Stray magnetic field compensation with a scalar atomic magnetometerJ. Belfi, G. Bevilacqua, V. Biancalana, R. Cecchi, Y. Dancheva, and L. Moi Citation: Review of Scientific Instruments 81, 065103 (2010); doi: 10.1063/1.3441980 View online: http://dx.doi.org/10.1063/1.3441980 View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/81/6?ver=pdfcov Published by the AIP Publishing Articles you may be interested in An optically modulated zero-field atomic magnetometer with suppressed spin-exchange broadening Rev. Sci. Instrum. 85, 045124 (2014); 10.1063/1.4872075 Real-time vector field tracking with a cold-atom magnetometer Appl. Phys. Lett. 102, 173504 (2013); 10.1063/1.4803684 In situ triaxial magnetic field compensation for the spin-exchange-relaxation-free atomic magnetometer Rev. Sci. Instrum. 83, 103104 (2012); 10.1063/1.4756046 Nanoscale magnetic field mapping with a single spin scanning probe magnetometer Appl. Phys. Lett. 100, 153118 (2012); 10.1063/1.3703128 High-sensitivity inductive magnetometer for pulsed magnetic fields Rev. Sci. Instrum. 70, 82 (1999); 10.1063/1.1149544

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Stray magnetic field compensation with a scalar atomic magnetometerJ. Belfi,a� G. Bevilacqua, V. Biancalana,b� R. Cecchi, Y. Dancheva, and L. MoiDepartment of Physics and CSC, CNISM UdR Siena, University of Siena, Via Roma 56, 53100 Siena, Italy

�Received 10 March 2010; accepted 8 May 2010; published online 23 June 2010�

We describe a system for the compensation of time-dependent stray magnetic fields using a dualchannel scalar magnetometer based on nonlinear Faraday rotation in synchronously opticallypumped Cs vapor. We detail the active control strategy, with an emphasis on the electronic circuitry,based on a simple phase-locked-loop integrated circuit. The performance and limits of the systemdeveloped are tested and discussed. The system was applied to significantly improve the detectionof free induction decay signals from protons of remotely magnetized water precessing in an ultralowmagnetic field. © 2010 American Institute of Physics. �doi:10.1063/1.3441980�

I. INTRODUCTION

High sensitivity magnetometry has a wide range of ap-plications in both fundamental and applied research. Nowa-days, the best performance in terms of sensitivity is providedby superconducting quantum interference devices �SQUIDs�and by atomic optical magnetometers �AOMs� operating inthe spin-exchange relaxation-free �SERF� regime. In bothcases, the impressive limit of 1 fT /�Hz has been surpassedexperimentally, when operating in volumes carefullyshielded against the environmental magnetic noise.1

Scalar magnetometers operating in nonvanishing mag-netic fields have also been reported to have a sensitivity asgood as a few tens of fT /�Hz when operating in shieldedenvironments. This value degrades to about 1 pT /�Hz inunshielded volumes.

The precise determination and control of magnetic fieldsis a key factor in atomic physics experiments,2–4 fundamentalresearch �such as determination of the neutron electric dipolemoment5,6�, ultralow-field �ULF�-NMR and imaging,7 andthe development of biomagnetic diagnostics.8,9

Field control is commonly attempted through both pas-sive shielding and active compensation approaches, andmany solutions are reported in literature. Different applica-tions require that the shielding/compensating systems havedifferent levels of noise rejection and respond in differentfrequency ranges.

Passive shielding can be achieved by surrounding themeasurement volume with soft-iron/large permeability mate-rial, or with thick conducting material �or superconductinglayers�, in which eddy currents are induced. The supercon-ducting approach has been studied and developed since the1970s �Refs. 9–11� and yields the highest field stability.

The efficiency of passive shielding decreases at low fre-quency, thus shields can be profitably coupled with activecompensation systems to improve the performance in thehertz and subhertz ranges. Active compensation has also

been proposed and studied for use in compensating peakedspectral components of the stray field, such as those at 50 or60 Hz due to power lines. Active compensation of determin-istic disturbances has also been attempted with a feed-forward approach.12 Nevertheless the most recent literaturerefers to feedback systems, which are suitable for attenuatingboth deterministic and random components.

Active compensation with a suitably extended band-width can also be applied in standalone configurations �i.e.,not combined with passive shielding�. This is of interest, forinstance, where easy access �no geometrical constraints� isrequired, and/or when large magnetically clean volumes areneeded, which would make the passive shield bulky and veryexpensive.

Naturally, the specifications �bandwidth and attenuationfactor� and the final performance of such active setups de-pend on the feedback design and on the sensors used to gen-erate the error signals. In the latter respect, a large variety ofsensors were reported in literature, ranging from the me-chanical ones �a sort of compass!� of decades ago13 to Hallsensors,14 magnetoresistances,2,15 fluxgates,6,16 SQUIDs,17,18

SERF-AOMs,19 and NMR-based magnetometers.20

A further approach frequently used to counteract mag-netic noise relies on the fact that noise sources, being at alarge distance from the measurement volume, produce arather homogeneous disturbance. Thus the use of differential�gradiometric� sensors makes it possible to reduce noisewhen measuring signals produced by nearby sources21 sothat the common mode signal can be attributed to magneticnoise and rejected.

In this paper we present the results obtained with a dualchannel scalar Cs AOM operating in microtesla field, whichwas used to stabilize the modulus of the bias field. This setupmakes it possible to detect the low-level time-dependentanomalies generated by small sources located close to one ofthe two sensors composing the differential magnetometer.Both sensors work in the same bias magnetic field, which isalso the field in which the anomaly source is immersed. Thestability of the bias field plays a key role when the magne-tometer is used to detect NMR free induction decay �FID�:

a�Present address: Department of Physics, University of Pisa.b�Author to whom correspondence should be addressed. Electronic mail:

biancalana@unisi.it.

REVIEW OF SCIENTIFIC INSTRUMENTS 81, 065103 �2010�

0034-6748/2010/81�6�/065103/7/$30.00 © 2010 American Institute of Physics81, 065103-1

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Stabilizing the field makes the nuclear precession occur at amore stable frequency, thus facilitating its analysis and longlasting averaging.

While the characteristics of the dual channel magneto-meter and of its applications to ULF-NMR detection22,23 andmagnetocardiosignal detection24 are reported elsewhere, herewe discuss the performance of a bias field stabilization sys-tem, focusing, in particular, on the effects of residual mag-netic noise in our simplified configuration, in which a pair ofscalar sensors are used to counteract the noise in the fieldmodulus by acting on a single component.

In the case of a self-oscillating magnetometer consideredas a field-to-frequency converter when the field compensa-tion is controlled by a voltage, the entire setup acts as avoltage-controlled oscillator. The whole compensation sys-tem can be regarded as belonging to a well known and ex-tensively studied class of electronic circuits: phase-lockedloops �PLLs�. From a practical and technical point of view,this analogy has two advantages: A complete feedbacktheory related to the PLL has already been developed andassessed, and the fact that a variety of integrated circuits arecommercially available simplifies the task of building highquality phase comparator circuits of several kinds, and de-signing loop filters. The system described in this paper uses acommercial PLL integrated circuit and a Programmable In-terface micro-Controller �PIC� used as a divider to generatean adjustable reference frequency. We foresee that furtherimprovements could be achieved by using digitally con-trolled PLL devices, which contain different kinds of phasecomparators and programmable loop filters.25

II. EXPERIMENTAL APPARATUS

The experimental apparatus is sketched in Figs. 1 and 2.Two sealed cells containing Cs and 90 Torr of Ne as a buffergas are heated to about 45 °C. The two cells are displaced by7 cm along the x direction. Each cell is illuminated by twooverlapping laser beams, both attenuated down to a few�W /cm2. One of them �the pump beam� is circularly polar-ized and broadly frequency modulated so to make it passperiodically in and out of the Doppler-broadened atomicresonance. The other beam �the probe beam� is linearly po-larized and unmodulated. The bias magnetic field �B0z� isperpendicular to the beams’ propagation axis �y�. Providedthat the pump beam excites atoms synchronously with theirLarmor precession, a precessing macroscopic polarization ofatoms is induced in both the cells. The probe beam polariza-tion undergoes a time-dependent Faraday rotation of the po-larization, which is detected by means of two balanced po-larimeters. The modulation frequency of the pump laser canbe scanned to detect and characterize the resonance, or beproduced by a pulse generator triggered by the polarimetricsignal of the main channel, as represented in Fig. 1. Thelatter configuration makes the system a self-oscillating mag-netometer. The plots show the resonance shape detected inscanned mode and the dependence of the oscillatingfrequency as a function of the loop dephasing introduced asa feedback delay. The merit factor of the resonance�Q=�0 /��FWHM� is about 130 and is consistent with the

maximum slope of the lower plot �� rad /110 Hz�. Thisvalue was optimized by adjusting several experimental pa-rameters such as cell temperature, laser intensities, and probedetuning. The value achieved is consistent with other opti-

FIG. 1. �a� Schematic of the self-oscillating optical atomic magnetometer�main arm�. A balanced polarimeter detects the atomic precession, generat-ing a signal that triggers the synchronous optical pumping. The same signalis phase compared with a stable reference signal to generate the error signal,which is fed back to stabilize the field �see Fig. 3�. The �b� plots show thein-phase and quadrature signals obtained in scanned mode. The �c� plotshows the relationship between loop dephasing and oscillation frequency inself-oscillating mode.

DAC

current ctrl

Analog

current controlB0

Active

compensation

+

x

y

z

HC

FIG. 2. Schematic of the apparatus �not to scale�. The transverse field com-ponents Bx,y are compensated by Helmholtz coils �HCs� supplied by numeri-cally controlled current generators. The Bz component �bias field� is analogi-cally adjusted and actively stabilized. The symbols � and � represent thecells of the main arm and secondary arm, respectively, while � representsthe sample. For the sake of clarity, one of the transverse HCs, and thegradient compensation quadrupoles are not shown.

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mized results reported in literature.26 Additional details ofthe self-oscillating setup are available in Refs. 22 and 23.

A. Coil dc supplies

Figure 2 represents �not to scale� the relative positions ofthe atomic sensors and compensation coils. A set of threemutually orthogonal, large �180 cm sides� Helmholtz pairsare used to control the magnetic field. Two pairs are used tocancel out the local magnetic field in the horizontal plane �xand y pairs�, while the third pair is used to partially compen-sate the z component, thus setting the value of the bias fieldto about 4 �T. Three separate current generators are used tosupply the compensation coils. The current supply for the zpair is analogically controlled by high quality potentiom-eters, while the currents for the x and y pairs are digitally�and automatically� controlled. The procedure used to zerothe transverse components of the field is based on the mini-mization of the self-oscillating frequency: The PLL circuit isdeactivated and a simplex procedure adjusts the currents inthe x and y coil pairs while evaluating and minimizing theatomic Larmor frequency.

B. Active PLL compensation

The analog signal generated by one balanced polarimeteris preamplified, filtered, and compared to a threshold value,in order to produce a synchronous square wave which trig-gers �possibly after frequency scaling� the self-oscillatingmagnetometer. The same signal is phase compared with astable reference signal in order to produce the error signal forthe PLL. The phase comparison is performed by phase-comparator-1 �PhC-1, exclusive or� or by phase-comparator-2 �PhC-2, edge-triggered three-state circuit� of aCD4046 integrated circuit.

A chain made of a one-pole/one-zero active filter fol-lowed by an adjustable-gain linear amplifier and by avoltage-to-current converter closes the loop, sinking a smallpart of the current supplying the z Helmholtz pair �seeFig. 3�.

III. PERFORMANCE ANALYSIS

The self-oscillating magnetometer converts the modulusof the magnetic field to frequency so that any modulation inthe modulus appears as a pair of sidebands in the powerspectral density of the magnetometric signal. The left plots inFig. 4 show typical spectra recorded without any shielding oractive compensation. Strong sidebands appear at the powerline frequency and its multiples, as a result of significantmagnetic noise produced by transformers and other electricaldevices. The amplitude of such sidebands varies in time, andin the case of the spectra shown amounts to �10 and−13 dBc for the main and secondary arm, respectively. Theright plots in Fig. 4 were recorded in the same conditions,apart from having activated the compensation system: The50 Hz sidebands are reduced to the broadband noise level�i.e., by about 60 dB� in the case of the main arm and by42 dB for the secondary arm. In addition, for both the armsthe carrier peaks emerge from much weaker pedestals, indi-

cating a significant rejection of low frequency noise. A quan-titative analysis of such noise reduction is made in thefollowing.

A. Short term

The analysis of the short-term performance consists inthe characterization of the residual phase noise in the polari-metric signals. The relatively low frequency operation of themagnetometer makes the direct digitization of these signalspossible, as well as the implementation of a full numericalprocedure for phase noise evaluation.

The data analysis reproduces, using numerical tools, thetypical setup of double balanced mixers �DBMs�, which arecommonly used in the phase noise characterization of rf andmicrowave devices �see Refs. 27 and 28 for an overview andmore in depth discussion�. Pairs of samples are digitized at50 kS/s �or less� with an amplitude resolution of 16 bit. Eachsample is a trace containing 213=8192 data points. In thesetraces two out of the three signals available are recorded: themain arm polarimetric signal, the secondary arm polarimetricsignal, and the PLL reference. Consequently, noise charac-terization can be achieved for the relative phase of the twoarms or for the absolute phase of each individual arm.

Once the digitized traces are transferred to the computer,the main single tones of the two traces are evaluated bymeans of a Lanczos–Grandke procedure29,30 in order to ex-tract their frequency and relative phase and to normalizetheir amplitude. Both traces are then interpolated by meansof fast Fourier transform �FFT�/zero-padding/FFT−1 tech-nique, up to an apparent sampling rate of about 1 MS/s ormore. The two digitized signals are then relative phaseshifted up to a dephasing of � /4. The precision of the phaseshifter is limited by the sampling rate 1 /dt� of the interpo-

VCC=6V

−

+

47M

1k5

−

+

+

−

10k

1k

39

R

2k7

10k

3k3

3k3

gain

OA1 OA2

OA3

−

+OA4

10k

Q1

10k

k3

9

phase

invert

82

R

330n

10

k

DC current

generator

PhC

zco

il

FIG. 3. Schematic of the loop filter and gain adjustment circuit. The phasecomparator �PhC� signal is filtered by OA1, amplified �and optionally in-verted� by OA2, and converted to current by OA3 and Q1. OA4 supplies alocal reference biased 3 V above the ground. Two green LEDs are used as amonitor for the Q1 work point. The Q1 collector sinks part of the HCcurrent, when the closed-loop operation is selected.

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lated trace ���=2��0dt��. This amount sets an upper limit tothe accuracy of the phase measurement described below. Theresulting quadrature traces are then multiplied element byelement. The procedure described so far implements thephase shifter and the mixing circuit of the DBM. The low-pass filter which follows the mixer in DBM setups is re-placed here by a single tone identification and subtractionroutine, tuned to the second harmonic of the previously de-tected frequency. The residual signal can thus be modeled as

g�t� = sin�2��0t + �1�t� + ���cos�2��0t + �2�t��

= 12 �sin�2 � 2��0t + �1�t� + �2�t� + ���

+ sin��� + �1�t� − �2�t��� , �1�

which, after having filtered out �actually subtracted� the com-ponent at 2�0, can be approximated as

g�t� �cos������1�t� − �2�t��

2�

�1�t� − �2�t�2

=���t�

2, �2�

where the two approximations consist in the linearization ofsin���� in the hypothesis ��=�1−�21, and in the as-sumption that ��=0, respectively. The phase difference���t� is finally analyzed by repeated FFT evaluations of itspower spectral density, which are then averaged to highlightthe main features of the spectrum. The above mentioned

quadrature error �� implies a systematic error �1−cos ���2

���4 /4 on the spectrum scale, which is negligible providedthat the interpolation produces a good level of oversampling,e.g., we used dt��1 / �100�0�⇒���60 mrad.

The phase noise spectra recorded by comparing differentcouples of traces are plotted in Fig. 5. Absolute phase noisein the main arm ��� appears together with the relative phase

FIG. 4. Power spectrum of the polarimetric signal. In �a� and �b�, no compensation is used and 50 Hz noise sidebands are clearly visible, with an amplitudeof about �10 and −13 dBc, respectively. Plots �c� and �d� are obtained with active compensation. Plots �a� and �c� refer to the main arm, while �b� and �d�refer to the secondary one. Plots �a� and �c� show that the 50 Hz sidebands disappear, meaning that an extinction of 60 dB is attained in the main arm. In thesecondary arm �plots �b� and �d��, the 50 Hz sidebands are attenuated by 42 dB. Thanks to the rejection of low frequency noise, in traces �c� and �d� the carrierpeak emerges from a much weaker pedestal with respect to �a� and �b�.

FIG. 5. Residual absolute phase noise in the main arm ���, and relativephase noise between the two arms in locked ��� and unlocked ��� fieldconditions �see text for details�. The right and left vertical axes refer tophase noise and magnetic sensitivity, respectively. The conversion betweenthem follows Eq. �3�.

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noise of the two arms, recorded in locked-field ��� andunlocked-field ��� conditions �right axis units�. The relativephase plots are used to infer the differential magnetometricsensitivity, which is evaluated by converting the power spec-tral density of the phase noise to magnetic field units �leftaxis units�,

�SB =�B

��S =

�0

�Q�S =

��FWHM

��S, �3�

where ��21.97�109 �rad /s� /T is the gyromagnetic factorof Cs.

The results shown in Fig. 5 demonstrate an ultimate limitof the residual noise of less than 1 pT /�Hz in the main arm,and of 2.5 pT /�Hz in the secondary one. Concerning thenoise at 50 Hz, residuals of 4 and 10 pT /�Hz appear in thetwo arms, respectively. This evaluation is consistent with therougher estimate derived from Fig. 4. It is important to pointout that the 50 Hz noise varies in time, thus preventing ac-curate comparisons. The peak at 150 Hz is not a third-ordersideband, but rather a consequence of the 50 Hz noise anhar-monicity, and is only partially attenuated in both arms as it islocated close to the cutoff frequency of the loop filter. Aspurious peak at 100 Hz is also occasionally observable inthe phase noise of both arms. This cannot be explained interms of a second order sideband �as it is definitely largerthan estimate J2

2�M� /J12�M��M2 /16 at the modulation index

M inferred from the first sideband amplitude�. This is, forexample, the case of the Fig. 4�d�.

The phase noise spectra can be compared to the noiselevel expected from the photodetectors and photocurrent am-plifiers. The photocurrent shot noise, transimpedanceJohnson noise, and preamplifier input current noise all con-tribute with similar amounts, resulting in about135 �V /�Hz at the output of each arm. This estimate, withan in-phase signal Vx�1 V, leads to a white phase noise aslarge as �2��Vy

2 /Vx2�20 nrad2 /Hz, consistently with the

values that appear in Fig. 5.The stabilization system reduces such phase fluctuations

by a factor of 4 in the main arm. This is achieved by thedesired and appropriate reaction to the magnetic noise and toan unwanted reaction to other noise sources. The latter is atthe expense of an increase in the magnetic noise in the sec-ondary arm. It is worth noting that no additional noise isintroduced, as demonstrated by the substantial agreement be-tween the plots ��� and ��� in Fig. 5. In short, the commonmode noise represented by magnetic fluctuations is counter-acted by the system, while phase fluctuations originatingfrom other noise sources are compensated by magnetic fieldadjustments in the main arm. In this way, part of the noisedetected in the main arm is transferred to the secondary armthrough an additional magnetic noise.

The data shown in Figs. 4 and 5 were obtained using asa phase comparator the PhC-1 �exclusive or�. We observedthat in spite of a larger residual noise at 2�0 in the errorsignal �partially transferred to the coil current�, PhC-1 gave alower background in the phase noise than PhC-2. This fea-ture is consistent with the well-known greater sensitivity tonoise of PhC-2. A cleaner polarimetric signal would probablyresult in favorable conditions for the use of a PhC-2 instead.

B. Long term

Even in ideal conditions, a test of the long term stabilityof the compensation system described cannot be performedby analyzing the drift or the Allan variance of the self-oscillating frequency: Such a test would instead characterizethe relative stability of the reference oscillator and the dataacquisition clock.

Evaluation of the long term stability, i.e., the amount ofthe noise contribution in the subhertz to the submillihertzrange, requires knowledge of the extent to which a certainvalue of the �locked� oscillation frequency corresponds to agiven value of the magnetic field. In fact, while the self-oscillation frequency is ideally set by the Larmor frequencyof atomic spins and thus by the modulus of the magneticfield, several subtle effects may induce slight deviations.These effects have been studied intensively as they are com-monly considered as sources of systematic errors in precisemagnetometry. Should they vary in time due to drifts in theatom-light interaction conditions, an apparent field driftwould appear and would be inopportunely compensated bythe active stabilization system. In conclusion, the long termstability would be set by the drifts of the systematic errors.

We point out several potential causes of drifts of system-atic errors, among which the most relevant are the cell tem-perature and the laser junction currents and temperatures,with their well-known effects on laser tuning. In particular,drifts in pump laser detuning result in slight phase deviationof the synchronous pumping, thus producing drifts in theself-oscillating frequency, in accordance with the behaviorshown in Fig. 1, plot �c�.

The relationship between loop dephasing and oscillationfrequency is a delicate feature of self-oscillatingmagnetometers.31 The loop dephasing is set by a time delayin the loop, whose conversion to dephasing depends on theoperating frequency. For this reason, stabilizing the fre-quency makes the magnetometer more reliable and robust.

It is worth noting that the long term drifts limit the per-formance of the magnetometer to a greater or lesser extentdepending on the specific application. We considered appli-cations in magnetocardiography23 and in ULF-NMR remotedetection.22 In the first case only drifts occurring during acardiac epoch are relevant, which are definitely negligiblecompared to the signal dynamics. In the case of NMR detec-tion, the low signal-to-noise ratio makes long lasting aver-ages necessary, so appropriate measurement requires that thefield drifts occurring during the whole average process causethe nuclear precession frequency to drift negligibly com-pared to the NMR resonance width. We considered averagingintervals ranging from several minutes to several hours, andthe nuclear resonance detected �at �n�180 Hz� had a widthof ��n=1 / �2�T2��100 mHz. The stability required canthus be written as �B /B=��n /�n5�10−4 to prevent anapparent reduction in T2 due to inappropriate averaging.

The latter consideration suggests that nuclear resonancecan be used as an independent and accurate magnetometer tostudy the long term performance of the active compensationsystem in terms of the Allan variance of the nuclear preces-sion frequency. This procedure requires the �n evaluated tobe determined with a very small margin of uncertainty ��n.

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In Sec. IV we report a practical example, where the best-fitprocedure gives a relative frequency uncertainty of ��n /�n

�1.6�10−6, meaning that the relative field drifts in thisamount, occurring in the averaging time �2800 s�, are appre-ciable. The relatively poor signal-to-noise ratio of our setupmakes this procedure difficult. Nevertheless it can be pro-posed as a characterization method for evaluating perfor-mance over the very long term, especially in conjunctionwith more sensitive magnetometers.

C. Residual noise due to field inhomogeneities

As discussed above, our system allows for completecontrol of all the dc components of the magnetic field, whiletracking only its modulus over time, and counteracting itsvariations through adjustments of the strongest �z� compo-nent. It is worth considering the limitations deriving fromsuch a configuration, with the specific aim of evaluating thecritical features that occur when the system is used in a dualchannel setup.

Let us assume that the system perfectly compensates thefield fluctuations in the main arm, and let us derive the re-sidual noise in the secondary arm. We will assume that thenoise B� n and the compensation field Bcz are homogeneous,while the bias field may differ due to static gradient: the fieldis B� 0 in the main cell and B� 1 in the secondary one, with B� 1

�B� 0+J�x�, J being the Jacobian of the field, and �x� theposition of the secondary cell with respect to the main one.Thus, perfect compensation in the main cell reads

�B� 0 + B� n + Bcz�2 = B� 02, �4�

while, concerning the secondary cell,

�B� 0 + B� n + Bcz + J�x��2 = B� 12 + ��B1

2� , �5�

With ��B12� representing residual noise. The difference be-

tween Eqs. �4� and �5� gives

2�B� n + Bcz� · �J�x�� = ��B12� . �6�

Solving Eq. �4� for Bc in the hypothesis B� 0=B0z gives

Bc = − B0 − B� n · z + B01 −Bn�

2

B02 1/2

� − B� n · z −Bn�

2

2B0,

�7�

where Bn� is the projection of the noise field on the xy plane.Thus, for small noise, in the first-order approximation, thefirst parenthesis in Eq. �6� reduces to 2B� � ·J�x�. In conclu-sion, using a scalar sensor to compensate homogeneous fieldnoise in a dual channel setup requires accurate preliminarycompensation of the gradient of the field components per-pendicular to the bias field, in our case �Bx /�x and �By /�x.The inhomogeneity of the parallel component �Bz /�x has adirect �i.e., first order� effect on the field modulus �and thusmust be compensated to force the secondary arm at the cen-ter of its resonance� but has no first-order effects on the noisecompensation efficiency.

IV. PERFORMANCE IN APPLICATION

The compensation system was used in an ULF-NMRexperiment, with measurements similar to those reported inRef. 22. Nuclear precession of remotely polarized water pro-tons was detected in a microtesla field. Figure 6 shows theFID signals and their power spectral density obtained afteraveraging 700 traces.

Active compensation of the bias field renders unneces-sary the frequency rescaling procedure described in Ref. 22,where NMR traces were interpolated and time rescaled tocompensate the nuclear Larmor frequency drift, which wasinferred from the trace-by-trace measured bias magneticfield.

Furthermore, operating in locked-field conditions im-proves the quality of the average trace, as can easily be rec-ognized by comparing the residual traces obtained as a dif-ference of the average signals with their best fits �modeled asan exponentially decaying sinusoid�. In the best-fit examplesshown in Fig. 6, activating the field stabilization systemleads to a reduction in the minimum �2 by a factor of 7. Inaddition, locking the field significantly reduces the low fre-quency noise, as can be clearly seen in the plot �d�.

V. CONCLUSION

We have presented and characterized a simple, eco-nomic, and reliable magnetic field compensation system. Thesystem is integrated in a setup working at a bias field of afew microtesla, which is measured by means of a dual chan-nel magnetometer, based on all-optical atomic scalar sensorsand operating as a self-oscillator.

The compensation system is driven by one of the twochannels and acts on the field component parallel to the bias,feeding back the current supplied to a Helmholtz pair. Itcounteracts the common mode magnetic field fluctuations,

FIG. 6. The plots on the left show FID signals �dots�, fitting curves �thingray lines�, and residuals �thick lines� in the time domain. The correspond-ing spectra are reported on the right. The upper plots are obtained withactive field stabilization, while the lower plots are obtained by tracking thefield and applying the methods described in Ref. 22. Time-domain traceswere improved �as in Ref. 22� by identification and subtraction of compo-nents at 50 Hz and its multiples, and linear filtering �first order Bessel filter,170–190 Hz bandpass�.

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thus facilitating difference-mode detection. The difference-mode signal reproduces the field anomalies generated by aweak source located in the proximity of one sensor.

The operating principle makes the compensation partial,meaning that only the modulus of the magnetic field is keptconstant. The limitations posed by this feature have beendiscussed, arriving at the conclusion that the compensationefficiency is mainly degraded by inhomogeneities of the fieldcomponents perpendicular to the bias direction, which there-fore require careful control.

Detailed data analyses were carried out to characterizethe short-term performance of the stabilization system and toevaluate the phase noise spectra, from which the stray fieldextinction ratio was inferred. Concerning the power-net dis-turbances, we demonstrated an extinction factor in excess of40 dB.

As a practical example, we reported results obtainedwith and without active compensation in an ULF-NMR ex-periment. The FID of remotely polarized water samples wasrecorded and averaged over long intervals. The possibility ofusing these NMR measurements to assess the long term per-formance of the compensation system was also considered.

ACKNOWLEDGMENTS

The authors are grateful to Emma Thorley of the UniSiTranslation Service for reviewing the manuscript.

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