Sub-nanotesla magnetometry with a fibre-coupled diamond sensor

Sensing small magnetic fields is relevant for many applications ranging from geology to medical diagnosis. We present a fiber-coupled diamond magnetometer with a sensitivity of (310 $\pm$ 20) pT$/\sqrt{\text{Hz}}$ in the frequency range of 10-150 Hz. This is based on optically detected magnetic resonance of an ensemble of nitrogen vacancy centers in diamond at room temperature. Fiber coupling means the sensor can be conveniently brought within 2 mm of the object under study.


I. INTRODUCTION
The sensing of magnetic fields using the nitrogen vacancy center (NVC) in diamond has seen rapid growth over the last decade due to the promise of high sensitivity magnetometry with exceptional spatial resolution [1,2] along with a high dynamic range [3]. The use of NVC ensembles rather than single centres improves sensitivity while degrading the spatial resolution [3][4][5][6][7][8][9][10][11][12][13]. Recent advancements have demonstrated ensemble sensitivities of 0.9 pT/ √ Hz for d.c. fields [14] and 0.9 pT/ √ Hz for a.c fields [15]. However, these results have been limited to systems that are bulky and are typically fixed to optical tables. In contrast, fibre-coupling provides a small sensor head that may be moved independently from the rest of the control instrumentation and thus offers the possibility of application in medical diagnostic techniques such as magnetocardiography (MCG) [16,17]. Most fiber-coupled diamond magnetometers have relied on using nanodiamonds/microdiamonds attached to the end of a fiber, achieving sensitivities in the range of 56000-180 nT/ √ Hz [18][19][20]. Utilising a two-wire microwave transmission line in addition to a fiber-diamond set-up was able to achieve a sensitivity of ∼300 nT/ √ Hz [21]. A fiber-based gradiometer approach was able to provide a sensitivity of ∼35 nT/ √ Hz with projected shot-noise sensitivities potentially allowing for MCG [22,23]. Using a hollow-core fiber with many nanodiamond sensors in a fluidic environment provided a sensitivity of 63 nT/ √ Hz per sensor and a spatial resolution of 17 cm [24]. Other compact magnetometers that use a fiber have demonstrated sensitivities in the ranges of 67-1.5 nT/ √ Hz [25][26][27]. The best sensitivity reported for a fibre-coupled diamond magnetometer so far is 35 nT/ √ Hz when sensing a real test field [22], and 1.5 nT/ √ Hz when estimating the sensitivity based on the signal-to-noise-to-linewidth using the slope of a resonance in the magnetic resonance spectrum [27]. Other diamond magnetometers which offer high portability whilst maintaining a compact structure have been demonstrated with a compact LED-based design achieving a minimum detectable field of 1 µT whilst offering minimal power consumption [28]. Here, a diamond-based fiber-coupled magnetometer with sub-nT sensitivity is presented. The key feature is the use of lenses to reduce optical losses from the fiber to the diamond and back as shown in figure 1b).
The NVC, when in its negative charge state, is a spin S = 1 defect that can be optically initialised into the m s = 0 ground state and possesses spin-dependent fluorescence giving rise to optically detected magnetic resonance (ODMR) [29]. The energy level diagram is shown in figure 1a). The Zeeman-induced splitting of the NVC leads to the detection of magnetic fields with high sensitivity where the sensitivity of the magnetometer scales with 1/ √ N , where N is the number of centers probed [30,31]. The zero-field splitting at room temperature is ∼2.87 GHz. Upon application of an external magnetic field Zeeman-induced splitting leads to sub-levels that are split by where g e = 2.0028 is the NVC g-factor, µ B is the Bohr magneton, B || is the projection of the external magnetic field onto the NVC symmetry axis (the 111 crystallographic direction) and h is Planck's constant. The energy levels are further split by the hyperfine interaction between the electron spin and 14 N nuclear spin (I = 1) by A ≈ 2.16 MHz. Under a continuous wave excitation scheme, which is employed in this paper, the photon-shot-noiselimited sensitivity of a diamond-based magnetometer is given by where ∆ν is the linewdith, C is the measurement contrast (the reduction in fluorescence when on resonance compared to when not on resonance) and I 0 is the number of collected photons off resonance [5,32].

Magnetometry is performed with the set-up shown in figure 1b). A Laser Quantum
Gem-532 with a maximum power output of 2 W is used to excite the NVC ensemble; for our experiments 1 W was used to reduce laser noise. The laser beam is passed through a Thorlabs BSF10-A beam sampler whereby approximately 1% is picked off and supplied to the reference arm of a Thorlabs PDB450A balanced detector to cancel out laser intensity noise; the illumination levels incident upon each photodiode is equal in the absence of microwaves.
The remaining (high-intensity) portion of the laser beam is focused into a custom-ordered For the second method known test fields were applied using a Helmholtz coil which was calibrated using a Hirst Magnetics GM07 Hall probe. The test fields were applied along (100) and the sensitivity was found to be (310 ± 20) pT/ √ Hz, as shown in figure 3. The worse sensitivity is due to this non-optimal test field orientation. For the targeted application the fields of interest will be applied along the (100) direction and thus the sensitivity using the second method is considered to be the true sensitivity. Our sensitivity improves on the value of 35 nT/ √ Hz previously obtained with a fiber coupled NVC magnetometer using applied test fields. The photon shot noise limit is calculated using equation 2 from the fluorescence which was measured to be 1.

IV. DISCUSSION
Our sensitivity is 310 pT/ √ Hz and thus we are a factor of ∼6 away from the shot-noise limit. This may be due to uncancelled laser and microwave noise some of which could be cancelled out through the implementation of a gradiometer which would also alleviate ambient magnetic noise from the environment [34,35]. To detect signals for MCG it is estimated that the sensitivity required would need to be over an order of magnitude beyond what we currently achieve [12,36].
The biggest limitation of our system is the collection efficiency in which significant improvements are expected as the conversion efficiency of green to red photons is calculated to be 0.03%. Improving this would also improve the excitation efficiency. Due to the high refractive index of diamond n d = 2.42, the majority of light emitted by the defects will undergo total internal reflection and thus the majority of emitted light will escape through the sides of the diamond [7]. A possible option for improvement would be an adaptation of the fluorescence waveguide excitation and collection [37] which reported a 96-fold improvement in the light collected. Another approach would be to surrounded the diamond with a total internal reflection lens to collect light from the diamond sides and focus it toward a small area [38], which would be easier to integrate with our system, leading to an enhancement of 56 in the photon collection when compared to a lossless air objective of 0.55 N.A. This would represent a photon enhancement of ∼30 for our system and assuming a shot-noise limited scaling the measured sensitivity would become ∼60 pT/ √ Hz.
Ferrite flux concentrators have demonstrated a ×254 improvement in the sensitivity for a diamond magnetometer [14] at a cost of degrading the spatial resolution due to concentrating the flux from a large area and directing it toward a diamond. Due to the constraints of our system integrating the design discussed in [14] is not straightforward and thus the enhancement to sensitivity will be smaller. A further improvement would be to use the dual-resonance technique [14] which would allow our system to be invariant to temperature fluctuations [39] which is essential for practical applications of our magnetometer. Another way to introduce temperature invariance into our system would be the use of double-quantum magnetometry [40,41]. This would also be compatible with the use of pulsed schemes such as Ramsey magnetometry which would offer significant improvements to the sensitivity of a magnetometer compared to continuous wave excitation schemes [5,9]. However, it should be noted that significantly more laser excitation power and more homogeneous microwave driving fields will be required to realize the potential benefits of Ramsey magnetometry [42][43][44].

V. CONCLUSION
In this work a fiber-coupled magnetometer that reaches a sensitivity of (310± 20) pT/ √ Hz over the frequency range of 10-150 Hz has been presented. The mobility of the system and the compact nature of the sensor head are designed to target the application of magnetocardiography with further improvements discussed to be able to reach higher sensitivities.  an Oxford Instrument Optistat cryostat. The concentration was determined to be 4.6 ppm for negatively charged NVC and 0.8 ppm for neutral NVC and was found from the intensities of the 637 nm and 575 nm zero-phonon line respectively [45]. FTIR data, figure 4b) were taken at room temperature using a Perkin Elmer Spectrum GX FT-IR spectrometer. The concentrations from FTIR were established to be 5.6 ppm for neutral substitutional nitrogen (N 0 s ) and 3 ppm for positively charged substitutional nitrogen (N + s ) [46,47].
Appendix B: Zero-crossing Slope vs. Modulation frequency The variation of the zero-crossing slope as a function of the modulation frequency is shown in figure 5. The expected trend of a decrease in the zero-crossing slope for higher modulation frequencies due to the finite repolarisation time of the NVC centre is followed [5,33,48].
Despite the continued increase of the zero-crossing slope at progressively lower modulation frequencies, the best sensitivity was achieved at a modulation frequency of 3.0307 kHz (data not shown), we attribute this to an increased susceptibility to noise at particularly low modulation frequencies nearer to DC. The maximum value of the zero-crossing slope at a modulation of 3.0307 kHz was 17.9 V/MHz which was slightly higher than the maximum in