As quantum technologies gain momentum, it has become abundantly clear that no physical platform is ideally suited to all practical tasks, a notion presently driving the study of hybrid devices integrating quantum systems with complementary functionalities 1,2 . Among them, much attention is being devoted to physical realizations where an individually-addressable qubit in the form of a trapped ion 3 , a cold atom 4 , or a molecule 5 interacts with a superconducting qubit or a superconductor electrode in an atom chip [6][7][8][9][10][11][12] . Of special interest are all-solid architectures integrating superconductors and adjacent spin-active color centers 13,14 because they can be exploited not only as long-lived quantum memories but also as interfaces, e.g., to convert microwave into optical photons. A closely related research frontsharing the same geometry and ultimately governed by the same physical principles -exploits color centers as local probes 15,16 . Scanning magnetometry of superconductors via nitrogen-vacancy (NV) centers in diamond is a paradigm example that has led to quantitative imaging of individual vortices with exquisite resolution [17][18][19] .

Although color centers in these applications are often seen as isolated systems, the host crystal contains co-existing spins [20][21][22][23][24] as well as fluctuating charges [25][26][27] whose dynamics are not immune to the superconductor vicinity. Diamond surfaces in particular feature various non-fluorescent paramagnetic centers -e.g., in the form of dangling bonds 28 -as well as shallow carriers, whose type and concentration largely depend on surface termination 29 . Here we use an alldiamond scanning probe to control the distance between an individual tip-hosted NV center and a high-criticaltemperature superconductor film. Upon application of pulsed control protocols, we find that close proximity to the superconductor extends the NV spin coherence lifetime. We develop a theoretical formalism that allows us to compare the impact of alternative noise sources near the diamond surface, and propose a superconductor-induced charge rearrangement process as the one responsible for our observations. We then build on the interplay between the superconductor and the NV environment to reconstruct a one-dimensional transverse relaxometry image across the superconductor boundary.

Diamond-based magnetometry builds on the singular properties of the NV center, a spin-1 system amenable to optical initialization and readout via a spin-selective excitation cycle 30 . For sufficiently low magnetic field amplitudes 𝐵 " , the energies corresponding to the 𝑚 $ = ±1 eigenstates change linearly with the field projection 𝐵 () along the NV symmetry axis. Correspondingly, opticallydetected magnetic resonance (ODMR) between one of these two levels and 𝑚 $ = 0 -insensitive to 𝐵 () -allows one to measure the field projection amplitude. Full vector magnetometry is made possible by comparing the responses from NVs oriented along different crystalline axes 31 .

Figure 1a lays out our experimental setup: The system uses confocal microscopy to monitor the fluorescence from an individual NV in an all-diamond probe, in turn connected to the tuning fork of an atomic force microscope (see Supplementary Information (SI), Section I and Figs. S1 andS2). The sample we study is a 500-nm-thick film of Tl 2 Ba 2 CaCu 2 O 8 (here referred to as TBCCO), a Spin dynamics of a solid-state qubit in proximity to a superconductor Richard Monge 1,2 , Tom Delord 1 , Nicholas Proscia 1 , Zav Shotan 1 , Harishankar Jayakumar 1 , Jacob Henshaw 1 , Pablo R. Zangara 1 , Artur Lozovoi 1 , Daniela Pagliero 1 , Pablo D. Esquinazi 3 , Toshu An 4 , Inti Sodemann 5,6 , Vinod M. Menon 1,2 , Carlos A. Meriles 1,2, * A broad effort is underway to understand and harness the interaction between superconductors and spin-active color centers with an eye on the realization of hybrid quantum devices and novel imaging modalities of superconducting materials. Most work, however, overlooks the complex interplay between either system and the environment created by the color center host. Here we use an all-diamond scanning probe to investigate the spin dynamics of a single nitrogen-vacancy (NV) center proximal to a high-critical-temperature superconducting film in the presence of a weak magnetic field. We find that the presence of the superconductor increases the NV spin coherence lifetime, a phenomenon we tentatively rationalize as a change in the electric noise due to a superconductor-induced redistribution of charge carriers near the NV site. We build on these findings to demonstrate transverse-relaxation-time-weighted imaging of the superconductor film. These results shed light on the complex surface dynamics governing the spin coherence of shallow NVs while simultaneously paving the route to new forms of noise spectroscopy and imaging of superconductors.

superconductor material with critical temperature 𝒯 , ≈ 105 K 32 . The film was patterned into square patches of variable size, allowing us to probe superconductivity-induced changes in the NV response as we approach the sample edges (SI, Section II and Fig. S3). We operate our system as an atomic-force or confocal fluorescence microscope, and isolate for inspection a corner in a representative TBCCO patch. Topographic imaging reveals a rugged surface (Fig. 1b), common for this type of sample 33 . On the other hand, Fig. 1c reproduces the NV magnetometry image of the same corner as derived from the frequency shift in the NV fluorescence dip under continuous microwave (mw) excitation (here referred to as "ODMR signal", see SI, Fig. S2). Operating at 𝒯 = 69 K -well below the critical temperature -we observe a reduction of the applied magnetic field on the superconductor film, an indication of (partial) magnetic shielding 15 . Note that the measured profile does not completely match the geometry of the film, a consequence of the varying magnetic field projection on the NV axis. Good agreement, however, can be attained with the help of a second, differently-oriented NV, which also allows us to recover a map of the field orientation (SI, Section III, and Fig. S4).

One of the advantages of NV sensing is the ability to implement time-resolved magnetometry protocols, of interest given the complementary information they convey. Figures 2a and2b present an example in the form of a Hahn-echo (HE) sequence on the NV spin probe. We observe "revivals" of the echo amplitude stemming from bulk 13 C spin precession at the applied magnetic field 34 , whose local value can be extracted from the time interval between them and the known 13 C gyromagnetic ratio. Comparing the NV response above the superconductor or the lanthanum aluminate (LAO) substrate, we observe not only a change in the revival period but also in their relative amplitudes, indicative of a varying spin coherence time. As shown in Figs. 2c and2d, this change stems from proximity to the superconductor, vanishing as we retract the tip by a few microns. Note that unlike the case in Fig. 2b, the revival peak times in Fig. 2c remain unchanged with distance, implying that the magnetic field amplitude is constant within the probed range. Further, we observe comparable fractional 𝑇 2 changes even in the absence of refocusing pulses -the so-called Ramsey protocol, see SI, Section IV and Fig. S5 -indicating that the TBCCOinduced noise reduction extends to near-zero frequencies.

Interpreting the above findings is involved since shallow carriers and surface paramagnetic centers create a complex environment featuring alternative electric and magnetic noise mechanisms [21][22][23]35 . Under the assumption of Gaussian, stationary processes leading to pure dephasing, the noise spectral density 𝑆 𝜔 relates to the "coherence functional" 𝜒 𝑡 via the integral equation 36

In the above expression, 𝐶 𝑡 denotes the in-phase probe spin coherence amplitude after a total time 𝑡 = 𝑛𝜏, and 𝐹 L 𝜔𝑡 is a filter function intrinsic to the n-pulse protocol 36 . It follows from Eq. (1) that different noise sources contribute to the spin coherence decay with characteristic transverse relaxation times that depend both on the mechanism at play and the control protocol in use.

In order to unravel the system dynamics, we start by considering magnetic interactions of the NV with its environment. Meissner shielding from magnetic fluctuations -arising, e.g., from reconfigurations of paramagnetic centers or transient currents on the diamond tip surfacecould hypothetically lead to longer coherences lifetimes, but numerical modeling indicates the effect is marginal (see SI, Section V and Figs. S6 andS7).

The NV center, however, is also sensitive to changes in the distribution of local charges due to the impact of electric fields on the electronic orbitals defining the ground spin triplet 37,38 . In particular, prior work with shallow NVs in bulk crystals has reported longer coherence lifetimes in the presence of liquids with high dielectric constants, suggesting that electric noise from reconfiguring surface charges can play a major role 25,39 . In the present case, the superconductor response can be modeled through a collection of "mirror charges" cancelling the transverse electric field on the TBCCO surface (Fig. 3a). Using a model of Ohmic conduction that accounts for the electrostatic energy cost of charge fluctuations (SI, Section VII), we write the electric noise density from surface carriers at the NV site as

where 𝜌 is the surface resistivity of the diamond tip, 𝛿 || is the NV-electric field coupling parameter, ε is the dielectric constant of diamond, and ℏ is the reduced Planck constant. Eq. (2) shows that 𝑆 5 𝜔 increases as the system becomes less conducting, precisely the case for oxygenterminated diamond (where the surface resistivity falls in the range 10 10 -10 17 Ohm [40][41][42] ). As shown in Fig. 3b, it also captures the 𝜔 b2 dependence observed via spin-lattice relaxation measurements 22,27,43 (impractical herein given the long time scales under cryogenic temperatures). In particular, we find for 𝜌~4×10 [\ Ohm a cutoff frequency 𝜔 c ≡ 8 3 𝜖𝜌𝑑 b[ ~170×10 \ rad•s -1 , consistent with the higher frequency range probed in these prior studies (0.1-1 MHz). Further, we calculate coherence lifetimes in the tens of microseconds, in qualitative agreement with our observations.

Unlike the case for magnetic noise (SI, Sections V and VI), Eq. (2) shows no explicit 𝑧-dependent shielding, a counterintuitive result that stems from the superconductorinduced self-screening of surface carriers (SI, Section VII). The presence of image charges in the TBCCO film, however, effectively lowers the inter-carrier repulsion energy on the tip wall closest to the sample, which correspondingly boosts the local surface carrier density (Fig. 3c). In the low-frequency regime applicable here (i.e., where 𝜔 ≲ 𝜔 c and 𝑆 5 𝜔 is frequency insensitive), the corresponding reduction of the diamond surface resistivity -inversely proportional to the carrier concentration -leads to a concomitant growth of the spin coherence lifetime, in qualitative agreement with our observations. This is explicitly shown in Fig. 3d where we combine Eqs. ( 1) and (2) to calculate the NV signal envelope under a Hahn-echo protocol; proximity to the superconductor halves the surface resistivity and leads to NV signal changes comparable to those seen in Fig. 2. Note that while a sheet of metal could arguably lead to similar effects, the magnetic 10,44 and electric 11,45 noise it produces -absent in a superconductor material [6][7][8][9][10][11][12] but strong in metallic systemsmay counter any coherence lifetime gains.

Interestingly, electric noise does not seem to be singlehandedly responsible for the observed NV spin dynamics. This is shown in Figs. 3e and3f where we extend the observations of Fig. 2 to determine the NV transverse relaxation time in the case of multi-pulse spin control protocols. These measurements provide complementary information as pulse trains with greater number of inversion pulses tend to probe a higher frequency range (in turn, a consequence of the short inter-pulse spacing required for efficient spin decoupling). We find qualitative agreement with our observations only if we also include a second, "background" noise source insensitive to the superconductor (faint black trace in Fig. 3b). One such contribution could be the noise created by surface paramagnetic defects (e.g., in the form of dangling bonds); note that since carriers trapped on the diamond surface can also give rise to paramagnetic centers 35 , our findings point to interwoven electric and magnetic (spin) noise sources of comparable impact.

The relation between NV coherence and superconductor proximity can, in principle, be exploited to reconstruct a 𝑇 2 -based map of the film. Similar to known techniques in magnetic resonance imaging 46 , this form of transverse spin relaxometry -based on the Hahn echo or tailored to multipulse trains -conveys valuable information, particularly because the timed structure of the protocol can be potentially adapted to expose dynamical processes not apparent through standard ODMR sensing 47 . In the present case, however, the implementation is challenging because supercurrents induced by the antenna on the TBCCO patch can screen or enhance the mw field as the NV probe moves from one point to the next. We gain a crude appreciation of the superconductor impact on the local mw through Fig. 4a, displaying the duration of the mw pulse required for NV spin inversion along a straight line partly overlapping with the superconductor film. After reaching a local minimumstemming from mw-field enhancement near the edges -we measure long pulse durations as we enter the superconductor, indicating mw field shielding.

To circumvent this problem, we modify our protocol so as to scale the mw power at each point during the scan in a way that maintains unaltered the duration (and net spin rotation) of all pulses in the sequence; unlike the case where pulses change their length, this strategy guarantees NV spin control over a constant excitation bandwidth (see SI, Section VIII and Fig. S8). Further, we shift at each point the mw frequency -varying across the probed section, Fig. 4b-so as to ensure resonant NV spin manipulation at all positions (Fig. 4c). Figure 4d shows the result: Consistent with our prior findings, we observe a growth of the spin coherence lifetime when the NV hovers above the superconductor film (right section of the plot). Intriguingly, we also find a local reduction of 𝑇 2,45 near the TBCCO edge, which roughly correlates with the observed mw and magnetic field minima across the same path. We hypothesize this change could stem from a surface charge redistribution away from the pillar as other sections of the cantilever -in the form of a 5 µm wide board, see Suppl. Fig. 2a -get close to the superconductor edge during the scan; additional work, however, will be necessary to gain a fuller understanding.

In summary, our results show that proximity to a superconductor can extend the coherence lifetime of a shallow NV center. This behavior is reminiscent of prior work on cold atoms or ions trapped near metallic and superconducting electrodes [6][7][8][9][10][11][12] , although the complex nature of the diamond surface makes the mechanisms at play herein especially hard to unravel. Our theoretical modeling suggests that electric noise plays a major role in the limit of high surface resistivity (the case for oxygen-terminated diamond 40,48 ). Longer coherence lifetimes are possible thanks to a superconductor-induced boost in the local surface electron density. We warn, however, that a detailed microscopic modeling of a resistivity change in a strongly insulating system is difficult, implying that the proposed noise suppression process must be seen as tentative. Other possible mechanisms contributing to a change in surface resistivity include the drop of the carrier thermal activation gap from screening 49 , and a screening-induced reduction of the disorder potential experienced by charge carriers.

Our results also reveal a close inter-dependence between electric and magnetic noise sources, one growing with the surface resistivity, the other with the surface conductivity (SI, Sections VI and VII). Further, analysis of the system response for different control protocols suggests spin noisecomparable in magnitude to the electric contribution but insensitive to the action of the superconductor -is also 1 G. Kurizki, P. Bertet, Y. Kubo present. Spin and electric noise are likely related because an increase in the electron surface density should be accompanied by the enhanced depletion of donors in the NV vicinity (most notably, neutral nitrogen) 48,50 . While N 0 is paramagnetic, N + ions are spin-less, implying this class of process also reduces the magnetic noise (without the need for Meissner shielding).

The interplay between the superconducting sample, the spin probe, and the environment created by its crystal host serves as the basis for new imaging modalities. Besides the transverse relaxation measurements shown here, proximity to the superconductor also produces changes in the shape and overall structure of the echo revival pattern, particularly in cases where the NV strongly couples to an adjacent 13 C spin (not shown here for brevity). This form of contrasttentatively attributed to Stark shifts produced by reconfigured donors around the NV 38 -could be advantageous as the required evolution time is comparatively shorter.

Looking forward, an interesting possibility is to exploit time-resolved NV magnetometry to derive the noise spectral density of the superconductor film from a comparison to a reference far away from its surface. This approach could help monitor thermally activated processes in type-II superconductors such as vortex creep and depinning 51 , or spectroscopically characterize spin-and charge-induced noise in superconducting qubits 52 . In the latter case, nmresolved decoherence imaging of Josephson junctions seems particularly attractive provided tip contamination stemming from close sample proximity -a problem herein, see SI, Section I -can be mitigated.

Figure S1 displays a block diagram of our instrument, a scanning probe-assisted confocal/ODMR microscope adapted to a closed-cycle cryo-workstation (Montana Instruments). Two independent nano-positioning stacks within the workstation chamber are used to vary the relative location of the sample and atomic force microscopy (AFM) probe. Optical excitation is carried out with a continuouswave (cw) laser at 532 nm and an acousto-optic modulator (AOM) to generate light pulses with ~10 ns temporal resolution. We use a 0.7-NA, 3.2-mm-working-distance objective for sample and/or scanning probe illumination and photon collection; although within the cryo-workstation main chamber, the objective sits on the thermally insulated, notcryo-cooled section of the platform, thus remaining near room temperature at all times. Under normal operation, fluorescence imaging is attained by scanning the sample while both the laser beam and the cantilever remain fixed.

Except for the objective, all control stages sit on an actively-cooled, floating breadboard featuring vibration isolation technology with a very low resonance frequency that effectively transforms rapid vibrations into very slow displacements. As a result, the distance between the tip and the sample surface remains virtually unchanged at all times, with relative displacements below 1 nm. Integrated control of all capabilities -including sample/tip positioning, magnetic resonance protocol design, and optical/atomic force microscopy -is attained via a home-made user interface.

Throughout the present experiments, we use commercial all-diamond scanning probes (Qnami), each hosting a few NV centers derived from 14 N implantation; these probes have the geometry of a "diving board" with a protruding, 200-nmwide cylinder hosting a few NVs ~10 nm from the scanning surface 53 (Fig. S2a). The spin probe fluorescence is coupled into a single mode fiber (also serving as the confocal pinhole, not shown) and detected with the help of a dichroic mirror and two avalanche photo-detectors (APD) in the Hanbury-Brown-Twiss geometry; a time-correlated photon counting module is used to establish photon correlations when required.

We implement NV ODMR through the use of a microwave signal generator (Rohde-Schwartz SMB100A) and a 30 W broadband amplifier (Mini-Circuits ZHL-16W-43-S+). For time-resolved measurements, we use a 3-nsrisetime switch to generate microwave pulses, which we deliver here with the help of a 25-µm-diameter copper wire overlaid on the sample surface. Unless otherwise noted, we monitor the 𝑚 $ = 0 ↔ 𝑚 $ = -1 transition. The optical contrast in these NVs typically reaches optimal values (12-15% for ODMR measurements) with photon counts of up to 400 Kcts•s -1 under saturation conditions thanks to preferential wave-guiding by the diamond pillar (Fig. S2b). The magnetic field from a permanent magnet allows us to spectroscopically separate NVs with different orientations. Figure S2c shows an example where individual NVs along two crystallographic axes can be seen.

We observe large variations in the Hahn-echo lifetime 𝑇 2,45 , ranging from ~5 µs to 75 µs depending on the tip we use. Further, we find NVs in these tips exhibit lower fluorescence at low temperatures and are prone to blinking and/or sudden quenching (especially below ~50 K, see Figs. S2d andS2e), a major complication that translates into frequent tip replacement. We hypothesize this problem stems from charge instabilities exacerbated under vacuum and cryogenic conditions.

The superconductor sample is a 500-nm-thick film of TBCCO 2212 on a 0.4 mm lanthanum aluminate (LAO) substrate 54 . The sample was manufactured by DuPont and kept in dry storage since fabrication. As shown in Fig. S3a, the superconductor substrate was diced into 5×5 mm² squares and patterned through optical lithography with negative resist (NR9-1000P Futurex Inc) and wet etching 55 (0.5 mol L -1 citric acid aqueous solution for 45 min, followed by a 5 s wash in an alkaline solution RD6 Futurex Inc which was found to lower re-deposition). The sample was kept under atmospheric pressure for weeks before use.

Resistance measurements provided by the seller gave a surface resistance below 3×10 -4 Ω at 80 K. In Fig. S3b, an estimation of 𝒯 , was performed by measuring the resistance of a 5×5 mm² square sample with two leads wire-bonded to the sample. A sharp drop of the resistance is visible between 90 K and 110 K corresponding to the superconducting transition. We attribute the residual resistance 𝑅 h = 23 Ω to the poor contact between the superconducting film and the measurement leads.

We have observed that close tip proximity to the TBCCO surface leads to fast degrading of the NV fluorescence, which first becomes unstable and ultimately decays to negligible values. We hypothesize this behavior derives from tip contamination, though we presently ignore the mechanisms at play. To extend the tip lifetime, all experiments herein are limited to working distances of 150 nm or more.

Magnetic fields whose component perpendicular to the surface exceeds values of order ~10 mT are seen to quench the superconducting state, the reason why we limit our measurements to lower field amplitudes. Throughout our experiments, we typically cool down the sample below 𝒯 , after applying the external magnetic field. Zero field cooling, however, produced no observable changes in the TBCCO response; the latter includes the magnetometry measurements discussed immediately below.

Since the relative orientation of the applied magnetic field can influence the NV spin coherence lifetime 56 , we make use of ODMR magnetometry to map the impact of the TBCCO film on 𝐁 " . In the experiments of Fig. S4, we choose an NV center -here denoted NV 9 -whose crystallographic orientation coincides with that of NV 1 in Fig. 1 of the main text. The magnetic field magnitude 𝐵 " and polar angle 𝜃 ()j relative to the axis of NV 9 , can be obtained from the 𝑚 $ = 0 ↔ 𝑚 $ = ±1 transition frequencies -respectively denoted 𝜈 b and 𝜈 l , see Fig. S4a -using the relations 57

and 𝐷 = 2.877 GHz is the NV zero field splitting at 69 K. Figure S4b shows the measured frequencies for NV 9 across the TBCCO area of interest (left-hand side plots) and the corresponding maps of the field amplitude and angle relative to NV 9 (respectively, upper and lower images on the righthand side). We find a ~20% reduction of the magnetic field amplitude above the TBCCO film, a finding consistent with prior observations in similar samples 58 . Further, we observe only a minor change of the field direction (not exceeding ~8 deg. in the region of the TBCCO patch), which allows us to conclude that the increase of the NV coherence lifetimes cannot stem from superconductor-induced changes in 𝐁 " .

A closer inspection of the magnetometry images in Fig. S4b shows they do not accurately reproduce the rectangular geometry of the TBCCO patch (dashed white line), a consequence of the limited sensitivity of Eqs. (S1) and (S2) to small field changes. Fortunately, however, the diamond tip used for these experiments hosts NVs with two different crystallographic orientations (Fig. S4a), implying that additional information can be attained by simultaneously monitoring the response of both NVs. One possibility is to make use of Eqs. (S1) and (S2) to find the angle 𝜃 ()[N between 𝐁 " and NV 10 . We also resort to the magnetic field 𝐁 y from an ancillary coil below the TBCCO film (Fig. S4c) to determine the orientation of the plane containing NV 9 and NV 10 relative to the sample surface. From these constraints, we find that in the absence of the superconductor, 𝐁 " forms an angle 𝜃 " N ≈ 43 deg. with the sample surface; by the same token, the angle relative to the plane formed by NV 9 and NV 10 -perpendicular to the sample surface, see Fig. S4d -is 𝜙 " N ≈ 32 deg.

Alternatively, we can combine the frequency shifts experienced by NV and

where 𝜀 is the angle between the two NV axes, here considered to be 70.5 deg., and 𝜃 ()j || denotes the angle between 𝐁 " || and NV 9 . Fig. S4e shows the results: The projected field amplitude (upper right-hand side image) recaptures the rectangular geometry of the TBCCO patch, including the attenuated Meissner shielding of 𝐁 " near the boundaries 59 . We warn, however, that the above equations consider 𝜈 l and 𝜈 b symmetrically distributed about the midpoint -a crude approximation in the present case -and must therefore be seen as approximate.

Lastly, we observe no substantial changes in the measured magnetometry images if we increase the tip separation to a few microns (not shown here for brevity). This observation indicates that the field varies comparatively slowly as a function of distance, and hence does not correlate with the observed changes in the NV spin coherent response.

Whereas Hahn-echo and CPMG measurements yield information on the magnetic noise spectrum over select frequency bandwidths, we can also probe the overall magnetic noise amplitude via the dephasing time under free evolution 60 , 𝑇 2 * . We perform 𝜋/2 -𝜏 -𝜋/2 Ramsey sequences at two locations, namely, away from and on the superconductor at a vertical distance of 150 nm. We follow the same procedure as in the echo sequences: At each location, we first evaluate the spin resonance and Rabi frequency; to mitigate slow instrumental fluctuations, Ramsey measurements are performed at a frequency close to (but slightly away from) resonance. Due to the NV hyperfine coupling to the native 14 N nuclear spin, Ramsey oscillations occur at three different frequencies corresponding to nuclear spin projections 𝑚 Ö = 0, ± 1. To model the NV response, we first perform for each measurement a lower power ODMR as shown in Figs. S5a andS5b. The relative position of each peak is subsequently used to set parameter bounds on the Ramsey signal fit. Figures S5c andS5d show the Ramsey oscillations measured off and on the superconductor along the corresponding fits. We obtain 𝑇 2 * values of 2.6 ± 0.4 µs and 2.0 ± 0.16 µs respectively on and off the superconductor, a 1.5-fold enhancement consistent with the Hahn-echo and CPMG observations in the main text.

The coherence of near-surface NV spins has been shown to be impacted by fluctuating magnetic fields created by surface paramagnetic impurities 21,22 . Close to the superconductor, the Meissner effect tends to suppress these fluctuations, therefore enhancing the NV spin coherence. As presented in Fig. S6a, we consider an ideal superconductor generating a Meissner field equal to the field generated by the mirror image of the source spin impurity relative to the superconductor interface assuming no penetration depth 61 . For an uncorrelated spin bath, the magnetic field noise spectral density can be written as 21,62 where 1/γ á and 𝐵 á respectively denote the correlation time and field created by the 𝑘-th spin. For a bath with homogeneous correlation time, the noise at a given frequency is proportional to the mean square magnetic field:

The effect of the Meissner effect on coherence is therefore obtained by calculating the ratio of the mean square field with and without the superconductor. For simplicity, we consider the NV spin and the bath spin impurities aligned along the zaxis, with the spin impurities homogeneously distributed on the surface of a cylindrical diamond pillar; we also assume the continuous limit, neglecting the effects of discrete spin impurity sites. Denoting 𝐵 âäh $y 𝑧/𝑑 the root-mean-square spin noise field in the presence of the superconductor, we show in Fig. S6b the suppression factor, 𝐵 âäh $y 𝐵 âäh , as a function of the relative distance to the sample surface 𝑧/𝑑. Stronger suppression of the noise is observed when the NV lies further away from the spin impurities, which favors a better screening by the superconductor. The tip-surface distance, 𝑧, in our experiments typically ranges from 150 to 350 nm. On the other hand, the NV spin probes we use have implantation energies of 6 and 12 keV. For those energies, simulations put the NV depth in the 5-13 nm and 11-23 nm ranges, respectively, but experiments performing AFM scans with the same commercial tips have found NV distances in the 60-79 nm range 53 . For a perfect superconductor, we therefore obtain a coherence enhancement due to Meissner shielding ranging from 0.003% (9 nm deep NV) to 3.5% at best (79 nm deep NV), more than one order of magnitude weaker than the observed effect.

In this section, we discuss the contribution to the local magnetic noise at the NV site created by orbital electric currents on the diamond surface (to which we assign a surface conductivity 𝜎). The model presented in Fig. S7a is based on a picture analogous to that shown in the main text: Current fluctuations give rise to magnetic field noise at the NV center via the Biot-Savart law. We model the superconductor as an ideal "current shield" meaning that for any physical current on the diamond surface there is an equal and opposite virtual current inside the superconductor at a distance 2𝑧 from the tip surface, hence guaranteeing a vanishing normal magnetic field at the TBCCO surface. While the above is a crude approximation that ignores the finite penetration depth in the superconductor, our calculation can be seen as the best-case scenario for the purposes of noise reduction.

We follow closely the derivations of Refs. [63] and [64] (see also Ref. [65] for a related discussion). We focus for concreteness on the fluctuations of the component of the magnetic field along the 𝑧-direction. Then Eq. (G12) of Ref. [64] can be rewritten as 𝐵 å 𝐪, 𝑡 = -𝑖 𝜇 N 2 𝑒 b|ë|í -𝑒 b|ë| íl2ì 𝑗 î (𝐪, 𝑡), S7

where the first term is the contribution to the field from a transverse current 𝑗 î fluctuation with wave-vector q on the diamond surface, and the second one is the contribution from the virtual image current in the superconductor. Following a similar analysis, one arrives at a slightly modified version of Eq. (G25) in Ref. [64], relating the magnetic field response function 𝜒 U ï ,U ï 𝜔 to the transverse conductivity of the diamond surface

In general, the transverse conductivity 𝜎 î can have a nontrivial dependence on wave-vector and frequency, but since in our case the NV center is located at a distance larger than the mean-free path and we probe frequencies much smaller than the scattering rate, it is legitimate to express 𝜎 î as the frequency and wave-vector-independent Drude conductivity, i.e., 𝜎 î ≈ 𝜎. Upon performing the integral in Eq. (S8), we write the magnetic field noise as where we used the NV spin gyromagnetic coupling 𝛾 ò and the fluctuation dissipation theorem to express the spectral density in terms of the magnetic response function, i.e., 𝑆 ñ = 𝛾 ò 2 𝜒 U ï ,U ï coth ℏ𝜔/2𝑘 ñ 𝒯 /2; Eq. (S9) results after taking the high-temperature limit.

Figure S7b shows the calculated ratio between the Hahnecho coherence lifetimes in the presence of magnetic noise, 𝑇 2,45 U 𝑧, 𝑑 𝑇 2,45 U 𝑧 → ∞, 𝑑 for different NV depths (including the change that derives from charge accumulation at the tip, see main text). We find that the fractional change of the transverse spin lifetime trends to values lower than those observed. Further, the low surface conductivity of oxygen-terminated diamond [40][41][42] of order 10 -6 -10 -14 s -1 , several orders of magnitude below the experimental results. While uncertainty in the composition of the diamond surface makes absolute comparisons difficult, the large discrepancy allows us to rule shielding of magnetic interactions as the leading mechanism in our observations.

Here, we discuss the contribution to the local electric noise at the NV center created by charge density fluctuations on the diamond surface. For our purposes, the TBCCO film can be crudely seen as an ideal metal whose action can be modeled through image charges, instantaneously cancelling the electric field component parallel to the superconductor surface (Fig. 4a). By solving the electrostatic Poisson equation, one arrives at the counterpart to Eq. (S7) relating the normal electric field to charge density fluctuations 𝜌 𝐪, 𝑡 of wave-vector q on the diamond surface 𝐸 å 𝐪, 𝑡 = 1 2𝜀 𝑒 b|ë|í -𝑒 b|ë| íl2ì 𝜌 𝐪, 𝑡 . S10

For simplicity, Eq. (S10) assumes the system is embedded in a medium of uniform dielectric constant 𝜀. Following an analysis similar to that in Ref. [64], one arrives at the counterpart to Eq. (S8), namely

Here, 𝜒′ † † 𝒒, 𝜔 is the imaginary (dissipative) part of the charge density susceptibility of the diamond surface. We model this response function assuming a two-dimensional hydrodynamic system based on Ohmic conduction. The charge density obeys the following equations 𝜕 £ 𝜌 + ∇ • 𝑗 = 0, S12 𝜕 £ 𝑗 + 𝛤 𝑗 = 𝐷 𝐸 hò ¶ß + 𝐸 ò®© , S13

where 𝐷 = 𝜂𝑒 2 /𝑚 N is the Drude weight for carriers of concentration 𝜂, charge 𝑒, and mass 𝑚 N , 𝛤 is the carrier scattering rate, 𝐸 hò ¶ß is the electric field generated by the fluctuating charge density itself, and 𝐸 ò®© is an external perturbing electric field introduced for purposes of computing the response functions. After Fourier transforming, these electric fields can be written as 𝐸 hò ¶ß = 𝑖𝑞𝑉 ë y≠AE ¶ 𝛿𝜌 ë , S14

𝐸 ò®© = -𝑖 𝒒𝜙 ë ò®© , S15

where 𝑉 ë where 𝑡 á is the time of the k-th pulse in the train, and 𝜏 ∂ is the pulse duration.

Heterogeneities in the amplitude and direction of the applied mw and magnetic fields -present in our system at temperatures below 𝑇 , -complicate the of sample scans on the sequential application of pulsed NV control sequences. To circumvent this problem, we implement a Rabi protocol at each location, which we then use to determine the local duration of the mw π-pulse (Fig. S8a). We also make use of our ODMR magnetometry images -literally, mapping the NV 𝑚 $ = 0 ↔ 𝑚 $ = -1 transition frequencies at all positions throughout the scanned area -to maintain the mw excitation on-resonance irrespective of the tip position. During the scan, we build on this information to adjust the mw frequency and amplitude so as to ensure that 53 A. Finco, A. Haykal, R. Tanos, F. Fabre, S. Chouaieb, W. Akhtar, I. Robert-Philip, W. Legrand, F. Ajejas, K. Bouzehouane, N. Reyren, T. Devolder, J-P. Adam, J-V. Kim, V. Cros, V. Jacques, "Imaging non-collinear antiferromagnetic textures via single spin relaxometry", Nat Commun 12, 767 (2021). the length of each pulse in the protocol remains unchanged (Fig. S8b). As the pulse length inversely relates to the excitation bandwidth, this strategy eliminates the signal distortions otherwise arising from the use of longer pulses with nominally equivalent spin rotations.