Towards a first observation of magneto-electric directional anisotropy and linear birefringence in gases

0.58 MB·English

by  Cécile Robilliard

1
Towards a first observation of magneto-
electric directional anisotropy and
linear birefringence in gases
1 C. Robilliard(a,b) and G. Bailly(a,b)
1
0
2

n
a
J
Abstract: Inthiscontribution toPSAS’2010wereport on recent progress on anexperiment

6 aimed at measuring small optical directional anisotropies byfrequency metrology inahigh

finesse ringcavity. Wefocuson our first experimental goal, themeasurement of magneto-
]
h electriceffectsingases. Afterareview of theexpected effectsin our set-up, wepresent
p the apparatus and themeasurement procedure, showing that wealready havethenecessary
- sensitivity tostartnovel experiments.
t
n
Re´sume´: Danscettecontribution a` PSAS’2010,nous pre´sentons lesavance´es re´centes sur
a
u une expe´rience destine´ea` mesurer de tre`sfaibles anisotropies optiques par me´trologie de
q fre´quence dans une cavite´ en anneau de haute finesse. Nouspre´cisons notre premier objectif
[ expe´rimental, lamesure d’effetsmagne´to-e´lectriques danslesgaz. Apre`s unerevue des effets

attendus dans notre montage, nous pre´sentons l’appareil et laproce´dure demesure, montrant
1
que nous avons de´ja` lasensibilite´ ne´cessaire pour commencer des mesuresnovatrices.
v
4
7 1. Introduction
1
1 Searchingforanisotropiesand,moregenerally,symmetryviolationsis oneofthe maintrendsof
.
1 contemporaryphysics,with the aim of testingthe presentmodelsandtheir proposedextensions.Let
0 usciteasexamplestestsofLorentzinvariance[1]ormeasurementsofatomicparityviolation[2].For
1 thispurpose,lowenergyprecisionexperimentsappeartobecomplementarytohighenergyphysics.
1
Inthiscontextwherepreciseopticalmeasurementsareneeded,highfinesseopticalcavitiesplayan
:
v importantrole.Indeed,theyallowforimpressivelysensitiveapparatus,reachingrelativesensitivitiesas
Xi lowas3 10−23/√Hzforkmlonggravitationalwavesinterferometers[3,4]or10−18to10−19 /√Hz
×
fortable-topvacuummagneticbirefringenceexperiments[5–7].
r
a Inthispaperwepresentthestatusofanewexperimentaimedatmeasuringverysmalldirectional
anisotropies thanks to a high finesse ring cavity. Our first objective is to measure effects induced in
gases by magneticand electric fields, whichwill allow us to optimize ourexperimentalset-up. Sub-
sequently,we shouldbe able to measurethese effectswith enoughprecision to be sensitive to small
contributionssuchas relativisticones,andalso to observethemin vacuum,which wouldbe thefirst
Received.Revisionreceived.Accepted.Revisionaccepted.
C.Robilliard(a,b)1andG.Bailly(a,b).
a Universite´ de Toulouse; UPS; Laboratoire collisions Agre´gats Re´activite´, IRSAMC; F-31062 Toulouse,
France.
bCNRS;UMR5589;F-31062Toulouse,France.
1Corresponding author (e-mail:email: [email protected]).
Proof/E´preuve
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2 Can. J.Phys.Vol.XX,2010
evidenceofthenonlinearityofvacuumduetoquantumeffects.Westartbydescribingtheseeffectsand
thewayweplantomeasurethem,afterwhichwepresentourapparatus,afirstversionofitbeingnow
fullyoperational.Wefinallyconcludebymentioningthenearfuturemeasurementstobemadeandthe
improvementsthatareunderwayontheset-up.
2. Optical anisotropies induced in gases by magnetic and electric fields
In the 1980’s, several authors performedsome methodical studies of the optical anisotropies in-
ducedinmaterialmedia,eithersolidsorfluids,byelectricandmagneticfields[8–11].Specialattention
wasdevotedtoeffectsbilinearinelectricandmagneticfields,whichhadneverbeenobserved.Indeed,
whiletheKerrandCotton-Moutoneffectsarisefromlowestorderinteractions,namelyelectricdipole
coupling,thebilinearmagneto-electriceffectsoriginatefromelectricquadrupoleandmagneticdipole
couplings.Thereforetheyareconsiderablysmaller,hencemoredifficulttoobserve.Ontheotherhand,
theyprovidemolecularresponsefunctionsthatarenotaccessibletoanyotherexperiment.
Our project aims at measuring two of these effects in usual gases, starting with nitrogen N :
2
magneto-electricdirectionalanisotropyandmagneto-electriclinearbirefringence.Bothoccurforcrossed
transversemagneticfields.
2.1. Directional anisotropy
Magneto-electric directional anisotropy can be induced in all media, including centrosymmetric
ones. It was first predicted by G. E. Stedman and coworkers [11] and observed for the first time in
a crystal by G. L. J. A. Rikken and coworkers[12]. This non-reciprocaleffect is supposed to be in-
dependentonlightpolarization.Althoughithasneverbeencalculatedinmaterialmedia,itsorderof
magnitudeisexpectedtobesimilartothatofthemagneto-electriclinearbirefringence.
TheschemeofthiseffectispresentedinFig.1(a).
(a)
E
k n
+
n -k
B -
(b) E B
-45°
+45°
kk n ! n
+ -
(c)
B E
45°
k n+ n- -k
Fig.1. (coloronline) (a)Schemeofthemagneto-electricdirectionalanisotropy. Theindexofrefractionoftwo
waves counterpropagating perpendicularly to the crossed applied electric and magnetic fields are different. (b)
Scheme of the magneto-electric Jones birefringence. (c) Scheme used in our experiment, converting the Jones
birefringenceintoadirectionalanisotropyeffect.
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C.RobilliardandG.Bailly 3
2.2. Linear birefringence
Similarlytodirectionalanisotropy,theexistenceoflinearbirefringenceisnotsubmittedtoanycon-
ditiononthemedium,andcanthereforebeinducedinatomicormoleculargasesbycrossedtransverse
electricandmagneticfields.Ithasonlybeenobservedinmolecularliquids[13],andneveringases.
Althoughour apparatusis aimed at measuringdirectionalanisotropies, we have found a scheme
which is sensitiveto linear birefringence.Actually,this schemeis based onthe equivalencebetween
magneto-electriclinearbirefringenceincrossedfieldsandmagneto-electricJonesbirefringence,which
occursforparallelfields[10].Figure2presentsthesymmetryargumentsprovingthisequivalence.The
proofisvalidundertheassumptionthatthesystemiscentrosymmetricintheabsenceofexternalfields,
andthatthemagneto-electricandJonesbirefringencesarebilinearinE andB.
(a)
(b)
n E n
v v
n n
h B h B
E
n-n = EB n -n= EB
v h h v
+
(c)
(d) n
v B
2E
n
n 45° h
v
n E
h 2B n-n = EB
v h
n-n =2 EB
v h
(e) E
n 45°
v
n
h B
n-n = EB
v h
Fig.2. (coloronline)Graphical proofoftheequivalencebetweenmagneto-electriclinearbirefringence(a)and
magneto-electric Jones birefringence (e).(a) Scheme of linear birefringence, whichisbilinear inE and B.nv
(resp. nh) istheindexof refractionof vertically(resp. horizontally) polarized light.(b)Situationafterrotating
E andB byπ/2:nv andnh aresimplyinverted.(c)isdeduced from(b)byB inversion. (d)Superpositionof
situations (a) and (c). This is valid because of the bilinearity of the linear birefringence in E and B, and also
becausethereisnobilinearbirefringenceinducedbyparallelEandBfieldswithaxisparallelandorthogonalto
thefields[10].(e)Afterrenormalisingthefields,oneobtainstheJonesconfiguration,withparallelEandBand
eigenaxisrotatedby±45◦withrespecttothefields.
As a consequence, measuring the linear or the Jones birefringence is simply a matter of conve-
nience.TheschemewearedevelopingusestheJonesconfigurationandconvertsJonesbirefringence
intoadirectionalanisotropyeffect,asshownonFig.1(b-c).
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2.3. Expected values
There is not much literature on magneto-electric effects in gases. Jones birefringence has been
computedinusualsimplegases[14]fornon-resonantlight,whileanalyticalcalculationsareavailable
foralcaline-earthatoms[15]andforalcalineatoms[16]inthevicinityofseveralatomicresonances.
Itisworthnotingthatforalltheresonancesstudied,directionalanisotropyisslightlylargerthanlin-
ear birefringence. Actually, both phenomena arise at the same developmentorder of the interaction
Hamiltonianandarethereforeexpectedtohavesimilaramplitudes.
Boththemagneto-electricdirectionalanisotropyandthemagneto-electriclinearbirefringencehave
been calculated in quantum vacuum [17,18], using the usual Euler-Heisenberg-Weisskopf effective
Lagrangian.
We summarizeinTable 1theexpected∆n infieldsB = 1 T andE = 1 V/m,fornon-resonant
light. For gases, the reference pressure is P = 1 bar and the temperature is T = 293 K. Conver-
sionfromvaluesobtainedindifferenttemperatureorpressureconditionsismadewithintheidealgas
approximation.The calculations have been performedfor λ = 632.8 nm, except for vacuum where
it is valid for all wavelengths. We take into account the equivalence between linear and Jones bire-
fringences:accordingto Fig. 2, we have∆nJones = n+45◦ n−45◦ = nB nE = ∆nMELB. The
− −
polarizationanglesforJonesbirefringenceareorientedcounterclockwisefromthefieldsdirectionas
judgedbyan observerinto whoseeye the lightis travelling;nB (resp. nE) isthe refractionindexin
theB (resp.E)direction.Concerningdirectionalanisotropy,wedefine∆nMEDA = n+ n− where
thepositivepropagationdirectionisalongE B. −
×
Medium Ref. η (T−1.V−1.m) η (T−1.V−1.m)
MELB MEDA
QuantumVacuum [17,18] 2.7×10−32 −6.7×10−32
HeliumHe [14] 1.6×10−24 -
HydrogenH [9] 3.4×10−23 -
NeonNe [14] 4.2×10−24 -
ArgonAr [14] 3.6×10−23 -
KryptonKr [14] 7.8×10−23 -
HydrogenH [14] 4.8×10−23 -
2
NitrogenN [14] 9.0×10−23 -
2
CarbonmonoxideCO [14] 1.4×10−22 -
Table1. Calculatedmagneto-electriclinearbirefringenceηMELB = nB −nE andmagneto-electricdirectional
anisotropy ηMEDA = n+ −n− ingasesandinvacuum, normalizedtoB = 1 T,E = 1 V/m.Forgases,the
referencepressureandtemperatureareP =1barandT =293K.Thecalculationsweremadeatλ=632.8nm.
Thenumbersin Table 1clearlyshow thatmeasurementsin gasesarenotbeyondpresenttechno-
logical possibilities, but are neverthelesschallenging.Indeed, using a magnetic field on the order of
10 T and an electric field on the order of 10 MV/m one gets ∆n 10−14 to 10−16 in gases and
∆n 7 10−24inquantumvacuum! ≃
≃ ×
Ourfirstmeasurementwillbethemagneto-electricdirectionalanisotropyofNitrogen,agaswhich
shouldexhibitalargeeffectwhilebeingeasytohandle.
Inthefollowing,wedescribeourapparatus.
3. Experimental set-up
Ourapparatusisdescribedindetailsin[19].Theexperiment,representedinFig.3,mainlyconsists
ofa squareresonantcavityoftotallengthL = 4L0 = 1.6 m, witha finessevaryingbetween15000
and50000dependingonmirrorpollution.
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3.1. Optical set-up
Twolightbeamsofidenticalfrequency,originatingfromthesameNd:YAGlaser,areinjectedinto
the cavity, respectively clockwise and counterclockwise. Their frequency is stabilized on the cavity
resonancefortheclockwiselaserbeam,followingthewell-knownPound-Drever-Hallscheme[20,21].
Anerrorsignalsimilartotheclockwiseoneisgeneratedinthecounterclockwisedirection:provided
thefrequencystabilizationisgoodenough,thiserrorsignalisproportionaltotheresonancefrequency
difference between the two counterpropagatingbeams. In order to improvethe signal-to-noise ratio,
we implement a homodyne detection scheme by modulating the effect to be measured at frequency
fmodandusingalock-inamplifier.
rEOM
phase mod. OI Laser
PZT TEC
AOM
fast slow
feedback feedback
!"#
% cw
Freq. servo
error signal

ccw PR
CW
PT
EOM CW
P
calibration
!"$
E & B
generation
!"$
P PT
CCW
measurement
signal PR
CCW
Fig. 3. Experimental set-up of our experiment (color online). The beampath is represented in red with arrows
indicatingthepropagationdirection,thefrequencystabilizationsysteminblueandthemeasurementsignalgener-
ationingreen.Thegreenblocksoneachcavityarmrepresenttherodsgeneratingthemagneticandelectricfields
formagneto-electro-opticmeasurementsingases.ThegreyEOMinsertedintheclockwisebeampathjustbefore
the cavity allows to calibrate the experiment. An optical isolator (OI) prevents feedback noise; the laser beam
frequencyisthenfrequency-shiftedwithanacousto-opticmodulator (AOM)inacat’seyeretroreflector.Ares-
onantelectro-opticmodulator(rEOM)providesthephasemodulationatfrequencyΩforthePound-Drever-Hall
frequency stabilization. The servo actuators are the laser thermo-electric cooler (TEC), the laser piezo-electric
transducer(PZT)andtheAOM.Thelightpolarizationiscontrolledallalongthebeampathbyhalfwave(λ/2)and
quarterwave(λ/4)retardationplates,andbypolarizers(P).Lightisinjectedintothecavitybothintheclockwise
(cw)andcounterclockwise(ccw)directions;thePR andPR (resp.PT andPT )photodiodesmonitor
cw ccw cw ccw
thereflected(resp.transmitted)powerinbothdirections.
Additionnally,bothbeamsaresuperimposednearlyallalongandthusexperiencethesamemirror
vibrationsandthesamefluctuationsofthepropagatingmedium.Wehavecheckedthatthespectraof
both error signals are identical, and the noise power spectral density in the relevantfrequencyrange
(a fewhundredsHertz)islessthananorderofmagnitudeovershot-noise.Verylownoisefrequency
stabilizationandefficientcommonmodenoiserejectionarethetwokeyelementsexplainingthegood
sensitivityofourapparatuseventhoughitisnotyetinavacuumchamber.
As explained in [19], we have performed sensisitivity tests thanks to a broadband electro-optic
modulator(EOM)placedontheclockwisebeamjustbeforethecavityandsuppliedwithasinusoidal
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voltageatfmod.Wehavecheckedthattheamplitudeoftheerrorsignalfrequencycomponentatfmod
isproportionaltotheamplitudeoffrequencymodulation,andthattheapparatusisabletodetectfre-
quencymodulationsaslowas200µHzrms.
3.2. External fields generation
We have built 4 rods generating crossed transverse magnetic and electric fields. These rods are
LE×B =20cmlongandhavea4 4mm2holeintheircenterforopticalaccess.
Themagneticfieldisgenerated×bytwopermanentNdFeBblockmagnetswithdimensions200
20 10mm3,surroundedbyapureironshellinordertoguidethefieldlinesandtoenhanceBbetwee×n
×
thetwooppositemagnets,asillustratedinFig.4(a).Wehavemeasuredthemagneticfieldprofilealong
the beam path with a Hall effect probe: it is presented if Fig. 4(b). All the 4 rods provide the same
magneticfieldwithinlessthan2%.Themagneticfieldisuniformandequalto0.185 Twithin2%in
the central 185 mm, and rises from 1% to 98% of its maximum value in 20 mm. In the transverse
direction,moving1mmawayfromthecenteroftheholeinducesafieldincreaseof1%.Toavoidthis
weadjusttherodscenteringaftereachcavityrealignment.
Theelectricfieldoriginatesfromapairofaluminumelectrodeswitharoughly4 4 mm2square
section,theanglesbeingroundedtopreventbreakdowns.Theyarestraightandparalle×lwithin0.3mm.
Theseelectrodesaresuppliedwithasinewavevoltagethrougha 2 kVhighvoltageamplifier.They
are 4 mm apart, symmetrically arranged with respect to the light±path. The electrodes are glued be-
tweenthemagnetswithanisolatingepoxyresin.Duringoperation,oneoftheelectrodesisgrounded
whiletheotherisconnectedtohighvoltage.Theelectrodescanbeswitchedeasilytoreversetheelec-
tric field direction. We operate the electrodes between 200 and 500 Hz, and we have checked that
the voltagedistortionis alwaysnegligibleatthese frequencies,evenwhenthefourrodsaresupplied
simulatneouslyatthemaximumvoltage.
Finally, our rods can provide fields Beff = 0.85 T and E = 0.35 MV/m rms on a total length
4LE×B = L2 = 0.8 m.Insuchfields, theorderofmagnitudeoftheexpectedfrequencymodulation
duetoJonesbirefringenceinNitrogen(seeTable1)isδn=2.7 10−17 rms,hence
×
δν = ν4LE×Bδn (1)
L
1
= 2.8 1014 2.7 10−17
× × 2 × ×
δν = 3.8mHzrms (2)
SinceourrodsprovidecrossedfieldsratherthanparallelonesasrequiredforJonesbirefringence,
we will measure the magneto-electric directional anisotropy in Nitrogen, and we therefore expect a
slightly differentvalue,probablya few times higherasfar as we can inferfromvariouscalculations
[15–18].Comparisonwiththepresent200µHzsensitivityofourapparatusmakesusconfidentthatwe
shouldachievesoonthefirstobservationofmagneto-electricdirectionalanisotropyinagas.
3.3. Measurement procedure
The procedure we follow for the measurements uses the broadband EOM to calibrate the error
signalbeforeandaftereachrun.Indeed,theconversionfactorbetweentheamplitudeoftheerrorsignal
frequencycomponentatfmod andthefrequencymodulationatthissamefrequencyisproportionalto
the intracavity power, which varies according to cavity misalignment and mirror pollution. On our
experiment,itdecreasedbyanorderofmagnitudeonatimescaleofthreemonthswithoutanycavity
realignmentnormirrorcleaning.
Besides, we periodically switch on and off the electric field, with a period TAM of a few tens
of seconds,to avoid longterm driftsof the zero field signal. Presently, our signalprocessing simply
Proof/E´preuve
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C.RobilliardandG.Bailly 7
(a)
Magnets
Pure iron
shell
E Electrodes
B
(b)
Fig.4. (coloronline)(a)Arrangementoftherodsgeneratingtheelectricandmagneticfields.Theholeforoptical
accessis4×4 mm2.(b)Profileofthemagneticfieldalongtheopticalpath.Thefieldwellinsidethemagnetis
0.185T,aswellastheeffectivefielddefinedasBeff =R Bdl/LE×B.
consistsofcalculatingforeachperiodPi (seeFig.5)thedifferenceδVi betweenthesignalwithE on
andtheonewithEoff,excludingthesettlingtimeduetothelock-inamplifiertimeconstantaftereach
electricfieldswitch.ThenwesimplyaverageovertheN periods:δV = N1 PNi=1δVi.
This way, we have reached a relative frequency sensitivity about 10−16/√Hz. In terms of the
expectedJonesbirefringenceδninNitrogen,thisamountstoaround10δn/√Hz.
4. Outlook and conclusion
Inconclusion,wehaveshownthatourapparatusisnowfullyoperationalandshouldprovideusin
thenearfuturewithameasurementofmagneto-electricdirectionalanisotropyingases.Atpresent,we
canmakeonlymeasurementsinharmlessgasessuchasairornitrogen,sinceouropticalcavityisnot
yetinatightgaschamber.
Thenextstepsinourexperimentaleffortwillbeontheonehandtheinstallationofthehighfinesse
cavity in a vacuumchamber.This will notonly allow us to performmeasurementsin variousgases,
but it will also avoid mirror pollution, which is presently the main reason for excess noise over the
quantumlimit.Thiswillalsobringsomeimprovementsinthermalandacousticinsulation.Alongwith
someworkconcerningthecavityfinesse,theopticalcouplingandthefrequencystabilization,thisstep
will allowustoreacha nearshot-noiselimitedrelativefrequencysensitivitybelow10−20/√Hz. On
theotherhand,we aredesigningnewrodswithhighermagneticandelectricfields,generatingeither
Proof/E´preuve
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8 Can. J.Phys.Vol.XX,2010
:; :<
)8)/3
7, -
)./012&3)/45645)* !
)8)/99
! $! #! "! !
%&’()*+,
Fig.5. Typicalsignal,obtainedforafrequencymodulationof6.5mHzrmsappliedtothecalibrationEOM.We
periodicallyswitchonandofftheelectricfield,withaperiodT .Thesignalprocessingconsistsofaveraging
AM
thedifferencebetweenthesignalwiththeelectricfieldonandthatwiththefieldswitchedoffovertheN periods.
crossed or parallel fields close to 2 T and to 3 MV/m. Such an apparatuswill open the way to very
precisemeasurementsofdirectionalanisotropies.
On the longer term, we want to use even higher fields, in order to observe the magneto-electric
directionalanisotropyinquantumvacuum,whichisanexperimentalchallenge.WithfieldsB =15T
andE =20MV/m,alreadydemonstratedinothergroups,thesignal-to-noiseratiowouldreachunity
within10h.
Besides,ourmeasurementsintheabsenceofexternalfieldsmayalsoprovidesometestsofLorentz
invarianceandconstrainsomeoftheproposedextensionstoStandardModel.
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