Removing inhomogeneous broadening in cross-relaxation spectra via hole burning

Technical Summary: Removing Inhomogeneous Broadening in Cross-Relaxation Spectra via Hole Burning

Problem Statement
Quantum relaxometry based on nitrogen-vacancy (NV) centers in diamond is a robust technique for detecting magnetic noise via longitudinal relaxation ($T_1$) measurements. Unlike magnetometry, which requires sophisticated spin-control pulses to filter environmental noise, relaxometry has minimal requirements on spin control, making it ideal for complex scenarios such as flexible nanodiamonds, cellular activity detection, and chemical reaction monitoring. However, a critical limitation of NV relaxometry is its poor spectral resolution. In cross-relaxation spectroscopy, where magnetic resonance signals are acquired by matching the energy splitting of the NV sensor with target spins, the spectral resolution is severely hindered by the inhomogeneous broadening of the NV ensemble. This broadening, determined by the dephasing rate ($\Gamma^*_2 = 1/T^*_2$), dominates the spectral line shape, preventing the clear resolution of distinct target spin frequencies. While dynamical decoupling techniques can mitigate this in magnetometry, they are incompatible with the cross-relaxation method.

Methodology
The authors propose and demonstrate a hole-burning technique to remove inhomogeneous broadening in cross-relaxation spectra. The core concept involves utilizing a weak radiofrequency (RF) pump field to selectively deplete a specific subset of NV centers during the relaxation measurement.

1. Mechanism: An RF field, tuned to a frequency within the inhomogeneous linewidth, drives specific NV centers from their initial "cold" state ($|0\rangle$) to thermal equilibrium. This "burns a hole" in the NV ensemble's spectral profile, effectively removing those specific sensors from contributing to the signal.
2. Probe: The cross-relaxation coupling between the remaining NV centers and the target spins serves as the probe.
3. Experimental Setup: The experiment utilizes an ensemble of NV centers located $\sim10$ nm below the surface of a diamond chip. The targets are nuclear spins (specifically $^{19}\text{F}$ and $^{1}\text{H}$) from a drop of Fomblin oil placed on the diamond surface.
4. Pulse Sequence: The measurement involves a standard cross-relaxation sequence where the NV energy splitting is tuned via an amplitude-modulated microwave field. A synchronized RF pump field is applied either continuously or turned on/off to induce the hole-burning effect. The population of the $|0\rangle$ state is measured to reconstruct the spectrum.

Key Contributions

• Theoretical Model: The authors developed a theoretical model describing the hole-burning spectrum as a convolution of the NV line shape and a Lorentz function. They derived an expression for the hole-burning signal contrast ($C_{HB}$) and the resulting linewidth ($\text{FWHM}_{HB}$), demonstrating that the linewidth is determined by the homogeneous decoherence rate ($\Gamma_2$) rather than the inhomogeneous rate ($\Gamma^*_2$).
• Experimental Demonstration: The team successfully applied this technique to detect nuclear spins on a diamond surface. By tuning the RF pump strength, they narrowed the cross-relaxation nuclear magnetic resonance (NMR) spectrum without sacrificing sensitivity.
• Resolution Enhancement: The method successfully resolved the resonance lines of $^{19}\text{F}$ and $^{1}\text{H}$ nuclear spins, which were indistinguishable in the normal cross-relaxation spectrum due to inhomogeneous broadening.

Results

• Spectral Resolution: In the normal cross-relaxation spectrum, the linewidth was approximately 0.23 MHz, dominated by the NV inhomogeneous broadening ($T^*_2 \approx 0.63 \, \mu\text{s}$). In the hole-burning spectrum, distinct peaks for $^{19}\text{F}$ (at 1.446 MHz) and $^{1}\text{H}$ (at 1.534 MHz) were clearly resolved.
• Linewidth Dependence: The measured hole linewidth showed a linear dependence on the square of the RF driving strength ($b^2$), consistent with the theoretical prediction ($\text{FWHM}_{HB} = 4\Gamma_2 + 0.6d^2t + 0.6b^2t$).
• Decoherence Rate Estimation: The data allowed for the estimation of the homogeneous decoherence rate $\Gamma_2$, yielding values between 1.4 and 1.8 kHz, significantly narrower than the inhomogeneous rate of 0.25 MHz.
• Contrast and Sensitivity: The hole-burning signal contrast saturated at approximately 1.2%, which corresponds to the intrinsic contrast of the $^{19}\text{F}$ spins in the bare cross-relaxation spectrum. This indicates that the signal loss due to the hole-burning process can be minimized, preserving the sensitivity of the relaxometry technique.

Significance
The paper claims that this hole-burning technique offers a straightforward method to eliminate inhomogeneous broadening in cross-relaxation spectra while retaining the inherent convenience and robustness of NV relaxometry. Unlike dynamical decoupling methods, this approach does not require complex pulse sequences, making it applicable to flexible nanodiamonds and large ensembles in complex biological or chemical environments. The authors note that while the current spectral resolution is comparable to ordinary dynamical decoupling methods ($\sim 1/T_2$), it remains inferior to correlation or heterodyne detection methods based on dynamical decoupling. However, the simplicity of the method and its compatibility with other color centers (e.g., in SiC or hBN) open new avenues for high-resolution magnetic resonance spectroscopy in scenarios where traditional high-resolution techniques are impractical. The authors suggest that future improvements could potentially be achieved by combining this method with target spin polarization to further enhance resolution.