Overcoming frequency resolution limits using a solid-state spin quantum sensor

TL;DR

The paper tackles the fundamental frequency-resolution limit in spectroscopy by introducing superresolution quantum sensing with a solid-state spin sensor (NV center) and a $^{15}$N memory. By selecting interrogation times that satisfy a superresolution condition and employing dynamical decoupling, the method yields finite Fisher information and an $\Delta\delta_r \propto t^{-2}$ scaling, enabling sub-kHz resolution in as little as $80\,\mu$s. The approach is demonstrated experimentally using two nearly identical incoherent signals, achieving high-contrast readout with single-shot nuclear memory readout that suppresses classical noise. These results open pathways for high-resolution nanoscale NMR and quantum-enhanced spectroscopy, with potential extensions to spin squeezing and higher-frequency domains.

Abstract

The ability to determine precisely the separation of two frequencies is fundamental to spectroscopy, yet the resolution limit poses a critical challenge: distinguishing two incoherent signals becomes impossible when their frequencies are sufficiently close. Here, we demonstrate a simple and powerful approach, dubbed {\it superresolution quantum sensing}, which experimentally resolves two nearly identical incoherent signals using a solid-state spin quantum sensor. By carefully choosing interrogation times that satisfy the superresolution condition, we eliminate quantum projection noise, overcoming the vanishing distinguishability of signals with near-identical frequencies. This leads to improved resolution, which scales as $t^{-2}$ in comparison to the standard $t^{-1}$ scaling. Together with a greatly reduced classical readout noise assisted by a nuclear spin, we are able to achieve sub-kHz resolution with a signal detection time of 80 microseconds. Our results highlight the potential of quantum sensing to overcome conventional frequency resolution limitations, with broad implications for precision measurements.

Figures (13)

• Figure 1: Principles of superresolution quantum sensing. (a) Single NV spin as a quantum sensor (with $\prescript{15}{}{\mathbf{N}}$ nuclear spin serving as a memory qubit) for detecting two incoherent oscillating signals with the frequencies $\{\omega_1,\omega_2\}$ (cf. Eq.\ref{['Eq:Hs-Main']}). (b) The transition probability, $P_t$, displays maximal difference for $\delta_{r}\equiv(\omega_1-\omega_2)/2=0$ (red curve) and $\delta_{r}=0.05\omega_{s}$ (blue curve), where $\omega_s=(\omega_1+\omega_2)/2$, in the neighboring region of $\omega_s t=2n\pi$ with $n$ positive integers, as indicated by the superresolution condition.
• Figure 2: Experimental protocol using a solid-state spin quantum sensor in diamond. (a) Resolving two angular frequencies, $\omega_1$ and $\omega_2$, is challenging as they approach each other. We shift $\omega_{i}\rightarrow\delta_{i}$ in the interaction basis by applying dynamical decoupling Degen2017Quantum with short $\pi$ pulses, separated by time $\pi/\omega_{\text{DD}}$. (b) Energy levels of the NV center electron spin with hyperfine coupling to its intrinsic $\prescript{15}{}{\mathbf{N}}$ nuclear spin. The $m_{S}= \pm1$ energy level degeneracy is lifted by a Zeeman splitting $2\gamma B_{z}$ with an external magnetic field $B_{z}$, aligned with the NV center symmetry axis. The hyperfine coupling further splits each electronic state into two sublevels, $m_{I}= \pm1/2$. The nuclear Zeeman splitting, $\gamma_{\text{n}}B_{z}$, is also taken into account due to the high magnetic field. (c) Sensing and readout sequence. The spin state of the sensor is mapped onto the nuclear spin with a controlled RF $\pi$ pulse, which can be repetitively read out by single shot readout (SSR) measurements. (d) The histogram of the fluorescence time trace obtained from SSR readout, which can be well fitted by two Gaussian distributions. The high and low counts correspond to a nuclear spin state $|\uparrow\rangle$, and $|\downarrow\rangle$, respectively, giving rise to a high readout fidelity of 99.69$\%$.
• Figure 3: Demonstration of superresolution condition. (a) The histograms of experimentally measured $P_t$ for frequency separation: $\delta_{r}/2\pi=[0, 0.5, 1, 2.5, 5]\,$kHz. They show apparent (negligible) distinguishability when the superresolution condition is (not) satisfied. (b) Simulated Fisher information, which determines the estimation uncertainty of $\delta_r$, shows gradually increased peaks when the interrogation time $t$ approaches $2n\pi/\delta_s$. Here we set $\delta_{s}=(2\pi) 50\,$kHz, $\widetilde{\Omega}=(2\pi)16.85$ kHz.
• Figure 4: (a) Signal contrast vs. total measurement time $t$ with dynamical decoupling with pulse separation $\tau=\pi/\omega_{\text{DD}}=200\,$ns for different $\delta_{r}$. The maximum difference is at the superresolution time of $t=2\pi/\delta_s=80\,\mu$s. (b) Experimental (red dots) and simulated (green line) signal contrast at $t=80\,\mu$s vs. angular frequency difference $\delta_r$. (c) Estimated vs. actual frequency separation $\delta_{r}$. We note that the approximation $\delta_r\ll \delta_s$ is not valid for $\delta_r=(2\pi)\,2.5$ kHz. (d) Estimation uncertainty $\Delta\delta_r$ vs. $\delta_r$, which is consistent with Eq. \ref{['Eq:Precision-Main']}. The green squares and line represent the measured and simulated $\Delta\delta_r$ at $t = 2.56\pi/\delta_s$ under the assumption of perfect measurements, corresponding to the data in Fig. \ref{['results_optimal_condition']}(a). The thin gray line shows simulations for an increased interaction time $t = 2.56\pi/\delta_s=102.4\,\mu$s. The red dots represent the experimental data at the superresolution condition. The red solid line shows the simulated $\Delta\delta_r$ derived from the theoretical Fisher information under the assumption of perfect measurements with the same experimental parameters. The red dashed line represents the simulated $\Delta\delta_r$ obtained by incorporating the decoherence of the quantum sensor Supplemental, showing excellent agreement with the experimental results.
• Figure S.1: Periodic control pulses to enhance signal detection. We assume the central frequency of two signals $\omega_{1,2}$ is $\omega_{s}$, and the periodic $\pi$ pulses are applied at a frequency of $\omega_{\text{DD}}=\omega_s+\delta_s$.