Fig. 2.

Spectroscopy of distant, weakly coupled ¹³C spins limited by sensor dissipation. a Spectra of the ¹³C target spins surrounding the NV center. For the upper (lower) ¹³C spectrum, the sensor spin is in its non-magnetic $∣m_{\mathrm{S}}=0⟩$ (magnetic $∣-1⟩$) state during the storage interval. Hence, the upper curve shows the bare Larmor frequency common to all ¹³C nuclear spins in the interaction range, whereas the lower curve shows additional hyperfine coupling offsets A _zz for the two distinct nuclei A ₁ and A ₂. The total phase-accumulation time is set to τ = 100 μs (orange parts of insets) and the RF π-pulse (dark green part of insets), which flips the ¹³C spins during the storage interval, is set to T _c = 430 μs (light green part of insets). b Increased sensing and storage time (τ = 200 μs and T _c = 860 μs, respectively) generate a close-up spectrum of target spins A ₁ and A ₂. Resonance frequencies are marked with vertical lines. c Possible angles θ and distances d of the two ¹³C spins relative to the NV center. The functions are $d\propto {\left(A_{zz}^{-1}∣3\mathrm{co}{\mathrm{s}}^2\theta -1∣\right)}^{1∕3}$. For the identified hyperfine interactions A _zz, 3 cos² θ − 1 is positive (dark line parts for θ = −54.7°…54.7°). d Variation of the overall phase-accumulation time τ for an RF π-pulse selective on the strongest coupled ¹³C spin (i.e., A ₁, ν _RF ≈ 16.373 MHz, T _c = 860 μs) reveals an oscillation, which confirms the coupling strength of 2.8 kHz. e For a phase-accumulation time of τ = 300 μs the memory state is almost maximally correlated with the flip of target A ₁. During storage, we perform a Ramsey oscillation on the ¹³C spin (inset shows two dark green π/2 pulses), which is converted into a memory signal and decays with the memory lifetime, limited by dissipation due to the sensor