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On-chip magnetic field sensing with Nitrogen-Vacancy (NV) centers in diamond requires scalable integration of 3D waveguides into diamond substrates. Here, we develop a sensing array device with an ensemble of shallow implanted NV centers integrated with arrays of laser-written waveguides for excitation and readout of NV signals. Our approach enables an easy-to-operate on-chip magnetometer with a pixel size proportional to the Gaussian mode area of each waveguide. The performed continuous wave optically detected magnetic resonance on each waveguide gives an average dc-sensitivity value of $195 \pm 3 {nT}/\sqrt{Hz}$, which can be improved with lock-in-detection or pulsed-microwave sequences. We apply a magnetic field to separate the four NV crystallographic orientations of the magnetic resonance and then utilize a DC current through a straight wire antenna close to the waveguide to prove the sensor capabilities of our device. We reconstruct the complete vector magnetic field in the NV crystal frame using three different NV crystallographic orientations. By knowing the polarization axis of the waveguide mode, we project the magnetic field vector into the lab frame.

At asymptotically late times ultrametricity can emerge from the persistent slow aging dynamics of the glass phase. We show that this suffices to recover the breaking of replica symmetry in mean-field spin glasses from the late time limit of the time evolution using the Keldysh path integral. This provides an alternative approach to replica symmetry breaking by connecting it rigorously to the dynamic formulation. Stationary spin glasses are thereby understood to spontaneously break thermal symmetry, or the Kubo-Martin-Schwinger relation of a state in global thermal equilibrium. We demonstrate our general statements for the spherical quantum $p$-spin model and the quantum Sherrington-Kirkpatrick model in the presence of transverse and longitudinal fields. In doing so, we also derive their dynamical Ginzburg-Landau effective Keldysh actions starting from microscopic quantum models., Comment: 17 pages, 4 figures, submitted version

Computing dynamical properties of strongly interacting quantum many-body systems poses a major challenge to theoretical approaches. Usually, one has to resort to numerical analytic continuation of results on imaginary frequencies, which is a mathematically ill-defined procedure. Here, we present an efficient method to compute the spectral functions of the two-component Fermi gas near the strongly interacting unitary limit directly in real frequencies. To this end, we combine the Keldysh path integral that is defined in real time with the self-consistent T-matrix approximation. The latter is known to predict thermodynamic and transport properties in good agreement with experimental observations in ultracold atoms. We validate our method by comparison with thermodynamic quantities obtained from imaginary time calculations and by transforming our real-time propagators to imaginary time. By comparison with state-of-the-art numerical analytic continuation of the imaginary time results, we show that our real-time results give qualitative improvements for dynamical quantities. Moreover, we show that no significant pseudogap regime exists in the self-consistent T-matrix approximation above the critical temperature $T_c$, an issue that has been under significant debate. We close by pointing out the versatile nature of our method as it can be extended to other systems, like the spin- or mass-imbalanced Fermi gas, other Bose-Fermi models, 2D systems as well as systems out of equilibrium., Comment: 21 pages, 11 figures

We consider continuous time-crystalline phases in dissipative many-body systems of atoms in cavities, focusing on the role of short-range interatomic interactions. First, we show that the latter can alter the nature of the time crystal by changing the type of the underlying critical bifurcation. Second, we characterize the heating mechanism and dynamics resulting from the short-range interactions and demonstrate that they make the time crystal inherently metastable. We argue that this is generic for the broader class of dissipative time crystals in atom-cavity systems whenever the cavity loss rate is comparable to the atomic recoil energy. We observe that such a scenario for heating resembles the one proposed for preheating of the early universe, where the oscillating coherent inflation field decays into a cascade of exponentially growing fluctuations. By extending approaches for dissipative dynamical systems to our many-body problem, we obtain analytical predictions for the parameters describing the phase transition and the heating rate inside the time-crystalline phase. We underpin and extend the analytical predictions of the heating rates with numerical simulations.

A field theory approach for the nonequilibrium relaxation dynamics in open systems at late times is developed. In the absence of conservation laws, all excitations are subject to dissipation. Nevertheless, ordered stationary states satisfy Goldstone's theorem. It implies a vanishing damping rate at small momenta, which in turn allows for competition between environment-induced dissipation and thermalization due to collisions. We derive the dynamic theory in the symmetry-broken phase of an $O(N)$-symmetric field theory based on an expansion of the two-particle irreducible (2PI) effective action to next-to-leading order in $1/N$ and highlight the analogies and differences to the corresponding theory for closed systems. A central result of this approach is the systematic derivation of an \emph{open-system Boltzmann equation}, which takes a very different form from its closed-system counterpart due to the absence of well-defined quasiparticles. As a consequence of the general structure of its derivation, it applies to open, gapless field theories that satisfy certain testable conditions, which we identify here. Specifically for the $O(N)$ model, we use scaling analysis and numerical simulations to show that interactions are screened efficiently at small momenta and, therefore, the late-time evolution is effectively collisionless. This implies that fluctuations induced by a quench dissipate into the environment before they thermalize. Goldstone's theorem also constrains the dynamics far from equilibrium, which is used to show that the order parameter equilibrates more quickly for quenches preserving the $O(N)$ symmetry than those breaking it explicitly., Comment: 24 pages, 15 figures, v2: published version

Diffusion noise represents a major constraint to successful liquid state nano-NMR spectroscopy. Using the Fisher information as a faithful measure, we theoretically calculate and experimentally show that phase sensitive protocols are superior in most experimental scenarios, as they maximize information extraction from correlations in the sample. We derive the optimal experimental parameters for quantum heterodyne detection (Qdyne) and present the most accurate statistically polarized nano-NMR Qdyne detection experiments to date, leading the way to resolve chemical shifts and $J$ couplings at the nanoscale., Comment: Matches published version. 20 pages and 7 figures. Comments very welcome

Spin-based applications of the negatively charged nitrogen-vacancy (NV) center in diamonds require efficient spin readout. One approach is the spin-to-charge conversion (SCC), relying on mapping the spin states onto the neutral (NV$^0$) and negative (NV$^-$) charge states followed by a subsequent charge readout. With high charge-state stability, SCC enables extended measurement times, increasing precision and minimizing noise in the readout compared to the commonly used fluorescence detection. Nano-scale sensing applications, however, require shallow NV centers within a few $\si{\nano \meter}$ distance from the surface where surface related effects might degrade the NV charge state. In this article, we investigate the charge state initialization and stability of single NV centers implanted $\approx \SI{5}{\nano \meter}$ below the surface of a flat diamond plate. We demonstrate the SCC protocol on four shallow NV centers suitable for nano-scale sensing, obtaining a reduced readout noise of 5--6 times the spin-projection noise limit. We investigate the general applicability of SCC for shallow NV centers and observe a correlation between NV charge-state stability and readout noise. Coating the diamond with glycerol improves both charge initialization and stability. Our results reveal the influence of the surface-related charge environment on the NV charge properties and motivate further investigations to functionalize the diamond surface with glycerol or other materials for charge-state stabilization and efficient spin-state readout of shallow NV centers suitable for nano-scale sensing., Comment: 9 pages, 5 figures

Despite the paradigmatic nature of the Fermi-polaron model, the theoretical description of its nonlinear dynamics poses challenges. Here, we apply a quantum kinetic theory of driven polarons to recent experiments with ultracold atoms, where Rabi oscillations between a Fermi-polaron state and a non-interacting level were reported. The resulting equations separate decoherence from momentum relaxation, with the corresponding rates showing a different dependence on microscopic scattering processes and quasi-particle properties. We describe both the polaron ground state and the excited repulsive-polaron state and we find a good quantitative agreement between our predictions and the available experimental data without any fitting parameter. Our approach not only takes into account collisional phenomena, but also it can be used to study the different roles played by decoherence and the collisional integral in the strongly interacting highly-imbalanced mixture of Fermi gases., Comment: 6 pages (main text), 14 pages (supplemental material), 3+5 figures

Diffusion noise is a major source of spectral line broadening in liquid state nano-scale nuclear magnetic resonance with shallow nitrogen-vacancy centres, whose main consequence is a limited spectral resolution. This limitation arises by virtue of the widely accepted assumption that nuclear spin signal correlations decay exponentially in nano-NMR. However, a more accurate analysis of diffusion shows that correlations survive for a longer time due to a power-law scaling, yielding the possibility for improved resolution and altering our understanding of diffusion at the nano-scale. Nevertheless, such behaviour remains to be demonstrated in experiments. Using three different experimental setups and disparate measurement techniques, we present overwhelming evidence of power-law decay of correlations. These result in sharp-peaked spectral lines, for which diffusion broadening need not be a limitation to resolution., Comment: 21 pages and 13 figures. Matches published version. Comments always welcome!

Modifying light fields at single-photon level is a key challenge for upcoming quantum technologies and can be realized in a scalable manner through integrated quantum photonics. Laser-written diamond photonics offers three-dimensional fabrication capabilities and large mode-field diameters matched to fiber optic technology, though limiting the cooperativity at the single-emitter level. To realize large cooperativities, we combine excitation of single shallow-implanted silicon vacancy centers via large numerical aperture optics with detection assisted by laser-written type-II waveguides. We demonstrate single-emitter extinction measurements with a cooperativity of 0.153 and a beta factor of 13% yielding 15.3% as lower bound for the quantum efficiency of a single emitter. The transmission of resonant photons reveals single-photon subtraction from a quasi-coherent field resulting in super-Poissonian light statistics. Our architecture enables single quantum level light field engineering in an integrated design which can be fabricated in three dimensions and with a natural connectivity to optical fiber arrays., Comment: 8 pages, 4 figures

In recent years, dynamical phase transitions and out-of-equilibrium criticality have been at the forefront of ultracold gases and condensed matter research. Whereas universality and scaling are established topics in equilibrium quantum many-body physics, out-of-equilibrium extensions of such concepts still leave much to be desired. Using exact diagonalization and the time-dependent variational principle in uniform martrix product states, we calculate the time evolution of the local order parameter and Loschmidt return rate in transverse-field Ising chains with antiferromagnetic power law-decaying interactions, and map out the corresponding rich dynamical phase diagram. \textit{Anomalous} cusps in the return rate, which are ubiquitous at small quenches within the ordered phase in the case of ferromagnetic long-range interactions, are absent within the accessible timescales of our simulations in the antiferromagnetic case, showing that long-range interactions are not a sufficient condition for their appearance. We attribute this to much weaker domain-wall binding in the antiferromagnetic case. For quenches across the quantum critical point, \textit{regular} cusps appear in the return rate and connect to the local order parameter changing sign, indicating the concurrence of two major concepts of dynamical phase transitions. Our results consolidate conclusions of previous works that a necessary condition for the appearance of anomalous cusps in the return rate after quenches within the ordered phase is for topologically trivial local spin flips to be the energetically dominant excitations in the spectrum of the quench Hamiltonian. Our findings are readily accessible in modern trapped-ion setups, and we outline the associated experimental considerations., Comment: Accepted version, 15 pages, 5 figures

We report the quantum calibration of a magnetic force microscope (MFM) by measuring the two-dimensional magnetic stray field distribution of the MFM tip using a single nitrogen vacancy (NV) center in diamond. From the measured stray field distribution and the mechanical properties of the cantilever a calibration function is derived allowing to convert MFM images to quantum calibrated stray field maps. This novel approach overcomes limitations of prior MFM calibration schemes and allows quantum calibrated nanoscale stray field measurements in a field range inaccessible to scanning NV magnetometry. Quantum calibrated measurements of a stray field reference sample allow its use as a transfer standard, opening the road towards fast and easily accessible quantum traceable calibrations of virtually any MFM., Comment: 31 Pages, 5 Figures

The negatively-charged NV$^-$-center in diamond has shown great success in nanoscale, high-sensitivity magnetometry. Efficient fluorescence detection is crucial for improving the sensitivity. Furthermore, integrated devices enable practicable sensors. Here, we present a novel architecture which allows us to create NV$^-$-centers a few nanometers below the diamond surface, and at the same time in the mode field maximum of femtosecond-laser-written type-II waveguides. We experimentally verify the coupling efficiency, showcase the detection of magnetic resonance signals through the waveguides and perform first proof-of-principle experiments in magnetic field and temperature sensing. The sensing task can be operated via the waveguide without direct light illumination through the sample, which marks an important step for magnetometry in biological systems which are fragile to light. In the future, our approach will enable the development of two-dimensional sensing arrays facilitating spatially and temporally correlated magnetometry., Comment: 10 pages, 4 figures

We consider an ensemble of ultracold bosonic atoms within a near-planar cavity, driven by a far detuned laser whose phase is modulated at a frequency comparable to the transverse cavity mode spacing. We show that a strong, dispersive atom-photon coupling can be reached for many transverse cavity modes at once. The resulting Floquet polaritons involve a superposition of a set of cavity modes with a density excitation of the atomic cloud. The mutual interactions between these modes lead to distinct avoided crossings between the polaritons. Increasing the laser drive intensity, a low-lying multimode Floquet polariton softens and eventually becomes undamped, corresponding to the transition to a superradiant, self-organized phase. We demonstrate the stability of the stationary state for a broad range of parameters., Comment: 19 pages, 9 figures

Fluorescent nanodiamonds containing negatively-charged nitrogen-vacancy (NV$^-$) centers are promising for a wide range of applications, such as for sensing, as fluorescence biomarkers, or to hyperpolarize nuclear spins. NV$^-$ centers are formed from substitutional nitrogen (P1 centers) defects and vacancies in the diamond lattice. Maximizing the concentration of NVs is most beneficial, which justifies the search for methods with a high yield of conversion from P1 to NV$^-$. We report here the characterization of surface cleaned fluorescent micro- and nanodiamonds, obtained by irradiation of commercial diamond powder with high-energy (10 MeV) electrons and simultaneous annealing at 800{\deg}C. Using this technique and increasing the irradiation dose, we demonstrate the creation of NV$^-$ with up to 25 % conversion yield. Finally, we monitor the creation of irradiation-induced spin-1 defects in microdiamond particles, which we associate with W16 and W33 centers, and investigate the effects of irradiation dose and particle size on the coherence time of NV$^-$.

We develop theoretically and demonstrate experimentally a universal dynamical decoupling method for robust quantum sensing with unambiguous signal identification. Our method uses randomisation of control pulses to suppress simultaneously two types of errors in the measured spectra that would otherwise lead to false signal identification. These are spurious responses due to finite-width $\pi$ pulses, as well as signal distortion caused by $\pi$ pulse imperfections. For the cases of nanoscale nuclear spin sensing and AC magnetometry, we benchmark the performance of the protocol with a single nitrogen vacancy centre in diamond against widely used non-randomised pulse sequences. Our method is general and can be combined with existing multipulse quantum sensing sequences to enhance their performance.

We present an efficient and very flexible numerical fast Fourier-Laplace transform, that extends the logarithmic Fourier transform (LFT) introduced by Haines and Jones [Geophys. J. Int. 92(1):171 (1988)] for functions varying over many scales to nonintegrable functions. In particular, these include cases of the asymptotic form $f(\nu\to0)\sim\nu^a$ and $f(|\nu|\to\infty)\sim\nu^b$ with arbitrary real $a>b$. Furthermore, we prove that the numerical transform converges exponentially fast in the number of data points, provided that the function is analytic in a cone $|\Im{\nu}|<\theta|\Re{\nu}|$ with a finite opening angle $\theta$ around the real axis and satisfies $|f(\nu)f(1/\nu)|<\nu^c$ as $\nu\to 0$ with a positive constant $c$, which is the case for the class of functions with power-law tails. Based on these properties we derive ideal transformation parameters and discuss how the logarithmic Fourier transform can be applied to convolutions. The ability of the logarithmic Fourier transform to perform these operations on multiscale (non-integrable) functions with power-law tails with exponentially small errors makes it the method of choice for many physical applications, which we demonstrate on typical examples. These include benchmarks against known analytical results inaccessible to other numerical methods, as well as physical models near criticality., Comment: 14 pages, 8 figures

We develop a non-equilibrium field-theoretical approach, based on a systematic diagrammatic expansion, for strongly interacting photons in optically dense atomic media. We consider the case where the characteristic photon-propagation range $L_P$ is much larger than the interatomic spacing $a$ and where the density of atomic excitations is low enough to neglect saturation effects. In the highly polarizable medium the photons experience nonlinearities through the interactions they inherit from the atoms. If the atom-atom interaction range $L_E$ is also large compared to $a$, we show that scattering processes with momentum transfer between photons are suppressed by a factor $a/L_E$. We are then able to perform a self-consistent resummation of a specific (Hartree-like) diagram subclass and obtain quantitative results in the highly non-perturbative regime of large single-atom cooperativity. Here we find important, conceptually new collective phenomena emerging due to the dissipative nature of the interactions, which even give rise to novel phase transitions. The robustness of these is investigated by inclusion of the leading corrections in $a/L_E$. We consider specific applications to photons propagating under EIT conditions along waveguides near atomic arrays as well as within Rydberg ensembles., Comment: accepted version, 32 pages + 10 pages appendix, 32 figures

The propagation of light in strongly coupled atomic media takes place through the formation of polaritons - hybrid quasi-particles resulting from a superposition of an atomic and a photonic excitation. Here we consider the propagation under the condition of electromagnetically-induced transparency and show that a novel many-body phenomenon can appear due to strong, dissipative interactions between the polaritons. Upon increasing the photon-pump strength, we find a first-order transition between an opaque phase with strongly broadened polaritons and a transparent phase where a long-lived polariton branch with highly tunable occupation emerges. Across this non-equilibrium phase transition, the transparency window is reconstructed via nonlinear interference effects induced by the dissipative polariton interactions. Our predictions are based on a systematic diagrammatic expansion of the non-equilibrium Dyson equations which is quantitatively valid, even in the non-perturbative regime of large single-atom cooperativities, provided the polariton interactions are sufficiently long ranged. Such a regime can be reached in photonic crystal waveguides thanks to the tunability of interactions, allowing to observe the interaction-induced-transparency transition even at low polariton densities., Comment: 6 pages and 4 figures in main text + 2 pages of supplemental material

Predictive high-fidelity finite element simulations of human cardiac mechanics co\-mmon\-ly require a large number of structural degrees of freedom. Additionally, these models are often coupled with lumped-parameter models of hemodynamics. High computational demands, however, slow down model calibration and therefore limit the use of cardiac simulations in clinical practice. As cardiac models rely on several patient-specific parameters, just one solution corresponding to one specific parameter set does not at all meet clinical demands. Moreover, while solving the nonlinear problem, 90\% of the computation time is spent solving linear systems of equations. We propose a novel approach to reduce only the structural dimension of the monolithically coupled structure-windkessel system by projection onto a lower-dimensional subspace. We obtain a good approximation of the displacement field as well as of key scalar cardiac outputs even with very few reduced degrees of freedom while achieving considerable speedups. For subspace generation, we use proper orthogonal decomposition of displacement snapshots. To incorporate changes in the parameter set into our reduced order model, we provide a comparison of subspace interpolation methods. We further show how projection-based model order reduction can be easily integrated into a gradient-based optimization and demonstrate its performance in a real-world multivariate inverse analysis scenario. Using the presented projection-based model order reduction approach can significantly speed up model personalization and could be used for many-query tasks in a clinical setting.

The Loschmidt echo (LE) is a purely quantum-mechanical quantity whose determination for large quantum many-body systems requires an exceptionally precise knowledge of all eigenstates and eigenenergies. One might therefore be tempted to dismiss the applicability of any approximations to the underlying time evolution as hopeless. However, using the fully connected transverse-field Ising model (FC-TFIM) as an example, we show that this indeed is not the case, and that a simple semiclassical approximation to systems well described by mean-field theory (MFT) is in fact in good quantitative agreement with the exact quantum-mechanical calculation. Beyond the potential to capture the entire dynamical phase diagram of these models, the method presented here also allows for an intuitive geometric interpretation of the fidelity return rate at any temperature, thereby connecting the order parameter dynamics and the Loschmidt echo in a common framework. Videos of the post-quench dynamics provided in the supplemental material visualize this new point of view., Comment: Accepted version. 7 pages with 4 Figures in main file. 3 pages including 2 Figures of supplemental material. 3 videos linked in the references of the main file

We discuss the phase diagram and the universal scaling functions of attractive Fermi gases at finite imbalance. The existence of a quantum multicritical point for the unitary gas at vanishing chemical potential $\mu$ and effective magnetic field $h$, first discussed by Nikoli\'{c} and Sachdev, gives rise to three different phase diagrams, depending on whether the inverse scattering length $1/a$ is negative, positive or zero. Within a Luttinger-Ward formalism, the phase diagram and pressure of the unitary gas is calculated as a function of the dimensionless scaling variables $T/\mu$ and $h/\mu$. The results indicate that beyond the Clogston-Chandrasekhar limit at $(h/\mu)_c\simeq 1.09$, the unitary gas exhibits an inhomogeneous superfluid phase with FFLO order that can reach critical temperatures near unitarity of $\simeq 0.03\, T_F$., Comment: 15 pages, 8 figures, JETP style, text overlap with arXiv:1608.00457, v2: published version

We construct the finite-temperature dynamical phase diagram of the fully connected transverse-field Ising model from the vantage point of two disparate concepts of dynamical criticality. An analytical derivation of the classical dynamics and exact diagonalization simulations are used to study the dynamics after a quantum quench in the system prepared in a thermal equilibrium state. The different dynamical phases characterized by the type of non-analyticities that emerge in an appropriately defined Loschmidt-echo return rate directly correspond to the dynamical phases determined by the spontaneous breaking of $\mathbb{Z}_2$ symmetry in the long-time steady state. The dynamical phase diagram is qualitatively different depending on whether the initial thermal state is ferromagnetic or paramagnetic. Whereas the former leads to a dynamical phase diagram that can be directly related to its equilibrium counterpart, the latter gives rise to a divergent dynamical critical temperature at vanishing final transverse-field strength., Comment: journal article, 15 pages, 12 figures. Final version

Sensing, localising and identifying individual nuclear spins or frequency components of a signal in the presence of a noisy environments requires the development of robust and selective methods of dynamical decoupling. An important challenge that remains to be addressed in this context are spurious higher order resonances in current dynamical decoupling sequences as they can lead to the misidentification of nuclei or of different frequency components of external signals. Here we overcome this challenge with engineered quantum sensing sequences that achieve both, enhanced robustness and the simultaneous suppression of higher order harmonic resonances. We demonstrate experimentally the principle using a single nitrogen-vacancy center spin sensor which we apply to the unambiguous detection of external protons.