time ordering in kicked qubits

Time ordering in kicked qubits

L. Kaplan,1 Kh. Kh. Shakov,1 A. Chalastaras,1 M. Maggio,1 A. L. Burin,2 and J. H. McGuire11Department of Physics, Tulane University, New Orleans, Louisiana 70118, USA

2Department of Chemistry, Tulane University, New Orleans, Louisiana 70118, USA(Received 23 June 2004; published 1 December 2004)

We examine time ordering effects in strongly, suddenly perturbed two-state quantum systems(kicked qubits)by comparing results with time ordering to results without time ordering. Simple analytic expressions are givenfor state occupation amplitudes and probabilities for singly and multiply kicked qubits. We investigate the limitof no time ordering, which can differ in different representations.

DOI: 10.1103/PhysRevA.70.063401 PACS number(s): 32.80.Qk, 42.50.2p, 42.65.2k

I. INTRODUCTION

There are two reasons to consider time ordering in kickedqubits. First, the behavior of a two-state quantum systeminteracting with a rapidly changing external field, i.e., a dia-batically changing qubit, may be described analytically.While such solutions were examined some 40 years ago inthe context of Landau-Zener transitions in atomic and mo-lecular reactions[1], relatively little attention has been paidto this problem in the context of more recent work usingtwo-state systems[2] ranging from quantum computing[3,4]to quantum control of atomic and molecular reactions[5] tomanipulation of matter waves[6], where this class of ana-lytic solutions may be useful. The second reason is that timeordering has been used recently[7–9] to formulate an under-standing of time correlation in multiparticle systems(or, inthe context of this paper, systems of interacting qubits). Thecentral question here is how one particle(or qubit) is con-nected with other particles(or qubits) in the time domain.This problem has previously been formulated[8] using sec-ond order perturbation theory, where observable time corre-lations between different particles arise from time orderingof weak, external interactions in atomic scattering[10]. Thekicked qubit gives us an opportunity to study the nature oftime ordering for a simple, analytically tractable system in astrongly nonperturbative regime.

Except for relatively simplee−iEjt/" phases, where theEjare simple eigenvalues of a time-independent Hamiltonian,there are only two ways in which time enters the time evo-lution of a quantum system. The first is through the explicit

time dependence of an external interactionVstd, and the sec-ond is through the constraint of time ordering imposed by thetime-dependent Schrödinger equation itself. This time order-ing, discussed below, imposes a causal-like constraint that

places operators such asHstnd¯ Hst2dHst1d in order of in-creasing time. This confining condition interrelates the influ-ence of the time parameterstn¯ t1. In second order pertur-bation theory it has been shown[10] that the time ordering

constraint has negligible effect if eitherVstd or its variationwith time is sufficiently small over the time of the experi-ment (perturbative or constant potential limit) or if the en-ergy levels of the system before perturbation are all nearlythe same(degeneracy limit). In either case, principal valuecontributions from energy fluctuations in short-lived interme-diate states vanish[11].

In this paper we formulate the problem of time ordering ina nonperturbative two-state quantum system, i.e., a qubit.After describing the basic formalism in Sec. II A, in Sec. II Bwe define the limit without time ordering and show that timeordering disappears in either the constant potential limit orthe limit of degeneracy of the two unperturbed states, where

Hst8d and Hst9d commute. We discuss the relationship be-tween time ordering and the adiabatic approximation. Ana-lytic solutions for singly and multiply kicked qubits are pre-sented in Sec. II C, with and without time ordering. In thecase of a single kick, we show how time ordering affects thetransition probability from one state to another. In the subse-quently discussed case of a double kick, any transition is dueentirely to time ordering effects. We discuss corrections forpulses of finite duration and in Sec. III provide calculationsillustrating our results. We present most calculations andsome key formulas in both the Schrödinger and intermediate(or interaction) pictures, and discuss some differences be-tween time ordering effects in the two pictures.

II. THEORY

A. Basic formulation

Consider a two-state system, whose states are coupled bya time-dependent external interaction, e.g., a qubit with “on”and “off” states. The time-dependent Hamiltonian for thissystem may be expressed as

Hstd = H0 + Vstd = F− DE/2 0

0 DE/2G + F 0 Vstd

Vstd 0G

= −DE

2sz + Vstdsx, s1d

whereDE=E2−E1 is the energy difference of the eigenstates

of H0. Heresx andsz are the usual Pauli spin matrices.Two simplifying, but removable, assumptions have been

made in the second equality. First, we assume that all of the

time dependence in the interaction operatorVstd is containedin a single real function oft, which is often justifiable onexperimental grounds[12–14]. Second, in this paper we as-sume for convenience that the interaction does not contain a

term proportional toH0. Obviously, an interaction operatorV

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having the form of a combination ofsx andsy (i.e., a com-plex time-dependent field) is equivalent to the above formafter rotation of coordinates. As we shall discuss later, there

are other choices of how to separateH into H0+V, and thesechoices have consequences.

For a qubit with “on” and “off” statesf 10g and f 0

1g, the

probability amplitudes then evolve according to

i"d

dtFa1

a2G = F− DE/2 Vstd

Vstd DE/2GFa1std

a2std G . s2d

The solution to Eq.(2) may be written in terms of the time

evolution matrixUstd as

Fa1stda2std G = UstdFa1s0d

a2s0d G = FU11std U12stdU21std U22std

GFa1s0da2s0d G ,

s3d

where an experiment is begun at a timet=0 and completed att=Tf. Since we assume the two-state system is closed,P1std+P2std= ua1stdu2+ ua2stdu2=1.

The time evolution operatorUstd may be expressed hereas

Ustd = T expS−i

"E

0

t

Hst8ddt8D= T expF−

i

"E

0

t S−DE

2sz + Vst8dsxDdt8G

= Ton=0

`s− i/"dn

n!E

0

t

Hstnddtn ¯ E0

t

Hst2ddt2E0

t

Hst1ddt1.

s4d

The only nontrivial time dependence inUstd arises from

time-dependentHstd and time orderingT. The Dyson time

ordering operatorT specifies thatHstidHstjd is properly or-dered:

THstidHstjd = HstidHstjd + ustj − tidfHstjd,Hstidg.

Time ordering imposes a connection between the effects of

Hstid andHstjd and leads to observable, nonlocal, time order-

ing effects[12–14] when fHstjd ,HstidgÞ0.

Pulses

In this paper we regardVstd as having the form of asmoothly varying pulse, or sequence of pulses, each of du-ration t and peaked atTk. We define phase angles

a =E0

Tf

Vst8ddt8/",

b = tDE/2",

vt = tDE/2". s5d

The anglea is a measure of the strength of the interactionVstd over the duration of a given pulse. In this paper we aremostly interested in the nonperturbative regime correspond-ing to aù1, so that substantial changes in the state occupa-tion probabilitiesP1 and P2 may occur. The angleb is a

measure of the influence ofH0 during the interaction intervalt. The anglevt is the phase accumulation of the propagation

due to H0 over a timet. The diabatic(kicked) limit corre-sponds tob!1, the perturbative limit corresponds toa!1,and the adiabatic(slow) limit generally corresponds tot→`.

B. Time ordering

Since time ordering effects can be defined as the differ-ence between a result with time ordering and the correspond-ing result in the limit of no time ordering, it is useful tospecify carefully the limit without time ordering. Removingtime ordering corresponds to replacingT→1 in Eq.(4). Thiscorresponds to the zeroth order term in an eikonal-like, Mag-nus expansion in commutator terms[15]. In the limit of notime ordering, a multiparticle time evolution operator factor-izes into a product of single particle evolution operators[8].

1. Limit of no time ordering

ReplacingT with 1 in Eq. (4), in the Schrödinger picturewe have

Ustd = T expS−i

"E

0

t

Hst8ddt8D→ o

n=0

`s− i/"dn

n! FE0

t

Hst8ddt8Gn

= on=0

`s− i/"dn

n! FH0t +E0

t

Vst8ddt8Gn

= on=0

`s− i/"dn

n!fsH0 + V

ˆ dtgn = e−iHˆt/" = U0std, s6d

where

Vˆt =E

0

t

Vst8ddt8 =E0

t

Vst8ddt8sx = a"sx,

Hˆ

=H0+Vˆ, and fH0,V

ˆ g terms are nonzero. By expanding in

powers offHst9d ,Hst8dg, it is straightforward to show that to

leading order inV andH0 the time ordering effect is given by

U − U0 . −1

2"2E0

t

dt9E0

t9dt8fHst9d,Hst8dg

= −1

2"2fH0,V0gE0

t

dt8st − 2t8dfst8d, s7d

where Vst8d=V0fst8d. This leading term disappears if thepulse centroidTk= t /2 and fst8d is symmetric aboutTk. Fur-

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thermore,U−U0 vanishes identically in the special cases of

Vst8d=0, Vst8d=V, or DE=0, as will be discussed below inSec. II B 2[16].

In general there is no simple analytic form for the exact

result Ustd. For the result without time ordering, we have

U0std = e+ivtsz−iasx = 3cosj + ivtsin j

j− ia

sin j

j

− iasin j

jcosj − ivt

sin j

j4 ,

s8d

where j=Îa2+svtd2. Here we have used the well knownidentity [17] eifsW ·n=1 cosf+ isW ·nsin f, following fromssW ·ndn=1 (or sW ·n) if n is even(or odd).

Similarly, in the intermediate, or interaction, picture,

UIstd=eiH0tUstd, and one has

UIstd = T expS−i

"E

0

t

VIst8ddt8D → expS−i

"E

0

t

VIst8ddt8D= o

n=0

`s− i/"dn

n! FE0

t

VIst8ddt8Gn

= on=0

`s− i/"dn

n!fVˆ Itgn

= UI0std, s9d

whereVIst8d=eiH0t8/"Vst8de−iH0t8/" andVˆ

It=e0t VIst8ddt8. For a

Gaussian pulse of the form discussed in Sec. III,

VIt = a"e−b2fsx cos 2vTk + sy sin 2vTkg s10d

and

UI0std = F cossae−b2

d − i sinsae−b2de−2ivTk

− i sinsae−b2de2ivTk cossae−b2

dG

s11d

as long as the measurement timet is after the completion ofthe pulse, i.e.,t−Tk@t.

It has been shown previously[10] that to second order in

perturbation theory,UI −UI0,fVIst9d ,VIst8dg, somewhat simi-

lar to the commutator in the Schrödinger picture above.From this we immediately see that time ordering effects donot appear until second order in a perturbative expansion in

a. Again, UI −UI0→0 in the special limitsVI →0, VI → V

ˆI, or

DE→0, to be discussed immediately below. However, we

will also find in Sec. III B 1 thatUI −UI0 vanishes in the

diabatic limit of a single ideal kick,Vstd,dst−Tkd, whereas

U−U0 is nonzero. Thus, in principle, the definition of thelimit of no time ordering depends on the picture(representa-tion) used. In the discussion we shall relate this difference to

the gauge choice of how one separatesH into H0+Vstd. Asshown in calculations presented below, this difference can benegligibly small under some conditions.

In both pictures, time ordering effects are associated withthe fluctuation of a time-dependent interaction about its timeaveraged value.

2. Relation to other limiting cases

Now we compare the limit without time ordering with thedegenerate, weakly varying potential, and adiabatic limits.The connection with the diabatic(kicked) limit appears in

Sec. II C, where we discuss the analytic solution forU in thecase of a short pulse.

We noted above that for a general pulse, there is no ana-

lytic solution for Ustd. However, in the limit when the un-perturbed states become nearly degenerate, i.e.,DE!" /Tf,we obtain

Ustd → UDstd = expS− iE0

t

Vst8ddt8/"D = e−iasx

= F cosa − i sin a

− i sin a cosaG . s12d

This result may be obtained[18] either from the coupledstate equations of Eq.(2), or from Eq.(4) with DE→0. Inthis degenerate limit the mathematical complexity of the qu-bit simplifies significantly, as may be seen by comparing Eq.(12) with Eqs.(15) and(17). Most of the complex time con-nections have been removed. We call this degenerate qubit adit. In such a dit if the phase anglea equalsp /2, then an“on” state is turned off and an “off” state is turned on withprobability 1. Then the dit is further reduced in complexity toa trivial classical bit.

We notice that Eq.(12) may also be obtained from Eq.(8)by taking v→0. Thus, time ordering effects vanish in thedegenerate limit.

A second situation in which time ordering effects generi-cally become small is the case of a constant or weakly vary-

ing potential uV−Vˆ u!" /Tf. Then the full evolution matrix

approaches the non-time-ordered expression given by Eq.(8)or Eq. (9). Clearly the perturbative limita!1 is a special

case of this, butU−U0 vanishes also for a large average

external potentialVˆ, as long as fluctuations aroundV

ˆare

small. Such a situation may sometimes be addressed more

transparently by absorbing the average part ofV into H0. In

the extreme caseV=0, Eq.(8) reduces to

U0std → e−iHˆ

0t/" = eivtsz = Feivt 0

0 e−ivtG , s13d

which of course agrees with the exact evolution matrixUstd.In summary, time ordering effects disappear either when

(i) a!1 or when (ii ) b!1 svTf !1d in the intermediate(Schrödinger) picture. The physics becomes especiallysimple in the overlap of regimes(i) and(ii ). There, one eas-ily finds that in the Schrödinger picture, for example,

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Ustd . U0std . F1 + ivt − ia

− ia 1 − ivtG . s14d

It can be shown that corrections and time ordering effectsstart atOsav2t2d and Osa2vtd for a symmetric pulse cen-tered atTk= t /2 [see Eq.(23)]. The situation is similar in theintermediate picture, except that time ordering effects vanishidentically at leading order ina, and begin atOsa2vtd only.

Strictly speaking, time ordering effects also vanish if(iii )fH0,Vg=0; however, this situation is of little practical inter-

est due to the fact that no transition or population transfer ispossible.

In addition to the degenerate, perturbative, and diabaticregimes, a fourth limit exists in which analytic solutions for

Ustd are generally available. In the adiabatic limit where theexternal interactionVstd changes slowly in time[19], it isuseful to define the instantaneous level splittingVstd=ÎsDEd2+4V2std and the accumulated phaseu=e0

t Vst8ddt8 /2". Then

Ustd → UAstd = F cosu cosf− + i sin u cosf+ cosu sin f− − i sin u sin f+

− cosu sin f− − i sin u sin f+ cosu cosf− − i sin u cosf+G . s15d

Heref±=ffstd±fs0dg /2 wherefst8d=tan−1f2Vst8d /DEg. Inthe special case whenVstd=Vs0d, relevant for a pulse,

UAstd

= 3cosustd + iDE

Vstdsin ustd − 2i

VstdVstd

sin ustd

− 2iVstdVstd

sin ustd cosustd − iDE

Vstdsin ustd 4 .

s16d

We note that this solution is similar mathematically to therotating wave approximation, which is widely used to de-scribe atomic transitions using external fields tuned to fre-quencies near the resonant transition frequency between twostates[19–21]. Inserting Eq.(15) into Eq. (2), one finds that

the leading correction is small when"Vst8dDE!V3st8d.When the splittingDE of the unperturbed qubit is not small,i.e., DEùV, then this adiabatic validity condition reduces to

the Landau-Zener criterion[22], namely, "Vst8d! sDEd2.The leading correction to Eq.(15) or Eq. (16) is then givenby nonadiabatic transitions at the avoided level crossings,whereVst8d and thus the level splittingVst8d goes through aminimum. More generally, for a pulse having a smoothshape, such as the Gaussian pulses discussed in Sec. III, the

criterion "Vst8dDE!V3st8d reduces to the union ofa!b2

andb!a2.Remarkably, the adiabatic regime overlaps both with the

degenerate limit(when b ,vTf →0 at fixeda) and with theperturbative limit (when a→0 at fixed b ,vTf). Thus, Eq.(15) reduces to either Eq.(12) or Eq. (13) as DE→0 orV→0, respectively. In these overlap regions, time orderingeffects are small. More generally, however, time orderingeffects in the purely adiabatic regime(a@1 and bù1 orb@1 and aù1) are large. Time ordering effects are alsolarge whena andb are both of order unity, where no simple

analytic solutions exist for the full evolution matrixUstd.

C. Kicked qubits

Now we consider time ordering in kicked qubits, i.e., thediabatic limit whereVstd,dst−Tkd. First, we present ana-lytic expressions[23,24] for a kicked qubit, i.e., a two-statesystem subject to an external interactionVstd that changesrapidly with respect toTDE=p /v=2p" /DE, the period ofoscillation of the free system. The corrections needed forfinite pulses will be briefly analyzed. We also discuss theextension to multiple kicks, using the double kick as an ex-ample. Finally, we consider the influence of time ordering inthese kicked systems.

1. Single kick

Here we consider a two-state system where the interactionVstd may be expressed as a sudden “kick” att=Tk, namely,Vstd=a"dst−Tkd. For such a kick the integration over time istrivial and the time evolution matrix in Eq.(4) becomes

UKstd = expSiDE

2st − Tkdsz/"D

3expS− iETk−e

Tk+e

Vst8ddt8sx/"DexpSiDE

2Tksz/"D

= Feivst−Tkd 0

0 e−ivst−Tkd GF cosa − i sin a

− i sin a cosaG

3FeivTk 0

0 e−ivTkG

= F eivtcosa − ieivst−2Tkdsin a

− ie−ivst−2Tkdsin a e−ivtcosaG s17d

for t.Tk. The second equality follows from the identitygiven below Eq.(8) above. As explained below, this solution

is valid when b!1 so that there is little effect fromH0

during the short time whenV is active. In the intermediate

picture,UI at any time after the kick is given by substituting

t=0 into Eq.(17), i.e., UI is independent oft.


063401-4

From Eqs.(3) and (17) we have for a kicked qubit ini-tially found in state 1

P1std = ua1stdu2 = uU11K stdu2 = cos2 a,

P2std = ua2stdu2 = uU12K stdu2 = sin2 a. s18d

The corresponding probabilities for a kicked qubit withouttime ordering are discussed below in Sec. II C 4.

2. Finite pulse corrections for a single pulse

When the pulse width is finite the corrections to Eq.(17)are Osbd and corrections to Eq.(18) are Osb2d. These cor-rections result from the commutator of the free Hamiltonian

H0 with the interactionV during the timet when the pulse isactive. This is related to the series expansion that arises inthe split operator method[25]. For example, in the case of arectangular pulse of widtht, the exact time evolution isgiven by

Urectangularstd = 3eivt−ibScosa8 + ibsin a8

a8D − ieivst−2Tkda

sin a8

a8

− ie−ivst−2Tkdasin a8

a8e−ivt+ibScosa8 − ib

sin a8

a8D 4 , s19d

wherea8=Îa2+b2. To leading order inb, i.e., in the widthof the pulse, the error in the kicked approximation is givenby

dUstd = Urectangularstd − UKstd = ibSsin a

a− cosaD

3Feivt 0

0 − e−ivtG . s20d

For a narrow pulse having a generic symmetric shape, theleading correction to the kicked approximation will still havethe form

dUstd = Ustd − UKstd = ibgsadFeivt 0

0 − e−ivtG , s21d

wheregsad is now a function that depends on the shape ofthe pulse. By comparing Eqs.(4) and(17) at leading order inDE, after some algebra one obtains

gsad =2

tE dt Fcos2SE

Tk

t

Vst8ddt8/"D − cos2sa/2dG .

s22d

ExpandingUstd of Eq. (4) and UKstd of Eq. (17) simulta-neously inDE andV, or equivalently inb anda, we find

dUstd = Ustd − UKstd =iDE

"E dt FSa

2D2

− SETk

t

Vst8ddt8/"D2GFeivt 0

0 − e−ivtG+

isDEd2

2E Vst8dst8 − Tkd2 dt8/"3

3F 0 eivst−2Tkd

e−ivst−2Tkd 0G , s23d

so the two leading correction terms scale asba2 andb2a.

3. Multiple kicks

A series of either identical or nonidentical pulses can eas-ily be handled by multiplication of several matrices of theform of Eq.(17). For example, one may consider a sequenceof two kicks of opposite sign at timest=T1 and t=T2,namely, Vkick antikickstd=a"dst−T1d−a"dst−T2d. Followingthe procedure given in Eq.(17) one obtains the time evolu-tion matrix for t.T2,

Ukick antikickstd = eivst−T2dszeiasxeivsT2−T1dsze−iasxeivT1sz =FeizscosvTs + i sin vTs cos 2ad eivst−2Tdsin vTs sin 2a

− e−ivst−2Tdsin vTs sin 2a e−izscosvTs − i sin vTs cos 2adG ,

s24d


063401-5

where z=vst−Tsd, T=sT1+T2d /2, and Ts=T2−T1.

As vTs→0, Ukick antikickstd reduces to Eq.(13). Again, UI isobtained by substitutingt=0 into Eq.(24).

For a double kick withV=0, we have from Eqs.(3) and(24)

P1std = ua1stdu2 = uU11kick antikickstdu2

= cos2 vTs + sin2 vTs cos2 2a,

P2std = ua2stdu2 = uU12kick antikickstdu2 = sin2 vTs sin2 2a.

s25d

The single kick result of Eq.(17) remains valid for two ormore kicks of combined strengtha1+¯ +an=a if the totalphase associated with the interkick free evolutionvsTn

−T1d is small. A mathematical analysis for multiple kicksseparated by arbitrary time intervals is straightforward, butnot included here. In the case of a periodic series of pulseswith periodT, the time evolution may be obtained by diago-nalizing the matrix of Eq.(17) and finding the Floquet eigen-states and eigenphases. The two Floquet eigenphases arethen given bye±ix, wherex=cos−1fcosa cosvTg.

4. Time ordering for single and multiple kicks or pulses

We first consider the case of a single kick or pulse. In theSchrödinger picture, time ordering effects are present evenfor a single ideal kick, specifically the time ordering betweenthe interaction and the free evolution preceding and follow-ing the kick. Thus, in the absence of time ordering, the time

evolution U0std is given by Eq.(8), which differs from the

exact expressionUKstd of Eq. (17) whena andvt are both

nonzero. The time ordering effectUKstd−U0std vanishes ineither the degenerate limitvt→0 or the perturbative limita→0. For smalla and vt and assumingTk= t /2, the timeordering effect at leading order takes the form of a sum ofO(asvtd2) andOsa2vtd terms. For a Gaussian-shaped pulse,

the transition probability in the Schrödinger picture withouttime ordering is given by the second equality of Eq.(29).

In the intermediate picture, time evolution without timeordering for an ideal kick is obtained by substitutingb=0into Eq. (11), and agrees perfectly with the exact expressionof Eq. (17), when the latter is transformed into the interme-diate picture. Thus, time ordering effects disappear for asingle ideal kick in the intermediate picture, in contrast withthe Schrödinger case. This is easily understood by consider-ing that in the intermediate picture, time ordering is only

between interactions at different times,VIst8d andVIst9d, not

between the interactionVst8d and the free Hamiltonian

H0st9d, as in the Schrödinger case. For a single ideal kick, allthe interaction occurs at one instant, and no ordering isneeded. Of course, for a finite-width pulse, i.e.,bÞ0, timeordering effects do begin to appear even in the intermediate

picture. To leading order,UIKstd−UI

0std=Osa2bd. We notethat the time ordering effect in the intermediate picture isindependent of the measurement timet, though it does de-pend on the pulse widtht through theb parameter. For aGaussian-shaped pulse, the transition probability in the inter-mediate picture without time ordering is given by the thirdline of Eq. (29).

We are now ready to examine the time ordering effect fora multipulse sequence, focusing on the pulse-antipulse sce-nario of Sec. II C 3. In the limit of no time ordering one hasP2

0std=0 in the Schrödinger picture as seen from Eqs.(3) and

(13) with V=0. In the intermediate picture, however, thetransition probability is nonzero even without time ordering.For a Gaussian pulse-antipulse sequence, one may show that

Vˆ

It=2a"e−b2sin vTsfsx sin2vT−sy cos2vTg, where

T=sT1+T2d /2. Note that fort−T2@t, VIt depends onT1 and

T2 but not t. Then, e−iVˆ

It/"=cosf2ae−b2sin vTs/2g

− ifsx sin2vT−sy cos2vTgsinf2ae−b2sin vTsg.

Consequently,

UI0std =F cosf2ae−b2

sin vTsg e−2ivT sinf2ae−b2sin vTsg

− e2ivT sinf2ae−b2sin vTsg cosf2ae−b2

sin vTsgG . s26d

We use this result in the next section to study the effect oftime ordering on the transfer of population from one state toanother.

We note that as eitherv→0 or a→0, one hasUI0std

→ f 1 00 1

g in contrast toU0std→f eivt 00 e−ivtg. For simplicity, we may

consider the case where each pulse is an ideal kick, i.e.,b=0. Then in the perturbative regime, expanding ina, onefinds UI 11

0 <1–2a2 sin2 vTs, which can be compared toe−ivtU11

kick antikick<1–2ia2e−ivTs sin vTs, so that except forspecial values ofv (including 0), these matrix elements dif-

fer by Osa2d. Similarly, UI 120 <2a sin vTse

−2ivT+Osa3dcompared to e−ivtU12

kick antikick<2a sin vTse−2ivT+Osa3d.

HenceUI andUI0 agree to leading order ina. This should not

be surprising, as time ordering has no effect at leading orderin perturbation theory.

III. CALCULATIONS

As an illustrative specific example we present in this sec-tion the results of numerical calculations for 2s→2p transi-


063401-6

tions in atomic hydrogen caused by a Gaussian pulse ofwidth t. The occupation probabilities of the 2s and 2p statesare evaluated by integrating two-state equations using a stan-dard fourth order Runge-Kutta method. This enables us toverify the validity of our analytic solutions for kicked qubitsin the limit t→0 and also to consider the effects of usingfinite-width pulses. In this system, the unperturbed levelsplitting is the Lamb shiftDE=E2p−E2s=4.37310−6 eV.The corresponding time scale is the Rabi timeTDE=2p" /DE=972310−12 s, which gives the period of oscilla-tion between the states.

For any practical system, the pulse durationt can neitherbe too large nor too small. Ift is larger thanTDE, then thepulse will not be sudden and the kicked approximation willfail. On the other hand, ift is too small, then the interactionwill have frequency components that couple the initial stateto other levels. Specifically, ift is less than 2p" / sE3p

−E2sd<10−15 s, then the interaction will induce transitionsinto the 3p level and the system will not be well approxi-mated by a two-state system. Also there is another constraintin our case. If the experiment lasts longer than the lifetime ofthe 2p state, 1.6310−9 s, then we lose population from ourtwo-state system, i.e., dissipation cannot be neglected. Simi-lar calculations can be done in many other applications, in-cluding, for example, Josephson junctions[4].

In the first part of this section we present results for thetarget state occupation probabilityP2 as a function of time.We shall examine how well the approximations we use aresatisfied for a 2s-2p transition caused by a pulse of finitewidth. We shall do this first for a single pulse and then for adouble pulse. In the second part of this section we examineeffects of time ordering. Here we shall evaluateP2std bothwith and without time ordering for pulses of finite width.This will be done in both the Schrödinger and intermediatepictures.

A. Pulsed two-state system

In our numerical calculations we use for convenience aninteraction of the formVstd=sa" /Îptde−st−Tkd2/t2

, i.e., aGaussian pulse centered atTk with width t. The evaluation ofthe integrated pulse strengtha in terms of the dipole matrixelement for the 2s-2p transition is discussed in a previouspaper[18]. Whent is small enough for the sudden, kickedapproximation to hold,Vstd→a"dst−TKd, and the analyticexpressions of Eqs.(17) and(24) apply. Here we shall deter-mine how the occupation probabilityP2std depends on thepulse widtht, to find where the kicked results are approxi-mately valid for finite pulses. We do this first for a singlepulse chosen so that an ideal kick would transfer the occu-pation probabilityP2std suddenly from zero to one att=Tk.Then we consider two equal and opposite pulses occurring attimes T1 and T2. We study this doubly kicked system as afunction of both pulse widtht and separation intervalTs=T2−T1.

1. Single pulse

In Fig. 1 we show results of a calculation for the prob-ability P2std that a hydrogen atom initially in the 2s state

makes a transition into the 2p state when strongly perturbedby a single Gaussian pulse applied att=Tk. We have ob-tained our results by numerically integrating the two-statecoupled equations,

i"a1 = −1

2DEa1 +

a

Îpte−st − Tkd2/t2

a2,

i"a2 =1

2DEa2 +

a

Îpte−st − Tkd2/t2

a1. s27d

Here the pulse is applied atTk=150 ps and we have chosena=p /2 so that in the limit of a perfect kick all of the popu-lation will be transferred from the 2s to the 2p state aftert=Tk.

In Fig. 1 one sees that the ideal kick results are verynearly achieved by choosingt to be a factor of 10−3 timessmaller than the Rabi timeTDE in which the population os-cillates between the 2s and 2p states. Whent /TDE<10−2, asmall deviation from an ideal kick can be seen in the figure.In this caseP2sTfd=0.9977. Whent /TDE<10−1, the transi-tion takes a few tenths of a nanosecond to occur and only82% of the population is transferred at 300 ps. The error inP2sTfd resulting from the kicked approximation grows asst /TDEd2,b2, as expected from Eq.(23).

2. A positive followed by a negative pulse

In Figs. 2 and 3 we show results of a calculation for theprobabilityP2std that a hydrogen atom initially in the 2s statemakes a transition into the 2p state when acted on by adouble Gaussian pulse. Two Gaussian-form pulses are ap-plied, the first att=T1 and the second att=T2. The separationinterval between pulses isTs=T2−T1. The final occupationprobability of the target state is measured att=Tf. The pulsesare opposite in sign, but otherwise identical, so the interac-

FIG. 1. Occupation probability of the target state as a functionof time for a qubit interacting with a single pulse. The heavy solidline corresponds tot=1 ps (almost an ideal kick), the thin dashedline to t=10 ps(where small deviations from an ideal kick occur),and the thin dotted line tot=100 ps(where the kicked approxima-tion is breaking down). The Rabi time for the oscillation betweenthe states is 972 ps.


063401-7

tion integrated over the whole intervalf0,Tfg is zero.Our results for this double pulse have been obtained by

numerically integrating

i"a1 = −1

2DEa1 + sa/Îptdfe−fst − T1d/tg2 − e−fst − T2d/tg2ga2,

i"a2 =1

2DEa2 + sa/Îptdfe−fst − T1d/tg2 − e−fst − T2d/tg2ga1.

s28d

In Fig. 2, the first pulse is applied atT1=100 ps and thesecond atT2=586 ps, giving a separation timeTs=486 ps=p /2v. From Eqs.(5) and(25) one sees that this is preciselythe value ofvTs required to yield complete transfer from the2s to the 2p state atT1 and then full transfer back to the 2sstate at timeT2 in the limit of an ideal double kick. More-over, we have chosen an integrated pulse strengtha=p /2 foreach pulse for the same reason. The parameters for Fig. 3 arethe same except thata=p /4, so that the 2p target state isfully populated afterT2 for an ideally kicked system.

As in Fig. 1 we see that whent /TDE,10−3, the kickedlimit is well satisfied, and whent /TDE<10−2, small devia-tions from an ideal kick can be seen in Fig. 2. In this caseP2sTfd=1.1310−5. Again deviations from the ideal kicklimit in the transfer probabilityP2 grow asst /TDEd2.

The results in Fig. 3 are for a double pulse that first takesthe population halfway from 2s to 2p, and then the rest of theway for an ideal kick-antikick sequence. Whent /TDE<10−2, small deviations from an ideal kick can be observedin the figure near both steps. At 700 ps,P2sTfd=0.999 34,i.e., the population is nearly, but not quite perfectly, trans-ferred to the target state. Whent /TDE<10−1, the transitiontakes a few tenths of a nanosecond to occur and only 80% ofthe population is transferred, i.e., the transfer is not ideal.

B. Time ordering

In this subsection we consider the more complex issue oftime ordering in 2s-2p transitions in atomic hydrogen causedby a single or double Gaussian pulse. Finite-width pulse ef-fects are again considered. The effect of time ordering isevaluated by comparing results of calculations with andwithout time ordering for the probabilityP2 of transferringan electron population from the 2s launch state to the 2ptarget state. Since the limit without time ordering is differentin the Schrödinger and intermediate pictures, we include re-sults for both pictures.

The equations including time ordering are given by Eqs.(27) and (28) above. The analogous equations without time

ordering are found by takingVstd→ V in the Schrödinger

picture andVIstd→ VI in the intermediate picture.

1. Single pulse

In Fig. 4 we show the effects of time ordering for a singlepulse. We have chosen our parameters so that in the limit ofan ideal kick the population is completely transferred fromthe 2s to the 2p state of hydrogen at timeTk, as describedabove. On the left hand side of Fig. 4 we show how theoccupation probability of the 2p target state varies as a func-tion of the Gaussian pulse widtht for three different valuesof the observation timeTf. For sharp pulses the exact transferprobability P2sTfd and the transfer probability without timeordering in the intermediate picturePI 2

0 sTfd are quite similar,but differences appear, as expected, whent /TDE=b /p be-comes large.

However in the Schrödinger picture there are very largedifferences between the results with and without time order-ing, P2sTfd and P2

0sTfd, even for an ideal kick. This occurs

FIG. 2. Occupation probability of the target state as a functionof time for a double pulse that returns the system to its initial statein the kicked limit. The heavy solid line corresponds tot=1 ps(almost ideal kicks), the thin dashed line tot=10 ps(where small deviations from ideal kicks occur), and thethin dotted line tot=100 ps(where the kicked approximation isbreaking down).

FIG. 3. Occupation probability of the target state as a functionof time for a double pulse that fully transfers population in thekicked limit. The heavy solid line corresponds tot=1 ps (almostideal kicks), the thin dashed line tot=10 ps (where small devia-tions from ideal kicks occur), and the thin dotted line tot=100 ps(where the kicked approximation is breaking down).


063401-8

because the energy splittingDE is nonzero, and for

Tf .a" /DE=aTDE/2p, the average potentialV=a /Tf be-comes smaller than the energy splittingDE. Thus, for a givenpulse, the influence of the potential necessarily decreases atlargeTf, and any transfer probability becomes exponentially

small. In effect, the free propagation before and after thepulse diminishes the effect of the pulse itself in theSchrödinger picture, when time ordering is removed. Thisbehavior contrasts with the intermediate picture result[Eq.(11)], where PI 2

0 sTfd depends onb=vt but not onTf, as

FIG. 4. Target state probability as a function of the pulse widtht (on the left), and as a function of the observation timeTf (on the right).Here TDE=2p" /DE=p /v is the Rabi time for oscillations between the states, whereDE=E2p−E2s. The heavy line denotes probabilityincluding time ordering, the dashed line denotes the probability in the intermediate picture without time ordering, and the dotted linerepresents the probability in the Schrödinger picture without time ordering. On the right, the lines begin at the midpoint of the pulse,Tf

=Tk. The Schrödinger results damp out for largeTf on the right as explained in the text.


063401-9

seen also on the left side of Fig. 4. The contrast is evident onthe right hand side of Fig. 4 where, after the pulse has diedoff, the value of P2

0sTfd dies out asTf increases, whilePI 2

0 sTfd approaches a constant.For a single narrow pulse one may compare the time or-

dered result for the transfer probability using the kicked ap-proximation[Eqs.(17) and (23)] with the exact expressionsin the absence of time ordering in the Schrödinger and inter-mediate pictures, given by Eqs.(8) and (11),

P2sTfd = sin2 a + Osa2b2d,

P20sTfd =

a2

a2 + svTfd2sin2Îa2 + svTfd2,

PI 20 sTfd = sin2sae−b2

d. s29d

These three equations are consistent with the numerical re-sults shown in Fig. 4. AsDE→0, vTf andb become small,time ordering effects disappear, and all three results coincideat sin2 a.

2. A positive followed by a negative pulse

Finally, we consider the role of time ordering in the caseof two equal and opposite pulses separated by an intervalTs.In the limit of ideal kicks, the kick-antikick evolution opera-tor has been expressed analytically above with and withouttime ordering in Eqs.(24) and(26). This yields for the prob-ability P2sTfd at timesTf after the second kick

P2sTfd = sin2 vTs sin2 2a + Osb2d,

PI 20 sTfd = sin2f2ae−b2

sin2svTsdg,

P20sTfd = 0. s30d

The existence of these analytic results is helpful in studyingthe role of time ordering inP2.

In Fig. 5 we compareP2 with PI 20 as a function of the

separation timeTs between the pulses. Here the measurementtime Tf is taken to be well after the second pulse has de-cayed,Tf −T2@t. We note that, as expected, in all cases theoccupation probability goes to zero as the two oppositepulses coalesce, i.e., asTs→0. On the left side we show theoccupation probability for a Gaussian pulse of widtht=10 ps for three values of the pulse strengtha, namely,a=p /2 corresponding to a kick that turns a qubit from off toon atT1 and back to off atT2, a=p /4 where a qubit is onafter T2, and an intermediate strengtha=3p /8. On the leftsidet /TDE=10−2, so that the kicked result is accurately ob-tained. We note thatPI 2

0 < P2, and time ordering effects van-ish, whenTs/TDE=vTs/p is an integer or half integer, con-sistent with Eq.(30). Away from these special values, largetime ordering effects are present. On the right side, the pulses

are quite broadst /TDE=10−1d, and the kicked approximationis clearly breaking down.

IV. DISCUSSION

In this paper we have studied the role of time ordering ina strongly perturbed two-state quantum system. We have de-fined the time ordering effect as the difference between acalculation with time ordering and one without time order-ing. This is the way correlation, entanglement, and nonran-dom processes are also defined[26]. In all of these cases it isuseful to define carefully the limit without the effect(timeordering, correlation, or nonrandomness). In the case of timeordering we have seen that the limit without time correlation

depends on how one separatesH into H0+V, i.e., what wecall a choice of gauge. In practice this choice sometimesrests on the choice of the time averaged interaction. Preciselythis same issue occurs in defining correlation[27], namely,the somewhat arbitrary choice of a mean field interaction. Asimilar problem can arise in defining entanglement[28] andnonrandom processes. In any case the problem is not new.

For clarity and simplicity we have chosenH0=const

3sz and V= fstdV0sx in this paper. However, other gaugechoices are possible and in some cases may be more sen-sible. Specifically, in many cases experimental conditionscan lead to a sensible gauge choice where, for example, theasymptotic state of an unperturbed atom is an eigenstate of

H0, and V corresponds to an external electric or magneticfield imposed on the atom during part of the experiment.

Such aV may sensibly containsz and/orsy components.We have shown above that the limit of no time ordering

depends on the picture(representation) used. While this dif-ference is small when time ordering effects are small, thedifferences can be large otherwise. Hence it might be arguedthat our time ordering analysis is useful primarily to deter-mine if time ordering effects are small or large. This argu-ment sometimes occurs in the use of correlation(although itis not common to use different representations for correla-tion). In any case the dependence on representation is notnew [29]. Convergence properties of the Magnus expansionin the Schrödinger and interaction pictures have been knownfor some time to differ widely[30]. While the Schrödingerpicture result(that in a sense corresponds to an especially

simple gauge choice ofH0=0) is formally easier to writedown, it often does not yield predictions as reliable as themore complete results found in the intermediate picture. Wehave illustrated this above with calculations and analysis of

U0 and UI0 for single and multiple pulses. The intermediate

picture takes maximum advantage of knowledge about the

eigenstates and spectrum ofH0. In other words, the interme-diate picture is generally more complete than theSchrödinger picture and often more sensible. It is useful to

separateH into H0+V in such a way as to include as much of

the problem as possible inH0, whose solutions are known. Inthe extreme limit of the Heisenberg picture(which can be

thought of as a gauge choice whereH=H0 and V=0), we


063401-10

haveUH=UH0 =1, and there is never any time ordering in the

time evolution.Here we have worked in the time domain and formulated

the question of time ordering by explicitly working with theDyson time ordering operatorT. Equivalently one may work

in the energy domain, as has been done recently in the con-text of atomic scattering to analyze experimental data andidentify time ordering effects[10,14]. A key transformationfor time ordering from the time domain to the energy domainis the Fourier transform of the step function, namely,

FIG. 5. Target state probability as a function of the separation timeTs=T2−T1 between the two pulses, for integrated pulse strengtha=p /2, p /4, and 3p /8. On the left the pulse width is a rather narrow 10 ps, while on the right the pulse width is 100 ps, where the kickedapproximation is breaking down. Again the heavy line denotes the exact result including time ordering, while the thin dashed line denotesthe intermediate picture result without time ordering. The Schrödinger picture result without time ordering is identically zero for all valuesof Ts anda.


063401-11

eeiE8tQstde−iEtdt=1/sE−E8+ ihd=pdsE−E8d+ iPvf1/sE−E8dg, where the effect of time ordering is associated withih, which gives rise to the principal value term. Theih car-ries the effect of the boundary condition on the Green’s func-tion in energy space. This is discussed in more detail else-

where [8,10]. Since fHst9d ,Hst8dg provides a connectionbetween interactions att9 and t8, the quantum time propaga-tor includes nonlocal effects in time. An example of a coun-terintuitive time sequence occurs in stimulated Raman adia-batic passage, where efficient and robust population transferis attained using two pulsed radiation fields in a three-levelsystem[31].

The time ordering effects considered in this paper are as-sociated with a sequential ordering of interactions. The nor-mal boundary condition imposed on the evolution operator isthat the sequence proceeds in the direction of increasing time(or alternatively decreasing time to study time reversal).Hence effects of time ordering are associated with a directionof the flow of time. In this paper we have not included dis-sipation; hence all amplitudes explicitly satisfy invarianceunder time reversal, e.g., in Eqs.(17) and(24) which includeeffects due to time ordering. Effects of time ordering havebeen observed in systems that satisfy time reversal invari-ance [12–14]. This means that observable evidence of thedirection of time (time ordering) can be obtained withoutviolating the symmetry of time reversal invariance.

V. SUMMARY

In this paper we have given a definition of time orderingin a strongly perturbed quantum system, namely, that timeordering is the difference between calculations with andwithout time ordering. This definition is similar to the defi-nition of correlation. In both cases effects arise from differ-ences between an instantaneous interaction and its averagedvalue. When the effect of time ordering is small, the depen-dence on representation is weak. However, when time order-ing effects are large, the difference between representationscan also be large. We have considered in detail time orderingfor qubits that are strongly and suddenly perturbed by anexternal interaction. We have illustrated our methods for a2s-2p transition in atomic hydrogen caused by a Gaussianpulse of finite width in time. Other diabatically changingqubits may also be analyzed with our methods. Simple ana-lytic expressions have been given for the occupation ampli-tudes and probabilities for kicked qubits, including singleand multiple kicks. We think that it should be possible to findanalytic solutions for correlated kicked qubits, so that timecoupled interacting qubits may also be studied analytically.

ACKNOWLEDGMENTS

We thank D. Uskov, C. Rangan, B. Shore, J. Eberly, and P.Berman for useful discussions. A.B. is supported by TAMSGL Fund No. 211093 through Tulane University.

[1] F. T. Smith, Phys. Rev.179, 111 (1969).[2] The rotating wave approximation has been extensively and

sensibly used. However, this approach is not well suited forexternal kicks that require a broad frequency spectrum.

[3] M. A. Nielsen and I. L. Chuang,Quantum Computation andQuantum Information(Cambridge University Press, Cam-bridge, U.K., 2000); J. I. Cirac and P. Zoller, Phys. Rev. Lett.74, 4091(1995).

[4] C. P. Poole, Jr., H. A. Farach, and R. J. Creswick,Supercon-ductivity (Academic Press, San Diego, 1995); C. H. van derWal, A. C. J. ter Haar, F. K. Wilhelm, R. N. Schouten, C. J. P.M. Harmans, T. P. Orlando, S. Lloyd, and J. E. Mooij, Science290, 773 (2000).

[5] Proceedings of Ann Arbor Conference on Building Computa-tional Devices using Coherent Control, 2004, edited by V. Ma-linovsky (unpublished); C. Rangan, A. M. Bloch, C. Monroe,and P. H. Bucksbaum, Phys. Rev. Lett.92, 113004(2004); D.J. Winelandet al., J. Res. Natl. Inst. Stand. Technol.103, 259(1998); G. Turinici and H. Rabitz, Chem. Phys.267, 1 (2001).

[6] G. Zabow, R. S. Conroy, and M. G. Prentiss, Phys. Rev. Lett.92, 180404(2004).

[7] J. H. McGuire, A. L. Godunov, S. G. Tolmanov, Kh. Kh. Sha-kov, R. Dörner, H. Schmidt-Böcking, and R. M. Dreizler,Phys. Rev. A63, 052706(2001).

[8] A. L. Godunov, J. H. McGuire, P. B. Ivanov, V. A. Shipakov,H. Merabet, R. Bruch, J. Hanni, and Kh. Kh. Shakov, J. Phys.B 34, 5055 (2001); A. L. Godunov and J. H. McGuire,ibid.

34, L223 (2001). Time correlation is defined as a deviation

from the independent time approximation, whereUstd=p j U jstd.

[9] J. H. McGuire and A. L. Godunov, Phys. Rev. A67, 042701(2003).

[10] J. H. McGuire, A. L. Godunov, Kh. Kh. Shakov, Kh. Yu. Ra-khimov, and A. Chalastaras, inProgress in Quantum PhysicsResearch, edited by V. Krasnoholovets(Nova Science, NewYork, 2004).

[11] H. L. Haselgrove, M. A. Nielsen, and T. J. Osborne, Phys. Rev.A 69, 032303(2004).

[12] H. Z. Zhao, Z. H. Lu, and J. E. Thomas, Phys. Rev. Lett.79,613 (1997).

[13] L. H. Andersen, P. Hvelplund, H. Knudsen, S. P. Moller, K.Elsener, K.-G. Rensfelt, and E. Uggerhoj, Phys. Rev. Lett.57,2147 (1986).

[14] H. Merabet, R. Bruch, J. Hanni, A. L. Godunov, and J. H.McGuire, Phys. Rev. A65, 010703(R) (2002).

[15] W. Magnus, Commun. Pure Appl. Math.7, 649 (1954); B. W.Shore, Theory of Coherent Atomic Excitation(Wiley, NewYork, 1990), Chap. 19.

[16] D. Uskov (private communication) points out that if we relaxthe assumption of Eq.(1) and consider the most general qubit

Hamiltonian Hstd= fxstdsx+ fystdsy+ fzstdsz, then U−U0<−s1/2"2doi j e0

t dt9e0t9 dt8ff ist9df jst8d− f ist8df jst9dgei jksk and the

condition of vanishing time ordering becomesf ist8d=ci fst8d. If

fst8d is constant, this is theVst8d=0 or Vst8d=V limit under


063401-12

rotation of coordinates. For nonconstantfst8d, the above con-dition corresponds to the degenerate limitDE=0 under rota-tion of coordinates.

[17] J. J. Sakurai,Modern Quantum Mechanics(Addison-Wesley,Reading, MA, 1985).

[18] Kh. Kh. Shakov and J. H. McGuire, Phys. Rev. A67, 033405(2003). In this paper,x=a" /Îpt.

[19] B. W. Shore, Theory of Coherent Atomic Excitation(Ref.[15]).

[20] L. Allen and J. H. Eberly,Optical Resonance in Two-levelAtoms(Dover, New York, 1987).

[21] P. Milonni and J. H. Eberly,Lasers(Wiley, New York, 1985).[22] E. Shimshoni and M. Gefen, Ann. Phys.(N.Y.) 210, 16

(1991); L. D. Landau and E. M. Lifshitz,Quantum Mechanics(Pergamon, Oxford, 1976).

[23] Expressions similar to ours have been noted by P. R. Bermanand R. G. Brewer(private communication); also see P. R. Ber-man and D. G. Steel, inHandbook of Optics, edited by M.Bass, J. M. Enoch, E. Van Stryland, and W. L. Wolf(McGraw-

Hill, New York, 2000), Vol. IV, Chap. 24.[24] X.-B. Wang and M. Keiji, Phys. Rev. Lett.87, 097901(2001).[25] C. Leforestieret al., J. Comput. Phys.94, 59 (1991); M. D.

Feit, J. A. Fleck, and A. Steiger, J. Comput. Phys.47, 412(1982); D. Tannor, Introduction to Quantum Mechanics: ATime-Dependent Perspective(University Science Books, Sau-salito, CA, 2004), Chap. 11.3.1.

[26] L. Mandel and E. Wolf,Optical Coherence and Quantum Op-tics (Cambridge University Press, Cambridge, U.K., 1995).

[27] J. H. McGuire,Electron Correlation Dynamics in Atomic Scat-tering (Cambridge University Press, Cambridge, U.K., 1987).

[28] H. Mabuchi(private communication).[29] D. Tannor, Introduction to Quantum Mechanics: A Time-

Dependent Perspective(Ref. [25]), Chap. 9.3.[30] W. R. Salzman, J. Chem. Phys.85, 4605(1986).[31] K. Bergmann, H. Theuer, and B. W. Shore, Rev. Mod. Phys.

70, 1003(1998).


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time ordering in kicked qubits

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