Non-invasive detection of animal nerve impulses with an atomic magnetometer operating near quantum limited sensitivity

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by  Kasper Jensen

Non-invasive detection of animal nerve impulses with an atomic magnetometer
operating near quantum limited sensitivity.
Kasper Jensen,1 Rima Budvytyte,1 Rodrigo A. Thomas,1 Tian Wang,1 Annette Fuchs,2 Mikhail V. Balabas,1,3
Georgios Vasilakis,1 Lars Mosgaard,1 Thomas Heimburg,1 Søren-Peter Olesen,2 and Eugene S. Polzik1
1Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, 2100 Copenhagen, Denmark
2Department of Biomedical Sciences, Faculty of Health and Medical Sciences,
University of Copenhagen, Blegdamsvej 3, 2200 Copenhagen N, Denmark
3Department of Physics, St Petersburg State University,
Universitetskii pr. 28, 198504 Staryi Peterhof, Russia
Magnetic fields generated by human and an- yields the magnetic field values which are much higher
6 imal organs, such as the heart, brain and ner- than that in an animal because the return currents in
1
vous system carry information useful for biologi- the surrounding tissue are not measured. Here we are
0
cal and medical purposes. These magnetic fields able to detect the nerve impulse with the sensor placed
2
are most commonly detected using cryogenically- beside the nerve, several millimeters away, the setting
n
cooled superconducting magnetometers. Here we compatible with in vivo studies.
a
J present the first detection of action potentials Sensitivityofatomicmagnetometers[2]improveswith
2 from an animal nerve using an optical atomic the number of atoms sensing the field, which for vapor
1 magnetometer. Using an optimal design we are magnetometers is defined by volume and temperature.
able to achieve the sensitivity dominated by the For example, femtoTesla sensitivity has been achieved
]
h quantum shot noise of light and quantum projec- with magnetometers operating at a temperature of sev-
p tion noise of atomic spins. Such sensitivity allows eralhundredof◦Cintheso-calledSERFregime[3]used
- us to measure the nerve impulse with a minia- also for medical applications [4–6]. Similarly high sen-
t
n ture room-temperature sensor which is a critical sitivity has been achieved at room temperature using
a advantage for biomedical applications. Position- much fewer atoms by means of quantum state engineer-
u
ing the sensor at a distance of a few millimeters ing [7] leading to operation beyond standard quantum
q
[ from the nerve, corresponding to the distance be- limits of sensitivity. Room temperature operation allows
tween the skin and nerves in biological studies, to place the sensor in contact with the skin or poten-
1
we detect the magnetic field generated by an ac- tially inside the human body. The close proximity of the
v
3 tion potential of a frog sciatic nerve. From the sensor to the source of magnetic field is a big advantage
7 magnetic field measurements we determine the as the magnetic field rapidly decreases with the distance
2 activity of the nerve and the temporal shape of fromthesource. Room-temperaturecesiummagnetome-
3 the nerve impulse. This work opens new ways ter has been used for medical applications [8], however,
0
towards implementing optical magnetometers as it operated far above quantum limits of sensitivity.
.
1 practical devices for medical diagnostics.
Here we use the approach of Ref. [7] for nerve impulse
0
The magnetic field generated around a signaling nerve measurements. Thesensitiveelementofthemagnetome-
6
1 fiber is of key interest both from a basic scientific and ter is cesium atomic vapour. Cesium has a high va-
: a clinical point of view. The transmembrane potentials por pressure such that high sensitivity can be reached
v
have been extensively measured with electrophysiologi- at room- or human body temperature. The magnetic
i
X cal techniques. Magnetic field measurements are insen- moment (spin) of atoms J = (J ,J ,J ) is prepared by
x y z
r sitive to the transmembrane currents as the fields from optical pumping in the x-direction, along the direction
a
the opposite currents in and out of the membrane can- of a bias field B [see Fig. 1(a)]. The magnetic field of
x
cel. Instead, magnetic field measurements allow for a the nerve B will create a transverse spin component
nerve
true measurement of the axon’s axial net current, which J = (J ,J ) which afterwards will rotate in the y-z
⊥ y z
is the depolarizing wavefront driving the action poten- plane at the Larmor frequency Ω=B /γ [see Fig. 1(b)],
x
tial. Magneticfieldrecordingsalsoallowfornon-invasive where γ =2.20·1010 rad/(s·T) is the cesium gyromag-
measurements of the conduction velocity of peripheral netic ratio. The J spin component is detected optically
z
nerveswhichisnecessaryfordiagnosticsofmultiplescle- bymeasuringthepolarizationrotationoftheprobelight.
rosis, myotonia and intoxication in patients. The magnetic field from the nerve is detected in two
The magnetic field of a nerve impulse was first mea- modalities, a continuous mode where the magnetic field
sured by Wikswo et al [1] using a combination of a su- asafunctionoftimeB(t)isdetected,andapulsedmode
perconductingSQUIDmagnetometerandatoroidalpick- where the Fourier component |B(Ω)| is detected. In the
up coil through which the nerve had to be pulled. This continuous mode the pump and probe light is continu-
method is not compatible with in vivo diagnostics and ously on. In the pulsed mode [see Fig. 1(c)], a pulse of
2
(a) (b)
x
z
(c)
pump &
probe
repump
B
nerve
S(t)
stimulation
FIG. 1: (a) Schematic of the experimental setup. Probe light propagates along the z axis. Half-wave plate λ/2, polarizing
beamsplitter(PBS)anddifferentialphotodetectionarecomponentsofpolarizationdetector. (b)Themagnetometerprinciple.
The amplitude of the collective atomic spin precession in z,y plane is proportional to B . Spin projection J is measured
nerve z
by probelightwith thesensitivitylimited by thequantumprojection spin noise (fuzzy circle). (c)The measurement sequence
for the pulsed magnetometer mode.
pumplightisfollowedbythepulseofmagneticfield,and age distance of 4 mm from the nerve axis which is close
finally the spins are detected with a pulse of probe light. to a typical distance for many medical applications.
Optical magnetometers are fundamentally limited by A frog sciatic nerve contains a few nerve bundles each
quantum spin-projection noise (PN) shown as the fuzzy withseveralthousandaxonsinside(seeMethodssection).
circle in Fig. 1(b). This limit has been reached for mag- The nerve is placed inside a plastic chamber where it
netic fields oscillating at hundreds of kHz in [7]. The can be kept alive in a saline solution for more than 5
techniquesallowingustoapproachthePNlimitedsensi- hours. The nerve is electrically stimulated from one end
tivity for nerve impulses whose frequency is much lower with a pair of gold electrodes [see Fig. 1(b)]. The stimu-
are described in Supplementary Material. For continu- lus triggers an action potential (a nerve impulse) propa-
ous measurements, the PN limited magnetic field uncer- gating along the nerve. As a reference measurement we
tainty ∆B normalized by the total measurement time perform an electrical recording of the impulse with an-
PN √ (cid:16) (cid:112) (cid:17) otherpairofelectrodes. Figure2(a)showstheelectrically
T yieldsthesensitivity∆B T ∼1/ γ T J /2
tot PN tot 2 x
√ recorded signals for different stimulation voltages. Fig-
in units of T/ Hz. T2 is the spin coherence time and ure2(b)showsthefrequencyspectraofthenervesignals
Jx =4NA is the total atomic spin for NA cesium atoms. and Fig. 2(c) shows the amplitude of the 700 Hz Fourier
Atroomtemperatureof22◦C,thecesiumatomicdensity component. The nerve is stimulated at t = 5.5 ms.
is 3.6×1016 m−3 which is the highest of all elements ap- The signature of the nerve signal is its non-linear be-
propriateforatomicmagnetometry. Thepulsedmeasure- havior with stimulation voltage, with a firing threshold
ment√has the PN-limited uncertainty of ∆|BPN(Ω)| = ataround0.7V.Forvoltagesabovethethresholdanerve
(cid:0) (cid:1)
1/ γ 2Jx if the magnetic pulse duration τ (cid:28)T2 [9]. impulseismeasuredwiththerecordingelectrodeswithin
Alongspin-coherencetimeT iscrucialforahighsen- the time interval t = 6.5−9.5 ms. We also observe a
2
sitivity. In this work we utilize a vapor cell with the stimulation artifact at t=5.5 ms [see inset in Fig. 2(a)]
inside surface coated with alkane [10, 11]. The coating which is proportional to the stimulation voltage.
protects atomic spin states from decoherence over many In parallel to the reference electrical recording the
thousands of wall collisions and provides Tdark = 15 ms nerve signal is detected optically using the pulsed mag-
2
which is longer than a typical nerve impulse duration netometer mode. The magnetometer is positioned near
τ ≈ 2 ms, as required for the ultimate sensitivity. The the middle part of the nerve separated only by a thin
inner diameter of the vapour cell is 5.3 mm and a wall microscopecoverslip. AsthenerveisbentinaU-shape,
thicknessof0.85mmallowsustohaveatomsatanaver- we mainly detect the field from the 5 mm section of the
3
(a) electrical recording (b) x 10(cid:237)(cid:24) electrical recording (c) x 10(cid:237)(cid:24) electrical recording
(cid:19)(cid:17)(cid:19)(cid:21)
s1t0ixm 10(cid:237)(cid:22)ulation artifact 4 0 V 4
8 0.8 V
V] (cid:19)(cid:17)(cid:19)(cid:20) 2(cid:23)6 3 1.6 V 3
al [ (cid:237)0(cid:21) saline solution
n (cid:24)(cid:17)(cid:24) (cid:24)(cid:17)(cid:24)(cid:21) (cid:24)(cid:17)(cid:24)(cid:23) (cid:24)(cid:17)(cid:24)(cid:25) 2 2 high K solution
g
si 0 0 V
(cid:19)(cid:17)(cid:27)(cid:3)(cid:57) 1 1
(cid:20)(cid:17)(cid:25)(cid:3)(cid:57)
(cid:237)(cid:19)(cid:17)(cid:19)(cid:20) 0 0
6 8 10 12 0 1000 2000 3000 0 0.5 1 1.5
(d) time [ms] (e) frequency [Hz] (f) stimulation voltage [V]
x 10−4 optical recording x 10(cid:237)(cid:24) optical recording optical recording
5 2
0 V 0 V
4
0.8 V 0.8 V
1.5
V] 3
al [ 0 1 saline solution
gn 2 high K solution
si
0.5 1
−5 0 0
11 13 15 17 19 0 500 1000 1500 0 0.5 1 1.5
time [ms] frequency [Hz] stimulation voltage [V]
FIG.2: Electricalandopticalmeasurementsofthenerveimpulsefordifferentstimulationvoltages. Theopticalmeasurements
were done in the pulsed mode using 1000 averages.The figures show the signals in time, the square-root of the power spectral
density PSD and the 700 Hz frequency component. The plotted electrical signals are after 10 times amplification. The
uncertainties on the data points in (c) are to small to be visible in the figure. The uncertainties on the points in (f) can be
estimatedasthestandarddeviation(0.20pT·ms)ofthedatapointswherethenervedidnotfire. Bydividingthenervesignal
(4.1 pT·ms) by the standard deviation we find the signal to noise ratio, SNR≈20.
(cid:112)
nerve closest to the magnetometer [see Fig. 1(b)]. The averages. As the SNR scales as 1/ N we find the
avg
axialioniccurrentinthis5mmsectioncanbeconsidered SNR ≈ 0.6 for a single shot, i.e., we should be able to
constant, as the action potential has a duration ≈2 ms, detect a nerve impulse in a single shot with just a minor
the velocity ≈ 40 m/s, and therefore an extent ≈ 8 cm improvement in SNR.
(cid:29)5mm. ThecircumferalmagneticfieldB fromthe
nerve Themagnetometercanalsobeoperatedinthecontin-
nerveisonaveragetransversetotheinitialspindirection
uous mode which allows for determination of the tempo-
J and will therefore create a transverse spin component
x ral shape of the magnetic field generated by the nerve,
[see Fig. 1(b)]. Figure 2(d) shows the magnetometer sig-
B (t). Themagnetometerresponsewasoptimizedby
nerve
nal, Fig. 2(e) shows the spectrum and Fig. 2(f) shows
matching its frequency response (a Lorentzian centered
the magnetic field Fourier component at the Larmor fre-
at the Larmor frequency with a full width at half max-
quency of 700 Hz. A clear threshold for the nerve fir-
imum 1/(πT )) with the spectrum of the nerve impulse
2
ing is observed confirming that the magnetometer is ca-
(see Fig. 2(b)). The bandwidth 1/(πT ) = 720 Hz was
2
pable of detecting the nerve impulse. From calibration
set by choosing suitable power levels for the pump, re-
measurements (see Methods section) we determine the
pump and probe lasers. Figures 3(a) and 3(b) show the
Fourier component of the magnetic field from the nerve
electricalandopticalsignalsrespectivelyasafunctionof
as |B (Ω)| = 4.1 pT·ms. Note that the stimulation
nerve time for different stimulation voltages. In both electri-
artifact which was observed in the electrical recording is
calandopticalmeasurementsweobservetwofeatures(A
notdetectedwiththepulsedmagnetometerasthestimu-
and B). Feature A is due to the stimulation as it starts
lation occours during the optical pumping [see Fig. 1(c)]
atthetimeofstimulationandincreaseslinearlywiththe
wheretheresponsetomagneticfieldsisstronglydamped.
stimulationvoltage. FeatureBisduetothenervesignal,
As a control experiment, we make the nerve inex- as it starts a few ms after the stimulation and only ap-
citable [12] by replacing the saline solution in the plastic pears above the threshold for the nerve firing (at 1 V or
chamber with a solution with high potassium concentra- greater). Figure 3(c) shows a comparision of the electri-
tion. Asexpected,weclearlyobservefrombothelectrical cal signal for 1 V stimulation and the detected magnetic
[Fig. 2(c)] and optical [Fig. 2(f)] measurements that the field B(t) as calculated by deconvolving the optical sig-
nerve signal is blocked by this solution. From Fig. 2(f) nal with the magnetometer response [see Methods]. The
we infer that the nerve impulse can be detected opti- temporal profiles of the electrical signal and the mag-
cally with a signal to noise ratio SNR ≈ 20 using 1000 neticfieldlookverysimilar; bothshowtheactionpoten-
4
(a) (b) (c)
x 10(cid:237)(cid:22) electrical recording 3x 10(cid:237)(cid:22) optical recording mV] (cid:20) 1.9 ms electrical recording
2 feature A 0 V feature A 0 V al [ 0
1 12 VV 2 12 VV sign(cid:237)(cid:20) (cid:20)(cid:3)(cid:57)
V] V]
gnal [ 0 gnal [ 1 d [pT](cid:20)(cid:24)(cid:19) 1.3 ms optical recording
si(cid:237)(cid:237)(cid:21)(cid:20) feature B si(cid:237)0(cid:20) feature B gnetic fiel(cid:237)0(cid:24) 7 pT (cid:20)(cid:3)(cid:57)
6 8 10 12 6 8 10 12 ma 6 8 (cid:20)(cid:19) (cid:20)(cid:21)
time [ms] time [ms] time [ms]
FIG. 3: Electrical and optical measurements of the nerve impulse for different stimulation voltages. The magnetometer was
operated in the continuous mode and the signals were averaged 5000 times. (a) Electrical and (b) optical measurements. (c)
Opticalsignalfor1Vstimulationandmagneticfieldcalculatedbydeconvolution. ForthesespecificmeasurementstheLarmor
frequency was 410 Hz and the coherence time 0.44 ms.
tialandastimulationartifact. Thenervewasstimulated tionofnerveimpulsesfromthefrogsciaticnervebymea-
for 50 µs [inset to Fig. 2] while the magnetic field from suring the magnetic field generated by the nerve with a
the stimulation (Fig. 3(c)) lasts for around 0.5 ms. This room-temperaturesensorwithnearquantumlimitedsen-
discrepancy is due to the rather slow response time of sitivity. Afewmm-sizedsensorwhichissensitiveenough
the magnetometer T =0.44 ms and low pass filtering of to detect sub-picoTesla fields at a distance of a few mil-
2
the optical signal in the data analysis. From the bottom limetersfrombiologicalobjectsmakesthemagnetometer
plate of Fig 3(c) we conclude that the nerve magnetic perfect for medical diagnostics in physiological/clinical
fieldhasa7pTpeak-to-peakamplitude(measuredatan areas such as cardiography of fetuses, synaptic responses
average distance of 4.5 mm) and that the nerve conduc- in the retina, and magnetoencephalography.
tion velocity is 38(9) m/s [9]. The effective axial ionic
currentisestimatedtobe0.16µA[9]whichisconsistent
with earlier measurements [13]. METHODS
From the data we find the experimental uncertainty
∆|B (Ω)| = 5.7 pT·ms for the pulsed mode and a Nerve preparation
exp √
sensitivity of 360 fT/ Hz in the continuous mode (see
Methods section). The quantum projection noise lim- This study was approved by the Animal Care and
ited uncertainty in the pulsed mode is ∆|BPN(Ω)| = Use Committee of Copenhagen University, and was con-
0.30pT·ms. Inthismodethelightisoffduringthenerve ductedinaccordancewiththeGuidingPrinciplesforthe
impulse duration τ ≈2 ms which satisfies τ (cid:28)Tdark. In CareandUseofAnimalsintheFieldofPhysiologicalSci-
2
the continuous mode, where the T2 = 0.44 ms matches ence. All efforts were made to minimize animal suffering
thener√veimpulsebandwidththePN-limitedsensitivityis and the number of animals used.
29 fT/ Hz. Thus the projection noise constitutes about Sciatic nerves were isolated from green frogs (Rana
10% of the total experimental uncertainty, the quantum Temporaria). The frogs were decapitated and the sciatic
photon shot noise contributes ≈ 50%, with the rest due nerves were isolated from spine and down to the knee.
totheuncompensatedlowfrequencyclassicalnoiseofthe Thenervesare7-8cmlongwithadiameterof1.3mmin
probe light and of the atomic spin. theproximalendandslightlythinnerinthedistalend. In
Projection noise dominated sensitivity can be reached theproximalendthereisonebundlethatdividestwiceso
by relatively straightforward steps, such as using multi- distally it is composed of three bundles. Figure 4 shows
plass vapor cells [14], by modest heating (increasing the an electron micrograph of the nerve where it is seen that
temperature from room-temperature 22◦C to the human the nerve bundles contain a few thousand of axons.
body temperature 37◦C will increase the sensitivity by a Throughout the dissection and the course of the ex-
factor of two [15]) or by employing a low finesse optical periments, the frog sciatic nerves were kept moist in
cavity [16]. Gradiometry with two cells with oppositely cold Ringers solution (also called saline solution) of 115
oriented spins allows for generation of nonclassical en- mM NaCl, 2.5 mM KCl, 1.8 mM CaCl , 1.08 mM
2
tangled states leading to sensitivity beyond the PN limit Na HPO · 2H O, 0.43 mM NaH PO · H O, adjusted
2 4 2 2 4 2
[7] as well as provides additional compensation of the to pH 7.1 [17]. Ringers solution approximates the ionic
ambient magnetic fields and classical fluctuations of the compositionoftheextracellularfluidsofthefrog. Ahigh
atomic spins. potassium concentration Ringers solution, in which all
In conclusion, we have performed non-invasive detec- NaCl was replaced by KCl with final K+ concentration
5
being 117mM and Na+ 0 mM and Cl− concentration re- Operation of the magnetometer
maining constant, was used to make nerves inexcitable
[12].
Themagnetometerisbasedonopticalread-outofspin-
polarizedatomicvapour. Thecesiumatomsareprepared
by an optical pumping with a pulse of circular polar-
ized light such that the total spin vector J=(J ,J ,J )
x y z
points in the x-direction, which is also the direction of a
bias field B [see Fig. 1(a)]. Any magnetic field perpen-
x
diculartox-direction(suchasthemagneticfieldfromthe
nerve) will create a transverse spin component J which
⊥
afterwards will precess around the bias magnetic field at
the Larmor frequency Ω = B /γ [see Fig. 1(b)], where
x
γ =2.20·1010 rad/(s·T) isthe cesiumgyromagnetic ra-
tio. The transverse spin component is detected optically
by measuring the polarization rotation of linearly polar-
ized probe light passing through the vapor cell using a
balanced polarimeter. The magnetic field from a nerve
can be detected in two modalities, a pulsed or a contin-
uous one. In the continuous mode the pump and probe
light is continuously on. In this case, the magnetome-
FIG. 4: Electron micrograph of a frog sciatic nerve. The mi-
crograph shows the cross section of the frog sciatic nerve on ter signal S(t) is proportional to the convolution of the
lowerfemurafteritsfirstdivision. Thetwonervebundlesare magnetic field with the magnetometer response function
(cid:110) (cid:111)
surroundedbyafasciaofconnectivetissue,andtheyareseen S(t) ∝ (cid:82)t e−Γ(t−t(cid:48))cos[Ω(t−t(cid:48))] B (t(cid:48))dt(cid:48), where
above sections of skeletal muscle. The diameters of the two t(cid:48)=0 y
we assumed the transverse field is along the y-direction.
nerve bundles are 0.59 and 0.44 mm and they each contain
1750 and 950 single axons with a minimal diameter of 7 µm. The relaxation rate Γ = 1/T2, which is the inverse
Theindividualmyelinatednervefibreshaveanaveragediam- of the spin-coherence time T , increases linearly with
2
eter of 16 µm. laser power and is in the limit of low power denoted
Γ = 1/Tdark. In the pulsed mode [see Fig. 1(c)],
dark 2
a pulse of pump light first initializes the atomic spins
along the x-direction, then the pulse of magnetic field
creates a transverse spin component, and finally the
Nerve chamber and electrical recording spins are detected with a pulse of probe light. In this
case, the magnetometer signal is a free induction de-
The nerves are kept in a 3D-printed plastic chamber cay S(t) = Asin(Ωt+θ)e−Γt, where θ is a phase and
during the experiments. The chamber is a 47 x 18 mm the amplitude A is proportional to the Fourier com-
block of 8.5 mm height that contains a longitudinal U- ponent of the magnetic field at the Larmor frequency:
shaped channel with a diameter of 2 mm in which the |B(Ω)| = (cid:12)(cid:12)(cid:82)0τBy(t)e−iΩtdt(cid:12)(cid:12), where τ is the duration of
nerve is placed. The channel allows maintenance of a the magnetic field pulse. For both pulsed and continu-
saturated water-vapor atmosphere in order to keep the ousmodes, theproportionalityconstantcanbefoundby
nerve moist. The front part of the chamber is placed applying a known calibration magnetic field.
close to magnetometer and it is covered by a microscope
glass cover slip of 0.13 mm thickness.
The chamber has 6 circular gold electrodes on each Calibration of the pulsed magnetometer
side. Theelectrodeshaveanouterdiameterof6mmwith
theholeinsideof1.5mm,whichfitsintothechannel. On The magnetometer is calibrated by applying a known
each side, the distance between electrodes is 5 mm. magnetic field. This calibration field is produced by a
The nerve was externally stimulated in the proximal coilpositionedinsidethemagneticshield;thefieldpoints
end by applying a short 50 µs square voltage pulse be- in the z-direction and has the temporal shape of a sin-
tweentwospatiallyseparatedelectrodessurroundingthe gle sinusoidal oscillation B (t) = B sin(Ω t) with
cal cal cal
nerve. The nerve signal was measured in the distal end amplitude B = 1 nT, frequency Ω = 2π ·700 Hz
cal cal
asapotentialdifferencebetweenapairoftwoelectrodes. and Fourier component |B (Ω )| = πB /Ω =
cal cal cal cal
The electrical signal was amplified 10 times and filtered 0.71 nT·ms.
(usinga3kHzlowpassanda10Hzhighpassfilter)and In the pulsed mode, the calibration field is applied in
recorded simultaneously with the magnetic field record- between the pump and probe pulses. The recorded mag-
ings. netometer signal (the free induction decay) is shown in
6
(a) (b)
(cid:237)(cid:21)
(cid:19)(cid:17)(cid:20) 10
off off
on 10(cid:237)(cid:22) on
(cid:19)(cid:17)(cid:19)(cid:24)
V] (cid:237)(cid:23)
al [ 0 10
n (cid:237)(cid:24)
g 10
si
(cid:237)(cid:19)(cid:17)(cid:19)(cid:24) (cid:237)(cid:25)
10
(cid:237)(cid:26)
(cid:237)(cid:19)(cid:17)(cid:20) 10
0 2 4 6 0 500 1000 1500
(c) time [ms] (d) frequency [Hz]
1
(cid:19)(cid:17)(cid:21) 10
V] on off
al [ (cid:19) on
gn 100
si detector signal with fit
(cid:237)(cid:19)(cid:17)(cid:21)
etic fieldnT](cid:19)1 on 10(cid:237)(cid:20)
gn [ deconvoluted signal 10(cid:237)(cid:21)
a
m (cid:237)(cid:20)
6 8 (cid:20)(cid:19) (cid:20)(cid:21) 0 500 1000 1500
time [ms] frequency [Hz]
FIG. 5: Measurements with and without the calibration field. (a,b) Pulsed mode: Ω = 700 Hz, Tdark = 15 ms. The PSD
2
is calculated using the first 8 ms of the recorded signal. (c,d) Continuous mode: Ω = 414 Hz, T = 0.44 ms. The PSD is
2
calculated using 37.1 ms of the recorded signal. The Larmor frequencies are marked in (b) and (d) with a dot.
Figure5(a). Thespectrum(calculatedasthesquare-root sponse function. Figure 5(c) top shows the detected
√
of the power spectral density PSD) is peaked at the signal together with a fit to the function S(t) ∝
(cid:110) (cid:111)
LarmorfrequencyoftheatomsΩ=2π·700Hz[Fig.5(b)]. (cid:82)t e−Γ(t−t(cid:48))sin[Ω(t−t(cid:48))] B (t(cid:48))dt(cid:48) from which we
t(cid:48)=0 z
Thepeakamplitudeisproportionaltothemagneticfield
(cid:112) can determine the Larmor frequency and the coherence
Fourier component: PSD(Ω) ∝ |B(Ω)|. The pro-
time. Using the fitted parameters we can perform nu-
portionality constant can be calculated from the data in
mericaldeconvolutionandbyscalingwiththeamplitude
Fig. 5(b) and the known Fourier component of the cali-
of the calibration field we can obtain the magnetic field
bration field. With this calibration we can calculate the
asafunctionoftimeB (t)[Figure5(c)bottom]. Wesee
Fouriercomponent|B (Ω)|ofthenervemagneticfield z
nerve (cid:112) that the deconvolution procedure works well as the de-
[seeFig.2(f)]fromthemeasuredpeakvalues PSD(Ω)
convolvedsignalresemblesasingle sinusoidaloscillation.
[see Fig. 2(e)].
Figure5(d)showsthecalculatednoisespectrumofthe
Figures5(a)and5(b)alsoshowthemagnetometersig-
deconvoluted signal. It also shows the noise spectrum
nal without the applied calibration field. The signal was
whenthecalibrationfieldwasoff. Inthiscasethenoiseat
√
averaged 1000 times before recorded. The noise at the
theLarmorfrequencyis11.4fT/ Hzwhen1000averages
Larmor frequency is 3900 times smaller than the signal
is used. The single shot magnetic field sensitivity is then
√
obtained with the calibration field, i.e., the calibration
360 fT/ Hz.
field is detected with a SNR = 3900 corresponding to
a minimal detectable magnetic field Fourier component
0.71 nT·ms/3900 = 0.18 pT·ms using 1000 averages. Acknowledgements
ThesingleshotminimaldetectableFouriercomponentis
then 5.7 pT·ms.
We would like to thank J¨org Helge Mu¨ller for helpful
discussions, Bing Chen for help during the initial stages
of the experiments, and Hannelouise Kissow for taking
Calibration of the continuous magnetometer the electron micrograph image of the nerve. This work
was supported by the ERC grant INTERFACE and by
In the continuous mode, the lasers are on dur- the DARPA project QUASAR. K. J. was supported by
ing the applied calibration field. In this case the the Carlsberg Foundation. R. A. T. was supported by
magnetometer signal is proportional to the convolu- CNPq/Brazil.
tion of the magnetic field with the magnetometer re-
7
SUPPLEMENTARY INFORMATION Dirac delta-function. These equations can be integrated
and the solutions are
MAGNETOMETER PRINCIPLE √ (cid:90) t
J(cid:48)(t)=e−ΓtJ(cid:48)(0)+ 2Γ e−Γ(t−t(cid:48))F (t(cid:48))dt(cid:48)
y y y
The total spin of the atomic ensemble is defined as t(cid:48)=0
(cid:90) t
J = (Jx,Jy,Jz). Here J is a quantum operator, and +γJx e−Γ(t−t(cid:48))[cos(Ωt(cid:48))Bz(t(cid:48))−sin(Ωt(cid:48))By(t(cid:48))]dt(cid:48),
thecomponentshavethecommutationrelation[Jy,Jz]= t(cid:48)=0
iJ . J is here defined as being unitless and equals the (7)
x
total angular momentum divided by the reduced Planck √ (cid:90) t
J(cid:48)(t)=e−ΓtJ(cid:48)(0)+ 2Γ e−Γ(t−t(cid:48))F (t(cid:48))dt
constant h¯. The equations of motion for the spin vector z z z
t(cid:48)=0
can be derived using the Heisenberg equation of motion (cid:90) t
−γJ e−Γ(t−t(cid:48))[sin(Ωt(cid:48))B (t(cid:48))+cos(Ωt(cid:48))B (t(cid:48))]dt(cid:48).
x z y
J˙(t)= 1 [J(t),H ]. (1) t(cid:48)=0
i¯h B (8)
The dot denotes the time-derivative and the bracket de- From these equations we can calculate the mean values
notes the commutator. The Hamiltonian describing the and noise proporties of the transverse spin components
coupling between the spin and the magnetic field is as a function of time.
H =h¯γB·J, (2)
B
Free precession
where γ is the gyromagnetic ratio which for the
cesium atom in the F = 4 ground state equals
Assume that the transverse spin has some mean value
2.2×1010 rad/(sec·Tesla). In vector form the equation
at t = 0 and that it is then left free to precess. At a
of motion reads
latertimetthetransversespincomponentintherotating
J˙(t)=γJ×B. (3) frame is
(cid:104)J(cid:48) (t)(cid:105)=(cid:104)J (0)(cid:105)e−Γt. (9)
In the experiment, the atoms are spin-polarized in the ⊥ ⊥
x-direction and located in a static magnetic field B
x We see that the mean value decays in time. In the lab
pointing in the x-direction. In the presence of a small
frame the transverse spin will perform a damped oscilla-
time-dependentmagneticfieldB (t)orB (t)pointingin
y z tion.
they-orz-direction,thespinvectorwillacquireatrans-
verse component J = (J ,J ) = |J |(cosθ,sinθ). We
⊥ y z ⊥
will assume that J is large compared to J and J , and
x y z Atomic response to a pulse of magnetic field
that J is independent of time. We now introduce spin
x
operators J(cid:48) and J(cid:48) rotating at the Larmor frequency
y z Consider the case where a magnetic field B (t) is ap-
Ω=γB : z
x pliedforadurationτ. Weassumethatτ (cid:28)T suchthat
2
(cid:18)J(cid:48)(cid:19) (cid:18) cosΩt sinΩt(cid:19)(cid:18)J (cid:19) anydecayofthespincomponentscanbeneglegted. Ifini-
y = y . (4) tiallythetransversespincomponentiszero(cid:104)J (0)(cid:105)=0,
J(cid:48) −sinΩt cosΩt J ⊥
z z we find
In the rotating frame, the equations of motion read
(cid:104)J(cid:48) (τ)(cid:105)=γJ (Re[B (Ω)],Im[B (Ω)]) (10)
⊥ x z z
J˙(cid:48)(t)=γJ [cos(Ωt)B (t)−sin(Ωt)B (t)]
y x √ z y and
−ΓJ(cid:48)(t)+ 2ΓF (t), (5)
y y
J˙z(cid:48)(t)=−γJx[sin(Ω√t)Bz(t)+cos(Ωt)By(t)] |(cid:104)J(cid:48)⊥(τ)(cid:105)|=Jxγ|Bz(Ω)|. (11)
−ΓJ(cid:48)(t)+ 2ΓF (t). (6)
z z HerewehavedefinedtheFouriercomponentofthemag-
netic field at the Larmor frequency as
Thetransversespincomponentwilleventuallydecay,and
we have therefore added decay terms in the above equa- (cid:90) τ
tions. ThedecayrateisdenotedbyΓandtheassociated B(Ω)= B(t(cid:48))e−iΩt(cid:48)dt. (12)
decay time is T = 1/Γ. We also added Langevin noise t(cid:48)=0
2
operators Fy(t) and Fz(t) with zero mean values and Similarly, if the magnetic field is applied along the y-
correlation functions (cid:104)Fy(t)Fy(t(cid:48))(cid:105) = var(Fy)δ(t − t(cid:48)), direction instead, the transverse spin component will be
(cid:104)F (t)F (t(cid:48))(cid:105) = var(F )δ(t−t(cid:48)) and (cid:104)F (t)F (t(cid:48))(cid:105) = 0,
z z z y z
where var(F ) = var(F ) = |J |/2 and δ(t−t(cid:48)) is the |(cid:104)J(cid:48) (τ)(cid:105)|=γJ |B (Ω)|. (13)
y z x ⊥ x y
8
We see that magnetic fields in y- and z-directions have (suchthatS (t)isalargequantity)andthattherotation
x
similar effects on the spins: the fields create transverse angleissmall,theoutputlightaftertheatomicensemble
spincomponentswithlengthsproportionaltotheFourier can be described by the equation
components of the magnetic fields at the Larmor fre-
Sout(t) = Sin(t)+aS (t)J (t)
quency. y y x z
For the specific case of a sinusoidal magnetic field = Sin(t)+aS (t)(cid:2)sin(Ωt)J(cid:48)(t)+cos(Ωt)J(cid:48)(t)(cid:3).
y x y z
B (t) = B sin(Ωt) applied for one period of oscil-
z 0 (17)
lation τ = 2π/Ω, we find |B (Ω)| = πB /Ω and
z 0
|(cid:104)J(cid:48) (τ)(cid:105)|=γJ (πB /Ω). Theparameteradescribesthecouplingstrengthbetween
⊥ x 0
theatomsandthelight[18]. TheStokesoperatorSout(t)
y
can be measured with polarization homodyning. There
Projection noise limited detection are several ways that one can extract information about
the transverse spin components and therefore about the
The measurement of the transverse spin component is magnetic field from the measured signal. One can for
fundamentally limited by the spin-projection noise orig- instance measure the mean value (cid:10)Sout(cid:11) or the power
y
inating from the Heisenberg uncertainty principle. This spectraldensityofthesignal. Thepowerspectraldensity
(cid:112)
uncertainty is ∆|J |= J /2. By equating the created (PSD) for a function x(t) is defined as
⊥ x
mean value given by Eq. (13) to the projection noise we
(cid:42)(cid:12) (cid:12)2(cid:43)
find the uncertainty on the magnetic field Fourier com- 1 (cid:12)(cid:90) T (cid:12)
S (ω) = (cid:12) x(t)e−iωtdt(cid:12)
ponent due to the projection noise: xx T (cid:12) (cid:12)
(cid:12) t=0 (cid:12)
∆|BPN(Ω)|=1/(cid:16)γ(cid:112)2Jx(cid:17). (14) 1 (cid:90) T (cid:90) T (cid:104)x(t)x(t(cid:48))(cid:105)e−iω(t−t(cid:48))dtdt(cid:48)
T
t=0 t=0
We can also calculate the uncertainty on the amplitude (18)
of an oscillating magnetic field due to the projection
noise. For a sinusoidal magnetic field with total dura- WewillshowbelowthatthePSD of(cid:10)Syout(t)(cid:11)ispropor-
tionτ equaltoanintegralmultipleoftheLarmorperiod, tional to the amplitude squared of the applied magnetic
theamplitudeB isrelatedtotheFouriercomponentby field.
0
|B(Ω)| = B τ/2. From this and Eq. (14) we find the
0
projection noise limited uncertainty on the amplitude:
Detection of a pulse of magnetic field
(cid:16) (cid:112) (cid:17)
∆B =1/ γ J /2τ , (15)
PN x
AssumethatapulseofmagneticfieldB (t)ofduration
z
τ is applied from t = −τ to t = 0. After the pulse, the
which is often called the minimal detectable field. The
spinshaveacquiredanon-zerotransversespincomponent
magnetic field sensitivity can be found by multiplying
(cid:104)J(cid:48) (0)(cid:105) ∝ |B (Ω)| as given by Eq. (11). At t = 0 the
∆BPN√with the square-root of the total measurement ⊥ z
spin will continue to precess until it decays as described
time T and setting T =τ =T :
tot tot 2
by Eq. (9). This spin vector can be measured using a
(cid:112) (cid:16) (cid:112) (cid:17) pulse of light with duration T and starting at the time
∆B T ∼1/ γ T J /2 . (16)
PN tot 2 x
when the magnetic field pulse ends. The mean value of
the measured signal is
Measuring the atomic signal (cid:10)Sout(t)(cid:11)=aS (t)(cid:2)sin(Ωt)(cid:10)J(cid:48)(0)(cid:11)+cos(Ωt)(cid:104)J(cid:48)(0)(cid:105)(cid:3)e−Γt
y x y z
=aS (t)|(cid:104)J(cid:48) (0)(cid:105)|sin(Ωt+θ)e−Γt, (19)
The atomic spin can be measured optically. Assume x ⊥
that a linearly polarized pulse of light is propagating in where θ is the polar angle of (cid:104)J(cid:48) (0)(cid:105). The amplitude
⊥
the z-direction through the atomic ensemble. The po- of the transverse spin vector can be extracted from the
larization of the light will be rotated by an angle pro- measurement by, for instance, a fit of the experimental
portional to Jz due to the Faraday paramagnetic effect. data to Eq. (19). Alternatively, one can calculate the
The polarization of the light is described using Stokes PSD of the signal. For x(t) = Asin(Ωt+θ)e−Γt we
operators Sx(t), Sy(t), and Sz(t) which have the unit of calculate that the peak value of the PSD is
1/time. Here S (t) = [Φ (t)−Φ (t)]/2 equals one half
the difference inxphoton flxux of x-yand y-polarized light. (cid:34)(cid:0)1−e−ΓT(cid:1)2 (cid:35)
S (Ω)=|A|2 +(cid:15)(θ,Γ,Ω,T) , (20)
Sy(t)refertothedifferencesof+45◦ and−45◦ polarized xx 4Γ2
light, and S (t) to the differences of right hand and left
z
hand circular polarized light. Assuming that the input where the second term (cid:15)(θ,Γ,Ω,T) is much smaller than
lightbeforetheatomicensembleiseitherxory-polarized the first term for our experimental parameters. We see
9
that seen in Fig 6. Due to misalignment of the pump and re-
pump laser beams with respect to the bias field B (see
x
Sxx(Ω)∝|(cid:104)J(cid:48)⊥(0)(cid:105)|2 ∝|Bz(Ω)|2, (21) Fig.1inthemaintext),onemayobserveafreeinduction
decay S (t) [see probe pulse B in Fig. 6] even if there is
B
and that the Fourier component of the magnetic field at no magnetic field. The pump/repump and probe pulses
the Larmor frequency can be extracted from the peak are therefore repeated and the signals from probe pulses
value of the PSD. A and B are subtracted giving the magnetometer signal
S(t)=S (t)−S (t). Theamplitudeofthismagnetome-
A B
ter signal will be proportional to the Fourier component
Continuous recording of the magnetic field of the magnetic field at the Larmor frequency |B(Ω)|.
Wewillnowdiscusshowonecanmeasurethemagnetic
field as a function of time. Assume that a magnetic field CONDUCTION VELOCITY
B (t)isappliedandthatlightiscontinuouslymonitoring
z
the atomic spin. If (cid:104)J (0)(cid:105)=0, then at a later time Thenerveconductionvelocitycanbecalculatedbydi-
⊥
vidingthedistancefromthestimulationelectrodestothe
(cid:90) t
(cid:104)J (t)(cid:105)=γJ e−Γ(t−t(cid:48))sin[Ω(t−t(cid:48))]B (t(cid:48))dt(cid:48). recording site [5(1) cm for optical recording and 7(1) cm
z x z for electrical recording] by the time interval between the
t(cid:48)=0
(22) stimulusartifactandthepeakofthenerveimpulse. The
The mean value of the measured signal will be earlier arrival of the nerve impulse for optical recording
compared to electrical recording [1.3(2) ms compared to
(cid:10)Syout(t)(cid:11)=aSx(t)(cid:104)Jz(t)(cid:105). (23) 1.9(1)ms]isconsistentwiththemagnetometerbeingpo-
sitioned in between the stimulating and recording elec-
From this we see that the measured signal (cid:10)Sout(t)(cid:11) is trodes. From the measurements Fig. 3(c) we calculate
y
proportional to the convolution of the magnetic field the conduction velocity of 38(9) m/s and 37(6) m/s for
Bz(t) with the function (cid:2)sin(Ωt)e−Γt(cid:3). Similarly, if the optical and electrical recording.
transverse magnetic field is pointing in the y-direction,
the signal is proportional to the convolution of B (t)
y
with the function (cid:2)−cos(Ωt)e−Γt(cid:3). The magnetic field ESTIMATE OF THE AXIAL IONIC CURRENT
asafunctionoftimecanbeextractedfromthemeasured
data using numerical deconvolution. The detected magnetic field is created by axial ionic
currentsinsidethenervebundle. Thereisaforwardcur-
rent inside the axons and a return current outside the
EXPERIMENTAL PROCEDURE axons. The magnetic fields from the forward and return
currents can cancel each other, the exact degree of can-
Three lasers denoted pump, repump and probe are cellation depends on the anatomy of the nerve, and the
used in the experiment. The pump laser is on resonance geometry of the experiment, such as the size of the mag-
with the cesium F = 4 → F(cid:48) = 4 D1 transition and has netic field sensor and the distance from the nerve to the
thewavelength895nm. Therepumplaserisonresonance sensor.
with the cesium F =3→F(cid:48) =2,3,4 D2 transitions (all Wecanestimatetheaxialcurrentinthenervefromour
arewithintheDopplerlinewidth)andhasthewavelength magnetic field measurements. We use a simple model,
852nm. Thesetwolasersareusedforopticalpumpingof where we assume that the ionic current is concentrated
the cesium atoms into the F = 4,m = 4 hyperfine sub- at the center of the nerve, and that the nerve produces
level and are thereby creating a high spin-polarization of a magnetic field similar to that of an infinitely long con-
the cesium vapor. The probe laser is 1.6 GHz higher in ducting wire. In the experiment, we detect the magnetic
frequencythanthecesiumF =4→F(cid:48) =5D2transition fieldaveragedoverthevolumeofthesphericalvaporcell.
and has the wavelength 852 nm. The 1.6 GHz detuning We calculate that the average magnetic field is equal to
is much larger than both the natural linewidth (5 MHz the field at the center of the vapor cell. The magnetic
FWHM) and the Doppler linewidth (380 MHz FWHM) field from an infinitely long wire is |B| = µ I/(2πr),
0
such that negligible absorption occurs. where µ is the magnetic permeability, I is the cur-
0
The pulse sequence in Fig 6 is used for detection of rent, and r is the radial distance from the wire. Using
the magnetic field from the nerve impulse B . The r =4.5 mm for the distance from the center of the nerve
nerve
atomsarefirstopticallypumpedusingpumpandrepump tothecenterofthevaporcell,wecalculatethatacurrent
light, then the magnetic field is present, and finally the of 0.16 µA will produce a magnetic field of 7 pT.
atoms are measured using probe pulse A. The optically Our estimate of 0.16 µA is smaller than the 0.4 µA
detected signal S (t) will be a free induction decay as which was estimated in previous work on the frog sciatic
A
10
pump & pump &
repump probe A repump probe B
B
nerve
S (t) S (t)
A B
stimulation
FIG. 6: Pulse sequence for measuring the nerve impulse.
nerve [13]. This is expected as in that work, the nerve temperature 19-channel magnetic field mapping device
was put in a large container with saline solution and the for cardiac signals. Applied Physics Letters, 95:173701,
magnetic field was measured by a coil with the nerve 2009.
[9] See Supplementary Material.
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