Ramsey correlation spectroscopy with phase cycling using a single quantum sensor

Abstract

Magnetic spectroscopy at the nanoscale provides unique insights into material properties and dynamics, with quantum sensors like nitrogen-vacancy (NV) centers being ideally suited for these scales. However, detecting low-frequency signals remains a challenge due to finite coherence times ($T_2^*$), as signals oscillating slower than $1/T_2^*$ decay before sufficient phase accumulation occurs. We present RESOLUTE (Ramsey corrElation SpectroscOpy puLse seqUence wiTh phasE cycling), a protocol that overcomes these limitations by combining Ramsey measurements with correlation spectroscopy. By storing accumulated phase as a population imbalance during a correlation period ($T_\mathrm{corr} < T_1$) between two sensing periods, RESOLUTE generates an effective coherence time $T_2^p > T_2^*$. This shifts the frequency-matching condition to the correlation time, enabling detection in the previously inaccessible spectral region between $1/T_1$ and $1/T_2^p$. We experimentally demonstrate an extension of the effective coherence time from $T_2^* = 0.38\,μs$ to $T_2^p = 5.1\,μs$, surpassing Hahn Echo measurements. The technique successfully detects $^{13}$C nuclear spin Larmor precession at fields as low as 49$\,$G ($\sim$50$\,$kHz). We further provide theoretical insight using Fisher information to characterize RESOLUTE's frequency estimation capabilities compared to existing protocols. Finally, by integrating adiabatic pulses and phase cycling, we demonstrate robust spin control and effective DC signal extraction. These advancements provide enhanced sensitivity to weak dipolar interactions, essential for single-molecule imaging and quantum sensing applications.

Figures (13)

• Figure 1: Schematic of the experimental theater. Left panel: The frequency domain of sensitivity to signals, with Ramsey covering very low frequencies limited by $1/T_2^*$, dynamical decoupling (DD) covering high frequencies and a gap which is covered by RESOLUTE, the pulse sequence we introduce below in the text (Sec. \ref{['sec:Correlation']}). Middle panel: A plot depicting the sensor qubit, the target qubit, having some dipole-dipole coupling between them, and a spin bath. Right panel: The combination of chirped pulses and our pulse sequence allows to significantly improve the readout contrast of target spins and achieving a narrower linewidth (Sec. \ref{['sec:Ad_Corr']}).
• Figure 2: RESOLUTE pulses sequence, filter functions and sensitivity. (a) Laser pulses for initialization and read-out of the NV center are in green, MW pulses for NV center driving in blue, and an MW pulse for the target electron spin in white with a dashed line. The different choices of letters give the full RESOLUTE with all phase combinations for signal extraction. (b) False-color map showing the sensitivity of the RESOLUTE filter function in Eq. \ref{['eq:ff']} as a function of the target signal frequency, $\omega$, and the correlation time, $T_\mathrm{corr}$. Here $\tau=5 \,\upmu\text{s}$. (c) A comparison between the sensitivity of Ramsey, Hahn Echo and RESOLUTE pulses sequences as a function of the frequency. The amplitude on each curve marks the sensitivity for fixed sequence parameters, while the color intensity of each curve corresponds to the filter's ability to detect and isolate signals provided some sensor parameters. Thus, the stronger the color, the more sensitive the sequence is, with a decay in sensitivity (color strength) for lower frequencies resulting from the typical decay time of the sensor ($T_2^*$ = 0.36$\,\upmu\mathrm{s}$ for Ramsey and $T_2 = 4.3\,\upmu\mathrm{s}$ for a Hahn Echo). For RESOLUTE (the blue curve is the line-cut from panel a marked with a dashed line), the sensitivity window depends on the largest correlation time allowed $T_\text{corr} < T_1$ (500$\,\upmu\mathrm{s}$ in the figure), which fixes the lowest frequency that can be detected, and the effective dephasing time $T_2^p$ (5.1 $\,\upmu\mathrm{s}$ in the figure) which in RESOLUTE represents a limit to the sensitivity to high frequencies. Note that all decay times considered are in agreement with our experimental measurements (see Fig. \ref{['fig:T2_compare']})
• Figure 3: (a) Signal of RESOLUTE (blue), Hahn Echo (red), and Ramsey (gray) from a single NV showing the difference in decoherence times. The Ramsey signal also showed oscillations rising from the strong hyperfine coupling to the nitrogen nuclear spin. The RESOLUTE was taken with a correlation time of 10$\,\upmu$s (b) RESOLUTE signals with fixed sensing time $\tau=6\,\upmu\mathrm{s}$ and changing correlation time $T_\text{corr}$, taken with $107.2 \pm 0.2$ G (blue), $64.3 \pm 0.1$ G (gray) and $49.5 \pm 0.3$ G fields (orange), shifted vertically for clarity. All three show oscillations at the frequency of the carbon nuclear spin Larmor frequency.
• Figure 4: Accumulated Fisher information for frequency detection for 500 repetitions of a RESOLUTE sequence, as a function of the target signal frequency. For comparison, we include the equivalent Fisher information using a Hahn Echo sequence, a simple Ramsey sequence, and the Correlated Ramsey measurements proposed in Refs. Herbschleb2022Oviedo2024. For both RESOLUTE and the Hahn Echo, we assume that the inverse frequency is matched by either $T_\text{corr}$ for RESOLUTE or $\tau_H$ in the case of the Hahn Echo. In both cases, we consider a $T_2^p$ ($T_2$ for the Hahn Echo) of $5\,\upmu\text{s}$. For the Ramsey and Correlated Ramsey, we consider a $T_2^* = 0.5 \,\upmu\text{s}$ and an optimal phase accumulation time $\tau_R = T_2^*$. In all cases, $T_1 = 1000 \,\upmu\text{s}$, and the overhead time in between measurements is $3\,\upmu\text{s}$. The shadowed area marks the region in which information is not sufficient for successful frequency estimation, delimited by the Rayleigh criterion from optics at $\Delta \omega^2 > 4/\omega^2$.
• Figure 5: The effect of chirped pulses. (a) RESOLUTE signal (blue) and DEER time signal (green), both taken with $2\,\mathrm{\upmu s}$ long chirp pulse with $Q=5$, as detected from an NV center with a strong dipolar coupling to an electron spin. The inset shows the fast Fourier transform of the both signals and the detected dipolar coupling as extracted from the fitted data. (b) RESOLUTE signal with fixed duration of $2\,\mathrm{\upmu s}$ chirp pulse and different adiabaticities as detected from the same NV-electron system reported in panel a. Error bars of each signal are presented in the lower right side of the plot.