Multichannel optical atomic magnetometer operating in unshielded environment

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by  Giuseppe Bevilacqua

Multichannel optical atomic magnetometer operating in unshielded environment
Giuseppe Bevilacqua, Valerio Biancalana,∗ and Piero Chessa
DIISM, University of Siena - Italy
Yordanka Dancheva
DSFTA, University of Siena - Italy
(Dated: March 2, 2016)
A multi-channel atomic magnetometer operating in an unshielded environment is described and
6 characterised. ThemagnetometerisbasedonD1 opticalpumpingandD2 polarimetryofCsvapour
contained in gas-buffered cells. Several technical implementations are described and discussed in
1
detail. The demonstrated sensitivity of the setup is 100fT/√Hz when operating in the difference
0
2 mode.

b
e I. INTRODUCTION sors.
F
Atomicmagnetometryis basedonthe detectionofthe

9 High sensitivity magnetometers operating in geomag- Larmor precessionof an optically pumped vapour in the
2 netic and lower level fields have a wide range of applica- fieldundermeasurement. Thebasicworkingprinciplesof
tionsinseveralbranchesofbothfundamentalandapplied atomic magnetometers date back to over a century ago
]
t research[1–3]. They areused to bothmeasure andtrack [18, 19], and the first pioneering implementations date
e large scale field variations, such as in geomagnetic sur- backseveraldecades [20]. Nevertheless,renewedinterest
d
- veys [4], and to detect local magnetic variations, such as in atomic magnetometry has arisen much more recently,
s in measurements of ultra-low-field NMR [5–7] and MRI thankstoprogressesintheknowledgeofopticalpumping
n
[8], in relaxometry [9, 10], in biomagnetic applications processes [21, 22], and to technical achievements regard-
i
. such as magneto-cardiography [11–14] and in magneto- ing laser sources and atomic sample manufacturing [23].
s
c encephalography[15,16]. Thedetectionofweaklocalsig- This set of circumstances has led to important advances
si nalscanbe performedinsmallshieldedvolumes,andin- in the last decade, so that atomic magnetometers now
y deedthisapproachispreferablewhensmallsizedsources representaversatileclassofsensorswithstate-of-the-art
h needto becharacterised. However,the existenceoflarge sensitivity[2,24,25]. IncontrasttoSQUIDmagnetome-
p sizedsamples(suchashumanbodiesinbiomagneticmea- ters,whicharevectorcomponentdetectorsthatmeasure
[
surements),the highcostsandthe delicatenessofshield- thefieldprojectionalongoneparticulardirection,atomic

2 ing devices may prompt the use of alternative methods magnetometers(at least in their basic configuration)are
v in order to reject the magnetic noise generated by far- scalar in nature, as they measure the modulus of the
8 located sources. magnetic field. A scalar sensor operating in a dominant
3 Multi-channelsensors[12,14]openthe wayfortheac- bias field is mainly sensitive to the field variation along
9
tive compensation of common mode magnetic signals as the bias field. Thus atomic magnetometers operating in
6
0 wellasthedirectmeasurementofdifferencemodesignals. non-vanishing fields are responsive to one component.
. Thus,eveniftop-levelsensitivitynormallyrequiresoper- Provided that the environmental (bias) field is rela-
1
ationinshieldedvolumes,relativelyhigh-sensitivitymag- tively homogeneous, this kind of magnetometers can be
0
6 netometry can be profitably performed in open volumes profitably organisedinto sets of two or more elements to
1 [7,17]withamoresimplifiedimplementationwhenusing produce a gradiometric response [12, 26], thus rejecting
: sensor operating in non-vanishing fields. Superconduct- common-mode magnetic noise and identifying signals as
v
ing quantum interference devices (SQUIDs) have tradi- weakasseveralordersofmagnitudesbelowthebiasfield.
i
X tionally provided the highest magnetometric sensitivity. The literature reports several kinds of setups, where
r Recently, the most sensitive atomic magnetometers in various parameters of the pumping radiation are modu-
a the spin-exchange-relaxation-free (SERF) regime, which lated, such as the polarisation [27], the intensity [28] or
operateinvanishingfields,havedemonstratedtheircom- the optical frequency [26]. A comparison of the different
petitiveness with SQUIDs, while also having the advan- approachesispresentedinRef.[29]. Thefrequencymodu-
tage of not requiring cryogenic cooling, and of being un- lationof the pumping radiationsimplifies the implemen-
affected by electric noise. Compared to SQUIDs and tation (no optical modulators needed), particularly in a
SERF, other atomic magnetometers have worse sensi- dual radiation setup.
tivity, although their performance remains better than Inthis paper wedescribe anall-opticalatomicmagne-
1 pT/√Hz, meaning that they are unrivalled by other tometerbasedonCsvapour,opticallypumpedbymeans
simpler devices such as Hall and magnetoresistance sen- of a frequency modulated laser source tuned to the D1
line, andprobedthroughthe rotationofthe polarisation
plane of a weak linearly polarised beam tuned to the
D line. The systemcontains two composite sensorsand
2
∗ [email protected]
theoutputsignalsareextractedfromtwocouplesofhalf-
2
beampolarimeters,thusobtainingafour-channelsystem. tionalongasingleopticalaxis)andthepumpsuppression
Depending on the channels selected and on choices at after the interaction. This is effectively accomplished by
the data analysis stage, this setup enables gradiometric means of an interference filter.
measurements of various orders and with several differ-
ently arranged baselines. The sensitivity demonstrated
is limited by residual magnetic noise at low frequencies,
and, when using the shortest baseline, is essentially de-
termined by the instrumental noise. In this latter con-
figurationthe sensitivity level (100 fT/√Hz in the range
20 Hz 100 Hz) is comparable to values reported in the
÷
literature for similar setups operating in a shielded envi-
ronment.
II. SETUP OVERVIEW
A. Optics
This magnetometer is the evolution of an apparatus Figure 1. Schematics of a single arm of the magnetometer.
describedpreviously[6,26]. Ithastwoidenticalarmsfor The D1L pumps the atoms and the D2L probes them. The
differential measurements, but also allows for quadruple two radiations are mixed in the 2 2 coupler, and led to
×
channeloperation. Eacharm(seeFig.1)containsanillu- thesensorviaapolarisation-maintainingfibre(PMF).Atthe
minator (ILL) providing two co-propagating radiations, PMF output, the illuminator (ILL) collimates the two radi-
ations into a single beam and adjusts their polarisation by
a Cs vapour cell, and a dual balanced polarimeter. One
means of a multi-order wave-plate (MOWP), which renders
of the radiations is resonant with the D transitions of
1 the polarisation of the pump circular, and leaves the polari-
the Cs atoms and is used to optically pump the atoms;
sation of the probe linear. The Cs cell is illuminated by the
the other radiation is near resonant with the D transi-
2 two co-propagating radiations. After interaction with theCs
tions and is used to probe the atomic spins precession.
vapour, thepump radiation is stopped by an interference fil-
The light generated by the pump laser D1L and by the ter (IF), while the probe impinges on a Wollaston polariser
probe laser D2L is coupled to a polarisation-maintaining (W) oriented at 45◦ with respect to the input polarisation
fibre (PMF). A proper mixture of the two lights is ac- plane. A dual balanced polarimeter (DBP) measures the ro-
complished using a 2 2 PM on-fibre mixer. Both light tationoftheprobepolarisationplaneforthetwohalvesofthe
sources are based on×solid-state continuous-wave lasers. beamseparately. Tothisend,theimbalancesinthephotocur-
The probe radiation is attenuated by means of an on- rents flowing in the two couples of series silicon photodiodes
areamplifiedbytrans-impedanceamplifiers(TIA).Theinset
fibreattenuator(Att)andiskeptataconstantfrequency,
representsafront viewof theprobebeams of boththearms,
while the pump radiation is left at the mW level, and is
wheretherelativeorientationsofthevariousbaselinesandof
broadly frequency modulated. In the proximity of each
thebias magnetic field are defined.
vapourcell,thetworadiationsarecollimatedintoabeam
of about 10 mm in diameter, their linear polarisation
is reinforced by a polarising cube, and finally a special
multi-order waveplate (MOWP) makes the pump radi-
ation circularly polarised, while leaving the probe one B. Sensor
linearly polarised.
For this purpose we took advantage of quartz’s prop- The atomic sensors are made of high quality sealed
erty of birefringence at the D and D transition wave- cells with flat windows, containing solid source Cs and
1 2
lengths in Cs. In fact, it is possible [30] to design a 23 torr of N as a buffer gas. The vapour density of the
2
multi-orderquartzwaveplatewitha nominalrelativede- Cs is increased by warming the cells up to about 45◦C
lay of 4.75 wavelengths for D radiation and 5.00045 by means of analternating currentheater. The heater is
1
wavelengths for D radiation. Thanks to their low or- madeoftwocoaxialcoilsofthincopperwirewoundinan
2
der, the relative delay of such waveplates is only weakly anti-inductive configuration. Possible spurious magnetic
dependent upon misalignment (less than 0.1 % for 1deg fields generated by the heater are rendered of negligible
misalignment). relevance [31] by supplying the coils at 50 kHz, a value
The MOWP technique was similarly employed in an- largely exceeding the atomic Larmor frequency.
other setup described in the literature [15], with several After interaction with the atomic vapour, the pump
practicaladvantagesthatareworthyofmention. Infact, beam is blocked by an interference filter (IF) [Thorlabs
the use of different transitions for the pump and probe 850/40, with a transmission of 98.5% at 852 nm and an
processes, facilitates both the alignment (the two radia- optical density of 4.2 at 894 nm] and the probe beam
tionsaremixedinonefibre,andpropagateaftercollima- polarisation is analysed by dual balanced polarimeters
3
(DBP) made of a 45◦ oriented Wollaston analyser (W)
and two couples of silicon photo-detectors. A similar
technique, based on segmented photodetectors, has re-
centlybeenappliedinahigherbuffer-gaspressurepotas-
siumsetup[32]. Eachcoupleofphotodetectorsmeasures
the polarisation of half a beam (a semicircle in the inset
of Fig. 1) by converting the polarisation rotation into a
photo-current imbalance, which is in turn amplified and
convertedintovoltagebylow-noisetrans-impedanceam-
plifiers (TIA). The signal extracted by a balanced po-
larimeter is proportionalto the sinus oftwice the polari-
sation rotation angle. At the maximal observed rotation
(about 0.15 rad) the linear approximation results in a
0.35% underestimation, which in the data elaboration is
neglected.
The operator may select the channels to be acquired
Figure 2. Optical frequency of the D1L (solid black) in re-
and the corresponding TIA outputs are simultaneously
sponse to a square wave modulation signal (dashed green).
digitised by means of a 16 bit data acquisition card. The green hatched band points out the time intervals when
The sensors work in a homogeneous magnetic field D1L goes into resonance with an atomic line (rightmost red
B~ , which is obtained by partially compensating the profile), and the blue dotted line represents the resulting
bias
environmental one and is oriented along the y axis. The pumping rate. The figure is not to scale, but is produced
in accordance with the simple model described by eq. 1. In
longitudinal field component (x) is zeroed, and thus the
thetextthemaximalandminimalfrequenciesreachedinthe
sample magnetisation, which is periodically reinforced
periodwillbereferredtoasfront-andback-frequency,respec-
along the optical axis, precesses in the xz plane. Small
tively.
variations in the field slightly change the resonant pre-
cession frequency, or (as discussed in the Appendix) the
precession phase when periodic resonant pumping is ap-
plied. changingthejunctioncurrent,thelaseremissionchanges
for two reasons. The first is related to the change of re-
fraction index and has a very quick (instantaneous) re-
III. PRINCIPLE OF OPERATION sponse. The second reason is a thermal change of the
cavity length, and is a much slower process. The latter
A. Laser sources and modulation of the optical introducesasecondcut-offfrequencyatΓ2 2π 30Hz.
≈ ×
frequency Weperformedadetailedcharacterisation,basedontech-
niques involving both atomic and interferometric refer-
ences [33]). The optical frequency response to a signal
The probe radiation is generated by D L, a single-
2
at ω can be modelled as the sum of two low-pass-filtered
mode, Fabry-Perot, pigtailed diode laser that is near-
terms,andtheinstantaneousopticalfrequencydeviation
resonant with the D line of Cs at 852 nm. This illu-
2
can be approximated as
minates the cell at a weak intensity, which is attenuated
down to a level of a few µW/cm2 in order to negligibly
perturb the atomic state measured. +∞
w Γ w Γ
1 1 2 2
The pump radiation is generated by D L, a single- ∆ν = + V +c.c. (1)
1 n
(cid:18)Γ +inω Γ +inω (cid:19)
mode, distributed feedback, pigtailed diode laser. Its nX=0 1 2 0
optical frequency is controlled through the junction cur-
rent and is periodically made resonant to the transition
where Γ and w are the cut-off frequencies and weights
2S ,F = 3 2P of Cs at 894 nm. The optical i i
1/2 g 1/2 of the thermal and input filters, respectively, and V are
| i → | i n
frequency of the pump laser is modulated using a square
thesquarewaveFouriercomponents. Moreprecisely,the
wave signal generated by a computer-controlled digital-
modulation signal
to-analogue converter.
Applying a square wave modulation to the junction
currentof D1L produces a modulation of the optical fre- V(t)= +∞V +c.c.= +∞ 2Asin(nπδ)eniωt+c.c. (2)
quency that cannot easily be described in terms of fre- n nπ
quency steps. At least three aspects have to be taken nX=0 nX=0
into accountin order to evaluate the response to sudden
changesinthe modulatingsignal. Firstofall,the modu- is a square wave with amplitude A and duty cycle δ.
lation input of the laser driver has a limited bandwidth. Typically, the conditions, Γ ω and Γ >ω hold. The
1 2
≪
This acts as a low-passfilter with a cut-off frequency set resulting optical frequency modulation is represented in
at about Γ =2π 25kHz in our case. In addition, when Fig. 2.
1
×
4
(Fig. 3b), the resonance amplitude grows again (high-
rate Zeemanopticalpumping is performed),but the res-
onance is much broader. Moreover, in this condition a
small red detuning results in an abrupt decrease in the
signal amplitude, due to hyperfine optical pumping to-
ward the unprobed F =3 ground state.
g
C. Sensitivity and optimal working conditions
The modulation duty cycle, amplitude and offset are
adjusted to ensure maximum sensitivity. As seen from
Fig.3,thebestsensitivityisachievedwhentheD Lfront
1
frequency is resonant with the 2S ,F =3 2P
1/2 g 1/2
| i→| i
hyperfine components of the D line. Concerning the
1
probe laser, its optimal frequency is found to be 2 GHz
bluedetunedwithrespecttothetriplet 2S ,F =4
Figure3. Amplitudeandwidthoftheobservedresonanceline 1/2 g
| i→
for a fixed D1L modulation span, as a function of the front 2P3/2 , which is an acceptable compromise in order to
| i
frequency. Asquarewavemodulationsignalwithadutycycle achievea goodpolarisationrotationsignalandsmall ex-
of 75% is applied. This produces (see Fig. 2 and eq. 1) the citation rate.
instantaneousD1Lfrequencyrepresentedbyblacklinesinthe Undertheseconditions,thepumplaserplaystworoles:
insets, having a span of 11.2 GHz. Two amplitude maxima it causes a strong hyperfine optical pumping of the Cs
are observed when either (a) the front frequency is resonant
atoms towards the ground state analysed, and simulta-
with the leftmost hyperfine component or (b) the back fre-
neouslypumpstheatomsintoaspecificZeemansublevel
quency is resonant with the rightmost one. In case (a) the
ofthatstate,duetoitscircularpolarisation. Thecombi-
resonancewidth isminimum,whichmakesthisconfiguration
nation of these two effects ensures a high amplitude and
the optimal working condition. The dashed lines correspond
tointermediatedetunings,atwhichnoclearresonancelineis smalllinewidthoftheresonance,alsothankstotheeffect
observed. known as light narrowing [34, 35]. The linewidth is also
narrowthankstothefactthatthehyperfinegroundstate
measured is weakly perturbed, because D L is either far
1
B. Atom-light interaction off-resonance or resonant with the other (depleted) hy-
perfine ground state.
The D L optical frequency deviation amounts to tens The Zeeman pumping produces magnetisation of the
1
of GHz, thus the radiation periodically goes in and out Cs vapour along the wave-vector (x direction). As the
of resonance with given transitions of the D line. magnetic field is transverse to the pump light propa-
1
Let us define the highest instantaneous optical fre- gation (y direction), the atomic state where atoms are
quencyasfront frequencyandcorrespondinglythelowest pumpedevolvesintime. Atamacroscopicscalethiscor-
one as back frequency (see Figs. 2 and 3). If the fre- responds to a magnetisation precession in the xz plane
quency deviation is kept constant, when the front and at an angular frequency ωL, which is set by the mag-
the back frequencies increase, the amplitude and the netic field modulus B and the gyromagnetic factor γ:
width of the magnetic resonance change, as shown in ωL =γB. Clearlydetectableprecessingmagnetisationis
Fig. 3. The probe laser is cw and tuned to be nearly obtained,againatthemacroscopiclevel,whenthepump
resonant with the triplet 2S ,F =4 2P , thus radiation interacts with the atoms synchronously with
1/2 g 3/2
itprobestheatomicpreces|sionoftheF i=→4|state.iWhen the precession.
g
the D L approaches the D line from the side of the
1 1
2S ,F = 3 2P ,F = 4 transition, the ampli-
1/2 g 1/2 e
| i → | i
tude of the resonance increases, while its width remains IV. OPERATION MODES, FIELD RANGE,
almost constant. The amplitude shows two maxima oc- RESONANCE LINEWIDTH
curring when the front frequency goes into resonance
with the transitions 2S ,F = 3 2P ,F = 3,4 The atomic precessionis detectedby analysingthe ro-
1/2 g 1/2 e
| i → | i
(seeFig.3a))andwhentheback frequencygoesinto res- tation of the polarisation plane of a weak linearly po-
onance with the 2S ,F = 4 2P (see Fig. 3b)), larised probe beam, propagating in the x direction. As
1/2 g 1/2
| → | i
respectively. isknown,thelinearpolarisationcanberegardedasasu-
When D L is near-resonantor resonantwith the F = perposition of two, σ+ and σ−, counter-rotating circular
1 g
4 (either with its front or back frequency), the reso- polarisationcomponents,whichexperienceoppositevari-
nance width increases dramatically untile the resonance ations of the refraction index (time dependent circular
is barely recognisable. In the rightmost region, when birifringence). Along the propagation,these two compo-
the back frequency is near-resonant with the F = 4 nents accumulate different phase delays, and superpose
g
5
into a linear polarisation rotated by an angle oscillating
at frequency set by the atomic precession. Beside the
different dispersions, the σ+ and σ− components expe-
rience also different levels of absorption (circular dichro-
ism),buttherelevanceofthisaspectislimitedbyappro-
priately detuning the probe laser from the centre of the
atomic resonance.
The polarimetric signal can be analysed as a function
oftheD Lmodulationfrequencyinordertocharacterise
1
the atomic magnetic resonance, in an operation mode
that is referred to as scanning mode and enables accu-
rateevaluationofboththe precessionfrequency(usedto
adjustB ) andthe resonancelinewidth observed. The
bias
latteristheparametertobeminimised(seebelow)inthe
proceduredevotedto reducing the fieldinhomogeneities.
For short-time measurements, it is instead possible to Figure 4. Polarisation rotation of D2L radiation (in-phase
modulate D L with a periodic signal near-resonantwith andquadraturecomponents)asafunctionoftheD1Lmodu-
1
lation frequency. Dots: values measured, line: best-fit curve,
the Larmor precession, and to infer the magnetic field
modelled with a Lorentzian profile. The best fit estimation
fromthedephasingofthepolarimetricsignal. Thisoper-
of HWHMis20 Hzundertheoptimalworking conditions, in
ationmode is referredto asfree-running. Itunavoidably ◦
whichthemaximumrotationof16 peak-to-peakisobserved.
suffers from slow drifts of the environmentalfield, which
Anarrowerline(10Hz)isobtainedatlowertemperaturesand
can eventually lead the system too far from the reso- lower excitation rates, at the expenses of a much smaller ro-
nance. Thisproblemiscounteractedinathirdoperation tation (0.3◦).
mode (locked mode). In this case one of the polarimet-
ric signals is used as an input for a phase-locked loop
(PLL) to stabilise the magnetic field [36]. The environ- tions are inferred from phase estimations of the polari-
mental noise is reduced within the loop bandwidth, and metricsignal. As showninthe Appendix, the phaseϕ(t)
this improves the noise rejection when performing dif- respondstothetime-dependentfieldδBk(t),accordingto
ferential measurements, as the spurious common mode a first order equation (see eq. (A8)), so that working at
noise terms are reduced. the centreof resonance(∆=0)its variationis promptly
The magnetometer operates in bias magnetic fields inferred from the measured ϕ as
ranging from 100 nT to 6 µT. Fig. 4 shows the in-phase
δB =γ−1[Γϕ +2f (ϕ ϕ )] (3)
and quadrature signals of the Cs magnetic resonance as i i S i+1− i−1
functions of the pump laser modulation frequency, as
f being the sampling frequency, δB the field variation
S i
recorded in the scanning mode. The linewidth measured
and ϕ the phase, sampled at the times t .
i i
(half-width at half maximum, HWHM) is of the orderof
20 Hz and is determined by the spin-exchange collisions
and power broadening contributions. V. NOISE AND HIGHER-ORDER
As shown in the Appendix, this linewidth sets the re- GRADIOMETRY
sponse bandwidth of this kind of sensors, which –with
theexceptionofsomespecificimplementations[37]–cor- As far located (noise) field sources produce mainly
responds to the cut-off frequency of their response. common-mode magnetic signals, their contribution can
The direction of the bias magnetic field is set with beeffectivelyrejectedbyperformingdifferentialmeasure-
the help of three pairs of 1.8 m side square Helmholtz ments. The principal arms have an adjustable base-line
coils (see Refs. [11, 36] for additional details). The 5.6 cm in the y direction, with a minimum of 5.6 cm.
magnetic field Jacobian, Gij = ∂Bi/∂xj, is zeroed us- Eacharmis inturndividedintwosecondarysectionsby
ing five quadrupole devices, taking specific care with the using segmented photodiodes, as discussed in Sec IIB.
∂By/∂xi elements, which cause static inhomogeneities The array elements are arranged in such a way as to
of δBk and thus most affect the performance. This measure the two halves of the laser spot independently,
is coarsely achieved with permanent magnets and then which enables gradiometric measurements with a fixed
finely adjusted with electromagnets. After such field baseline of 0.5 cm, as well as higher-order gradiometric
inhomogeneity compensation, the resonance narrowing measurements.
is limited –under operative conditions– by the spin- The N buffer gas contained in the sensor cells
2
exchangecollisionsandpumpandprobeexcitationrates. quenches the fluorescence light avoiding radiation trap-
As a matter of fact, when operating at very low Cs den- ping, and slows down the thermal motion of Cs atoms,
sities and excitation rates (at about 10◦ C), resonances makingitdiffusive. The diffusionconstantofCs inN is
2
as narrow as 6 Hz are recorded. about D=2.4 cm2/s at 23 Torr [38]. Over a time inter-
In the free-running and locked modes the field varia- val as long as the spin relaxation time (1/2πΓ 8 ms),
≈
6
thelinearsuperpositionthatminimisethevarianceofthe
combinationintheabsenceofasignalsource. Informula,
whenacquiringNchannels,theNsignalss (t)withk=0,
k
N−1
1,..., N-1 are linearly combined in S(t) = a s (t),
k k
kP=1
wherea issetto1,anda (k=1,...,N-1)aredetermined
0 k
n
as the ones minimising S(t) 2 = S(t )2, t being
i i
k k { }
iP=1
the sampling times. Depending on the application, the
optimalcoefficients a can be determined after having
k
{ }
band-pass-filtered the signals s (t), in order to achieve
k
the best noise reduction in a specific spectral range.
VI. CONCLUSION
Figure5. Noisepatternsofthedifferencebetweentwosignals We have developed and characterised a multichannel
obtained in two operation modes. The traces show the RMS opticalatomic magnetometer based on a dual arm setup
values of the power spectral density averaged over 10 mea- andsegmenteddetectors,enablingpolarimetricmeasure-
surements. The three spectra correspond to the operation ments in transversesubsections of the probe beam, for a
modes reported in theinset. totaloffourindependentchannelsandvariousalignment-
dependent gradiometric arrangements. The setup de-
scribedmakesitpossibletoperformdifferentialmeasure-
diffusive displacements of the order of ∆x =1.4 mm are ments and gradiometric detection over various baselines
thusexpectedtooccur: adistancedefinitelyshorterthan anddifferentorders. Theenvironmental(commonmode)
thebeamradius. Theshortbaselineoperationisthusre- noise-rejection permits the achievement of a noise level
sponsive to field inhomogeneities. as low as a 100 fT/√Hz .
If the illuminator and the dual polarimeter (ILL and
DBP, respectively, in Fig. 1) of an arm are rigidly
mounted, so that they rotate synchronously around the
Appendix A: Polarimetric signal phase
optical axis, gradiometric detection is performed along
an arbitrary direction ξ of the yz plane (see the inset in
This appendix is devoted to explaining the conditions
Fig. 1) and the system responds with the short baseline
and the consequent approximations considered in order
to ∂B /∂ξ. The long baseline is oriented along the y
y
to infer time-dependent magnetic field variations from
direction, thus the four channel operation enables mea-
surements of second order terms ∂2B /∂ξ∂y, which (de- the phase of the polarimetric signal.
y
pending onthe alignment)canbe variedfrom∂2B /∂y2 We refer to the geometry of the experiment reported
y
to ∂2B /∂y∂z. in Fig. 1 and we assume that the magnetic field is a su-
y
perpositionoftwoparallelterms: adominantstaticterm
Hereafter, let us consider the case in which the three
(bias) and a small periodic one
baselines are parallel, i.e. each couple of channels re-
sponds proportionally to ∂B /∂y. When combining the
y B(t)=(B +δB (t))yˆ,
signalsfromtwochannelsdifferentcouplescanbechosen, bias k
and correspondingly different residual noise is observed.
with δB (t+2π/Ω)=δB (t).
As expected, the bigger the baseline the higher is the k k
The Larmor equations for the magnetization M =
residual noise, as shown in Fig. 5.
(M ,M ,M ) are
When applying the PLL field-stabilisation system [36] x y z
the noise in the y direction is actively compensated,
M˙ = ΓM +(ω +ω (t))M +f(t)
which decreases the noise in the frequency range from x − x L 1 z
0 to 200Hz, that is the bandwidth ofthe PLL.The ulti- M˙y = ΓMy
−
mate estimated magnetometer sensitivity, which results M˙ = ΓM (ω +ω (t))M ,
z z L 1 x
from the superposition of detection noise and residual − −
magnetic noise, is of the order of 100 fT/√Hz, as can where ω =γB and ω (t)=γδB (t) are the Larmor
L bias 1 k
be seen in Fig. 5 from measurements conducted over a frequenciescorrespondingtothebiasfieldandtoitsvari-
0.5 cm baseline. ation, respectively. In our case ω is in the kHz range
L
More generally, instead of recording the difference be- while the damping Γ and Γ are in the 10 Hz range so
k ⊥
tween two signals, it is possible to combine three or four we assume they are equal Γ = Γ = Γ. The forcing
k ⊥
signals in order to achieve optimal noise cancellation. term f(t) is due to the laser optical pumping and can
This task is performed by determining the coefficients of becalculatedexactly,intheapproximationofweaklaser
7
power (details will be provided in a forthcoming paper). the phase of M (t) = A(t)e−iωt so it is better to
x
ℜ
This term can be expanded in Fourier space as work out an equation for A(cid:2)(t) directl(cid:3)y. Assume that the
solution M (t) has the form outlined in (A5) and that
+
+∞ A(t) = a(t)ei(π/2+ϕ(t)). The phase ϕ(t) (the excess of
f(t)= fneinωt phase from exact resonance) is the quantity monitored
X−∞ inthe experiment. Insertingthis ansatzineq.(A1), after
some algebra we find
where ω is the laser modulation frequency. The coeffi-
cients fn satisfy f−n =fn∗ andforoddn itcanbe shown a˙ = Γa+ f−1 cosϕ (A6a)
− | |
that f =if .
n n aϕ˙ = (∆+ω )a+ f sinϕ, (A6b)
| | 1 −1
Introducing M =M +iM the relevant equation of − | |
+ z y
motion becomes where we have introduced the detuning ∆ ω
L
ω and coherently approximated f(t) f ≡e−iωt =−
−1
M˙ =[ Γ i(ω +ω (t))]M +f(t). (A1) f eiπ/2e−iωt. The laser operates at lo≈w power,so the
+ − − L 1 + | −1|
modulus of f Γ and the a(t) adiabatically follows
−1
| |≪
The steady-state (Γt 1) solution can be written as
≫ f
−1
a(t) | |cosϕ,
M (t)=e−(Γ+iωL)t−iθ1(t) te+(Γ+iωL)′+iθ1(t′) f(t′)dt′, ≈ Γ
+
Z0 giving the dynamical equation for the phase
(A2)
where θ1(t) = 0tω1(t′)dt′. Introducing the Fourier ex- ϕ˙ =−∆−ω1(t)−Γtanϕ. (A7)
pansion R
This,underourexperimentalconditionscanbelinearized
+∞ as
eiθ1(t) = G einΩt (A3)
n
X−∞ ϕ˙ = ∆ ω1(t) Γϕ, (A8)
− − −
(in the casethat B (t) is sinusoidalthe G arethe usual which describes the evolution of the phase as the sum of
1 n
Bessel functions) following some algebra we find adampingterm( Γϕ)andaforcingterm( ∆ ω (t)).
1
− − −
The steady-state (Γt 1) solution is now elementary
≫
+∞ G∗ G f
M+(t)= Γ+iωn−+simnωm+inΩeisΩteimωt. ϕ(t)= ∆ + te−Γ(t−t′)ω (t′)dt′. (A9)
s,n,Xm=−∞ L Γ Z 1
0
(A4)
Intheexperimentthemodulationfrequencyωisresonant For a sinusoidal signal ω (t)=wcos(Ωt+φ) one finds
1
with ω so that the term with m = 1 is the dominant
L
−
one. Finally we obtain the expression ∆ w
ϕ(t)= + cos(Ωt+φ arctan(Ω/Γ)). (A10)
Γ √Ω2+Γ2 −
+∞ G∗ G f
M (t) n−s n −1 eisΩte−iωt
+
≈s,nX=−∞Γ+i(ωL−ω)+inΩ (A5) ACKNOWLEDGMENTS
A(t)e−iωt,
≡
The authors acknowledge the valuable technical sup-
whichshows,asexpected,thatthemagnetizationfollows port of Leonardo Stiaccini and the financial support
the forcing term with a complex amplitude A(t), which of the Italian Ministry for Research (FIRB project
has the periodicity of the δB (t) field. RBAP11ZJFA005). The authors thank Emma Thorley
k
Althoughthisexpressionisexact,thedoublesumhides of Language Box (Siena) for revising the English in the
the interpretation. Moreover, the experiment monitors manuscript.
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