Upload
phamtuyen
View
222
Download
5
Embed Size (px)
Citation preview
ChitraRangan
DepartmentofPhysics,UniversityofWindsorWindsor,Ontario,Canada
Windsor, Ontario Canada - The rose city
Framework• System:Rydbergwavepacket(RWP):
SuperposiConofenergyeigenstatesexhibiCngKeplerianmoConaboutanucleus(‐1/rpotenCal).DynamicalCme(driK/field‐freeevoluCon)~10ps.
• Control:Terahertzhalf‐cyclepulse(HCP):Approximatelyanimpulse.Controls(~1ps)aremuchfasterthanthedriK.
D.You,R.R.Jones,D.R.Dykaar,andP.H.Bucksbaum,OpCcsLeWers18,290(1993).
• Decoherence:Negligiblesourcesofdecoherence.CoherenceCme~6ns.CoherentevoluCon.
Outline
• IntroducContoRWPs&HCPs• Case1:ControlwithsingleHCP
– OpCmalcontrol
• IfCmepermits…Controlwithtwokicks
• Case2a:Controlcanbetreatedsemiclassically– NewmethodfordeterminingthewidthofanRWP
• Case2b:Controlcanonlybetreatedquantummechanically– Removing&inserCngcoherencesfromasubspace
3DCoulombproblem
• Sphericalcoordinates• SeparaConofvariables
• Energyeigenstates:
€
i ˙ ψ = −∇2ψ
2−
1rψ
€
ψnlm (r,θ,φ) = Rnl (r)Pl (θ)Fm (φ)
Hψnlm (r,θ,φ) = Enψnlm (r,θ,φ) =12n2
ψnlm (r,θ,φ)
~Laguerrepolynomials SphericalharmonicsYlm
Energyleveldiagram
€
En =−12n2
; ΔEn =1n3
launchstate
νh
Theenergylevelspacingsare~THzDynamicalCmescales~10ps
n=15‐30Rydbergstates
ThestatevectorisacoherentsuperposiConofhighlyexcitedstatesofaoneelectronatom–aRydbergwavepacket
Typicalstates:n~25statesofanalkaliatom
€
|ψ〉 = cn |φn 〉n∑
€
|ψ(t)〉 = cne−iEnt |φn 〉
n∑
SculpBngaRydbergwavepacket
€
|ψ〉 = cn |φn 〉n∑
T. C. Weinacht, J. Ahn, and P. H. Bucksbaum, Phys. Rev. Lett. 80, 5508 (1998).
BoundandconBnuumwavepackets
€
|ψ(t)〉 = cne−iEnt / |φn 〉
n∑
€
|ψ(t)〉 = dE∫ c(E)e−iEt / |φ(E)〉
Control:THzHalf‐cyclepulse
• UnipolarelectromagneCcfield• FWHM~0.5ps
• Shortunipolarlobe,thenlongnegaCvetail
1.0 2.0
1.0
ImpulseapproximaBon
• WhenTHCP<<Tdynamical
• HCP~‘impulse’indirecConofpolarizaCon
• AtomsdonotfeeleffectofnegaCvetail
• Cantransfermomentumtoafreeelectron
ControlequaBon
• WeassumethefirstkickQ1isalongthequanCzaConaxis(z‐axis)–2Dproblem
• Thesecondkickcaneitherbealongthe(z‐axis)–sClla2Dproblem;ormakesomeangletothez‐axis–3Dproblem
• Comparisonstoexperimentsinalkaliatoms,potenCalslightlydifferentfromCoulomb
€
i ˙ ψ ( r ,t) = −∇2ψ( r ,t)
2−
1rψ( r ,t)
+ Q1δ(t1) r ⋅ ˆ n 1 + Q2δ(t2) r ⋅ ˆ n 2( )ψ( r ,t)
Numericalapproaches• Truncatedbasisofsphericalharmonics
1. EssenCalstatesbasis–energytruncaConusefulforstudyingboundstate‐to‐stateproblems– Ifimpulseisinthez‐direcCon– NumericaldiagonalizaConofimpulseoperator
RanganandMurray,Phys.Rev.A72,053409(2005)
KicksinarbitrarydirecBons
• Perpendicularkicks,impulsedeliveredalongthearbitraryaxis
• Rotatethedesiredaxisontothez‐axis,kickalongthenewz,andthenrotatebackusingD–matrices
2. RepresentaConusingtruncatedradialgridandsphericalharmonics(r‐lbasis)
– FreeevoluConbyimplicitpropagator(for2D)
– Inversionofapenta‐diagonalmatrixforeachvalueofl
– TimesteplimitedbystrengthofkickComputa8onalPhysics,Koonineq.(7.30)
Numericalapproaches
Numericalapproaches
3.RepresentradialwavefuncConviacollocaConusing(symmetrysuited)LaguerrefuncCons– Boyd,RanganandBucksbaum,JournalofComputaConalPhysics,188,56(2003)
• PropagateusingChebychevpropagator– H.Tal‐EzerandR.Kosloff,J.Chem.Phys.81,3967(1984)
• Fast,butnotsogoodforstudyingCoulombproblem
• Energyspectrumbywindowmethod:
• Finalstatehasbothbound(discrete)andunbound(conCnuum)components
SchaferandKulander,Phys.Rev.A42,5794(1990)
Calculatespectrumoffinalstate
CoherentinteracBonsofHCPswithRWPs
Q
0.0014a.u. Information storage & retrieval
0.0023a.u. Synthesizing eigenstates
0.0046a.u. Quantum search algorithm
0.01a.u. Detecting angular momentum
Higher Stabilization, imaging, chaos, … 0.02a.u. HCP assisted recombination
Need all Q Impulsive momentum retrieval
Ionizing away from the atomic core
Atomic units: e = me = ħ = 1
Controlmechanism‐quantumImpulsive interaction
Propagator U = eiQz
• To first order, HCP couples l→l±1
• As Q increases:
<l> increases
max(Σn|<nl|Ψ>|2) goes towards higher ‘l’
〉Ψ=〉Ψ initialiQz
final |e|
Controlmechanism‐classicalImpulsive interaction HCP boosts the momentum of the electron
pf=pi+Q
Ef=pf2/2
Ef = Ei + piQ + Q2/2
HCP also provides a torque to the bound
electron increasing its angular momentum
CaseI.PerformingaquantumalgorithminaRydbergatomusinganHCP
Singlekick• Collaborators:PhilBucksbaumandgroup:JaeAhn,JoelMurray,HaidanWan,SantoshPisharody,JamesWhite
Conversion of phase information to amplitude information
−
−
−
−
=
−
+−
+−
+−
+−
N41
N41N43N41
1
111
N21N2N2N2
N2N21N2N2N2N2N21N2N2N2N2N21
/
///
////
////////////
Average Average
(before IAA) (after IAA)
Grover, Phys. Rev. Lett. 79, 325 (1998)
ψi ψT
Controltrick–knowtheatom
ImpulsiveinteracBon
€
|Ψfinal 〉 = eiQz |Ψinitial 〉
〉〈= l,n|e|'l,m iQz
Matrixelement
)Q(f 0'ml
nl
f npm
p (Q
)
Q (a.u)
Q0
nop → nop
no±1,p→nop
no±2,p→nop
ArrangewavepacketsuperposiContoobtaindesiredoutcome
BothimpulsemodelandrealisCcmodeloftheHCPgiveexcellentagreementwiththeexperiment.
BeforeHCP
t=2.1ps
(000010)
t=4.2ps
(000100)
t=4.7ps
(001000)
Phys.Rev.LeW.86,1179(2001) Phys. Rev. A, 66, 22312 (2002)
HCPcanperformaquantumsearch
1.0 2.0
1.0
Shi & Rabitz (1988, 1990), Kosloff et al (1989), …
Find the control field E(t), 0 ≤ t ≤ T
Initial state:
Target functional:
Cost functional:
Constraint: Schrödinger’s equation
Introduce Lagrange multiplier: |λ(t)〉
Maximize unconstrained functional:
€
|Ψ(t = 0)〉 =| 24 p〉+ | 25p〉− | 26p〉+ | 27p〉+ | 28p〉+ | 29p〉
2|)T(|p26| 〉Ψ〈
dt|)t(E|)t(T
0
2∫ l
.c.c0)t(|))t(E,t(H)t(| +=〉Ψ+〉Ψ•
ι
maximize
minimize penalty
parameter
))t(|))t(E,t(H)t(|)t((dtRe2dt|)t(E|)t(|)T(|p26|JT
0
T
0
22 〉Ψ+〉Ψλ〈∫−∫−〉Ψ〈=•
ιl
UseKrotov’salgorithmwithTannor’supdaterule
Rabitz (1991); Krotov (1983, 1987); Rabitz (1998); Tannor (1992)
Krotovmethod:J=terminalpart+non‐terminalpart
=G(t=0,t=T)+0∫TdtR(t)
TheopCmalE(t)maximizesbothparts.
UsingmodifiedobjecCve:
Changeinthefieldatthek+1thiteraCon:
• T≈8ps,nt=1000• WavepacketpropagaCon:split‐operatormethod
• EssenCalbasisof187states:21≤n≤31,l<17• Basisstatescalculatedbyagrid‐basedpseudopotenCalmethod
€
ΔE(t) =−ιl(t)
〈λk (t) | z |Ψk+1(t)〉
Input: Shaped wave packet (phase information)
(010000)
(001000)
(000100)
(000010)
‘Optimal’ shaped
terahertz pulse
Output: Field ionization spectrum (amplitude information)
Design a shaped broadband terahertz pulse than can optimize the performance of the search algorithm
ShapeofterahertzpulseiscalculatedbyintroducingamodifiedtargetfuncConal
|ψi>=|010000>+|001000>+|000100>+|000010>(|001000>=|24p>+|25p>‐|26p>+|27p>+|28p>+|29p>)
Target=|<25p|ψ(T)>|2+|<26p|ψ(T)>|2+|<27p|ψ(T)>|2+|<28p|ψ(T)>|2
→maximize
Each‘subspace’evolvesindependentlyundertheinfluenceofthesameterahertzfield.
Changeincontrolfieldatk+1thiteraCon:
Independentsubspacemodel
AgainuseKrotov’salgorithmwithTannor’supdaterule
UsingmodifiedobjecCve:
Changeinthefieldatthek+1thiteraCon:
GivesopCmalpulseforinversionaboutmeanalgorithmforalliniCalstateswithinasubspace
Alsocalled:state‐independentcontrol,mulC‐operatorcontrol,opCmizingaunitarytransformaCon,opCmizingaquantumoperator,W‐problem,…
Phys. Rev. A, 64, 33417 (2001)
Database: |24p〉 to |29p〉 of Cs
Guess pulse: Half-cycle pulse
Optimal pulse: Shaped THz pulse
Phys. Rev. A, 64, 33417 (2001)
With increasing database size, the amplification of the marked bit tends to a value of 4.
〉−〉++〉=〉〉+〉+〉+〉−〉+=〉Ψ p262p29p24p29p28p27p26p25p24i |)|(||||||||
A half-cycle pulse destroys the localized wave packet while leaving the extended eigenstate untouched.
localized wave
packet delocalized eigenstate
4p26p26p262 2i
2ff ≈〉Ψ〈〉Ψ〈〉−≈〉Ψ |||/|||;||
ExampleIIa.TwokicksKickstrengthsarelowsothattheproblemiscompletelybound,i.e,noionizaCon
Collaborators:
Phil Bucksbaum’s group
(when at the University of Michigan)
Joel Murray
Santosh Pisharody
Haidan Wen
Hiding and retrieving coherence from within a subspace
Delay bet. HCP 2 and reference pulse τ
measure populations and
find correlations
Ψref(0) HCP 1
t1 chosen so that correlations go away
t
Ψref(τ) HCP 2
t2-t1 varied
Finite subsystem: n=26-31, p-states |k〉 of cesium.
Measure: Correlations between state populations after time delay
Amplitude of correlation is a measure of the ‘recoverability’ of stored phase coherence.
26p
27p
τ (in ps)
Storage/hiding of phase coherence Murray et al., Phys. Rev. A 71, 023408 (2005).
Ψref(τ)
Correlations provide information regarding the phase coherence between the various eigenstates.
Amplitude of correlation is a measure of the ‘recoverability’ of stored coherence.
Delay between HCP and reference pulse τ
measure populations and
find correlations
Ψref(0) Ψkicked
Choose phase structure of wave packet when kicked at t1
t
26p 27p 28p 29p 30p
27p
28p
29p
30p
31p
τ (in ps)
Coherent process
Population ‘hid’ mainly in degenerate ‘d’-states
Delay bet. HCP 2 and reference pulse τ
measure populations and
find correlations
Ψref(0) HCP 1
t1 chosen so that correlations go away
t
Ψref(τ) HCP 2
t2-t1 varied
Use second HCP
26p 27p 28p 29p 30p
27p
28p
29p
30p
31p
τ (in ps)
‘kick-kick’ physics
Phys. Rev. A 74, 043402 (2006)
Quantum control, use degeneracy of ‘p’ & ‘d’ states
CaseIIb.Kick‐kickKickstrengthsarehighenoughthatconCnuum
statesareinvolved
Collaborator:JeffreyRau(nowdoingaPhDatU.Toronto)
• MoCvaCon:‘kick‐kick’experimentsofZiebelandJones,PhysRevA68,023410(2003)
l=1,m=0angulardistribuCon
SchemaBcRadialwavepacketcreatedbylaser
excitaCon,definesthez‐axisRadialwavepacket(ConCnuum)
Measure/calculate
• IonizedfracCon(integralofalltheposiCveenergycomponents)
• RecombinaConfracCon(1‐IonizedfracCon)a.k.a.survivalprobability
RecombinaBonaZerfirstz‐kick
Impulse
Outward
Beforekick AKerkick
IonizaCon
RecombinaCon
RecombinaBon:suppressionizaBon
RecombinaConRegion
CompleteIonizaCon
~40%survivalinrange
Q1=0.02isfixed,Q2isvaried
Kicks~iniBalmomentumSpliyng
Q1is0.02Q2=0.02
• kickstrengthneartheiniCalradialmomentum
-p0 +p0
pf = ±p0+Q = ±p0+(p0+Δ)
Z 0
Producestwopeaksinthesurvivalcurve
Summary• ImpulseoperatorcanperforminversionaboutaverageinRydbergatom:Phys.Rev.LeW.86,1179(2001)
• AugmentedopCmalcontroltoopCmizeaquantumalgorithm:Phys.Rev.A,64,33417(2001)
Kick‐kickcontrol:
• Out‐of‐subspacetrajectorycontrol–informaConhidingandretrieval,protecCngcoherence:Phys.Rev.A,74,43402(2006).
• QIPwithangularmomentumstates:Phys.Rev.A72,053409(2005);Phys.Rev.A68,53405(2003).
• HCPassistedionizaConandrecombinaCon–J.G.Rau&C.Rangan(unpublished).
AcknowledgementsResearchgroup:
Dr.TaiwangCheng(PDF)Ms.SomayehMirzaee(MSc)Mr.DanTravo(UG)
Ms.MaggieTywoniuk(UG)
Mr.PatrickRooney(PhD@UM)Mr.AminTorabi(MSc)Mr.JohnDonohue(UG)
Mr.MustafaSheikh(UG)Supported by
BiopSys: NSERC Strategic Network on Bioplasmonic Systems
Trapped‐ioncontrolwork• Finite(approximate)controllabilityoftrapped‐ionquantumstates
(evenbeyondtheLamb‐Dickelimit)IEEETrans.Aut.Control,v.55,pp.1797‐1805(2010).
• Onlyeigenstatecontrollabilityispossibleinspin‐halfcoupledtotwoharmonicoscillators(cannotuseLaw‐Eberlyschemesforgates)QuantumInformaConProcessing,v.7,pp.33‐42(2008).
• BichromaCccontrolbytruncaCngtheHilbertspace:Phys.Rev.LeW.,92,113004(2004).
• Spin‐halfcoupledtofiniteharmonicoscillatoriscontrollable;quantumtransfergraphs:J.Math.Phys.,v.46,art.no.32106(2005).
• Ifann‐qubitsystemhasasymmetricdistribuConoffield‐freeeigenenergies,thesystemcanbecontrolledbyonly2n(2n+1)elementsofthesp(2n)algebra:Phys.Rev.A,76,33401(2007).