Quantum Memory Enhanced Multipoint Correlation Spectroscopy for Statistically Polarized NMR

Abstract

Nuclear magnetic resonance spectroscopy with solid-state spin sensors is a promising pathway for the detection of nuclear spins at the micro- and nanoscale. Although many nanoscale experiments rely on a single sensor spin for the detection of the signal, leveraging spin ensembles can enhance sensitivity, particularly in cases in which the signal merely originates from statistically polarized nuclear spins. In this work, we introduce multipoint correlation spectroscopy, that combines the advantages of two well-established methods -- correlation spectroscopy and quantum heterodyne detection -- to enable temporally efficient measurements of statistically polarized samples at the nanoscale with spin ensembles. We present a theoretical framework for this approach and demonstrate an experimental proof of concept with a nitrogen vacancy center in diamond. We achieve single hertz uncertainty in the estimated signal frequency, highlighting the potential applications of the technique for nanoscale nuclear magnetic resonance.

Figures (3)

• Figure 1: (a) CS measurement scheme with ancilla based memory. (I) The memory spin $n$ is initialized before (II) acquiring an initial phase $\phi_0^{(j)}$ with the sensor spin $e$, where $(j)$ denotes the $j$-th repetition of the experiment or the $j$-th NV in an ensemble, $j=1,\dots, N$ ($j$ omitted for simplicity in the figure) and mapping it onto the memory spin by using a $\text{C}_\text{e}\text{NOT}_\text{n}$ gate. In a subsequent phase measurement (III) the phase $\phi^{(j)}_k$$\left( k \in \{ 1,...,M\} \right)$ is acquired, which is correlated to the initial phase by performing a $\text{C}_\text{n}\text{NOT}_\text{e}$ gate. The phases $\phi_k^{(j)}$ depend on the signal phase $\xi^{(j)}_{\mathrm{s}, k}$ at the beginning of the respective phase acquisition. (b) QDyne measurement scheme. Only a sensor spin $e$ is used to acquire the phases $\phi_k$ at a constant rate. Commonly, the protocol is performed for more than $M+1$ phase acquisitions in order to improve the SNR. (c) MCS scheme. Similar to CS, a sensor and memory spin are used to establish correlations between an initial phase $\phi_0$ and the measurement phases $\phi_k$ by storing and retrieving $\phi_0$ on the memory spin. $M+1$ phase acquisitions are performed during one sequence, which can be repeated $N$ times. (d) Comparison of the SNR of the three protocols vs. the number $N$ of NV centers in an ensemble, where photon shot noise is assumed to be the main noise source. SNR of MCS and CS (QDyne) protocols are normalized to the one of MCS (QDyne) with a single NV center. $\text{SNR}_{\text{Qdyne}}$ does not change with $N$ while SNR of both CS and MCS $\propto \sqrt{N}$. MCS is favorable by factor $f_T\propto \sqrt{M}$ to CS as its data acquisition lacks long waiting times.
• Figure 2: (a) Level scheme of an NV center. Electronic spin levels $m_\mathrm{s} = 0, \pm 1$ are split by the zero field splitting $D$ and Zeeman interaction $\pm \gamma_\mathrm{NV} B_z$, creating non-degenerate spin levels. Only the subsystem $m_\mathrm{s} = 0, -1$ is considered. Hyperfine $A$, quadrupolar $P$ and nuclear Zeeman $\gamma_\mathrm{N} B_z$ interaction with the intrinsic $^{14}\mathrm{N}$ nuclear spin splits the electron spin states into three additional sublevels $m_\mathrm{I} = 0, \pm 1$. (b) Pulsed ODMR measurement reveals the hyperfine structure of the NV center with intrinsic 14N for the $m_\mathrm{s} = 0 \leftrightarrow m_\mathrm{s} = -1$ transition. $89.1 +- 1.8%$ of the nuclear spin population is in the $m_\mathrm{I}=0, +1$ states. The dashed lines show the corresponding microwave transitions in (a), which are separated by the hyperfine coupling of approximately 2.1 MHz. The protocol was demonstrated, using the red (light blue) transition for RF (selective and strong MW) pulses.
• Figure 3: (a) Experimental scheme of MCS with an NV center in diamond. First, the nitrogen nuclear spin is initialized using selective MW (blue) and RF (red) $\pi$ pulses. The signal is then probed for the first time, using XY8-1 (purple). The acquired phase is mapped onto the nitrogen nuclear spin, by utilizing an RF $\pi$-pulse. Subsequently, the phase acquisition is performed $M = 1991$ times, correlating the acquired phase to the phase stored in the population of the nitrogen nuclear spin by applying a selective MW pulse. The sequence is repeated for $N=597894$ times. (b) Experimentally measured NV center fluorescence response for a $14.28 +- 0.04µ T$ signal at $1M Hz$ when running the sequence in (a). Error bars have been omitted for visiblity purposes and the inset shows a zoom in on the data, with error bars corresponding to photon shot noise. (c) Power spectral density of the measured signal. A fit to the signal reveals the detected undersampled frequency $f_\mathrm{u} = 4.2954 +- 0.0013k Hz$ with a full width at half maximum of $59.9 +- 2.1Hz$ corresponding to a resolution limit due to the nuclear spin lifetime $\widetilde{T}_{1, \mathrm{nuc}}$ under laser illumination.