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).