Quantum Relaxometry Under Continuous Wave Excitation

AI summary

Overview

Research area: Quantum sensing and spin physics, specifically nitrogen-vacancy (NV) center relaxometry in diamond, with extension to nanodiamond biosensing.

Technical level: Intermediate. Readers should be comfortable with the idea of spin–lattice relaxation (T₁), optically detected magnetic resonance (ODMR), and lock-in detection, but the paper's core argument is a first-order low-pass filter model that is accessible with basic signal-processing intuition.

Scope (one sentence): The paper introduces and validates a frequency-domain continuous-wave relaxometry (FDR) protocol that extracts the ground-state spin–lattice relaxation time T₁ of NV centers from microwave-amplitude-modulated cw-ODMR, benchmarked against pulsed time-domain relaxometry (TDR) in bulk diamond, nanodiamonds, and Mn²⁺-containing aqueous samples.

What This Paper Is About

Measuring T₁ with conventional pulsed protocols is slow and awkward: each cycle needs a laser pulse, an optional microwave π-pulse, a dark delay, a readout pulse, and a wait long compared with T₁ for full relaxation, repeated up to a million times. This confines pulsed relaxometry to a narrow window of intermediate T₁ values and makes it nearly impractical for nanodiamonds, where NV axes are randomly oriented and signals are weak and noisy.

The authors ask whether T₁ can instead be read out in the frequency domain, by modulating the microwave amplitude at low frequency in a continuous-wave ODMR experiment and analyzing how the fluorescence response rolls off. Their goal is a quantitative, parameter-free, and much faster route to T₁ that works from tens of microseconds to hundreds of milliseconds.

Key Contributions

1. A frequency-domain relaxometry (FDR) protocol for spin qubits. The authors show that the lock-in in-phase (X) and quadrature (Y) responses of cw-ODMR under microwave amplitude modulation follow a first-order low-pass form, X ∝ 1/[1+(ω_m T_eff)²] and Y ∝ −ω_m T_eff/[1+(ω_m T_eff)²], with an effective time constant T_eff = T₁/(1+2PT₁). T_eff is read directly from the modulation frequency at which Y peaks, T_eff = 1/(2πf_m).

2. An extrapolation recipe that recovers the intrinsic "dark" T₁. A rate-equation treatment gives 1/T_eff ≈ 1/T₁ + 2P + Γ_p, rewritten as 1/T_eff ≈ 1/T₁ + αP_MW + βI, so measuring T_eff versus microwave power and laser intensity and extrapolating to vanishing drive yields T₁ from purely continuous-wave data.

3. Experimental validation against pulsed relaxometry in bulk diamond, plus a broadband temperature demonstration. FDR is benchmarked on two CVD single crystals and shown to track T₁ from 60 μs to 200 ms (more than three orders of magnitude), including a temperature sweep where the relaxation rate saturates below ~120 K and T₁ reaches ~200 ms.

4. Application to quantitative nanoscale relaxometry and biosensing. FDR is applied to 50, 70 and 100 nm nanodiamonds, used to measure water-induced T₁ changes, and used to detect micromolar Mn²⁺ in aqueous solution, with a claimed speed-up over pulsed methods.

Main Findings

• Bulk T₁ values agree between methods. FDR gives T₁ = (4.75 ± 0.14) ms for the IN3x3 sample and (5.55 ± 0.20) ms for the DNV sample; TDR gives (4.15 ± 0.30) ms and (6.27 ± 0.85) ms respectively. For DNV the two agree within error; for IN3x3 the pulsed value is slightly shorter, attributed to suboptimal pulse settings caused by a shortened T₂* at high NV⁻ concentration.

• FDR is more precise and faster in this comparison. The measurement error for pulsed TDR was two to four times larger than for FDR, even though the total experiment time was at least six times longer in this specific case.

• Microwave saturation behavior differs by sample. For IN3x3, T_eff evolves nonlinearly with microwave power at all laser intensities and plateaus at a relatively low power (−30 dBm); for DNV the plateau occurs at P_mw = 25 dBm because its larger size reduces the effective microwave power experienced by the spins.

• Long T₁ is far easier to reach in the frequency domain. In the DNV temperature dependence, the relaxation rate drops sharply on cooling and saturates below ~120 K, with T₁ reaching ~200 ms at cryogenic temperatures. At 10 K, a 48-hour pulsed TDR measurement reached only SNR ≈ 1, giving T₁ = 468 ms with an uncertainty of 480 ms (unconstrained fit) or T₁ = 223 ms with an uncertainty of 107 ms (constrained fit).

• Nanodiamond T₁ values are unexpectedly long. Extrapolation gives T₁ of 1.6 ms, 0.8 ms and 0.7 ms for ND100, ND70 and ND50. The authors note that values above 1 ms had previously been seen only in isotopically purified or surface-modified material, while typical reports are a few hundred microseconds.

• Smaller particles saturate at higher microwave fields. Because relaxation is faster in smaller nanodiamonds, the onset of saturation shifts to higher microwave fields, producing an unexpected ordering of T_eff with particle size at elevated drive strengths.

• Water reduces T₁ in a size-dependent way. Repeated water addition and evaporation produced systematic, reproducible T₁ reductions of ~6 ± 2% for ND100, ~10 ± 4% for ND70, and ~50 ± 10% for ND50. Confocal PL spectra show ~50% enhancement of NV⁻ emission near the zero-phonon line at 637 nm upon water addition, interpreted as a water-induced revival effect stabilizing shallow NV⁻ centers.

• Micromolar Mn²⁺ detection. Exposure to 5 μL of 500 μM aqueous Mn²⁺ (S = 5/2) reduced T₁ by a factor of ~2.5 for ND100 and ~6.3 for ND50. For comparison, the paper cites a factor of 4 for Gd³⁺ (S = 7/2) at the same concentration in state-of-the-art quantum relaxometry probes. ND70 is presented as the best compromise, with T₁ changing by a factor of ~5 while being more robust to environmental fluctuations.

• Large speed-up for nanodiamond sensing. For ND70, a high-SNR measurement was completed within 2 min, versus ~1440 min for the conventional time-resolved protocol shown in the paper, corresponding to a speed-up of ~2.5–30 times relative to optimized protocols in the literature.

• Low optical and thermal load. The wide-field geometry delivers 0.04–1.1 W/cm², about five orders of magnitude below the ~10⁵ W/cm² typical of diffraction-limited confocal nanodiamond relaxometry, and no measurable microwave-induced heating was found.

Methodology in Plain English

The authors treat the relevant NV⁻ transition as an effective two-level system driven resonantly by microwaves, with longitudinal relaxation at rate 1/T₁. Instead of pulsing the microwaves on and off in discrete timed steps, they leave laser and microwaves on continuously and modulate only the microwave amplitude — switching it on and off symmetrically at a duty cycle of 0.5, at frequencies from 1 Hz to 20 kHz. A lock-in amplifier isolates the first harmonic of the resulting fluorescence modulation and records its in-phase (X) and quadrature (Y) components.

The key idea is intuitive: if the modulation is slow compared with the spin relaxation, the spin population tracks the drive and the response is flat. If the modulation is fast, the population cannot keep up and the response rolls off. The crossover happens at a characteristic frequency set by an effective time constant T_eff, which behaves exactly like a first-order low-pass filter.

T_eff is not T₁ itself, because microwave driving and optical pumping both add extra relaxation channels. So the researchers measure T_eff at several microwave powers and several laser intensities and extrapolate the rate 1/T_eff linearly to zero laser intensity, and to vanishing microwave power where applicable, to isolate the intrinsic "dark" T₁. For nanodiamonds, where low-power signals become too noisy, they instead perform "one-shot" measurements at moderate microwave and laser powers where T_eff is dominated by T₁ and SNR is high, focusing on differences in T₁ rather than absolute values.

Experiments used a home-built setup based on an integrating sphere, a 520 nm fiber-coupled laser delivering up to 300 mW to a ~6 mm spot, a silicon photodetector with long-pass filtering, a transimpedance amplifier, and a lock-in amplifier, with microwaves near 2870 ± 70 MHz applied through a coplanar waveguide. Pulsed TDR comparison measurements used a 1 ms polarization pulse, a π-pulse of typically 250 ns, a variable delay, and a 5 μs readout pulse, with and without the π-pulse.

Why This Matters

For the research community, the paper reframes cw-ODMR modulation — widely used simply to boost magnetometry and thermometry sensitivity — as a quantitative relaxometry tool with a defined physical model and a clear extrapolation procedure. Its claim that previously reported short nanodiamond T₁ values may reflect limitations of non-optimized pulsed protocols rather than intrinsic physics is a notable challenge to the field's assumptions. The demonstration that FDR keeps working where pulsed methods fail (48 hours of TDR at 10 K yielding SNR ≈ 1, versus a workable FDR value near ~200 ms) matters for anyone characterizing long-lived spin qubits.

Real-world applications suggested or supported by the work:

• Biological sensing with fluorescent nanodiamonds: detecting paramagnetic species such as Mn²⁺ at micromolar concentrations in aqueous media, relevant to spin-marker biosensing.
• Quantum computing hardware characterization: long, well-characterized relaxation times are needed for qubit performance and error correction, and FDR provides a practical metrology route to them.
• Quantum communication and memories: robust memories and repeaters depend on long T₁, and the authors expect the approach to extend to silicon carbide defects with T₁ on the minute scale.
• Nanodiamond probe optimization: the size-dependent sensitivity and water-susceptibility data give a practical basis for choosing particle size (ND70 as a compromise, ND50 for non-biological settings) for a given sensing task.

Industry relevance: the method requires only a continuous-wave ODMR setup with a modulated microwave source and lock-in detection, avoiding fast pulse switching and precise timing electronics. That lowers the barrier for compact, lower-cost relaxometry instruments, for quality control of nanodiamond batches used as sensors or biomarkers, and for characterization tools in solid-state qubit development. The low optical intensity in the wide-field geometry (0.04–1.1 W/cm² versus ~10⁵ W/cm²) is also relevant to photobleaching and light-induced damage in biological samples.

Future Directions

• Extending FDR to confocal and single-particle measurements. The work was demonstrated on ensembles; the authors flag optimization for confocal measurements on individual nanoparticles or photonic nanostructures, where lock-in detection from single NV centers has already been shown.
• Transferring the protocol to other spin platforms and resonance methods. The authors explicitly suggest silicon carbide divacancy and transition-metal defects, whose T₁ can reach the minute scale, and note the approach could apply to PDMR, ESR, and other magnetic resonance techniques beyond ODMR.
• Pushing past the ~200 ms ceiling. FDR works optimally for T₁ up to ~200 ms, where low-frequency drift and 1/f noise in the sub-hertz range remain negligible; how to suppress those technical noise sources for even longer relaxation times is left open.
• Disentangling environmental effects in small nanodiamonds. The strong water sensitivity of ND50 (a ~50 ± 10% T₁ change) complicates calibration, and the paper leaves open how to stabilize or model these fluctuations for reliable quantitative sensing.

Target Audience

Experimental quantum sensing researchers working with NV centers and other solid-state spin defects; groups building relaxometry, magnetometry, or thermometry instruments who want a faster alternative to pulsed T₁ measurements; nanodiamond biosensing and biomedical imaging researchers; and instrument developers in quantum technology who need practical metrology for long-lived spin qubits. Readers focused on theory will find the treatment light — the core model is a first-order filter — so the paper is best suited to experimentalists and application-oriented engineers rather than to theorists.

Authors’ abstract

Quantum relaxometry is one of the most successful applications of nitrogen-vacancy (NV) centers in diamond and, more broadly, solid-state spin qubits, enabling ultrasensitive detection of magnetic noise and paramagnetic species via measurements of the spin-lattice relaxation time $T_1$. Conventional pulsed protocols, however, probe $T_1$ efficiently only over a limited temporal range, which restricts the scope and throughput of the technique. Here we introduce a continuous-wave quantum relaxometry protocol that operates in the frequency domain. By measuring the frequency response of the optically detected magnetic resonance signal under low-frequency microwave amplitude modulation, we extract $T_1$ from the characteristic response time of the spin system. The method enables efficient $T_1$ measurements spanning more than three orders of magnitude -- directly demonstrated from 60 $μ$s to 200 ms in our experiments -- across a broad temperature range and under substantial ensemble inhomogeneity. We further show that this protocol enables quantitative relaxometry-based sensing in nanodiamonds, achieving a substantial speed-up over the pulsed methods and offering a practical approach to optimizing nanodiamond size for enhanced sensitivity.