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RabicycleFromWikipedia,thefreeencyclopedia
Agreatvarietyofphysicalprocessesbelongingtotheareasofquantumcomputing,condensedmatter,atomicandmolecularphysics,andnuclearandparticlephysicscanbeconvenientlystudiedintermsoftwolevelquantummechanicalsystems.Inthiscase,oneoftheeffectsisrepresentedbytheoscillationsbetweenthetwoenergylevels,asforexample,electronneutrinoemuonneutrinoflavourneutrinooscillations.Inphysics,theRabicycleisthecyclicbehaviourofatwostatequantumsysteminthepresenceofanoscillatorydrivingfield.Atwostatesystemhastwopossiblestates,andiftheyarenotdegenerate(i.e.equalenergy),thesystemcanbecome"excited"whenitabsorbsaquantumofenergy.
Theeffectisimportantinquantumoptics,nuclearmagneticresonanceandquantumcomputing.ThetermisnamedinhonourofIsidorIsaacRabi.
Whenanatom(orsomeothertwolevelsystem)isilluminatedbyacoherentbeamofphotons,itwillcyclicallyabsorbphotonsandreemitthembystimulatedemission.OnesuchcycleiscalledaRabicycleandtheinverseofitsdurationtheRabifrequencyofthephotonbeam.
Thismechanismisfundamentaltoquantumoptics.ItcanbemodeledusingtheJaynesCummingsmodelandtheBlochvectorformalism.
Forexample,foratwostateatom(anatominwhichanelectroncaneitherbeintheexcitedorgroundstate)inanelectromagneticfieldwithfrequencytunedtotheexcitationenergy,theprobabilityoffindingtheatomintheexcitedstateisfoundfromtheBlochequationstobe:
,
where istheRabifrequency.
Moregenerally,onecanconsiderasystemwherethetwolevelsunderconsiderationarenotenergyeigenstates.Thereforeifthesystemisinitializedinoneoftheselevels,timeevolutionwillmakethepopulationofeachofthelevelsoscillatewithsomecharacteristicfrequency,whoseangularfrequency[1]isalsoknownastheRabifrequency.ThestateofatwostatequantumsystemcanberepresentedasvectorsofatwodimensionalcomplexHilbertspace,whichmeanseverystatevector
isrepresentedbytwocomplexcoordinates.
where and arethecoordinates.[2]
Ifthevectorsarenormalized, and arerelatedby
.Thebasisvectorswillberepresentedas and
Allobservablephysicalquantitiesassociatedwiththissystemsare2
2Hermitianmatrices,whichmeanstheHamiltonianofthesystemisalsoasimilarmatrix.
Contents
1Howtoprepareanoscillationexperimentinaquantumsystem2DerivationofRabiFormulainanNonperturbativeProcedurebymeansofthePaulimatrices3RabioscillationinQuantumcomputing4RabioscillationinAmmoniamaser5Seealso6References7Externallinks
Howtoprepareanoscillationexperimentinaquantumsystem
Onecanconstructanoscillationexperimentconsistingoffollowingsteps:[3]
(1)Preparethesysteminafixedstatesay
(2)Letthestateevolvefreely,underaHamiltonianHfortimet
(3)FindtheprobabilityP(t),thatthestateisin
If
wasaneigenstateofH,P(t)=1andtherearenooscillations.Alsoiftwostatesaredegenerate,everystateincluding
isaneigenstateofH.Asaresulttherearenooscillations.SoifHhasnodegenerateeigenstates,neitherofwhichis
,thentherewillbeoscillations.TheseprobabilitiesofoscillationsaregivenbyRabiFormula.OscillationsbetweentwolevelsarecalledRabioscillation.ThemostgeneralformoftheHamiltonianofatwostatesystemisgiven
here, and arerealnumbers.Thismatrixcanbedecomposedas,
Thematrix isthe2 2identitymatrixandthematrices
arethePaulimatrices.Thisdecompositionsimplifiestheanalysisofthesystemespeciallyinthetimeindependentcasewherethevaluesof
and areconstants.Considerthecaseofaspin1/2particleinamagneticfield
.TheinteractionHamiltonianforthissystemis
.Where
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where isthemagnitudeoftheparticle'smagneticmoment,
isGyromagneticratioand
isthevectorofPaulimatrices.HereeigenstatesofHamiltonianareeigenstatesof
thatis and .Theprobabilitythatasysteminthestate
willbefoundtobeinthearbitrarystate isgivenby .Letsystem
initially isinstate thatiseigenstateof , .Thatis
.HereHamiltonianistimeindependent.
SobysolvingtimeindependentSchrdingerequation,wegetstateaftertimetisgivenby
,whereEisthetotalenergyofsystem.Sothestate
aftertimetisgivenby
.Nowsupposespinismeasuredinthexdirectionattimet,theprobabilityoffindingspinupis
givenby where isacharacteristicangularfrequencygivenby
whereithasbeenassumedthat
.[4]SointhiscaseprobabilityoffindingspinupstateinXdirectionisoscillatoryintimetwhen
systemisinitiallyinZdirection.SimilarlyifwemeasurespininZdirectionthenprobabilityoffinding
ofthesystemis .Inthecase ,thatiswhenthe
Hamiltonianisdegeneratethereisnooscillation.SowecanconcludethatiftheeigenstateofanabovegivenHamiltonianrepresentsthestateofasystem,thenprobabilityofthesystembeingthatstateisnotoscillatory,butifwefindprobabilityoffindingthesysteminotherstate,itisoscillatory.Thisistrueforeventime
dependentHamiltonian.Forexample
,theprobabilitythatameasurementofsysteminYdirectionattimetresultsin
is
,whereinitialstateisin .[5]
ExampleofRabiOscillationbetweentwostatesinionizedhydrogenmolecule.Ionizedhydrogenmoleculeiscomposedoftwoproton
and
andoneelectron.Thetwoprotonsbecauseoftheirlargemassescanbeconsideredtobefixed.LetuscallRbethedistancebetweenthemand
and thestateswheretheelectronislocalisedaround or
.Assume,atacertaintime,theelectronislocalisedaboutproton
.AccordingtotheresultsofprevioussectionweknowitwilloscillatebetweenthetwoprotonswithafrequencyequaltotheBohrfrequencyassociatedwithtwostationarystate
and ofmolecule.
Thisoscillationoftheelectronbetweenthetwostatescorrespondstoanoscillationofthemeanvalueoftheelectricdipolemomentofthemolecule.Thuswhenthemoleculeisnotinastationarystate,anoscillatingelectricdipolemomentcanappear.Suchanoscillatingdipolemomentcanexchangeenergywithanelectromagneticwaveofsamefrequency.Consequently,thisfrequencymustappearintheabsorptionandemissionspectrumofIonizedhydrogenmolecule.
DerivationofRabiFormulainanNonperturbativeProcedurebymeansofthePaulimatrices
LetusconsideraHamiltonianintheform .
Theeigenvaluesofthismatrixaregivenby and
.Where and .sowecantake
.
Noweigenvectorfor canbefoundfromequation: .
So .
Usingnormalisationconditionofeigenvector,
.So .
Let and .so .
Soweget .Thatis .Takingarbitraryphaseangle ,wecanwrite
.Similarly
.
Soeigenvectorfor eigenvalueisgivenby .
Asoverallphaseisimmaterialsowecanwrite .
Similarlywecanfindeigenvectorfor valueandweget .
Fromthesetwoequations,wecanwrite and .
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Letattimet=0,systembein ThatIs .
Stateofsystemaftertimetisgivenby .
Nowasystemisinoneoftheeigenstates or
,itwillremainthesamestate,howeverinageneralstateasshownabovethetimeevolutionisnontrivial.
Theprobabilityamplitudeoffindingthesystemattimetinthestate
isgivenby .
Nowtheprobabilitythatasysteminthestate
willbefoundtobeinthearbitrarystate isgivenby
Bysimplifying .........(1).
Thisshowsthatthereisafiniteprobabilityoffindingthesysteminstate
whenthesystemisoriginallyinthestate
.Theprobabilityisoscillatorywithangular
frequency
,whichissimplyuniqueBohrfrequencyofthesystemandalsocalledRabifrequency.Theformula(1)isknownas
Rabiformula.Nowaftertimettheprobabilitythatthesysteminstate
isgivenby ,whichisalso
oscillatory.ThistypeofoscillationsbetweentwolevelsarecalledRabioscillationandseeninmanyproblemssuchasNeutrinooscillation,ionizedHydrogenmolecule,Quantumcomputing,Ammoniamaseretc.
RabioscillationinQuantumcomputing
Anytwostatesystemiscalledqubit.LetaSpinhalfsystembeplacedinaclassicalmagneticfieldwithperiodiccomponentsuchas.TheHamiltonianofsystem(proton)ofmagneticmoment
infield isgiven
,where andb , isgyromagneticratioofproton.Onecanfind
eigenvalueandeigenvectoroftheHamiltonianbyabovementionedprocedure.Letattimet=0thequbitisinthestate
,attimetitwillhaveaprobability
ofbeingfoundinstate givenby where
.ThisisthephenomenonofRabioscillation.Theoscilltion
betweenthelevels and hasmaximumamplitudefor thatisatresonance.So
.Togofromstate tostate itissufficientto
adjustthetimetduringwhichtherotatingfieldactssuchas or
.Thisiscalleda pulse.Ifatimeintermediatebetween0and ischosen,we
obtainasuperpositionof and .Inparticular ,wehavea pulse:
.Thisoperationhascrucialimportanceinquantumcomputing.
Theequationsareessentiallyidenticalinthecaseofatwolevelatominthefieldofalaserwhenthegenerallywellsatisfiedrotatingwaveapproximationismade.Thenistheenergydifferencebetweenthetwoatomiclevels,
isthefrequencyoflaserwaveandRabifrequency
isproportionaltotheproductofthetransition
electricdipolemomentofatom andelectricfield ofthelaserwavethatis
.Insummary,Rabioscillationsarethebasicprocessusedtomanipulate
qubits.Theseoscillationsareobtainedbyexposingqubitstoperiodicelectricormagneticfieldsduringsuitablyadjustedtimeintervals.[6]
RabioscillationinAmmoniamaser
Inmolecularphysics,awellknownexampleofatwostatesystemisanitrogenatominthedoublewellpotentialofanammoniamolecule.ThetunnelsplittingoftheammoniaNH3moleculefindsapplicationintheammoniaMASER,anacronymforMicrowaveAmplificationofStimulatedEmissionofRadiation,similarlyasLASERisanacronymforLightAmplification.WhilemagneticinteractionsoftheNH3moleculesarenegligible,themoleculesdipolemomentsareverysensitivetoexternalelectricfields.Aprominentcauseforthesplittingofenergylevelsisthetunneleffect.AsanexampleweregardtheammoniamoleculeNH3.Ifwesqueezethenitrogenatomalongthezaxisthroughtheequilateraltriangleformedbythehydrogenatoms,thentheNatomseesadoublewellpotentialV(z).TheNatomcanbeoneithersideoftheplaneoftheHatoms.HencetheNH3moleculeformsatwostatesystemwithtwobasisstatesorspatialeigenstates
and
withthenitrogenatomononesideortheotheroftheplaneofhydrogenatoms,anditsenergygroundandfirstexcitedstates
and
arethesymmetricandantisymmetricquantumsuperpositionsofthespatialeigenstates(ignoringrotationalandvibrationalstates).ThesystemcanchangefromLtoRandbackbecausethereisacertainprobabilityamplitudeAthatthenitrogenatomtunnelsthroughthecentralpotentialbarriertotheotherwell.Theammoniamaserisbaseduponthetransitionbetweentheenergyeigenstates,whichmayalsobedescribedastheRabioscillationbetweenthespatialeigenstates.However,theinversiontransitionisseenonlyatlowgaspressure.Asthegaspressureisincreased,thetransitionbroadens,shiftstolowerfrequencyandthenquenches(thefrequencygoestozero).Theammoniamoleculeappearstoundergospatiallocalisationasaresultofinteractionwiththeenvironment.Thiswouldbeofimmensetheoreticalinterest.Inchemistryandintheclassicalworldgenerally,enantiomorphicmoleculeswithdistinguishablespatialeigenstates
and
arealwaysfoundintheirspatialeigenstates(classicalbehaviour)ratherthantheirenergygroundstates(quantumbehaviour).Whilstammoniaisnotenantiomorphic,itdoesappeartoshowbothbehaviours,quantumatlowpressureandclassicalathighpressure,ifthequenchingisconsideredtobeadirectobservationoflocalisationorcollapseofthewavefunctionintoaspatialeigenstate.Withinthecontextofthedecoherenceprogramme,ithasbeentreatedquantitativelyinthatway.Thebroadening,shiftandquenchingoftheRabioscillationaresimplyconsequencesofimpactsandmaybedescribedwithintheframeworkofanoscillatorsubjecttowhitenoisefromtheenvironment.Thereisnoevidenceforlocalisationontospatialeigenstates.[7]
SeealsoAtomiccoherenceBlochsphereLaserpumpingOpticalpumpingRabiproblemVacuumRabioscillationNeutralparticleoscillation
References
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07/05/2015 RabicycleWikipedia,thefreeencyclopedia
http://en.wikipedia.org/wiki/Rabi_cycle 4/4
1.
EncyclopediaofLaserPhysicsandTechnologyRabioscillations,Rabifrequency,stimulatedemission(http://www.rpphotonics.com/rabi_oscillations.html)2.
Griffiths,David(2005).IntroductiontoQuantumMechanics(2nded.).p.353.3.
SourenduGupta(27August2013)."Thephysicsof2statesystems"(http://theory.tifr.res.in/~sgupta/courses/qm2013/hand5.pdf)(PDF).TataInstituteofFundamentalResearch.4.
Griffiths,David(2012).IntroductiontoQuantumMechanics(2nded.)p.191.5.
Griffiths,David(2012).IntroductiontoQuantumMechanics(2nded.)p.196ISBN97881775823076.
AShortIntroductiontoQuantumInformationandQuantumComputationbyMichelLeBellac,ISBN97805218605677.
"au:Herbauts_Iin:quantphSciRateSearch"(https://scirate.com/search?q=au:Herbauts_I+in:quantph).SciRate.
QuantumMechanicsVolume1byC.CohenTannoudji,BernardDiu,FrankLaloe,ISBN9780471164333AShortIntroductiontoQuantumInformationandQuantumComputationbyMichelLeBellac,ISBN9780521860567TheFeynmanLecturesonPhysicsVol3byRichardP.Feynman&R.B.Leighton,ISBN9788185015842ModernApproachToQuantumMechanicsbyJohnSTownsend,ISBN9788130913148
Externallinks
AJavaappletthatvisualizesRabiCyclesoftwostatesystems(laserdriven)(http://www.itp.tuberlin.de/menue/lehre/owl/quantenmechanik/zweiniveau/parameter/en/)extendedversionoftheapplet.Includeselectronphononinteraction(http://www.itp.tuberlin.de/menue/lehre/owl/quantenmechanik/elektronphononwechselwirkung/parameter/en/)
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Categories: Quantumoptics Atomicphysics
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