Magnetic sensors capable of detecting nanoscale volumes of spins allow for non-invasive, element-specific probing. The error in such measurements is usually reduced by increasing the measurement time, and noise averaging the signal. However, achieving the best precision requires restricting the maximum possible field strength to much less than the spectral linewidth of the sensor. Quantum entanglement and squeezing can then be used to improve precision (although they are difficult to implement in solid-state environments). When the field strength is comparable to or greater than the spectral linewidth, an undesirable trade-off between field strength and signal precision occurs. Here, we implement novel phase estimation algorithms on a single electronic spin associated with the nitrogen-vacancy defect centre in diamond to achieve an ∼8.5-fold improvement in the ratio of the maximum field strength to precision, for field magnitudes that are large (∼0.3 mT) compared to the spectral linewidth of the sensor (∼4.5 µT). The field uncertainty in our approach scales as 1/T(0.88), compared to 1/T(0.5) in the standard measurement approach, where T is the measurement time. Quantum phase estimation algorithms have also recently been implemented using a single nuclear spin in a nitrogen-vacancy centre. Besides their direct impact on applications in magnetic sensing and imaging at the nanoscale, these results may prove useful in improving a variety of high-precision spectroscopy techniques.

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Non-invasive magnetic field sensing using optically -detected magnetic resonance of nitrogenvacancy (NV) centers in diamond was used to study spatial distribution of the magnetic induction upon penetration and expulsion of weak magnetic fields in several representative superconductors. Vector magnetic fields were measured on the surface of conventional, Pb and Nb, and unconventional, LuNi2B2C, Ba0.6K0.4Fe2As2, Ba(Fe0.93Co0.07)2As2, and CaKFe4As4, superconductors, with diffraction -limited spatial resolution using variable -temperature confocal system. Magnetic induction profiles across the crystal edges were measured in zero-field-cooled (ZFC) and field-cooled (FC) conditions. While all superconductors show nearly perfect screening of magnetic fields applied after cooling to temperatures well below the superconducting transition, Tc, a range of very different behaviors was observed for Meissner expulsion upon cooling in static magnetic field from above Tc. Substantial conventional Meissner expulsion is found in LuNi2B2C, paramagnetic Meissner effect (PME) is found in Nb, and virtually no expulsion is observed in iron-based superconductors. In all cases, good correlation with macroscopic measurements of total magnetic moment is found. Our measurements of the spatial distribution of magnetic induction provide insight into microscopic physics of the Meissner effect.

The lower critical magnetic field, Hc1, of superconductors is measured by using ensembles of NV-centers-in-diamond optical magnetometry. The technique is minimally invasive, and has subgauss field sensitivity and sub-µm spatial resolution, which allow for accurate detection of the vector field at which the vortices start penetrating the sample from the corners. Aided by the revised calculations of the effective demagnetization factors of actual cuboid -shaped samples, Hc1 and the London penetration depth, λ, derived from Hc1 can be obtained. We apply this method to three well-studied superconductors: optimally doped Ba(Fe1−xCox)2As2, stoichiometric CaKFe4As4, and high-Tc cuprate YBa2Cu3O 7−δ . Our results are well compared with the values of λ obtained using other techniques, thus adding another non-destructive and sensitive method to measure these important parameters of superconductors.

Unconventional superconductivity often emerges in close proximity to a magnetic instability. Upon suppressing the magnetic transition down to zero temperature by tuning the carrier concentration, pressure, or disorder, the superconducting transition temperature Tc acquires its maximum value. A major challenge is the elucidation of the relationship between the superconducting phase and the strong quantum fluctuations expected near a quantum phase transition (QPT) that is either second order (i.e. a quantum critical point) or weakly first order. While unusual normal state properties, such as non-Fermi liquid behavior of the resistivity, are commonly associated with strong quantum fluctuations, evidence for its presence inside the superconducting dome are much scarcer. In this paper, we use sensitive and minimally invasive optical magnetometry based on NV-centers in diamond to probe the doping evolution of the T = 0 penetration depth in the electron-doped iron-based superconductor Ba(Fe1−xCox)2As2. A non-monotonic evolution with a pronounced peak in the vicinity of the putative magnetic QPT is found. This behavior is reminiscent to that previously seen in isovalently-substituted BaFe2(As1−xPx)2 compounds, despite the notable differences between these two systems. Whereas the latter is a very clean system that displays nodal superconductivity and a single simultaneous first-order nematic-magnetic transition, the former is a charge-doped and significantly dirtier system with fully gapped superconductivity and split second-order nematic and magnetic transitions. Thus, our observation of a sharp peak in λ(x) near optimal doping, combined with the theoretical result that a QPT alone does not mandate the appearance of such peak, unveils a puzzling and seemingly universal manifestation of magnetic quantum fluctuations in iron-based superconductors and unusually robust quantum phase transition under the dome of superconductivity.

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We report combined experimental and theoretical analysis of superconductivity in CaK(Fe1−xNix)4As4 (CaK1144) for x=0, 0.017, and 0.034. To obtain the superfluid density ρ=[1+ΔλL(T)/λL(0)]−2, the temperature dependence of the London penetration depth ΔλL(T) was measured by using a tunnel-diode resonator (TDR) and the results agreed with the microwave coplanar resonator (MWR) with the small differences accounted for by considering a three orders of magnitude higher frequency of MWR. The absolute value of λL(T≪Tc)≈λL(0) was measured by using MWR, λL(5K)≈170±20 nm, which agreed well with the NV centers in diamond optical magnetometry that gave λL(5K)≈196±12 nm, which agreed well with the NV centers in diamond optical magnetometry that gave λL(5K)≈196±12 nm. The experimental results are analyzed within the Eliashberg theory, showing that the superconductivity of CaK1144 is well described by the nodeless s± order parameter and that upon Ni doping the interband interaction increases.

We present an experimental method to perform dual-channel lock-in magnetometry of timedependent magnetic fields using a single spin associated with a nitrogen-vacancy (NV) color center in diamond. We incorporate multi-pulse quantum sensing sequences with phase estimation algorithms to achieve linearized field readout and constant, nearly decoherence-limited sensitivity over a wide dynamic range. Furthermore, we demonstrate unambiguous reconstruction of the amplitude and phase of the magnetic field. We show that our technique can be applied to measure random phase jumps in the magnetic field, as well as phase-sensitive readout of the frequency.PACS numbers: 07.55. Ge,85.75.Ss,76.30.Mi The coherent evolution of a quantum state interacting with its environment is the basis for understanding fundamental issues of open quantum systems 1 , as well as for applications in quantum information science and technology 2 . Traditionally in these fields, the extreme sensitivity of coherent quantum dynamics to external perturbations has been viewed as a barrier to be surmounted. By contrast, quantum sensors have emerged that instead take advantage of this sensitivity; recent examples include electrometers and magnetometers based on superconducting qubits 3 , quantum dots 4 , spins in diamond [5][6][7][8] and trapped ions 9 .The nitrogen-vacancy (NV) defect center in diamond ( Fig. 1(a)) shows great promise as an ultra-sensitive solid-state magnetometer and magnetic imager because it features potentially atomic-scale resolution 6 , wide temperature range operation from 4 K -700 K 10 , and long coherence times that allow for high magnetic field sensitivity 11 . Recent demonstrations include nanoscale magnetic imaging 7,12,13 , coupling to nano-mechanical oscillators [14][15][16] , detection of single proximal nuclear spins [17][18][19][20] and nanoscale volumes of external electron and nuclear spins [21][22][23][24] .Magnetometry with diamond spin sensors detects the frequency shift of the NV spin resonance caused by the magnetic field via the Zeeman effect. Highly sensitive quantum sensing techniques use multi-pulse dynamical decoupling (DD) sequences 3,6,9,[25][26][27] that are tuned to the frequency of a time-dependent field. The resulting fluctuating frequency shift is rectified and integrated by the pulse sequence to yield a detectable quantum phase, while effectively filtering out low frequency noise from the environment ( Fig. 1(b)). Another advantage of these DD sequences is that they make the magnetometer insensitive to instabilities such as drifts in temperature or applied bias magnetic field.However, these state of the art quantum sensing methods also have significant drawbacks: the dynamic range is limited by the quantum phase ambiguity 28,29 , the sensitivity is a highly nonlinear function of field amplitude requiring prior knowledge of a working point for accurate deconvolution, and the classical phase of the field has to be carefully controlled to obtain accurate field ampli-Illustration of experimental setup f...