is part of the documented T1 protocol and is especially important for long scans.
6. Fit the population-recovery curve
After reference subtraction, plot the population signal against τ. For one dominant relaxation channel and one resolved population, use a single-exponential model such as S(τ) = S∞ + A exp(−τ/T1), where S∞ is the long-delay baseline and A is the initial contrast. Fit the raw counts or a statistically appropriate transformed signal with uncertainties, and report the confidence interval for T1.
Inspect residuals rather than accepting a single exponential automatically. Multiple relaxation channels, unresolved ensemble subgroups, cross-relaxation or pulse imperfections can produce systematic residuals. If the residuals show structure, compare a physically motivated multi-component model and state that the reported lifetime is an effective or component-specific value. Do not conceal multiple channels by forcing one number.
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How to measure Hahn-echo T2 step by step
1. Set the calibrated π/2 and π pulses
Use the ODMR resonance and Rabi data to set both pulse durations. The first and last π/2 pulses create and read a transverse superposition; the central π pulse reverses the phase evolution accumulated during the first half of the sequence. Pulse-area errors reduce contrast and can mimic a shorter T2, so verify the Rabi calibration at the beginning and end of a long measurement.
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2. Run the Hahn-echo sequence
- Apply a calibrated π/2 pulse.
- Allow free evolution for τ/2 with the microwave and optical fields off.
- Apply the calibrated resonant π pulse.
- Allow a second free-evolution interval of τ/2.
- Apply the final π/2 pulse with the phase selected for the desired bright or dark output state.
- Apply the optical readout pulse and record spin-dependent fluorescence.
Here τ is the total free-evolution time. Some laboratories instead label each free interval as τ; state your convention explicitly so that another reader does not interpret the fitted time by a factor of two.
3. Convert fluorescence to normalized echo contrast
Use bright and dark reference measurements, or an equivalent normalization, to convert photon counts into a dimensionless spin contrast. Keep the optical readout timing fixed while sweeping τ. Average repeated shots at each point and randomize or interleave delay values when drift could correlate with the sweep order.
4. Fit the echo envelope
Plot normalized contrast against total evolution time. A stretched exponential, for example C(τ) = C0 exp[−(τ/T2)n] + Cbaseline, is useful when the noise spectrum or an ensemble of defects produces non-single-exponential decay. Report the fitted stretch exponent n, baseline treatment and confidence interval as well as T2. A single exponential is appropriate only when it describes the residuals and the physical system supports that simplification.
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Choosing delay ranges, averaging and fit models
When T1 is unknown
Begin with a reconnaissance sweep spanning several decades of τ, constrained by the instrument’s minimum timing and the sample’s expected stability. Once the roll-off is located, concentrate points around the interval where the signal changes most rapidly and retain long-delay points to establish S∞. If the curve has not approached its baseline, the fitted T1 will be correlated with the baseline and should be reported as poorly constrained.
When T2 is unknown
Use short delays to establish the zero-time contrast and progressively longer delays until the echo approaches the baseline. Include enough points before the first visible decay to identify pulse errors and enough points after the decay to constrain the offset. Excessively long delays waste averaging time and can introduce drift that the fit interprets as a second decay component.
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Assess fit quality
- Plot residuals versus delay, not only the fitted curve.
- Use count-based uncertainties or independently estimated shot-to-shot variance.
- Check whether fitted T1 or T2 changes substantially when the first or last delay points are removed.
- Compare single- and multi-component models only when the additional component has a physical interpretation and is supported by the data.
- Report parameter confidence intervals and the fit window.
Why measured T1 and T2 vary between samples
Temperature and magnetic field
T1 is strongly environment-dependent. A study spanning 5 to 475 K and magnetic fields from 0 to 630 G reported an Orbach activation energy of 73(4) meV below room temperature. Under low-temperature conditions, T1 can range from milliseconds to minutes depending on field and sample conditions. Those values describe the measured samples and field-temperature regimes, not a universal NV specification.
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NV concentration, substitutional nitrogen concentration and other nearby paramagnetic defects open additional relaxation and dephasing channels. In an ensemble, unresolved subgroups can have different lifetimes, producing non-single-exponential curves. In a single-defect experiment, local spins and surface noise can dominate even when the bulk crystal is clean.
Field orientation and cross-relaxation
The magnitude and direction of the bias field determine which transitions are resonant and whether the defect approaches a level crossing with another spin species. Cross-relaxation can sharply change T1 near such resonances. A field sweep can therefore reveal more than a single lifetime measurement.
Magnetic noise and nanomagnets
T2 reflects fluctuations that change the transition frequency during the experiment. One room-temperature NV study reported T2 values of 1.23 ± 0.29 μs, 1.13 ± 0.17 μs and 3.48 ± 0.01 μs for different nanomagnet configurations, attributing the differences to dephasing from fluctuating magnetization. These are example configurations, not benchmarks that every defect should reach.
Use T1 as a form of spin spectroscopy
Instead of measuring T1 at one field, sweep the magnetic field and record T1(B) while keeping optical, microwave and timing conditions controlled. A demonstrated NV experiment used field-dependent T1 to map electron-spin resonances of P1 centers. A dip or other feature in T1(B) can indicate cross-relaxation with a nearby environmental spin, making relaxometry a spectroscopy tool as well as a lifetime measurement.
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For a field sweep, record the exact field magnitude, direction, sweep order and dwell or averaging time. Repeat features in both sweep directions when drift or hysteresis from a magnetic sample could otherwise mimic a resonance.
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Common failure modes and corrections
The ODMR line is ambiguous
Overlapping orientations, hyperfine features or field-dependent shifts can cause the pulse to address more than one transition. Increase spectral resolution, document the selected line and adjust the bias-field direction or magnitude. If multiple transitions remain excited, treat the result as an ensemble response rather than a single-defect lifetime.
Rabi oscillations do not produce stable π and π/2 times
Check microwave amplitude stability, cable connections, pulse-generator timing, resonance detuning and sample position relative to the wire or stripline. Recalibrate at the operating power and frequency. A pulse calibrated at one location or power cannot be assumed valid after the geometry changes.
The T1 reference subtraction drifts
Interleave the π and no-π sequences on a short cycle, monitor laser and detector counts, and reject or flag shots with focus loss or saturation. A reference collected only before or after the scan cannot remove drift occurring during the measurement.
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Verify that the apparent decay is not caused by an incorrect π pulse, detuning, phase-cycle error or a readout normalization problem. Measure a Rabi trace and a zero- or near-zero-delay echo, then repeat with shorter evolution times. If the corrected contrast still collapses rapidly, investigate magnetic noise, surface spins, nearby nanomagnets and field alignment.
The fit depends on the chosen model
Show the data and residuals for each plausible model, state the fit interval and report uncertainty. A stretched exponential can summarize a broad noise distribution, but its exponent should not be interpreted as a universal material constant. For ensembles, identify whether the fitted number is an effective lifetime or a resolved component.
What to report so the measurement is reproducible
- Defect species and charge state, such as NV−, and whether the measurement uses one defect or an ensemble.
- Crystal orientation and the addressed defect orientation.
- Temperature, magnetic-field magnitude and direction, and whether the field was swept.
- ODMR transition frequency and how it was selected.
- Optical wavelength, initialization and readout durations, and the fluorescence detection window.
- Microwave delivery structure, π and π/2 durations, pulse phases and the Rabi-calibration method.
- Delay range, number of points, repetitions or averaging time, and shot-order or interleaving scheme.
- Fluorescence normalization, bright or dark references and any common-mode subtraction.
- Fit function, baseline, fit interval, weighting method, confidence intervals and residual diagnostics.
- Whether the reported value is T1, Hahn-echo T2 or free-induction T2*.
These details allow readers to distinguish a change in the defect’s environment from a change caused by pulse calibration, optical drift, field alignment or data analysis.
Quick Recap
Practical comparison: which measurement should you run?
| Your question | Measurement | What the result tells you |
|---|---|---|
| How quickly does prepared spin population return to equilibrium? | T1 sequence with π pulse and variable dark delay | Longitudinal population lifetime and relaxation channels |
| How long does phase coherence survive when static offsets are refocused? | Hahn echo with π/2–τ/2–π–τ/2–π/2 | Transverse coherence under the experiment’s refocusing and noise conditions |
| What is the inhomogeneous dephasing time without refocusing? | Ramsey or free-induction experiment | T2*, which must be reported separately from Hahn-echo T2 |
| Is a nearby spin species causing relaxation? | Field-dependent T1(B) sweep | Cross-relaxation features and environmental-spin resonances |
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