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Diamond is a promising platform for quantum information processing as it can host highly coherent qubits that could allow for the construction of large quantum registers. A prerequisite for such devices is a coherent interaction between nitrogen vacancy (NV) electron spins. Entanglement between dipolar-coupled NV spin pairs has been demonstrated, but with a limited entanglement fidelity and its error sources have not been characterized. Here, we design and implement a robust, easy to implement entangling gate between NV spins in diamond and quantify the influence of multiple error sources on the gate performance. Experimentally, we demonstrate a record gate fidelity of $F=(96.0 \pm 2.5)$ % under ambient conditions. Our identification of the dominant errors paves the way towards NV-NV gates beyond the error correction threshold.

The surface conductivity of hydrogen-terminated diamond is a topic of great interest from both scientific and technological perspectives. This is primarily due to the fact that the conductivity is exceptionally high without the need for substitutional doping, thus enabling a wide range of electronic applications. Although the conductivity is commonly explained by surface transfer doping due to air-borne surface acceptors, there remains uncertainty regarding the main determining factors that govern the degree of band bending and hole density, which are crucial for the design of electronic devices. Here, we elucidate the dominant factor influencing band bending by creating shallow nitrogen-vacancy (NV) centers beneath the hydrogen-terminated diamond surface through nitrogen ion implantation at varying fluences. We measured the photoluminescence and optically detected magnetic resonance (ODMR) of the NV centers, as well as the surface conductivity, as a function of the nitrogen implantation fluence. The disappearance of the conductivity with increasing nitrogen implantation fluence coincides with the appearance of photoluminescence and ODMR signals from negatively charged NV centers. This finding indicates that band bending is not exclusively determined by the work-function difference between diamond and the surface acceptor material, but by the finite density of surface acceptors. This work emphasizes the importance of distinguishing work-function-difference-limited band bending and surface-acceptor-density-limited band bending when modeling the surface transfer doping, and provides useful insights for the development of devices based on hydrogen-terminated diamond., Comment: 50 pages, 18 figures

A nitrogen-vacancy (NV) center in a diamond enables the access to an electron spin, which is expected to present highly sensitive quantum sensors. Although exploiting a nitrogen nuclear spin improves the sensitivity, manipulating it using a resonant pulse requires a long gate time owing to its small gyromagnetic ratio. Another technique to control nuclear spins is a conditional rotation gate based on dynamical decoupling, which is faster but unavailable for nitrogen spins owing to the lack of transverse hyperfine coupling with the electron spin. In this study, we generated effective transverse coupling by applying a weak off-axis magnetic field. An effective coupling depends on the off-axis field; the conditional rotation gate on the nitrogen-14 spins of an NV center was demonstrated within 4.2 {\mu}s under an 1.8% off-axis field and a longitudinal field of approximately 280 mT. We estimated that a population transfer from the electron to nitrogen spins can be implemented with 8.7 {\mu}s. Our method is applicable to an ensemble of NV centers, in addition to a single NV center.

Transform-limited photon emission from quantum emitters is essential for high-fidelity entanglement generation. In this study, we report the coherent optical property of a single negatively-charged lead-vacancy (PbV) center in diamond. Photoluminescence excitation measurements reveal stable fluorescence with a linewidth of 39 MHz at 6 K, close to the transform-limit estimated from the lifetime measurement. We observe four orders of magnitude different linewidths of the two zero-phonon-lines, and find that that the phonon-induced relaxation in the ground state contributes to this huge difference in the linewidth. Due to the suppressed phonon absorption in the PbV center, we observe nearly transform-limited photon emission up to 16 K, demonstrating its high temperature robustness compared to other color centers in diamond., Comment: 13 pages,4 figures

We report the narrow inhomogeneous distribution of the zero-phonon line from tin-vacancy (SnV) centers in diamond and the overlap of spectra from multiple separated SnV centers. Photoluminescence excitation spectroscopy measurements at a cryogenic temperature showed that SnV centers exhibit stable fluorescence and linewidths close to the Fourier transform-limited linewidth. The inhomogeneous distribution was as low as ~4 GHz, which enabled the observation of Sn isotope-dependent resonant frequencies. Owing to the narrow inhomogeneous distribution, we observed multiple SnV centers showing identical photons with almost the same wavelength and linewidth. Identical SnV centers were also observed even in different diamond samples, confirming the reliable fabrication of the high-quality SnV centers.

Nanoscale nuclear magnetic resonance (NMR) signals can be measured through hyperfine interaction to paramagnetic electron sensor spins. A heterodyne approach is widely used to overcome the electron spin lifetime limit in spectral resolution. It uses a series of modified Hahn echo pulse sequences applied coherently with precession signal resulting in a subsampled NMR signal. Due to challenges with applying high electron Rabi frequencies its application is limited to low fields, thus the full potential of the method is not yet exploited at high magnetic fields, beneficial for NMR. Here we present heterodyne detection utilizing a series of phase coherent electron nuclear double resonance sensing blocks which extends nanoscale NMR protocols to arbitrary magnetic fields. We demonstrate this principle on a single NV center, both with an intrinsic $^{14}$N and a weekly coupled $^{13}$C nuclear spin in the bath surrounding single NV centres. We compare our protocol to existing heterodyne protocols and discuss its prospects. This work paves the way towards high field nanoscale heterodyne NMR protocols with NV centres which is crucial for reducing sample volumes and improving chemical resolution., Comment: 27 pages, 11 figures

Qubit technologies using nitrogen-vacancy color centers in diamonds require precise knowledge of the centers, including the number of emitters within a diffraction-limited spot and their orientations. However, the number of emitters is challenging to determine when there is finite background, which affects the precision of resulting quantum protocols. Here we show the photoluminescence (PL) intensity and quantum correlation (Hanbury Brown and Twiss) measurements as a function of polarization for one- and two-emitter systems. The sample was made by implanting low concentrations of adenine (C5H5N5) into a low nitrogen chemical vapor deposition diamond. This approach yielded well-spaced regions with few nitrogen-vacancy centers. By mapping the PL intensity and quantum correlation as a function of polarization, we can distinguish two emitter systems from single emitters with background, providing a method to quantify the background signal at implanted sites, which might be different from off-site background levels. This approach also provides a valuable new all-optical mechanism for the determination of one or two emitter systems useful for quantum sensing, communication, and computation tasks.

The nitrogen-vacancy colour centre in diamond is emerging as one of the most important solid-state quantum systems. It has applications to fields including high-precision sensing, quantum computing, single photon communication, metrology, nanoscale magnetic imaging and biosensing. For all of these applications, a high quantum yield of emitted photons is desirable. However, diamond samples engineered to have high densities of nitrogen-vacancy centres show levels of brightness varying significantly within single batches, or even within the same sample. Here we show that nearby nitrogen impurities quench emission of nitrogen-vacancy centres via non-radiative transitions, resulting in a reduced fluorescence quantum yield. We monitored the emission properties of nitrogen-vacancy centre ensembles from synthetic diamond samples with different concentrations of nitrogen impurities. While at low nitrogen densities of 1.81 ppm we measured a lifetime of 13.9 ns, we observed a strong reduction in lifetime with increasing nitrogen density. We measure a lifetime as low as 4.4 ns at a nitrogen density of 380 ppm. The change in lifetime matches a reduction in relative fluorescence quantum yield from 77.4 % to 32 % with an increase in nitrogen density from 88 ppm to 380 ppm, respectively. These results will inform the conditions required to optimise the properties of diamond crystals devices based on the fluorescence of nitrogen-vacancy centres. Furthermore, this work provides insights into the origin of inhomogeneities observed in high-density nitrogen-vacancy ensembles within diamonds and nanodiamonds., Comment: 22 pages, 14 figures (4 figures in the main text, 10 figures in the supplementary information)

Nanodiamonds containing negatively charged nitrogen-vacancy (NV$^-$) centers are versatile room-temperature quantum sensors in a growing field of research. Yet, knowledge regarding the NV-formation mechanism in very small particles is still limited. This study focuses on the formation of the smallest NV$^-$-containing diamonds, 5 nm detonation nanodiamonds (DNDs). As a reliable method to quantify NV$^-$ centers in nanodiamonds, half-field signals in electron paramagnetic resonance (EPR) spectroscopy are recorded. By comparing the NV$^-$ concentration with a series of nanodiamonds from high-pressure high-temperature (HPHT) synthesis (10 - 100 nm), it is shown that the formation process in 5 nm DNDs is unique in several aspects. NV$^-$ centers in DNDs are already formed at the stage of electron irradiation, without the need for high-temperature annealing. The effect is explained in terms of "self-annealing", where size and type dependent effects enable vacancy migration close to room temperature. Although our experiments show that NV$^-$ concentration generally increases with particle size, remarkably, the NV$^-$ concentration in 5 nm DNDs surpasses that of 20 nm-sized nanodiamonds. Using Monte Carlo simulations, we show that the ten times higher substitutional nitrogen concentration in DNDs compensates the vacancy loss induced by the large relative particle surface. Upon electron irradiation at a fluence of $1.5 \times 10 ^{19}$ e$^-$/cm$^2$, DNDs show a 12.5-fold increment in the NV$^-$ concentration with no sign of saturation. These findings can be of interest for the creation of defects in other very small semiconductor nanoparticles beyond NV-nanodiamonds as quantum sensors.

Quantum sensors are keeping the cutting-edge sensitivities in metrology. However, for high-sensitive measurements of arbitrary signals, limitations in linear dynamic range could introduce distortions when sensing the frequency, magnitude and phase of unknown signals. Here, we overcome these limitations with advanced sensing protocol that combines quantum phase-sensitive detection with heterodyne readout. We present theoretical and experimental investigations using nitrogen-vacancy centers in diamond, showing the ability to sense radio signals with a 98 dB linear dynamic range, a 31 pT/Hz$^{1/2}$ sensitivity, and arbitrary frequency resolution. Further, we perform the quantum-assisted distortion-free audio signal (melody, speech) sensing with high fidelity. The methods developed here could broaden the horizon for quantum sensors towards applications in telecommunication, where high-fidelity and low-distortion at multiple frequency bands within small sensing volumes are required., Comment: 30 pages, 7 figures

Nonlinear spectroscopy is widely used for studying physical systems. Conventional nonlinear optical spectroscopy and magnetic resonance spectroscopy, which use classical probes such as electromagnetic waves, can only access certain types of correlations in a quantum system. The idea of quantum nonlinear spectroscopy was recently proposed to use quantum probes such as entangled photons to achieve sensitivities and resolutions beyond the classical limits. It is shown that quantum sensing can extract arbitrary types and orders of correlations in a quantum system by first quantum-entangling a sensor and the object and then measuring the sensor. Quantum sensing has been applied to achieve nuclear magnetic resonance (NMR) of single atoms and the second-order correlation spectroscopy has been adopted to enhance the spectral resolution. However, quantum nonlinear spectroscopy (i.e., the measurement of higher-order correlations) of single nuclear spins is still elusive. Here we demonstrate the extraction of fourth-order correlations of single nuclear spins that cannot be measured in conventional nonlinear spectroscopy, using sequential weak measurement via an atomic quantum sensor, namely, a nitrogen-vacancy center in diamond. We show that the quantum nonlinear spectroscopy provides fingerprint features to identify different types of objects, such as Gaussian noises, random-phased AC fields, and quantum spins, which would be indistinguishable in second-order correlations. The measured fourth-order correlation unambiguously differentiates a single nuclear spin and a random-phased AC field. This work constitutes an initial step toward the application of higher-order correlations to quantum sensing, to examining the quantum foundation (by, e.g., higher-order Leggett-Garg inequality), and to studying quantum many-body physics., Comment: 23 pages and 4 figures

A key challenge in cardiology is the non-invasive imaging of electric current propagation occurring in the cardiovascular system at an intra-cardiac scale. A promising approach for directly mapping the current dynamics is to monitor the associated stray magnetic field. However, in this magnetic field approach, the spatial resolution deteriorates significantly as the standoff distance between the target and the sensor increases. Existing sensors usually remain relatively far from the target and provide only centimetre-scale resolution because their operating temperature is not biocompatible. Here we demonstrate millimetre-scale magnetocardiography of living rats using a solid-state quantum sensor based on nitrogen-vacancy centres in diamond. The essence of the method is a millimetre proximity from the sensor to heart surface, which enhances the cardiac magnetic field to greater than nanoteslas and allows the mapping of these signals with intra-cardiac resolution. From the acquired magnetic images, we also estimate the source electric current vector, flowing from the right atria base via the Purkinje fibre bundle to the left ventricular apex. Our results establish the solid-state quantum sensor's capability to probe cardiac magnetic signals from mammalian animals and reveal their intra-cardiac electrodynamics. This technique will enable the study of the origin and progression of myriad cardiac arrhythmias including flutter, fibrillation, and tachycardia., Comment: 17 pages, 4 figures

The ability to follow the dynamics of a quantum system in a quantitative manner is of key importance for quantum technology. Despite its central role, justifiable deduction of the quantum dynamics of a single quantum system in terms of a macroscopical observable remains a challenge. Here we show that the relation between the readout signal of a single electron spin and the quantum dynamics of the single nuclear spin is given by a parameter related to the measurement strength. We determine this measurement strength in independent experiments and use this value to compare our analysis of the quantum dynamics with experimental results. We prove the validity of our approach by measuring violations of the Leggett-Garg inequality.

Nitrogen vacancy (NV) centers in diamond have developed into a powerful solid-state platform for compact quantum sensors. However, high sensitivity measurements usually come with additional constraints on the pumping intensity of the laser and the pulse control applied. Here, we demonstrate high sensitivity NV ensemble based magnetic field measurements with low-intensity optical excitation. DC magnetometry methods like, e.g., continuous-wave optically detected magnetic resonance and continuously excited Ramsey measurements combined with lock-in detection, are compared to get an optimization. Gradiometry is also investigated as a step towards unshielded measurements of unknown gradients. The magnetometer demonstrates a minimum detectable field of 0.3-0.7 pT in a 73 s measurement by further applying a flux guide with a sensing dimension of 2 mm, corresponding to a magnetic field sensitivity of 2.6-6 pT/Hz^0.5. Combined with our previous efforts on the diamond AC magnetometry, the diamond magnetometer is promising to perform wide bandwidth magnetometry with picotesla sensitivity and a cubic-millimeter sensing volume under ambient conditions., Comment: 20 pages, 12 figures

Magnetic field sensors that exploit quantum effects have shown that they can outperform classical sensors in terms of sensitivity enabling a range of novel applications in future, such as a brain machine interface. Negatively charged nitrogen-vacancy (NV) centers in diamond have emerged as a promising high sensitivity platform for measuring magnetic fields at room temperature. Transferring this technology from laboratory setups into products and applications, the total size of the sensor, the overall power consumption, and the costs need to be reduced and optimized. Here, we demonstrate a fiber-based NV magnetometer featuring a complete integration of all functional components without using any bulky laboratory equipment. This integrated prototype allows portable measurement of magnetic fields with a sensitivity of 344 pT/ SqrtHz.

Diamond quantum sensors are sensitive to weak microwave magnetic fields resonant to the spin transitions. However the spectral resolution in such protocols is limited ultimately by sensor lifetime. Here we demonstrate a heterodyne detection method for microwaves (MW) leading to a lifetime independent spectral resolution in the GHz range. We reference the MW-signal to a local oscillator by generating the initial superposition state from a coherent source. Experimentally we achieve a spectral resolution below $1\ \rm{Hz}$ for a $4\ \rm{GHz}$ signal far below the sensor lifetime limit of kilohertz. Furthermore we show control over the interaction of the MW-field with the two level system by applying dressing fields, pulsed Mollow absorption and Floquet dynamics under strong longitudinal radio frequency drive. While pulsed Mollow absorption leads to highest sensitivity, the Floquet dynamics allows robust control independent from the systems resonance frequency. Our work is important for future studies in sensing weak microwave signals in wide frequency range with high spectral resolution.

Quantum metrology makes use of coherent superpositions to detect weak signals. While in principle the sensitivity can be improved by increasing the density of sensing particles, in practice this improvement is severely hindered by interactions between them. Using a dense ensemble of interacting electronic spins in diamond, we demonstrate a novel approach to quantum metrology. It is based on a new method of robust quantum control, which allows us to simultaneously eliminate the undesired effects associated with spin-spin interactions, disorder and control imperfections, enabling a five-fold enhancement in coherence time compared to conventional control sequences. Combined with optimal initialization and readout protocols, this allows us to break the limit for AC magnetic field sensing imposed by interactions, opening a promising avenue for the development of solid-state ensemble magnetometers with unprecedented sensitivity., Comment: 7 + 34 pages, 4 + 10 figures. V2: Improved sensitivity number and presentation

We experimentally and theoretically investigate the Hahn echo decay curve for nitrogen vacancy centers in diamond with different spin concentrations. The Hahn echo results show a non-exponential decay for low spin concentrations, while an exponential decay is dominant for high spin concentrations. By fitting the decay curve with a theoretical model, we show that both the amplitude and correlation time of the environmental noise have a clear dependence on the spin concentration. These results are essential for optimizing the NV center concentration in high-performance quantum devices, particularly quantum sensors.

Quantum technology relies on proper hardware, enabling coherent quantum state control as well as efficient quantum state readout. In this regard, wide-bandgap semiconductors are an emerging material platform with scalable wafer fabrication methods, hosting several promising spin-active point defects. Conventional readout protocols for such defect spins rely on fluorescence detection and are limited by a low photon collection efficiency. Here, we demonstrate a photo-electrical detection technique for electron spins of silicon vacancy ensembles in the 4H polytype of silicon carbide (SiC). Further, we show coherent spin state control, proving that this electrical readout technique enables detection of coherent spin motion. Our readout works at ambient conditions, while other electrical readout approaches are often limited to low temperatures or high magnetic fields. Considering the excellent maturity of SiC electronics with the outstanding coherence properties of SiC defects the approach presented here holds promises for scalability of future SiC quantum devices.

Electron spins in solids constitute remarkable quantum sensors. Individual defect centers in diamond were used to detect individual nuclear spins with nanometer scale resolution, and ensemble magnetometers rival SQUID and vapor cell magnetometers when taking into account room temperature operation and size. NV center spins can also detect electric field vectors, despite their weak coupling to electric fields. %even that of an isolated fundamental charge, despite their weak coupling to electric fields. Here, we employ ensembles of NV center spins to measure macroscopic AC electric vector fields with high precision. We utilize low strain, $^{12}$C enriched diamond to achieve maximum sensitivity and tailor the spin Hamiltonian via proper magnetic field adjustment to map out the AC electric field strength and polarization and arrive at refined electric field coupling constants. For high precision measurements we combine classical lock-in detection with aspects from quantum phase estimation for effective suppression of technical noise. Eventually, this enables $t^{-1/2}$ uncertainty scaling of the electric field strength over extended averaging periods, enabling us to reach a sensitivity down to $10^{-7}$ V/$\mu$m.

We investigate thermalization dynamics of a driven dipolar many-body quantum system through the stability of discrete time crystalline order. Using periodic driving of electronic spin impurities in diamond, we realize different types of interactions between spins and demonstrate experimentally that the interplay of disorder, driving and interactions leads to several qualitatively distinct regimes of thermalization. For short driving periods, the observed dynamics are well described by an effective Hamiltonian which sensitively depends on interaction details. For long driving periods, the system becomes susceptible to energy exchange with the driving field and eventually enters a universal thermalizing regime, where the dynamics can be described by interaction-induced dephasing of individual spins. Our analysis reveals important differences between thermalization of long-range Ising and other dipolar spin models., Comment: 6 + 14 pages, 4 + 8 figures

Quantum sensors have recently achieved to detect the magnetic moment of few or single nuclear spins and measure their magnetic resonance (NMR) signal. However, the spectral resolution, a key feature of NMR, has been limited by relaxation of the sensor to a few kHz at room temperature. The spectral resolution of NMR signals from single nuclear spins can be improved by, e.g., using quantum memories, however at the expense of sensitivity. Classical signals on the other hand can be measured with exceptional spectral resolution by using continuous measurement techniques, without compromising sensitivity. To apply these techniques to single-spin NMR, it is critical to overcome the impact of back action inherent of quantum measurements. Here we report sequential weak measurements on a single $^{13}$C nuclear spin. The back-action of repetitive weak measurements causes the spin to undergo a quantum dynamics phase transition from coherent trapping to coherent oscillation. Single-spin NMR at room-temperature with a spectral resolution of 3.8 Hz is achieved. These results enable the use of measurement-correlation schemes for the detection of very weakly coupled single spins., Comment: Article: 16 pages, 4 figures Supplementary: 23 pages, 9 figures

We experimentally demonstrate a simple and robust optical fibers based method to achieve simultaneously efficient excitation and fluorescence collection from Nitrogen-Vacancy (NV) defects containing micro-crystalline diamond. We fabricate a suitable micro-concave (MC) mirror that focuses scattered excitation laser light into the diamond located at the focal point of the mirror. At the same instance, the mirror also couples the fluorescence light exiting out of the diamond crystal in the opposite direction of the optical fiber back into the optical fiber within its light acceptance cone. This part of fluorescence would have been otherwise lost from reaching the detector. Our proof-of-principle demonstration achieves a 25 times improvement in fluorescence collection compared to the case of not using any mirrors. The increase in light collection favors getting high signal-to-noise ratio (SNR) optically detected magnetic resonance (ODMR) signals hence offers a practical advantage in fiber-based NV quantum sensors. Additionally, we compacted the NV sensor system by replacing some bulky optical elements in the optical path with a 1x2 fiber optical coupler in our optical system. This reduces the complexity of the system and provides portability and robustness needed for applications like magnetic endoscopy and remote-magnetic sensing., Comment: 6 pages, 8 figures

Temperature sensing with nitrogen vacancy (NV) centers using quantum techniques is very promising and further development is expected. Recently, the optically detected magnetic resonance (ODMR) spectrum of a high-density ensemble of the NV centers was reproduced with noise parameters [inhomogeneous magnetic field, inhomogeneous strain (electric field) distribution, and homogeneous broadening] of the NV center ensemble. In this study, we use ODMR to estimate the noise parameters of the NV centers in several diamonds. These parameters strongly depend on the spin concentration. This knowledge is then applied to theoretically predict the temperature sensitivity. Using the diffraction-limited volume of 0.1 micron^3, which is the typical limit in confocal microscopy, the optimal sensitivity is estimated to be around 0.76 mK/Hz^(1/2) with an NV center concentration of 5.0e10^17/cm^3. This sensitivity is much higher than previously reported sensitivities, demonstrating the excellent potential of temperature sensing with NV centers., Comment: 17 pages, 4 figures, 1 table

Superradiance is the archetypical collective phenomenon where radiation is amplified by the coherence of emitters. It plays a prominent role in optics, where it enables the design of lasers with substantially reduced linewidths, quantum mechanics, and is even used to explain cosmological observations like Hawking radiation from black holes. Hybridization of distinct quantum systems allows to engineer new quantum metamaterials pooling the advantages of the individual systems. Superconducting circuits coupled to spin ensembles are promising future building blocks of integrated quantum devices and superradiance will play a prominent role. As such it is important to study its fundamental properties in hybrid devices. Experiments in the strong coupling regime have shown oscillatory behaviour in these systems but a clear signature of Dicke superradiance has been missing so far. Here we explore superradiance in a hybrid system composed of a superconducting resonator in the fast cavity limit inductively coupled to an inhomogeneously broadened ensemble of nitrogen-vacancy (NV) centres. We observe a superradiant pulse being emitted a trillion of times faster than the decay for an individual NV centre. This is further confirmed by the non-linear scaling of the emitted radiation intensity with respect to the ensemble size. Our work provides the foundation for future quantum technologies including solid state superradiant masers.

We designed a loop-gap microwave resonator for applications of spin-based hybrid quantum systems, and tested it with impurity spins in diamond. Strong coupling with ensembles of nitrogen-vacancy (NV) centers and substitutional nitrogen (P1) centers was observed. These results show that loop-gap resonators are viable in the prospect of spin-based hybrid quantum systems, especially for an ensemble quantum memory or a quantum transducer., Comment: 5 pages, 4 figures

Longitudinal relaxation is the process by which an excited spin ensemble decays into its thermal equilibrium with the environment. In solid-state spin systems relaxation into the phonon bath usually dominates over the coupling to the electromagnetic vacuum. In the quantum limit the spin lifetime is determined by phononic vacuum fluctuations. However, this limit was not observed in previous studies due to thermal phonon contributions or phonon-bottleneck processes. Here we use a dispersive detection scheme based on cavity quantum electrodynamics (cQED) to observe this quantum limit of spin relaxation of the negatively charged nitrogen vacancy ($\mathrm{NV}^-$) centre in diamond. Diamond possesses high thermal conductivity even at low temperatures, which eliminates phonon-bottleneck processes. We observe exceptionally long longitudinal relaxation times $T_1$ of up to 8h. To understand the fundamental mechanism of spin-phonon coupling in this system we develop a theoretical model and calculate the relaxation time ab initio. The calculations confirm that the low phononic density of states at the $\mathrm{NV}^-$ transition frequency enables the spin polarization to survive over macroscopic timescales.

Nanomechanical sensors and quantum nanosensors are two rapidly developing technologies that have diverse interdisciplinary applications in biological and chemical analysis and microscopy. For example, nanomechanical sensors based upon nanoelectromechanical systems (NEMS) have demonstrated chip-scale mass spectrometry capable of detecting single macromolecules, such as proteins. Quantum nanosensors based upon electron spins of negatively-charged nitrogen-vacancy (NV) centers in diamond have demonstrated diverse modes of nanometrology, including single molecule magnetic resonance spectroscopy. Here, we report the first step towards combining these two complementary technologies in the form of diamond nanomechanical structures containing NV centers. We establish the principles for nanomechanical sensing using such nano-spin-mechanical sensors (NSMS) and assess their potential for mass spectrometry and force microscopy. We predict that NSMS are able to provide unprecedented AC force images of cellular biomechanics and to, not only detect the mass of a single macromolecule, but also image its distribution. When combined with the other nanometrology modes of the NV center, NSMS potentially offer unparalleled analytical power at the nanoscale., Comment: Errors in the stress susceptibility parameters present in the original arXiv version have been corrected

Understanding quantum dynamics away from equilibrium is an outstanding challenge in the modern physical sciences. It is well known that out-of-equilibrium systems can display a rich array of phenomena, ranging from self-organized synchronization to dynamical phase transitions. More recently, advances in the controlled manipulation of isolated many-body systems have enabled detailed studies of non-equilibrium phases in strongly interacting quantum matter. As a particularly striking example, the interplay of periodic driving, disorder, and strong interactions has recently been predicted to result in exotic "time-crystalline" phases, which spontaneously break the discrete time-translation symmetry of the underlying drive. Here, we report the experimental observation of such discrete time-crystalline order in a driven, disordered ensemble of $\sim 10^6$ dipolar spin impurities in diamond at room-temperature. We observe long-lived temporal correlations at integer multiples of the fundamental driving period, experimentally identify the phase boundary and find that the temporal order is protected by strong interactions; this order is remarkably stable against perturbations, even in the presence of slow thermalization. Our work opens the door to exploring dynamical phases of matter and controlling interacting, disordered many-body systems., Comment: 6 + 3 pages, 4 figures

In nanoscale metrology applications, measurements are commonly limited by the performance of the sensor. Here we show that in nuclear magnetic resonance (NMR) spectroscopy measurements using single nitrogen-vacancy (NV) centers in diamond, the NV sensor electron spin limits spectral resolution down to a few hundred Hz, which constraints the characterization and coherent control of finite spin systems, and furthermore, is insufficient for high resolution NMR spectroscopy aiming at single molecule recognition and structure analysis of the latter. To overcome the limitation, we support an NV electron spin sensor with a nuclear spin qubit acting as quantum and classical memory allowing for intermediate nonvolatile storage of metrology information, while suppressing the deleterious back-action of the sensor onto the system under investigation. We demonstrate quantum and classical memory lifetimes of 8 ms and 4 minutes respectively under ambient conditions. Furthermore, we design and test measurement and decoupling protocols, which exploit such memory qubits efficiently. Using our hybrid quantum-classical sensor device, we achieve high resolution NMR spectra with linewidths of single spins down to 13 Hz. Our work is therefore a prerequisite for high resolution NMR spectroscopy on nanoscopic quantum systems down to the single level., Comment: 6 pages paper, 8 pages appendix, 7 figures

Statistical mechanics underlies our understanding of macroscopic quantum systems. It is based on the assumption that out-of-equilibrium systems rapidly approach their equilibrium states, forgetting any information about their microscopic initial conditions. This fundamental paradigm is challenged by disordered systems, in which a slowdown or even absence of thermalization is expected. We report the observation of critical thermalization in a three dimensional ensemble of $\sim 10^6$ electronic spins coupled via dipolar interactions. By controlling the spin states of nitrogen vacancy color centers in diamond, we observe slow, sub-exponential relaxation dynamics and identify a regime of power-law decay with disorder-dependent exponents; this behavior is modified at late times owing to many-body interactions. These observations are quantitatively explained by a resonance counting theory that incorporates the effects of both disorder and interactions., Comment: 16+31 pages, 4+12 figures, major updates on experiments, theory, and data analysis

We study the depolarization dynamics of a dense ensemble of dipolar interacting spins, associated with nitrogen-vacancy centers in diamond. We observe anomalously fast, density-dependent, and non-exponential spin relaxation. To explain these observations, we propose a microscopic model where an interplay of long-range interactions, disorder, and dissipation leads to predictions that are in quantitative agreement with both current and prior experimental results. Our results pave the way for controlled many-body experiments with long-lived and strongly interacting ensembles of solid-state spins., Comment: 15 pages, 8 figures

We design and implement 3D lumped element microwave cavities for the coherent and uniform coupling to electron spins hosted by nitrogen vacancy centers in diamond. Our design spatially focuses the magnetic field to a small mode volume. We achieve large homogeneous single spin coupling rates, with an enhancement of the single spin Rabi frequency of more than one order of magnitude compared to standard 3D cavities with a fundamental resonance at \SI{3}{GHz}. Finite element simulations confirm that the magnetic field component is homogeneous throughout the entire sample volume, with a RMS deviation of 1.54\%. With a sample containing $10^{17}$ nitrogen vacancy electron spins we achieve a collective coupling strength of $\Omega=\SI{12}{MHz}$, a cooperativity factor $C=27$ and clearly enter the strong coupling regime. This allows to interface a macroscopic spin ensemble with microwave circuits, and the homogeneous Rabi frequency paves the way to manipulate the full ensemble population in a coherent way., Comment: 3 figures

Magnetic resonance with ensembles of electron spins is nowadays performed in frequency ranges up to 240 GHz and in corresponding magnetic fields of up to 10 T. However, experiments with single electron and nuclear spins so far only reach into frequency ranges of several 10 GHz, where existing coplanar waveguide structures for microwave (MW) delivery are compatible with single spin readout techniques (e.g. electrical or optical readout). Here, we explore the frequency range up to 90 GHz, respectively magnetic fields of up to $\approx 3\,$T for single spin magnetic resonance in conjunction with optical spin readout. To this end, we develop MW resonators with optical single spin access. In our case, rectangular E-band waveguides guarantee low-loss supply of microwaves to the resonators. Three dimensional cavities, as well as coplanar waveguide resonators enhance MW fields by spatial and spectral confinement with a MW efficiency of $1.36\,\mathrm{mT/\sqrt{W}}$. We utilize single NV centers as hosts for optically accessible spins, and show, that their properties regarding optical spin readout known from smaller fields (<0.65 T) are retained up to fields of 3 T. In addition, we demonstrate coherent control of single nuclear spins under these conditions. Furthermore, our results extend the applicable magnetic field range of a single spin magnetic field sensor. Regarding spin based quantum registers, high fields lead to a purer product basis of electron and nuclear spins, which promises improved spin lifetimes. For example, during continuous single-shot readout the $^{14}$N nuclear spin shows second-long longitudinal relaxation times., Comment: 7 pages, 4 figures

Polarizing nuclear spins is of fundamental importance in biology, chemistry and physics. Methods for hyperpolarizing 13C nuclei from free electrons in bulk, usually demand operation at cryogenic temperatures. Room-temperature approaches targeting diamonds with nitrogen-vacancy (NV) centers could alleviate this need, but hitherto proposed strategies lack generality as they demand stringent conditions on the strength and/or alignment of the magnetic field. We report here an approach for achieving efficient electron->13C spin alignment transfers, compatible with a broad range of magnetic field strengths and field orientations with respect to the diamond crystal. This versatility results from combining coherent microwave- and incoherent laser-induced transitions between selected energy states of the coupled electron-nuclear spin manifold. 13C-detected Nuclear Magnetic Resonance (NMR) experiments demonstrate that this hyperpolarization can be transferred via first-shell or via distant 13Cs, throughout the nuclear bulk ensemble. This method opens new perspectives for applications of diamond NV centers in NMR, and in quantum information processing., Comment: 8 pages, 4 figures