n-n" oscillationsbeyondthequasi-freelimitor

n-n" oscillationsinthepresenceofmagneticfield

E.D.DavisNorthCarolinaStateUniversity

Basedon:Phys.Rev.D95,036004(withA.R.Young) INTworkshopINT-17-69W

Contents

• Magneticfield(inthepropagationregion):muchadoaboutnothing?

• Propagationregiondynamics:mappingtospinorevolutionproblem

• Perturbativeandnon-perturbativeanalysisinnoisyfields

• ExploringtheuseofNMRquantumcontrolprotocols

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Themagneticfield:muchadoaboutnothing?

“ForthemagneticshieldinggeometriesofboththepreviousILLexperimentandtheproposedexperiment,… thedominantcomponentoftheresidualmagneticfieldinsidetheshieldisthecomponentalongtheaxisoftheshield.TheinternalshieldforthepreviousILLexperimentstronglysuppressedtransversecomponentsofthemagneticfieldandrenderedthelongitudinalcomponentsufficientlyuniformthatitcouldbelargelycompensatedbyahomogeneousexternalfieldgeneratedbyacoilwrappedontheoutsideoftheshield.Oncethismajorcomponenttotheresidualfieldwasremoved,anothersetofcoilswereabletotrimouttheresidualtransversefields.Currentloopsforshielddemagnetization,anactivecompensationsystemforexternalmagneticfieldvariations(includingtransversefields),internalmagnetometry,and removaloflargeexternalsourcesofmagneticfieldgradientswerealsorequiredtoensuremaintenanceofthequasi-freecondition.SincethisexperimentwasperformedagreatdealhasbeenlearnedaboutlargevolumemagneticshieldtechnologyinthecourseofR&Dperformedforexperimentswhichsearchfortheneutronelectricdipolemoment[125].”

FromD.G.PhillipsIIetal.,Phys.Rep.621,1:

I.Altarev etal.,Rev.Sci.Instrum.85,075106:“Amagneticallyshieldedroom withultralowresidualfieldandgradient”

shieldidealization =optimaladaptationofshield’smagnetizationtoexternalfields

Magneticallyshieldedpropagationregion

?

Theidealcase(nojoints) 3

𝑙 = 300m

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WhatdoILLfieldprofileslooklike?(Afteradailyshieldidealization)

Unavoidableirregularinhomogeneities thatchange

randomlyaftereachidealization

(T.Bitter,NIMA309,521)

“…fieldvariationsaremuchworsethanwithnoshieldatall.Ateveryjointandateveryweldingseem[sic](threepertubesegment)themagneticfluxleaksoutanddistortstheearthfieldconsiderably.”(T.Bitteretal,NIMA309,521)

“Thespikesvisibleatthebeginningandattheendofthefieldprofiledonotnoticeablydisturbtheneutron-antineutronoscillationprocess,sowedidnotbothertofurthersuppressthem.”

10nT

Whatconstraintsareplacedoncontroloftheresidualmagneticfieldwithapropagationregionofincreasedlength𝑙?

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Ldependence:reasonforanxiety?S.Dickersonetal.,Rev.Sci.Instrum.83,065108

Axialfieldinside14segmentshield(Peaksat13joints)

Effectofshieldlength𝑙 onambientfield(Simulation)

Data

Simulation(offsetby150mG)

Dashedline:1.2m shieldSolidline:8.5m shield(samelengthas14segmentshield)

Radiusofshields

Increasein peakheights duetoincreaseinmagneticfieldchanneledwithinshield

ApproximatelyquadraticdependenceonL525/10/2017 E.D.Davis:INT-17-69W

Propagationregiondynamics:mappingtospinorevolutionproblem

• Startingpoint:Pauli-Schrödingerequationinrestframe ofsystem

𝑖ℏ𝑑𝑑𝑡

𝜒,(𝑡)𝜒,"(𝑡)

= ℏ/0�⃗� 3 𝜔5 𝛿𝕀0𝛿𝕀0 −/0�⃗� 3 𝜔5

𝜒,(𝑡)𝜒,"(𝑡)

• Tolowest orderin𝛿,𝑛" detectionprobability

𝑃," 𝑡 = 𝜹𝟐𝒕𝟐×/0@𝑑𝑡A

𝑡 @𝑑𝑡AA

𝑡 Tr 𝑈0E(𝑡A)

0𝑈0(𝑡AA) 0

F

G

F

G• where

𝑈0 𝑡 = expK −L0@𝑑𝑡

A�⃗� 3 𝜔5(𝑡A)F

G

Larmorfrequencyvector𝜔5(𝑡) = −𝛾𝐵(𝑡) nn" interaction

Time-orderedexponential

Quasi-freeestimate Phaseaveragingsuppressionfactor

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Phaseaveragingsuppression? Exampleof‟longitudinalnoise”:𝐵 = 𝑏 𝑡 𝑘Q

§ Average of𝑃," overensembleofneutronsstudied

𝑃," = 𝜹𝟐𝒕𝟐× @ 1 − 𝑥 cos 𝝎𝑳 𝑡𝑥 exp −𝝌 𝑡𝑥 𝑑𝑥Z/

[/§with

𝜒 𝜏 = @sin𝜔𝜏2𝜔2

0

𝑺𝝎𝑳 𝝎 𝑑𝜔`

G

• wherepowerspectraldensity

𝑺𝝎𝑳 𝝎 = /a@ 𝜔5(𝑡 +

𝜏2) − 𝜔5 𝜔5(𝑡 −

𝜏2) − 𝜔5

`

G

cos𝜔𝜏𝑑𝜏

StationaryGaussianprocess

7

Identical factorcontrolsfreeinductivedecay inNMR

CanquantumcontrolsprotocolswithinNMRcombatphaseaveragingsuppression?

Fouriertransformofauto-correlationfunctionofrandomprocess𝜔5(𝑡)

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§ Treat𝑃," asfunctionalofrandomprocess 𝜔5

§ Minimalist stochasticmodel:assume𝜔5 𝑡 is

widesensestationary

(Duetoactivecompensation)

§ To2nd order insmall𝜔5,

𝑃,"(𝑡)𝛿0𝑡0 = 1 − /

/0 𝜔5 0𝑡0 − @ 𝑑𝜔`

G

𝑆de(𝜔)𝐹(𝜔𝑡)𝜔0

Widesensestationary?• Expectationvalue 𝜔5 timeindependent• Auto-correlationfunctions

𝐶de(Lh) 𝑡/, 𝑡0= 𝜔5,L 𝑡/ − 𝜔5,L 𝜔5,h 𝑡0 − 𝜔5,h

• dependon𝜏 = 𝑡0 − 𝑡/ only

Perturbativetreatmentofnoisymagneticfield(inshieldedregion) Ignorehiccups

8

𝑆de 𝜔 = j 𝑆de,L

�

Llm,n,o

(𝜔) Filterfunction 𝐹 𝑥 = 2 1 − sinc0 m0

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Quasi-freepropagationefficiency𝜂 (asinNIMA309,521)

𝜂 ≡𝑃," 𝑡 r𝛿0 𝑡0 r

= @𝑃,",r(𝑣)𝛿0𝑙0 𝑛 𝑣 𝑑𝑣

`

rtuv

@ 𝑣[0𝑛 𝑣 𝑑𝑣`

rtuv

w

𝜂 = 1 − //0 𝜔5 0 𝜇 [y

𝜇 [0𝑙0 − @ 𝑑𝜔

`

G

𝑆de(𝜔)𝜔0 @ 𝑣[0𝐹

𝜔𝑙𝑣 𝑛(𝑣)

𝑑𝑣𝜇([0)

`

rtuv

9

Quasi-freeestimateof 𝑃,"(𝑡) r

Involvesaverage of 𝑃,"(𝑡) overaxialspeeddistribution𝑛 𝑣 ofneutrons

Minimumspeedtoavoidcollisionswithdriftvessel

Substitutet in𝑃,"(𝑡) by𝑙/𝑣

Averageover𝑛 𝑣

Weakfield

𝜇({) = @ 𝑣[0𝑛 𝑣 𝑑𝑣`

rtuv

Flightpathlength𝑙:apparentquadraticdependence25/10/2017 E.D.Davis:INT-17-69W

𝑙-dependenceof𝜂 (weakfields)

Ingredients

10

Markovian noise⟶ 𝜂 = 1 − }}~ 𝜔5

0 + 2𝛽� ��� 𝜎5

0 � ��� �~

𝑙0

𝜼 versus𝒍:Bmean = 2� (5nT)=Brms

𝛽� ≈ 1

𝛽�

𝑙� = 0

𝑙� = 10𝑚

Approximatelylinear!𝑣�L, = 𝑙/(0.5s) White

noise

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Non-perturbative treatmentofwhiteGaussianlongitudinalnoise

11

Canobtainexact 𝑃,"(𝑡) inclosedanalyticform (functionalintegralsGaussian)

Inference:perturbativeestimatetooconservative25/10/2017 E.D.Davis:INT-17-69W

Influenceofanelementary‟quantumcontrolprotocol”

12

Rabi-likefieldconfiguration 𝐵 = 𝐵�𝑘Q + 𝐵� cos𝜔𝑡 𝚤̂ + sin𝜔𝑡 𝚥̂ (Phys.Rev.D91,096010)

𝑃," 𝑡𝛿0 =

𝜔�,50

𝜔�,50 + 𝜔 − 𝜔�,50𝑡0

4

+oscillatoryterms

Demanduniformity of𝐵� ⟶ otherquantumcontrolprotocols(Rev.Mod.Phys.76,1037)25/10/2017 E.D.Davis:INT-17-69W

Carr-Pound-Meiboom-Gill-likespin-flipsequence

§ Foruniform‟holding”field,

𝑃," 𝑡 = 𝛿0𝑡0×sinc0𝜔5𝑡2𝑛

• for𝑛‟bang-bang”spinflips• (Flipsseparatedbyinterval∆𝑡 = 𝑡/𝑛with1stflipattime= }

~∆𝑡)

§ Filterfunctionishigh-passfilterwithcutofffrequencyapproximatelyequalto𝝎𝒏 ≡

0,F

13

Filterfunction

𝜔 = /0𝜔,

𝜔 = 𝜔,

𝜔 = 2𝜔,

𝑛 = 10 flips

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Conclusions

• RevisitedthemagneticfieldstudiesoftheILLn-n" experimentwithaviewtoestablishingscalingwithsizeofthesystem.NOTaproblem.

• Synthesiswithquantuminformationmethodologiesdevelopedsince

theILLexperimentcouldbeproductive

• HelpfulinsearchformirrorneutronsatHFIR?

Thankyou!

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