⚛️ quantum physics

Quantum Relaxometry Under Continuous Wave Excitation

This paper introduces a continuous-wave quantum relaxometry protocol that overcomes the temporal limitations of conventional pulsed methods by extracting spin-lattice relaxation times ( $T_1$ ) from frequency-domain responses, thereby enabling efficient, high-throughput sensing across a broad range of timescales and temperatures, including in nanodiamonds.

Original authors: Vladimir Verkhovlyuk, Chayma Bouchair, Oleg A. Anisimov, Anton Pershin, Adam Gali

Published 2026-08-11

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CC BY 4.0

Original authors: Vladimir Verkhovlyuk, Chayma Bouchair, Oleg A. Anisimov, Anton Pershin, Adam Gali

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to listen to a whisper in a crowded, noisy room. In the world of quantum physics, scientists often use tiny, defective spots inside diamonds—called nitrogen-vacancy (NV) centers—as super-sensitive microphones to hear the faintest magnetic whispers from the surrounding world. These defects act like tiny compass needles that can be read with a laser. To figure out how "noisy" the environment is, scientists measure how long these compass needles take to settle down after being jostled. This settling time is called $T_1$ . Think of it like a spinning top: if it spins for a long time before wobbling and stopping, the room is very quiet; if it stops immediately, the room is chaotic. This measurement, known as quantum relaxometry, is a powerful tool for detecting magnetic fields, temperature changes, and even chemical reactions in biology. However, for a long time, the only way to measure this "settling time" was to give the top a sharp tap, wait, and check if it was still spinning. This "pulsed" method is like trying to time a race by starting a stopwatch, stopping it, waiting for the runner to rest, and then starting again. It works great for runners who finish in a few seconds, but if the runner takes an hour to finish, you'd have to wait an hour between every single start-stop check, making the whole process agonizingly slow and inefficient.

This paper introduces a clever new way to measure that settling time, turning the slow, stop-and-start race into a smooth, continuous flow. The researchers, led by Vladimir Verkhovlyuk and colleagues, developed a "continuous-wave" method that works in the frequency domain. Instead of tapping the diamond and waiting, they gently wiggle the microwave energy hitting the diamond at different speeds, like shaking a box of marbles back and forth. By watching how the diamond's light response changes as they change the speed of the shake, they can figure out exactly how long the marbles (the spins) take to settle down. They call this Frequency-Domain Relaxometry (FDR).

The team tested this new trick on two types of diamond samples: large, solid blocks and tiny, nano-sized diamonds (smaller than a human hair). They found that their new method could measure settling times ranging from incredibly fast (60 microseconds) to incredibly slow (200 milliseconds). That is a span of more than 1,000 times, covering a huge range that the old "pulsed" method struggled with. In fact, for the slowest samples, the old method was so inefficient that it would take days to get a reliable result, whereas the new method did it in minutes. The researchers showed that this new approach gives the same accurate results as the old, trusted methods but is much faster and less prone to errors caused by imperfect timing.

The real magic, however, happens when they applied this to the tiny nanodiamonds. These are the size of viruses and are used to sense things inside living cells. The team discovered that these tiny diamonds are surprisingly sensitive to water. When they added water to the nanodiamonds, the settling time changed significantly, and this effect was even stronger in the smallest particles. They then used this sensitivity to detect manganese ions (a type of metal) dissolved in water. The tiny diamonds reacted strongly to the manganese, changing their settling time by a factor of up to 6.3. This suggests the method could be a super-fast way to detect specific chemicals in biological environments. The researchers also noted that because their method uses a wide beam of light rather than a tight, intense laser focus, it is much gentler on the samples, reducing the risk of burning or damaging the delicate biological materials they are studying.

In short, this paper doesn't just offer a slightly faster way to do the same old thing; it opens the door to measuring things that were previously too slow or too difficult to catch. By switching from a "tap-and-wait" strategy to a "shake-and-listen" strategy, the authors have created a tool that is robust, fast, and ready to be used in complex environments like living cells, potentially speeding up our ability to understand the magnetic and chemical secrets of the microscopic world.

Technical Summary: Quantum Relaxometry Under Continuous Wave Excitation

Problem Statement

Quantum relaxometry, which utilizes the spin–lattice relaxation time ( $T_1$ ) of nitrogen-vacancy (NV) centers in diamond to detect magnetic noise and paramagnetic species, is currently limited by conventional pulsed time-domain relaxometry (TDR). While TDR is effective for intermediate $T_1$ values, it faces significant challenges:

Temporal Range Limitations: Accurate measurement requires waiting for full spin relaxation between pulses. This makes measuring long $T_1$ values (e.g., at low temperatures) prohibitively slow, often requiring hours to achieve adequate signal-to-noise ratios (SNR).
Nanoscale Constraints: In nanodiamonds, pulsed protocols struggle with low optical collection efficiency, microwave control fidelity, and spin coherence. The random orientation of NV axes in ensembles makes well-defined $\pi$ -pulses difficult to implement, and the resulting signals are often contaminated by NV $^0$ background or charge-state dynamics.
Measurement Artifacts: Pulsed measurements are susceptible to errors from imperfect pulse timing, residual illumination, and the need for precise pulse sequencing, which becomes technically demanding as device dimensions shrink.

Methodology: Frequency-Domain Relaxometry (FDR)

The authors introduce a continuous-wave (CW) quantum relaxometry protocol operating in the frequency domain. Instead of time-resolved pulse sequences, this method utilizes low-frequency microwave amplitude modulation combined with lock-in detection.

Principle: The NV spin system is driven by continuous laser and microwave fields. The microwave amplitude is modulated at a frequency $\omega_m$ . The fluorescence response is demodulated using a lock-in amplifier to extract the in-phase ( $X$ ) and quadrature ( $Y$ ) components.
Theoretical Basis: The system is modeled as a first-order low-pass filter. The complex frequency response $H(\omega_m)$ $H (ω_{m})$ is dominated by a single slow mode characterized by an effective time constant $T_{\text{eff}}$ $T_{eff}$ .
- $T_{\text{eff}}$ is related to the intrinsic relaxation time $T_1$ and drive-dependent rates (microwave power $P_{\text{MW}}$ and laser intensity $I$ ) via the relation:
  $\frac{1}{T_{\text{eff}}} \approx \frac{1}{T_1} + \alpha P_{\text{MW}} + \beta I$
- By measuring $T_{\text{eff}}$ across varying microwave powers and laser intensities, the intrinsic "dark" $T_1$ is extracted by linear extrapolation to zero power.
Experimental Setup: The protocol uses a CW-ODMR setup where the microwave frequency is fixed at resonance. The modulation frequency $f_m$ is swept (1 Hz to 20 kHz). $T_{\text{eff}}$ is determined either from the half-decay point of the $X$ component or, more conveniently, from the frequency $f_m$ where the $Y$ component reaches its maximum ( $T_{\text{eff}} = 1/2\pi f_m$ ).

Key Results

The authors validated the FDR method against conventional TDR in both bulk diamond and nanodiamond samples.

Bulk Diamond Validation:
- Measurements on two CVD single-crystal samples (DNV and IN3x3) showed excellent agreement between FDR and optimized TDR protocols.
- Speed and Precision: FDR achieved comparable or better precision than TDR in significantly less time. For long relaxation times (e.g., $\sim 200$ ms at cryogenic temperatures), TDR failed to yield reliable values even after 48 hours due to SNR limitations, whereas FDR successfully measured these values.
- Temperature Dependence: The method successfully mapped the temperature dependence of $1/T_1$ , identifying the phonon-driven saturation below $\sim 120$ K.
Nanodiamond Applications:
- Size-Dependent $T_1$ : The method measured intrinsic $T_1$ values for nanodiamonds of 50 nm, 70 nm, and 100 nm. The results (0.7 ms to 1.6 ms) were longer than typical literature values, suggesting previous pulsed protocols may have been suboptimal or limited by non-ideal pulse conditions in inhomogeneous ensembles.
- Environmental Sensing: The technique demonstrated sensitivity to paramagnetic ions ( $Mn^{2+}$ $M n^{2 +}$ ) in aqueous environments.
  - Water Effect: The presence of water reduced $T_1$ due to a "water-induced revival effect" stabilizing shallow NV centers, with smaller particles (ND50) showing the largest relative change ( $\sim 50\%$ ).
  - Ion Detection: Exposure to 500 $\mu$ M $Mn^{2+}$ caused significant $T_1$ reduction (factor of $\sim 2.5$ for ND100 and $\sim 6.3$ for ND50).
- Efficiency: A measurement on ND70 with high SNR was completed in 2 minutes, compared to $\sim 1440$ minutes required for a comparable TDR measurement. This represents a speedup of 2.5 to 30 times over optimized literature protocols.
- Optical Load: The wide-field FDR configuration distributes power over a millimeter-scale spot, resulting in an intensity ( $0.04–1.1$ W/cm $^2$ ) five orders of magnitude lower than typical confocal nanodiamond relaxometry, minimizing photobleaching and thermal load.

Significance and Claims

The paper claims that the FDR protocol establishes a parameter-free, robust, and quantitative route to measuring $T_1$ in realistic experimental environments, particularly where pulsed methods fail.

Broad Applicability: The method is effective across a $T_1$ range spanning more than three orders of magnitude (60 $\mu$ s to 200 ms) and is applicable to bulk crystals, nanodiamond ensembles, and potentially other solid-state spin qubits (e.g., in silicon carbide).
Overcoming Limitations: By eliminating the need for fast pulse switching and precise timing, FDR overcomes the technical bottlenecks of nanoscale measurements and the time penalties associated with long relaxation times.
Practical Impact: The approach offers a practical metrological tool for characterizing long relaxation times critical for quantum computing (qubit performance, error correction) and quantum communication (robust quantum memories). It enables rapid, sensitive relaxometry in biologically relevant environments without the thermal and optical damage associated with high-intensity confocal excitation.

The authors conclude that this frequency-resolved implementation bridges the gap between continuous-wave sensing and quantitative relaxometry, providing a faster and more efficient alternative to time-domain protocols for a wide range of quantum sensing applications.

Technical Summary: Quantum Relaxometry Under Continuous Wave Excitation

Problem Statement

Methodology: Frequency-Domain Relaxometry (FDR)

Key Results

Significance and Claims

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