Spin-orbit coupled interferometry with ring-trapped Bose--Einstein condensates

0.62 MB·

by  John L. Helm

Spin-orbit coupled interferometry withring–trapped Bose–Einstein condensates.
J. L. Helm,1 T. P. Billam,2 A. Rakonjac,1 S. L. Cornish,1 and S. A. Gardiner1
1Joint Quantum Center (JQC) Durham-Newcastle, Department of Physics,
Durham University, Durham DH1 3LE, United Kingdom
2Joint Quantum Center(JQC) Durham-Newcastle, Department of Mathematics and Statistics,
Newcastle University, Newcastle upon Tyne, NE1 7RU, United Kingdom
(Dated:January10,2017)
Weproposeamethodofatom-interferometryusingaspinorBose–Einstein(BEC)andthewell-established
experimentaltechniqueoftime-varyingmagneticfieldsasacoherentbeam-splitter. Ourprotocolcreateslong-
livedsuperpositionalcounterflowstates,whichareoffundamentalinterestandcanbemadesensitivetoboththe
Sagnaceffectandmagneticfieldsonthesubmicro-Gaussscale.Wesplitaring-trappedcondensate,initiallyin
them =0hyperfinesub-level,intosuperpositionsofbothinternalspinstateandcondensatesuperflow,which
7 f
are spin-orbit coupled. After interrogation a relative phase accumulation can be inferred from a population
1
transfer to the m = 1 states. Our protocol maximises the classical Fisher information, and owing to the
0 f ±
adiabaticsplittingitmaybeagoodcandidatefordevelopingquantumenhancedinterferometryschemes. We
2
presentnumericalandanalyticaltreatmentsofoursystem.

n
a
J Theendeavourtooptimallyapplymatter-waveinterferom- (a)
s] 9 esgetrrnayvsiihttiyavsewgraeovnteaetr[iao1tn5eda–l1m7[]1a–nd6ye]t,percgotirpoaovnsiatpalsrtiooatnoncadlo/plisnr.oertotOitayplpti[em7s–ao1lf4p]urloatrtnoad-- Bt/B()(0)-011...000 BBqz
a cols should fully exploit the characteristics of BECs and 1.0(b) n+1
g
- shouldaimtobeHeisenberglimitedthroughentanglementor ni0.5 n0
nt number-squeezing[18,19],andhavegloballyequivalentclas- 0.0 n−1
sicalandquantumFisherinformation(CFI,QFI)[20]. These
a (c)
qu wrehqeuriereamceonmtsmaroens-aptaistfihegdeboymtehteryscihseumseedd,eesncrsiubreindgineRnteafn.g[3le]-, χmax1.0 χχ−−11,,01
/
t. mentviathespatialoverlapofthearmsoftheinterferometer χi,j0.0 χ0,1
a and maximising QFI throughthe schemes spatial symmetry. 0.0 0.2 0.5 0.8 1.0
m Opticaltrappingallowsustoexploitinternalatomicspinde- Timet/tf
- greesoffreedom,inparticularbyallowingsimultaneoustrap-
d FIG.1. (coloronline)Schematicofspin-orbitcoupledinterferome-
pingofatomsindifferentmagneticsublevels[21,22]. Ause-
n try(SOCI)procedure. (a) B-fieldrampingscheme. Yellow(outer)
fulmethodofcontrollingsuchspinorcondensatesistopologi-
o shadedregionshighlightthebeam-splittingprocessesduringwhich
calvorteximprinting[23–28],wherethetextureofexternally
c thezbiasfield(B)isrampeddownorup,whileblue(inner)shaded
z
[ applied time-varying magnetic fields (B-fields) is embedded
regionsshowthephaseunpinningprocessesduringwhichthemagni-

inthecondensate’sspinand,therefore,itsphase.TheB-fields tudeofthequadrupolarfield(B )isrampeddownorup.Numerically
1 q
v usedinthisimprintingprocessarecommontothemajorityof calculatednorms,(b),andoverlaps,(c),ofthespinorcomponentsin
4 coldatomexperiments,andarethereforereadilyavailable. thez-quantisedbasisforaδ=πinterferometryrunendingattimetf.
5 InthisLetterweproposeamethodofmatter-waveinterfer-
1 ometry which splits a repulsively interacting spinor conden-
2
sate into a superpositionof both spin- and superflow-states, Thisamountstosimplyrampingdown(andrampingup)thez
0
whicharespin-orbitcoupled.Thisspin-orbitcoupledinterfer- biasfield.Wethenshowthatanaccumulatedphase-difference
.
1 ometry(SOCI) procedureusestopologicalvorteximprinting between the counterflowingcomponentstranslates to a pop-
0 asabeam-splitterinthemomentumandspinbases,achieved ulation transfer from the field-insensitive eigenstate to the
7
through the use of the time varying B-field [Fig. 1]. The strong/weak-field-seeking eigenstates after a second beam-
1
“arms”oftheinterferometerarenotnecessarilyspatiallysep- splitting procedure,and discuss how this populationtransfer
:
v arate, allowing for the creation of a common-path interfer- canbemeasured.
i
X ometer,whichisinsensitivetoavarietyofperturbingfactors A spin F condensate can most conveniently be treated as
due to its intrinsic symmetry. A noteworthy application of a system of 2F +1 coupledBECs. Here we will work only
r
a the common-path method is zero-area Sagnac interferome- in the z-quantised (ZQ) representation, where all spin states
try (ZASI) [29, 30], often discussed as an alternative to the are labelled with reference to the z-axis. For F = 1/2 our
Michelson geometry for optical gravity-wave detection [31– vectorvaluedorderparameterΨcanbedescribedin−termsof
33]. Wefirstpresentthecoupledequationsofmotiondescrib- theavailablespin-states,givingΨ = 1 Ψ j . Theeigen-
ingthe spinorcondensate,modifiedto accountfora rotating vectors j satisfy F j = j j , and arej=−si1mpjl|yithe different
z P
frameofreference. Wethenconsidertheeffectofthe B-field possible|vialuesofthe|zi-comp|oinentofthespin. Here~F /2is
z
in isolation, and howthisappliesto splitting(andrecombin- thestandardzdirectionangularmomentumoperatorintheZQ
ing)thewavefunctionintoaspin-orbitcoupledsuperposition. basis,resultinginspin-stateeigenvectors 1 =(1,0,0)T, 0 =
| i | i
2
(0,1,0)T, and 1 = (0,0,1)T. Inorderto considertheef- ing number. This requirement is satisfied by either an anti-
|− i
fectofrotations,weintroduceanangularmomentumtermde- Helmholtz(aH)orIoffe-Pritchard(IP)coilconfiguration. We
scribingtheenergyofaconstantrotationcharacterisedbythe considerthegeometricallysimplerIPconfiguration,whichis
angularvelocityvectorΩ. Assuch,ourfulldynamicalequa- quadrupolarinthe x-yplane,butnotethatthesamephenom-
tionsare enashouldbeobservableusingtheaHconfiguration. TheIP
B-fieldtextureis bestdescribedin cylindricalspatialcoordi-
∂ ~2
i~ Ψ = 2+V i~(r Ω) +g Ψ Ψ Ψ natesρ,φ,z. TheCartesiancomponentscanthenbewrittenas
∂t j "−2m∇ − × ·∇ n † # j B = (B (ρ)cos(φ), B (ρ)sin(φ),B),wherethequadrupolar
IP q q z
−
+ g F¯ F µ g B F Ψ , (1) field Bq(ρ) = b′ρvarieslinearlywithρandthezbiasfieldis
s B F
· − · j spatiallyuniform.Hence
nh i o
where the local spin vector F¯ has components F¯ =
α B B eiφ/√2 0
Σ Ψ Ψ kF j . Here we have atomic mass m, Bohr mag- z q
j,k ∗k jh | α| i B F=B e iφ/√2 0 B eiφ/√2. (2)
niinnettetohrnaecµFtBio=anns1dtrhmeynapgnetihrffiognlde).agnyTdrhoesmpsaicnga-ntstepetirincinrigantttieeorramgcFtsio(a−nre1s/tth2reefnongroth8r7mRgabl. ·  q −0 Bqe−iφ/√2 q−Bz 
n s The(spatiallydependent)eigenvectorsofEq.(2)are
TheV termdescribesexternallyimposedopticaltrappingpo-
tentials. We consider a ring-trapped condensate with V = T
B = B B eiφ, √2B , B B e iφ /2B (3)
mω2[(ρ R )2 + z2]/2, where ρ = x2+y2, giving a ra- |± i ± z ± q ∓ z −
dial⊥trapp−ing0frequencyω and majorpradius R0 (which will and (cid:16)(cid:2) (cid:3) (cid:2) (cid:3) (cid:17)
be described in terms of t⊥he harmonic oscillator length a ).
This potential simplifies our numerics, but the analysis on⊥ly Z = B eiφ, √2B ,B e iφ T /√2B (4)
q z q −
| i −
requires that the potential should be rotationally symmetric, (cid:16) (cid:17)
with an optical plug covering ρ = 0 to prevent undesirable where B = (B2 + B2)1/2. The + B and B eigenvectors
q z | i |− i
vortexdynamics.Gravityistakentoactinthezdirectionand denotethestrong-andweak-field-seekingstateswitheigen-
could be included in the trapping potential without altering values B,while Z isassociatedwiththezeroeigenvalueof
± | i
its symmetry, and so we do not consider it further. For our B F.Throughtheseeigenvectorswecanseethefundamental
·
numericsweconsiderexperimentalparameterscomparableto technique behind the imprinting method of Ref. [23]; vary-
thosedescribedinRefs.[34,35],however,forfasternumerics ing B (viab)and B overtime,thecondensateremainsina
q ′ z
wetaketheradialtrappingfrequencytobeω =2π 80Hz, giveneigenvectorofB F,buttransfersbetweenthem states,
f
themajorradiusoftheringtobeR =5a =6⊥.02µm×,andthe accumulatingangularm· omentumintheprocess. Someradial
0
numberof87RbatomstobeN =104. The⊥normalbinaryscat- dynamicscanoccurasthe B-fieldvarieswithtime,butthese
teringg = 4π~2(a +2a )/3misa standardnon-linearterm dynamicsseparateoutfromthebehaviourdescribedbyB F,
n 0 2
·
basedonthetotallocalnorm,whereallspin-componentsmay andsoarenotaddressedbyouranalytics.Later,whentreating
collidebuttheirspinsdonotchangeoraremerelyexchanged, theSagnaceffect,theeigenvaluesaremodifiedslightlybythe
whileg =4π~2(a a )/3mdescribesspin-flippingcollisions angularvelocitytermofEq.(1). Again,itcanbeshownthat
s 2 0
−
whichcanresultinpopulationtransfer,althoughthetotalspin thedominantspindynamicsdecouplefromtheradialdynam-
mustbeconserved. Herea anda arethescatteringlengths ics. Thisimplicationofspin-gaugesymmetryisconfirmedby
0 2
associatedwithbinaryelasticcollisionsinthetotalspin0and thefull3Dnumerics.
2 channels respectively. Our numerics are for 87Rb in the To achieve the counterflow state (i.e., beam-splitting) we
F = 1 manifold,where a = 101.8a a = 100.4a [27], mustinitiallyprepareourcondensateinthe 0 spin-statewith
0 B 2 B
∼ | i
anda istheBohrradius. Fortheseinteractionstrengths,our some dominating initial bias field in the z direction (B
B z
| | ≫
systemisferromagnetic(g < 0),anditsenergyisminimised B (R )). This constitutes the Z state. The 0 initial state
s q 0
formaximal F¯ . Thesplittingproceduregeneratesacounter- c|an be|achieved through RF-p|umiping a 1| i(weak-field-
flowstatefor|w|hich F¯ =0,andsominimisestheHamiltonian seeking)condensate[36]. Thiscanbecarr|ie−doiutaftertrans-
| |
energyofanti-ferromagnetic(g > 0)systems[21]. Assuch, fertoanopticaltrap,wheremagnetictrappingisnolongerre-
s
thecounterflowstateconstitutesametastableexcitedstatefor quired.Alternatively,astrongx(ory)biasfieldcouldbeused
ferromagneticsystems,butwouldbeastablegroundstatefor to adiabatically transfer the atoms into a weak-field-seeking
antiferromagnetic systems (e.g., 23Na in F = 1 or 87Rb in 0 (ZQ) state, then diabatically transferring these to the Z
| i | i
F = 2). However, since 87Rb is only weakly ferromagnetic state byquicklyswitchingoffthe x bias(thisprocesscan be
(g is comparativelysmall) the state’s instability has a long somewhatdestructiveifthe xbiasistoolarge). Withourini-
s
| |
associatedtimescale,andsotheg termcanbeignoredinthe tialconditionnowfixedinthe Z eigenstate,thecounterflow
s
| i
analytics.Wekeeptheg terminthenumerics,observingonly state is obtainedby ramping B down to zero over a period
s z
| |
smalldeviationsfromtheanalyticalresult T [Fig. 1,] splitting the condensate into a superposition of
s
The dominant dynamics of the system can be understood spin up andspin down, as can be seen from Eq. (4). Inour
fromtheeigenvectorsoftheB Foperator,inturndetermined numerics, the quadrupolarfield is characterised by an initial
·
bythetextureofthemagneticfield.Ourfundamentalrequire- gradientb = 3.7G/cm, whilethe initialz biasfield isset to
′
ment is that the B-field should have a non-trivial topology, B = 50 mG. We select the ramp-downperiod T = 22 ms,
z s
such that a curve encircling the origin has non-zero wind- selectedtobeoneorderofmagnitudegreaterthantheLarmor
3
precession time TL = 2π~/(µBgFb′R0) = 2.2 ms, ensuring (a) χ+1,0 (d) χ+1,0 1.0
the spins follow the B-field adiabatically. As the atoms in a C
T
givenspinstatehaveanassociatedflowfield,thecondensate T/I max 0.8
χ
isnowinasuperpositionalcounterflowstate. Wenotethat,for /
B B (R ),duringalinearramp-downofdurationT the (b) χ+1,−1 (e) χ+1,−1 χi,j 0.6
z| bz|ia≫sw|oquld0b|edominantformostoftheramp-down,ansdso TC ap
theeffectivetransferperiodwouldbetooshort. Assuch,we T/I verl 0.4
o
snthheionsutiilnadllocyuadrreencfuuamlyliyenrgcichtsaoioulssteionaagvsaomisduodiotitaahbblrayatimschppi-frdtoeocdwesansnedws.irteWhsceaanalceehdxiehpvyoe-- T/TIC(c) χ−1,0 (f) χ−1,0 Spatial 0.2
perbolic tangent function. Ramping too quickly can lead to 0.0
Timet/t Timet/t
radialexcitations, which do notnecessarily disruptthe split- f f
ting,butcanpreventthespatialoverlapofthecounter-flowing
FIG.2. (coloronline)Numericalsimulationdemonstratingoverlap
components. Duringcounterflowthesystemisstilldescribed
χ betweenthe i , j spin-statesduringcounterflowinterferometry.
bythe Z eigenstate,andsoifwenowreturn B toitsinitial i,j |i| i
| i z (a,b,c)showtheoverlapforSagnacinterferometrywithvaryingin-
value(oranyothervalueofsuitablylargemagnitude)theen-
terrogation times T giving varying accumulated phases δ . (d,e,f)
I S
tirecondensatewillreturnto 0 ,assumingnophase-shifting showtheZeeman-energyinterferometrycasewithphasedifference
| i
has occurred. We will use this method for recombinationin δ . Allaxesrunovertherange[0,1]. Theoverlapintegralisscaled
Z
ourinterferometryprotocol. bythemaximalvalueχ ,achievedwhenthecondensateisequally
max
Withourbeam-splittingandrecombinationprotocolsestab- splitandhasthesamespatialprofileinbothcomponents.
lished, we can now consider the impact of a relative phase-
shift, consistentwith theapproachdescribedinRef.[3]. Ar-
tificiallyimprintinga relativephasedifferenceδbetweenthe
unconstrained[37, 38]. In this high Ω˜ regime our transfor-
spin up and spin down components after the split (at some z
mationisnolongervalidastheassumedrotationalsymmetry
time when B = 0), we canre-writeourcounterflowstate as
z breaks when vortices enter the condensate. We will discuss
acombinationofallthreeeigenstatesofB F. Theweighting
ofthesestatesisdeterminedbyδ: · theimplicationsofthisequivalencybetween B˜z andΩ˜z later.
Suchatransformedsystemhasanalogoustransformedeigen-
Ψ = 1 ei(φ+δ/2),0, e−i(φ+δ/2) T (5) vinecthtoersZ. TehiigsemnsetaantesothfaBt,raFm,apnindgthdeorwefnor|Bezd|o→no0t,wexepaercetsatnilyl
| i √2(cid:16)− (cid:17) phase|diifference between·the 1 components to accumu-
1+cos(δ) 1 cos(δ) +B + B late.Inordertoobserverelativ|e±phaiseaccumulation,wemust
= Z i − | i |− i .
r 2 | i− r 2 √2 ! have a superpositional counterflow state in the absence of a
quadrupolarfield. Carefully ramping down B with B = 0
q z
Ramping Bz back up implements the recombination, after achievesthisaimandunpinsthephasesofthecounterflowing
whichthe B eigenvectorsarethe 1 spinstates, while components,butifthereremainssomesmallresidualcontri-
|± i |± i
the Z eigenvectoristhe 0 spinstate. Hence,formallypro- butionB , 0thenthesystemwill(eventually)returnto 0 ,
jecti|ngi our final state ont|oithe zero spin state via the 0 0 as dictatezRdby Eq. (4), and counterflowis lost. To avoid t|hiis
| ih |
projector(or, equivalently Z Z) we obtainan interferomet- restorativeeffectwe mustchoose the ramp-downcurvesuch
| ih |
ricsignalbasedonthecondensatefractioninthe 0 state,i.e., thattheB switch-offisdiabaticinsomesense. Forexample,
Ψ0(tf)2dr = [1+cos(δ)]/2,wheretf isthetim| eiwhenour thiscurveqcouldbesmoothlydecayingatfirstandthencutoff
| |
iRnterferometryprotocolends. Thepopulationsofthedifferent instantaneouslybeforethepointwhereB h10 B ,orfully
q zR
spincomponentscanbeobservedbyapplyingafieldgradient continuousbutrampedovera suitably fast time×scale. Inour
inthezdirection,resultinginStern–Gerlachseparation. numerics,theresidualfieldwaseitherzerooroftheorderof
Wenowsimulateprospectiveinterferometryapplicationsof micro-Gauss, and so T = 22 ms was suitably fast to avoid
s
thisprocedure,whereΩzorBzarenonzeroduringtheinterro- recombinationinto 0 . The key consideration should be re-
| i
gationcounterflowperiodTI. Assumingthecounterflowstate ductionoftheradialdynamicsandheatingassociatedwithdi-
is well described by Eq. (5) (particularly that it is rotation- abaticprocesses,notingthatthesmallertheresidualfield,the
allysymmetric),wemaycombinetherotationalandZeeman smallertheassociatedZeemanenergy,andsothelessdanger
termsofEq.(1)byapplyingthei~(r Ω) operatortothis of heating. We also highlight that this restorative effect re-
× ·∇
state. Theeffectoftherotationissimplytooffsetthestrength quirestheresidualfieldtosatisfy B B (R ) h 1.57mG.
zR q 0
≪
of the z bias field, which appears reduced (increased) as the ThisupperboundscaleslinearlywithR andb,andcaneasily
0 ′
coordinatesystem rotates with (against) the magnetic dipole beraised. Inthecompleteabsenceofamagneticfieldthehy-
precession. This gauge transformation can be expressed as perfinestatesbecomedegenerate. Ingeneralthisleadstoun-
B˜z B˜z +Ω˜z, where B˜ = µBgFB/~ω and Ω˜ = Ω/ω are desirablespin-flips,whichbecomemoreenergeticallyallow-
dim→ensionless quantities. Experimenta⊥lly, B˜z B˜q ⊥ Ω˜z able as B 0. We have previouslynoted that spin-flipping
is typically achievable. Indeed, if Ω˜ & 0.2 th≫e syste≫m nu- collisions →are suppressed in the superpositional counterflow
z
cleates vortices, and as Ω˜ 1.0the centrifugalforceover- state. Another possible source of undesirable spin-flips is
z
→
comestheopticaltrappingandthecondensate’sexpansionis stray fields. However, assuming the field can be controlled
4
(a) (e) 1.0 field-seeking. These oscillations can be seen in the overlap
C integralsshownin Figs.1(c)and2, howevertheydonotaf-
T
T/In n 0.8 fecttherecombinationasthe radialdynamicsdecouplefrom
(b+)1 (f+)1 nt()i 0.6 trhaemepiignegntvheectzorbsiaosffiBel·dFb.aTckhetoreacofimnibteinlaatrigoenvisalauceh,iethvuesdrbey-
C m
T/ or encodingthephaseinformationaspopulationsinthedifferent
TIn0 n0 aln 0.4 spinstates[Eq.(5)].
(c) (g) Tot Results of numerical simulations (using CUDA [39]) are
TC 0.2 displayed in Figs. 2 and 3. We performed full interferome-
/
TIn n try procedures for both rotational- and field-sensing scenar-
(d−)1 Timet/tf (h−)1 Timet/tf 0.0 ios, fixing Ω˜z and B˜zR to be either 0.04 or zero respectively.
n+1 Different phases accumulated over varying the interrogation
nt()if nn−01 tdiemnessitTyIo.vFeirgla.p2idnitsepglaraylsχthi,ej(tf)ul=ltimΨeeiv2oΨlujt2iodnrofofrtheeacdheninsitteyr--
| | | |
nnf ferometry run. In both cases, weRobserve good overlap dur-
PhaseδS/2π=TI/TC PhaseδZ/2π=TI/TC 0 ingthecounterflowphasefortheadiabaticsplittingprocedure
outlined above. When the quadrupolar field is restored, os-
FIG. 3. (color online) Numerical simulation demonstrating the re- cillations are evidentas a result of the condensate now pop-
sponseofthesystemtodifferentinterrogationperiodsT,resultingin
I ulating the B eigenstates. After the recombinationthese
differentaccumulatedSagnacphasesδ andZeeman-energyphases |± i
S oscillations are no longer present, confirming the robustness
δ .(a-c,e-g)showhowthenormsofeachspinorcomponentvarywith
Z of the procedureimplied by the separation of radial dynam-
time.Thex-axisisscaledbythetotalruntime(t),whichisincreased
f icsintheanalyticaltreatment. Thetimeevolutionofthetotal
toaccumulatealargerphasedifferenceδ .(d,h)showfinalinterfer-
ometrycurvesatthefinaltimetf.AllaxeSs,Zrunovertherange[0,1]. norm ni = |Ψi|2dr is displayed in Fig. 3. Population os-
cillations ouRtlined by Eq. (5) are clearly evident at the end
of the interferometry procedure. The interferometry cosine
curves [Fig. 3(d,h)] are offset slightly as a result of minor
phase accumulation during the quadrupolar ramp-down pe-
on the sub mG scale, such processes have a long associated riod. Furthermore,thecurvesdeviateslightlyfromatrueco-
timescaleandcanbeignored. Finally, wenotethatquantum sine due to the minor instability introduced by the g term.
s
and thermal fluctuations may be another source of undesir- This was determined numerically by performing runs with-
able spin-flips, but such analysis is beyond the scope of this out this term, producing smoother curves. We note that the
Letter. Astrategytoavoidundesirablespin-flipswouldbeto equivalencybetween Ω˜ and B˜ require thatcare be taken in
z z
purposefullyretainanonzeroB . Oncethequadrupolarfield experimentalmeasurements.Thevalueof0.04usedinthenu-
zR
isabsent,the 1 componentscanevolvefreely,andsoac- mericscorrespondssimultaneouslytoΩ = 2π 6.4Hzand
z
cumulateaSa|g±naciphaseδ forΩ , 0,oraZeeman-energy B = 4.5µG.Assuch,itshouldberelativelystr×aightforward
S z z
phase δ for B , 0. The magnitude of this phase can be to make single-shot field measurements on the micro-Gauss
Z zR
quantified in terms of either the ring’s enclosed area or the scale, as large rotations should be absent. Similarly, it will
interrogationperiodT [3]. Allowingeachcomponenttoper- benecessarytoknowtheprofileofatime-dependentangular
I
formtheequivalentofonefullcirculationaroundtheringwill velocity, so that a signal can be extractedfrom a time-series
causetheaccumulationofaSagnacphaseδ = 2AΩ m/~,as ofmeasurementswhichmayincludeexceedinglysmallback-
S z
theringwillbedoubly-enclosedbythecounterflowingatoms. ground B-fields. Alternatively,anadditionalRF-pumpcould
Theparticlevelocityaroundtheringisgivenbythevelocity- beappliedhalfwaythroughtheinterrogationphase,swapping
fieldofavortex,v = (~/mρ)φˆ,andsowedeterminethetime thecomponentsintotheoppositespinstatesandallowingany
for a single particle to fully circumnavigate the ring (such accumulatedZeemanphasestounwindduringtheremaining
that ρ = R ) to be T = 2πR2m/~ = 2Am/~ = 156 ms. halfoftheinterrogation.
0 C 0
The phase accumulated for an arbitrary interrogation time To conclude, we have first outlined a new technique for
T is then δ = (T /T )2Ω Am/~ = Ω T . The same ar- generating long-lived superpositional counterflow states, a
I S I C z z I
guments apply for the Zeeman-energy phase under the sub- quantum-mechanicalstate of fundamentalinterest. We have
stitution Ω (µ /~)B. After interrogation, restoring the then explored the application of this state to interferometry.
z B z
→
quadrupolarB-fieldprojectsourphase-shiftedwave-function Our setup maximises the classical Fisher information [20],
onto the eigenstates of B F. Over the interrogation period and relies on adiabatic processes and so may be a candidate
·
the populations in the m (ZQ) states do not change. How- for large scale many-body quantum enhanced interferome-
f
ever, phasesareaccumulatedwith the effectthatthe popula- try [18, 19]. Further investigationswill explorethe effect of
tions in the B , Z basis are changedupon restorationof quantumfluctuationsandapplicationstoZASI.
|± i | i
the quadrupolar field, and so the weighting of these eigen- ThedatapresentedinthisLettercanbefoundinRef.[40].
states after interrogation differs from the initial weighting. We thank A. L. Marchant, R. Bettles, C. Weiss, and S.
This restoration and projection induces some radial oscilla- HaineforusefuldiscussionsandtheUKEPSRC(grantnum-
tionsasthe B eigenstatesarerespectivelystrongandweak- berEP/K03250X/1).
|± i
5
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