Diagrammatic many-body theory for atoms in high-intensity laser fields Part I

Vol. 9, No. 5/May 1992/J. Opt. Soc. Am. B 627

Diagrammatic many-body theory for atoms in high-intensitylaser fields. Part I

Lars Jonsson* and Goran Wendin*

Laboratory for Atomic, Molecular, and Radiation Physics, Department of Physics, University of Illinois at Chicago,RO. Box 4348, Chicago, Illinois 60680

Received December 14, 1990

Diagrammatic many-body theory has been used extensively in the treatment of electron-electron correlationsin atomic photoionization at low intensities. We discuss how it can be used at high intensities also. The stan-dard diagrammatic formalism is derived for time-independent interactions by using adiabatic switching. Thisformalism can be formally derived for harmonic fields also but leads to unphysical results for strong fields. Ifthe amplitude function of the laser field is an exponential function, however, the standard derivation still ap-plies. If the adiabatic limit is not taken, this time-dependent formalism can be used for any intensity. Inpractice one is still limited to slow turn-on, but, assuming slow turn-on, we show that for a discrete system thelinked expansion, when summed to infinite order, gives the Floquet state that is adiabatically connected to theground state. When continua are included, we find that qualitatively correct results are obtained only if oneuses an expansion of the time derivative of the wave function instead of an expansion of the wave function it-self. We also show that the imaginary part of the self-energies must be excluded from the energy denomina-tors, since they otherwise lead to unphysical results. These results are found by treating model systems forwhich the exact solutions are known and for which the diagrams can be evaluated to any order.

1. INTRODUCTION

In the past few decades the development of powerful lasersystems has taken atomic physics into intensity regimesfor which linear response and lowest-order perturbationtheory give results that disagree with experiments. Thisis true, for instance, for the recent experiments withabove-threshold ionization and harmonic generation.'The laser intensities used in these experiments are typi-cally 10" W/cm2 for 1-eV photons and 1016 W/cm2 for 5-eVphotons. At present the highest reported intensity is1019 W/cm2 at a 248-nm wavelength (5 eV).2 The pulselength in the last case was a few hundred femtosecondsbut is typically -1 ps.

The theoretical treatment of atoms interacting with in-tense laser fields has essentially developed along threedistinct lines':

(a) Nonperturbative treatment of one-electron atoms.Early attempts were made by Keldysh,' and his approachhas been discussed and improved by many authors.4 Inrecent years the methods using Floquet states7 '10 and theresolvent operator formalism"' 4 have been the most suc-cessful. Gavrila and coworkers1516 presented a theorythat is valid in the high-frequency regime.

(b) Perturbative many-body calculations. Diagram-matic many-body theory has been used extensively.'17-2

These calculations, however, have not presented any truehigh-intensity results (infinite number of photon interac-tions), although such calculations have been attempted.2 'Other methods2 9 that have been used for many-electronatoms are density-functional theory"'32 and quantum-defect theory.33-35

(c) Direct numerical solution of the Schrodinger equa-tion in time and space." The Schrodinger equation has

been solved numerically for both single- and many-elec-tron atoms.

In this paper we discuss the possibility of unitingmethods (a) and (b) above. That is, we ask whether it ispossible to find a many-body formulation that is nonper-turbative in the laser field. We use diagrammatic atomicmany-body theory, which will give nonperturbative resultsif the diagrams are summed to infinite order. Althoughthe aim is to investigate many-electron effects, in thispaper we treat only one-electron systems. We do so be-cause the problems connected with the time dependenceof the laser field appear even though many-electron ef-fects are disregarded. To make the effects of the timedependence as clear as possible, we want to use thesimplest systems possible. These are invariably one-electron systems. All diagrams that we discuss in thispaper, however, will be present in any many-body problem.The individual interaction of each electron with the laserfield is described by the diagrams that we treat below.These are a subset of all diagrams and, if correlation isimportant, will not suffice for a proper description of thedynamics. Nevertheless, despite that limitation, it is im-portant to know how to treat these diagrams, and the nat-ural starting point is therefore the one-electron problems,even though the aim is higher.

By summing the diagrams to infinite order we meannot an explicit numerical summation of the perturbationseries but a formal summation, so that the whole set ofdiagrams is replaced by a nonperturbative expression thatcan be used at any intensity. The Taylor expansion ofthis expression for low intensities must then be identicalto the perturbation series given by evaluating the dia-grams. The expression does not have to be an algebraicfunction but could, for example, be an integral equation

0740-3224/92/050627-19$05.00 © 1992 Optical Society of America

L. Jnsson and G. Wendin

628 J. Opt. Soc. Am. B/Vol. 9, No. 5/May 1992

that can be solved numerically or a continued fraction ex-pansion that can be calculated numerically to any degreeof accuracy by an appropriate truncation. In effect, sum-ming a set of diagrams to infinite order means findingsome kind of analytic continuation of the correspondingperturbation series. A typical example of such a summa-tion is the screened photon interaction in the randomphase approximation, in which the unscreened dipole op-erator is replaced by an effective screened operator, whichis found by solving an integral equation. This procedureis then a summation to infinite order in the electron-electron interaction. We want to find similar techniquesto sum the photon interactions to infinite order. Theearly paper by Gontier et al." shows that it is possible touse perturbation theory in this way. Using the resolventoperator formalism, they showed that the general nonper-turbative solution for an atom in a laser field can be foundby summing all the higher-order terms.

Our task is twofold. First, we must find a diagram-matic formulation that gives correct high-intensity resultswhen summed to infinite order. Second, we must findsome way of actually doing the summation for the physi-cally interesting systems. In this paper we deal mainlywith the first problem. It is not a good idea to spend timetrying to find good summation techniques unless we knowthat the final result will at least be qualitatively correct.We therefore start by treating systems for which the exactsolution is known and for which the diagrams can beevaluated to any desired order. If we can show that thesummed diagrams give the same results as the analyticalsolutions, then it is time to go on to more complex sys-tems. At first, therefore, we do not worry about the factthat the summation technique used in this paper cannotbe readily used in the general case.

The standard diagrammatic formulation is due toGoldstone,3 7 who derived the diagrammatic rules for time-dependent fields by using adiabatic switching and theGell-Mann-Low theorem.3 " This theorem states thatthe linked expansion is nondivergent in the adiabaticlimit. The final results of this formalism are time inde-pendent, and the adiabatic switching is just a convenientmathematical tool. In the case of an atom interactingwith a laser field one can formally make the same deriva-tion by using adiabatic switching.3 9 This approachgives the diagrammatic many-body theory that has beenwidely used to treat atomic many-electron effects in weakfields. This theory was used at first to treat only one-photon absorption but is now used for multiphoton pro-cesses also. Our ambition is now to go one step furtherand to sum the diagrams to infinite order, therebyto obtain true high-intensity results. In a few recentpapers"," we discussed whether this summation can bedone. We found that this standard formulation gives riseto unphysical results when summed to infinite order. Inparticular, we found that the ground-state width is sub-tracted from the excited-state widths and is not added asit should be. This outcome shows that the adiabaticallyswitched formalism is not a good starting point for treat-ing high-intensity effects. This conclusion is not surpris-ing, since adiabatic switching is not compatible with afinite ionization rate. The use of adiabatic switchingtherefore implies that ionization can be neglected. Theadiabatic assumption, however, is used only as the last

step in the derivation of the diagrammatic formalism.The important step of transforming time integrals to en-ergy denominators assumes only that the interaction isexponentially switched. This means that the formalismalso applies to a harmonic field with a finite exponentialturn-on. Since we do not take the adiabatic limit, we nowobtain an explicitly time-dependent formalism. It is thisgeneralized Goldstone formalism that we use below. Inprinciple, if we can sum these diagrams to infinite order,we will get the exact solution to the problem of an atom inan exponentially turned-on laser field. Such a field goesthrough all intensities, since it diverges for infinite times,and the formalism must therefore contain the high-intensity solutions also. This generalized Goldstone for-malism is therefore a good starting point for discussingdiagrammatic methods at high intensities.

We find below that in practice we can sum the diagramsonly when the turn-on rate is small, but for that case wedo obtain results that are valid at all intensities. We firstdiscuss discrete systems, for which we find that the dia-grams when summed give the Floquet state40 that is adia-batically connected to the ground state. This result isillustrated by treating the two-level system in the rotating-wave approximation (RWA), which is done in Section 2.In Section 3 we discuss how to treat damped systemsproperly, which leads us to make some further changes inthe formalism.

The fact that the slow turn-on of a harmonic interactiongives the Floquet state for a discrete system is wellknown. The significance of the present study is that weshow how this result can be derived by explicitly summingthe diagrammatic expansion. Our interest is not to proveor to illustrate the general theorem but to find diagram-matic methods that work in practice. Our aim is there-fore not to discuss the general and formal aspects of theresults but to learn how one should deal with diagramsconcretely. This is also our motivation for investigatingboth the linked and the unlinked diagrams. When doingthis we find that the unlinked expansion cannot be usedin practice to derive high-intensity results. From a gen-eral point of view this restriction is caused by the secularterms (i.e., terms that contribute to the overall phase fac-tor of the wave function), and the complications that aredue to their existence are well known. On the otherhand, the unlinked expansion is derived from linear am-plitude equations, while the amplitude equations corre-sponding to the linked expansion are nonlinear. Forpractical applications, for which numerical properties ofthe underlying equations could be important, the unlinkedexpansion might be more convenient than the linked one.Furthermore, for the one-electron problem the unlinkeddiagrams in each order are fewer and easier to evaluatethan the linked ones. The best choice is therefore notobvious, and we investigate both.

Finally, let us discuss what is required by a usefuldiagrammatic formulation. First, it must reduce tothe standard formulation when the interaction is timeindependent. At present we are concerned with the prob-lems that are due to the laser field only. We must,however, make sure that we do not change the formalismso much that it does not remain valid when we reintroducethe electron-electron interaction. Here it is importantthat the diagrams be linked so that there are no diver-



gences in the adiabatic limit. The idea is to use adiabaticswitching for the electron-electron interaction and afinite switching rate for the laser field. One could, how-ever, have a formalism that is linked with respect to theelectron-electron interaction but not with respect to thelaser interaction. Another important requirement isthat, when a given diagrammatic structure, such as aproper self-energy, appears repeatedly in a diagram, itshould have the same value each time it appears. This isnecessary if we want to sum the series to infinite order,since this usually means finding repeated structures. Ifthe expansion does not fulfill this requirement, the wholeidea of self-energies would break down.

The formulation of a time-dependent theory in which agiven structure is independent of where it appears is asomewhat delicate matter, because the diagrams are timeordered. If a structure is independent of where it ap-pears, it is also in some sense time independent, or at leastis dependent only on instantaneous values of the parame-ters describing the atom and the driving field. Thusdamping effects, as opposed to energy shifts, turn out tobe incorrectly described if the standard formulation isused. It is reasonable to let the energy shifts of theatomic states be functions of the instantaneous laser in-tensity, but doing the same for the ionization probabil-ity or the widths of the atomic levels leads to unphysicalresults.

Further discussion and a summary of the results arepresented in Section 4. In Appendix A we have shownhow to evaluate the linked diagrams under the assump-tion of a slowly turned-on laser field.

2. NONRESONANT TWO-LEVEL SYSTEM

In this section we disregard effects that are due to damp-ing and discuss systems with a finite number of discretestates. We show that the generalized formalism that wepresent below gives the Floquet state of the system whenthe diagrams are summed to infinite order. Rather thanpresenting a formal proof of this conclusion, we illustrateit for a particular case. We discuss the two-level systemin the RWA.4" We choose this case because the Floquetsolutions are known and because in this case we canevaluate the diagrams to any order. The method thatwe have used to evaluate the diagrams is presented inAppendix A, and we give only the final expressions in themain text.

Initially we assume that the amplitude function is anexponential, but this assumption will be relaxed in theend. It is true that the mathematics is valid only for anexponential amplitude function, but, on the other hand, ifthe turn-on is slow, it is reasonable to assume that an ex-ponential turn-on should not give results that differ inany fundamental way from those of any other smooth am-plitude function; therefore there must be a way to expressthe final results so that they are independent of the func-tional form of the amplitude function. This step, how-ever, must be done with some care, and one must be awarethat this procedure may introduce further approxima-tions. To be more concrete, let r be the exponentialturn-on rate, so that the interaction Hamiltonian is pro-portional to exp(Ft). Then we might naively make thesubstitution exp(rt) -> A(t), where A(t) is an arbitrary am-

plitude function, and thereby get a more general formula-tion. As we show below, this substitution would giveincorrect results. For instance, if the factor exp(rt) is di-vided by r, then the correct generalization would rather beexp(rt)/r --> fA(t)dt. Our approach will be to solvesuch problems as we go along, and our initial ambition isthat all physical results should behave qualitatively cor-rectly. The approximations involved in the derivationwill be evaluated quantitatively once we have a consistentformulation.

The purpose of the Gell-Mann-Low procedure is to re-move factors that diverge in the adiabatic limit ( , 0),and the result is the linked diagrammatic expansion. Inour case we must assume that the turn-on is slow, but wewill not go so far as the adiabatic limit. We can, there-fore, work both with the linked and the unlinked expan-sion, and we discuss both below. Eventually, when weintroduce the electron-electron interaction again, wemust make sure that the diagrams are linked with respectto this interaction. This requirement does not entail thatthe diagrams must be linked with respect to the photoninteraction also. It is no problem to have differentturn-on rates for the two interactions and to take the adia-batic limit of one but not the other.

The two-level system that we discuss in this section isshown in Fig. 1. The upper level has index 1, and thelower level has index 0. The detuning from resonance, A,is defined by

A = E - Eo - hco, (1)

where co is the laser angular frequency. In the rest of thispaper we set h = 1 and make no distinction between ener-gies and frequencies. We take the interaction Hamilto-nian to be

H = 2D exp(Ft)cos wt, (2)

where D is the total dipole operator times half the fieldstrength at t = 0, r is a constant turn-on rate, and t is thetime. The dipole operator is assumed to have only nondi-agonal matrix elements different from zero, so D00 =DI, = 0 and Dio = Do,* 0. If we let a, and ao be theamplitudes of the upper and the lower states, respectively,we get the following set of equations from the Schrodingerequation in the interaction representation:

iaj = D10oexp[i(A - i)t] + exp[i(A + 2 - if)t]}ao,

(3a)

iao = Dolexp[-i(A + iF)t] + exp[-i(A + 2w + ir)t]}al.

(3b)

El t_ -_ X

1'W

EoFig. 1. Two-level system. El and Eo are the excited-state andground-state energies, respectively; w is the laser angular fre-quency; A is the detuning from resonance.

L. Jbnsson and G. Wendin


In the RWA we disregard the rapidly oscillating termsand get

t = Dio exp[i(A - i)t]ao, (4a)

UitO = Do, exp[-i(A + if)t]ai (4b)

or, in integral form,

ta = -i f Do exp[i(A - i)t']aodt', (5a)

ao = 1 - i Do, exp[-i( + f)t']aidt', (5b)

the discussion of what these diagrams yield for our two-level system.

A. Unlinked Expansion in the Rotating-WaveApproximationThe unlinked expansion is just the straightforward ex-pansion of U in Goldstone's derivation, while in our case itcorresponds to a direct iteration of Eqs. (5). The dia-grams for ao are shown in Fig. 2, and the correspondingseries is

D2 exp(2rt)ao 1 + (A - i) (-i2f)

where the initial conditions are a, = 0 and ao = 1.When F = 0, Eqs. (4) can be solved analytically. T

are two independent solutions, which we write as42T

Aill) + AolO) and "PB) = Bill) + BoJO) where

A- = D[D2

+ (181 _ Q)2]1/2

= sDo,(l - 1)ID[D

2+ (181 Q)21 2

B = - sD 1 o[D2

+ (181 + fl) 2]1/2

exp[i(8 - s)t],

exp[i(-8 - s)t],

exp[i(8 + sfl)t],

D4 exp(4rt)(A - ir) (-i2r) (A - i3r) (- i4r) + (9)

hereA) = (Recall that the sign of a diagram is given by the following

rule: Each vertex, internal hole line, and closed loopgives one minus.) When we discuss the diagrams we usethe words photon absorption and photon emission (readingthe diagrams from bottom to top). These terms referonly to the visual appearance of the wavy lines and do not

(6a) indicate that the laser field is quantized. We also de-scribe the creation of an electron-hole pair as a V vertex,and the annihilation of an electron-hole pair will be calleda A vertex. To shorten the notation, we define x =D2 exp(2t)/A2 and y = r/A, which yields

B0 = = 81 + l exp[i(-[D 2 + (181 + fl)2] 1/2 ex ai- + sfl)t] . (6b)

Here D = IDo1, 8 = A/2, s = A/IAI, and fl = (2 + D2)1/

2.

The coefficients are chosen so that Al, Bo - 1 andAO, B, - 0 when D -> 0. These states are the Floquet orquasi-energy states adiabatically connected to the upperand the lower states, respectively. Below we show howthese solutions appear when the relevant Goldstone dia-grams are summed to infinite order under the assumptionof slow turn-on. However, we must first make the con-nection to the many-body formulation used by Goldstone.

From a many-body point of view we are dealing with aone-electron atom with one core state and one excitedstate. In the ground state 10) the electron is in thecore state; in the upper state 11) the electron is in the ex-cited state, and there is a hole in the core state. The par-ticle (electron) and the hole creation operators are denotedpt and ht, respectively. We then obtain

Ii) = pth1o)

ao =1I+ (1 - iy)(-i2y)

x2

+ +(1 - icy) (-i 2 y) (1 - iffy) (-i4y) +

(10)

The major obstacles when we are summing this series arethe numbers in front of y in the denominators. Thesenumbers count the total number of interactions with thephoton field up to the corresponding point in the diagram.If we had to keep track of these numbers to get usefulresults, we would have to stop here, since in our simplemodel the accounting would already be too cumbersome.These numbers also destroy the notion that any givenstructure should be independent of its position in a dia-gram. As we see from Fig. 2, the a diagrams consist ofan increasing number of repetitions of the bubble diagramappearing in the second term. If the number in front of ywere important, every bubble would be unique, and theformalism would not be workable.

(7)

and, for the dipole operator,

D = Doihp + D1optht . (8)

We denote the total wave function in the interaction pic-ture by 18(t)), and the time development operator, U isdefined by 1T(t)) = U(t, t') (t')). Further, we haveI T(t)) = a l) + a ), which gives a = ( IU(t,-oo) 0) and

a = (01U(t, - ) 10), where we use the initial conditionlT(-o)) = ). These relations make the connection

between Goldstone's formulation in terms of U and ourformulation in terms of the amplitudes a and a. Forthe derivation of the Goldstone diagrammatic rules werefer the reader to his well-known paper.3 7 We now begin

+

Fig. 2. Unlinked diagrams for the ground-state amplitude in theRWA.

L. Mnsson and G. Wendin

I +


Let us see what we get if we disregard y in the non-diverging denominators:

x x2 x3

-i2y 2! (- i2) 2 + 3! (-i2y)3 +

/ ix) = e iD2 exp(2ft)= exP\2 = expL 2

iD 2 iD 2 t:- exp + A . (11)

For finite times Eq. (11) gives the lowest-order energyshift -D 2/A and the usual diverging phase. Equation (11)is clearly a low-intensity result, despite the fact that itrepresents a summation to infinite order. One can, how-ever, derive high-intensity results from the series inEq. (10).

If we write a = 1 + ax + 2 x 2 + ... and In a=f31 x + /32 x2 + ... and again assume that y is small,we get43

1 + iY131 = a, X (12a)

2= 2-1-a ~ iy + 5(iy)2 1b32 =a2 - 2 (-_i2y) ' (12b)

/33 = a3- -(a 2 ,01 + 2a 1l3 2 )3

8(iy)2 + 88(iy)3 (12c)3(-i2y)3

,04 = a4 - -(a 33 1 + 2a 2132 + 3a133)4

_ 10(iy)3 + 186(iy)4 (12d)(i2 Y)4

When we sum the first and the second terms in each orderseparately, we obtain

il x2 2 x3 5x4 ln(ao)i = (x - 2 + 3 4

= (-1 + x' - 2x'2 + 5x'3 -.. )dx'

i l 1- (1 + 4x')"2dx'2y 2x'

At= -i -_3s[82 + D2 exp(2Ft')]1/2 }dt', (13)

x 5X2 11x3 93x4

ln(ao)2 = -+ - --+ + 2 4 3 8

1 + (1 + 4x)"2

= n{ 4x + [1 + (1 + 4x)1/2]2}1/2,

- lnF 181 + [82 + D2 exp(2Ft)]" 2 11=I(D2 exp(2Ft) + {181 + [82 + D2 exp(2rt)] 1/2})1/2J

(14)

where the given expressions were found by inspection.Taken together, Eqs. (13) and (14) yield

Fig. 3.RWA.

Unlinked diagrams for the excited-state amplitude in the

181 + [2 + D2 exp(2Ft)] 2

a (D2 exp(2rt) + {181 + [2 + D2 exp(2t)]1 2}2)1 /2

x exp(-i L {8 - s[82 + D2 exp(2rt')]1/2}dt')

(15)

and, by an identical evaluation of the diagrams for a,shown in Fig. 3,

-sDo exp(iAt + t)a1 = (D2 exp(2rt) + {11 + [2 + D2 exp(2t)]1 /2}2)1/2

x exp(i| f.{3 l S[2

+ D2

exp(2rt')] 1/}dt).

(16)

These expressions are almost identical to the Floquet so-lution JTB) in Eqs. (6), the only difference being the timeintegral in the exponential. We see that in this case theassumption that y is small does not lead to a result that isvalid only at low intensities but to the Floquet state that isadiabatically connected to the ground state. That the so-lution to Eq. (5) can be written in this way is nothing new;the important point is that this solution is what the dia-grams yield and that it is not a low-intensity result. Themethod used, on the other hand, is intractable, and wetherefore leave the unlinked diagrams for now.

B. Linked Expansion in the Rotating-WaveApproximationThe linked expansion given by the Gell-Mann-Low theo-rem is a diagrammatic expansion of the wave function I'F),defined by

D) U(t, -m) 10)(o0u(t, -X) 10)

(17)

Gell-Mann and Low3" showed that the diagrams in the ex-pansion of l1D) are free from divergences in the adiabaticlimit. This is because the ground state does not appearas an intermediate state in the linked expansion, whichis true whether one takes the adiabatic limit or not.In ordinary time-dependent perturbation theory theGell-Mann-Low procedure corresponds to factoring outthe secular and normalization terms. How to treat thesekinds of term properly and related problems have beenthoroughly reviewed by Langhoff et al.44 In the discus-sion below we adopt some of their methods and notation.

L. J6nsson and G. Wendin


We have shown above that an = (nlU(t, - o) 10). For thetwo-level system n = 0 or n = 1, but the discussion belowis valid for any discrete system.44 We now define thecoefficient bn as the ratio an/ao, which gives us

-a (nIU(t, -0)10) (8b = = ((0 -)0) = (naft)' (18)ao (0IU(t, -00)0)-(l,

so b are the amplitudes of lcF) in the basis set In). Thelinked expansion is the diagrammatic expansion of theseamplitudes, which we get from comparing Eqs. (17)and (18).

When forming the amplitude ratios b, we lose some in-formation. It is the full amplitudes an that we want, andthese we get by multiplying by a, since by definitionan = a0bn. At low intensities multiplication is no prob-lem, since one assumes that laol = 1 at all times. That is,ground-state depletion is neglected, and then Ian = bnl.The phase of a is still unknown, but that is why we aredetermining the b coefficients. When the interaction isadiabatically switched, the phase of a diverges; when cal-culating the amplitude ratios instead, we get rid of thisdiverging phase, which has no physical significance. Athigh intensities we cannot disregard the ground-statedepletion, and the phase of a0 is highly significant, since itgives the energy shift of the ground state. We musttherefore recover the information lost by forming theratios bn-

In the interaction representation the Hamiltonian is

H = exp(iHot)V exp(- iH0 t)exp(Ft), (19)

where, in our case, V = 2 cos wt and Ho is the unper-turbed Hamiltonian, so that Hln) = En). In the time-independent formalism the ground-state energy shift, E,is given by AE = (1AFI), which in our case gives

AE = (0I1HIF) = VOn exp[i(Eo - E)t + t]bn- (20)n

From the equation for a, we obtain, further,

i1 = Von exp[i(E - E)t + Ft]ann

= {_V0n exp[i(Eo - E)t + t]bn ao = AEa0 . (21)

Thus

a = exp(-i AEdt). (22)

This result shows that AE contains all the informationthat we lost by forming the ratios b. Further, Eq. (20)shows that AE can be calculated once the b amplitudesare known. With the additional step of calculating AEthe amplitudes a are fully known once the b are known.Finally, let us derive the equation satisfied by bn:

i&n i= a oaibn =2 = Y nbm - AE bnao a0 2 I

= H + Hnmbm-AEbn,m 0o

where H is given by Eq. (19) and we have used the factthat bo = 1. We see by the last step in Eq. (23) that theground state will not appear as an intermediate state, asrequired. The price paid for this is the appearance of theterm containing AE. This term is quadratic in b, sinceAE is linear in bn. This nonlinearity in the equation forbn is reflected in the appearance of the saw-tooth struc-ture (see below) in the linked diagrams.

Returning to the two-level system in the RWA, we get

ib = Dio exp(iAt + t) - Do, exp(-iAt + t)b 2 , (24)

where we drop the index on b = a/ao, and

AE = Do, exp(-iAt + t)b. (25)

We can try to derive the perturbation expansion for b alge-braically by rewriting Eq. (24) as an integral equationand then iterate it, with the initial condition being thatb(-oo) = 0. This gives

b -iDjo exp(iAt' + t')dt'

; r~~~~~~~~~~t+ iDolf exp(-i,&t' + Ft')b2 dt'

D10 exp(iAt + t) DoDjo2 exp(iAt + 3t)A - ir + (A - i3r) (A - fr)2

+

(26)

We see that the calculation quickly becomes cumbersomebecause of the quadratic dependence on b in the secondintegral. Note, however, that the denominators are non-diverging in the adiabatic limit, which is the signature ofthe linked expansion.

Goldstone derived the linked expansion by purely dia-grammatic methods. We do not repeat this derivation indetail, but, to illustrate the procedures involved, we give ashort outline. We have shown above that the linked ex-pansion is the perturbation series for the ratio b = a/ao.The unlinked graphs for a and a are shown in Figs. 2and 3. We derive the linked expansion directly by form-ing the ratio between these two diagrammatic expansionsby the process of factorization. We start with the seconddiagram in the expansion of a and rewrite it as a sum ofnew diagrams, as illustrated in Fig. 4(a). The factoriza-tion comes from the rule that the product of two diagramsis equal to the sum of all time orders of the two diagrams.What we do in Fig. 4(a) is to add the diagrams that weneed to factorize the single diagram on the left-hand side,and then we subtract the same diagrams to make theequality hold. Finally, we redraw the two last diagramsso that they become linked. Note that when we do this allthe vertices and their time order are unchanged. All wedo is to redraw the bubble as a saw-tooth. This alterationdoes not change the numerical value of the diagram otherthan the change of sign that is due to the removal of thebubble. This sign change is compensated for by thechange in sign in front of the diagram. Now we dothe same thing with the next order in a, making surethat each term is a product of a linked diagram and a dia-gram from the expansion for a. The result is shown inFig. 4(b), where we have introduced the convention that,when different vertices are drawn with the same horizon-

L. Mnsson and G. Wendin


= y+V S

+t

+S

V= x+\X+(b)

Fig. 4. (a) Factorization of the third-order diagram. (b) Factor-ization of the fifth-order diagram. In the last diagram all timeorders of the vertices should be taken. The same is true for theopen diagram in the penultimate term.

only because of our shortened notation. All time ordersof the vertices should be taken, so the second diagramrepresents the sum of 2 time-ordered diagrams, and thethird diagram represents a sum of 16 time-ordered dia-grams. We do not discuss how these diagrams can beevaluated in the main text; all the derivations needed are

(a) included in Appendix A.Returning, for a moment, to the series in Eq. (26), we

see that once again there are different numbers in front ofthe 17s in the denominators. Just as in the unlinked case,it will be impossible to take these into account properly.We therefore make the same approximation as before, sothat A - iNP -r A for arbitrary N. With this simplifica-tion all the linked diagrams can be evaluated, which iswhat we show in Appendix A.

The series for b, given by the diagrams in Fig. 5, nowbecomes [see Eq. (A14) below]

-Dio exp(iAt + Ft)A

x (1 - x + 2x 2 - 5x 3 + 14x 4 - 42x5

+ 132X6 - 429x7 + ... )

-Djo exp(iAt + t) (1 + 4x)]2-A L 2x l

-sDlo exp(iAt + Ft)181 + [82 + D2 exp(2rt)] 12 (27)

where x = D2 exp(2rt)/A2 , as before. We see that thefinal expression is exactly the ratio between a, and a inEqs. (16) and (15). AE is given by Eq. (25) and becomes

AE = 3 - s[8 2 + D2 exp(2rt)] /2. (28)

Fig. 5. Linked diagrams for the ratio between the excited-stateand ground-state amplitudes in the RWA.

tal position (the same time), all their time orders shouldbe taken. This means, for example, that the last diagramin Fig. 4(b) represents the sum of the 16 different timeorders of its vertices.

The factorization is done for each order of the a, dia-grams. Finally, we take the ratio a,/ao. When we takethe sum of the factorized diagrams for a,, we see thatevery linked diagram is multiplied by exactly the seriesfor ao. This factor of ao in the numerator is canceled byao in the denominator, so we are left with the linked dia-grams only, which are the expansion of b. This series isshown in Fig. 5. All these diagrams (except the first) areso-called exclusion-principle-violating diagrams,45 becausethey all contain many electron-hole pairs excited simulta-neously in the same states, despite the fact that we areworking with a fermion system. This property is a gen-eral feature in all linked expansions, which in our casemakes the diagrams harder to evaluate. However, sincethe ground state does not appear as an intermediate state,none of them will diverge in the adiabatic limit. This situ-ation was the goal of the derivation. Note also that wehave drawn only one diagram in each order, but this is

In Fig. 6 we show the diagrammatic representation of AEin the RWA. These energy diagrams are evaluated withthe same rules as for the amplitudes. The only differenceis that there is an extra overall minus, and the final-stateenergy denominator should be excluded. These rulesfollow directly from the definition of AE.

With the definition of AE in Eq. (22), Eq. (28) now givesthe same exponential factor as in Eqs. (16) and (15).These results for b and AE are what the linked diagramsgive when the imaginary parts are neglected in all energydenominators. It is clear from the comparisons madeabove that they are related to the same slow-turn-on solu-tion that we derived with the unlinked diagrams. We getthe same ratio and energy shift, but we do not get thecorrect normalization. To obtain the correct normaliza-tion also, we cannot neglect the imaginary parts of theenergy denominators. In the unlinked case we had to

+

+A+

Fig. 6. Linked diagrams for the ground-state energy shift inRWA. When evaluating these diagrams one should exclude thefinal-state energy denominator. There is also an extra overallnegative sign.


X


keep two terms in the expansion around small [seerelations (12)-(14)] to get the correct normalization. Thesame is true for the linked diagrams, but such a calcula-tion is unnecessary. We now have the ratio between thetwo amplitudes and can calculate the norm, J, directlythrough

2 .=.k = 1 + = 1 + b12

laol2 aol 2

= D2 exp(2Ft) + {181 + [2 + D2 exp(2rt)] 112}2 (29)

{18 + [2 + D2 exp(2Ft)] 112}2

We now get the full solution from b, AE, and X through

b a, = - exp -i AEdt)

X ( I )I

1 (ao = - exp -iJAEdt)

(30a)

(30b)

b = -Dio exp(iAt + rt) & be exp(-i2nwt),A -

(31)

where the factor -D 1o/A is introduced for future conve-nience. When the turn-on rate F is small, all the b willbe slowly varying functions of time. The diagrams con-tributing to a given b are all those that have a netabsorption of 2n + 1 photons (counting emission as nega-tive absorption).

From the definition of AE in Eq. (20), we obtain,further,

E -D' exp(21t) (b6 + bn i)exp(-i2ncot),

with the slowly varying part being

-D 2 exp(2t)- - - +AEo =- (bo + b 1) = -Ax(bo + b 1).

(32)

(33)

which are identical to Eqs. (15) and (16).From the discussion above we infer, without further

proof, that for a general discrete system we can find theFloquet state that is adiabatically connected to the groundstate through the following procedure: Sum the linkeddiagrammatic expansion, neglecting all the imaginaryparts of the energy denominators. This step gives theamplitude ratios. Calculate the ground-state energyshift, which is known as a function of these ratios. Makean explicit normalization. Taken together, these stepsgive the full Floquet solution. Looking back at thederivations that we made, we also conclude that this is thebest that one can do with diagrams of the kind we arediscussing. Since we are forced to neglect the imaginaryparts of the energy denominators, we cannot relax the as-sumption of slow turn-on; but when this is a good approxi-mation we do get results that are valid at high intensities.

C. Counterrotating TermsBefore we look at damped systems, we briefly discuss thetwo-level system without the RWA, that is, when therapidly oscillating terms are not neglected. The methodthat we use to evaluate the diagrams still works, and wecan still evaluate all the individual diagrams we need. Asbefore, this evaluation is performed in Appendix A, andwe give only the final series in the main text. We seebelow that the expansions are much harder to sum di-rectly. These calculations show that, even for an approxi-mate treatment of a two-level system, the method of directsummation is impracticable, so another summation tech-nique must be found.

All the diagrams describing effects outside the RWAhave the same structure as the ones under the RWA. Thedifference is that we can now create an electron-hole pairwith a photon emission and annihilate an electron-holepair with a photon absorption. In the diagrams for theamplitude ratio b there are always an odd number of photoninteractions. The final-state exponential time factor ofany of these diagrams is exp{i[E - Eo - (2n + 1)w]t+ Mt} = exp[i(A - 2nco)t + M], where n is any in-teger and M is the total number of photon interactions inthe diagram. Using this factor, we define the Fourier co-efficients b through

To shorten the equations below, we find it convenient tointroduce the ratios q, defined by

q. = A/(A-2nw). (34)

Let us first calculate the correction to the slowly vary-ing part of the ground-state energy shift that is due to thecounterrotating terms. In terms of the q, the RWA im-plies that qo = 0, which is consistent with the RWA ap-proximation A/wc << 1. From this point of view the firstcorrection to the energy consists of the diagrams propor-tional to q and ql, and we have chosen to treat the onesproportional to q-1 . To calculate AEo, we need b0 and b6-.The diagrams for the amplitudes that give contributionsproportional to q are shown in Fig. 7. Figure 7(a) gives[Eq. (Al6a)]

(35)b- l=q-1,

and Fig. 7(b) gives [Eqs. (Al7a)-(Al7e)]

b = -2qlx(1 - 3x + 10x 2 - 35x3 + ... )

-4q-lx(1 + 4x)"2 [1 + (1 + 4X)1 2]

Equations (35) and (36) give

(1 + 4x)1/2

V

(36)

(37)

(a)

(b)

Fig. 7. Diagrams contributing to (a) b 1 and (b) go. These dia-grams are needed to calculate the lowest-order correction to theRWA energy that is due to the counterrotating terms. In (b) alltime orders of the vertices should be taken.



which, together with Eq. (33), yields

- D 2 exp(2ft)AEo = A\ERwA + (A + 2o){1 + [4D2 exp(2t)/A2 ]}"12

(38)

which for low intensities corresponds to the Bloch-Siegertshift.4 ' Going one step further, we could include thecontributions to o and b-1 that are proportional to anypower of q-1, but, still ignoring all the qn,,Ol, we obtain[Eqs. (Al6a)-(Al6e)]

L = q_, - (2q1 2 + q1 3)x

+ (2q_1 2+ 8q_13 + 6q_1

4 + 2q_15)X2

- (4q.1 + 20ql + 46q_ + 40q-15 + 20q- 6

+ 5q_,7 )x3

+ (10q_12

+ 60q_,3

+ 180q1 4

+ 320q-15

+ 300q1 6 + 180q_17 + 70q_18 + 14q_1 9 )x4-

(39)

and [Eqs. (Al7a)-(Al7e)]

= 1 - (1 + 2q-,)x + (2 + 6q_1 + 8q_12 + 2q_13)x2

- (5 + 20q_1 + 40ql + 46q_, + 20ql 4

+ 4q_,5)x3 + (14 + 70q_1 + 180q_12 + 300q_13

+ 320q _4

+ 180q ,5 + 60q 6 + 10q- 7)x

4- ....

(40)

These expressions can hardly be summed by inspection.Including more of the qn ratios gives even worse expres-sions, and the conclusion is that even for the two-levelsystem we get stuck almost immediately if we try to usedirect summation.

We have also summed two other classes of diagrams,which are shown in Figs. 8 and 9. They represent contri-butions to b, and b2 and are related to the generation ofthe third and the fifth harmonics by the two-level system.If we sum the diagrams in Fig. 8, assuming that qn,,o = 0,we get

-4qx

1 [1 + (1 + 4x)" 2][1 + 4qx + (1 + 4x)" 2]' (4)

if we sum the diagrams in Fig. 9, assuming thatqno,1,2 = 0, we get

b- + 4q~x + (1 + 16qlq2x2

[1 + 4q + 4x) 2 ][1 + 4q2 x + (1 + 4x)" 2]

X + + 4q,x - 1+4x)1/2}-.4 + 3q,[(1 + 4x)1/2 - 1] + (1 +

(42)

We do not perform these derivations in detail. Our pur-pose is to illustrate the complexity of the expressions evenfor relatively simple diagrammatic structures. It is clearthat a totally different summation technique is needed.

D. Energy-Shift EquationWe have shown above that the direct summation tech-nique is too cumbersome. In this section we derive theRWA results for the two-level system in another way. Weperform this derivation by starting again from theSchrodinger equation (in the RWA) for the amplitudes a,and ao, which is given by Eqs. (5).

In the discussion of the linked expansion we defined thequantity AE through

ao = exp(-i AEdt) . (43)

The real part of AE gives the ground-state energy shift,while the imaginary part gives the ground-state norm.We found above, however, that when we disregard theimaginary parts of the energy denominators we do not getthe correct normalization. In fact, AE turned out to bereal, a solution that corresponds to no ground-state deple-tion at all. This result is not correct, and we solved thisproblem by making an explicit normalization the last stepin our solution. This is the way that the diagrams forceus to proceed, and to adjust to this requirement we rede-fine AE to be real. We then must also introduce the nor-malization X, which gives

exp (-iJ AEdt)a0 = X AE, X real. (44)

Here both AE and X are time dependent, even though wehave not shown this dependence explicitly. E and X areunknown, but we use them as if they were known. Theidea is that from Eq. (20) we get AE as a function of theamplitude ratios bn. Then if these amplitude ratios,which are given by the diagrams, are functions of AE,Eq. (20) becomes an equation for AE. If we can solve thisequation, we will have found an implicit way of summingthe original diagrams to infinite order.

Using Eq. (44) in the equations for the amplitudes a,and ao [Eqs. (5)], we get the following expression for theamplitude ratio b:

b = -iX exp(iJ AEdt) f Dio

exp(iAt' + R' - i f, AEdt")x dt'. (45)

; +X+X~~4- + .Fig. 8. Diagrams contributing to bl. All time orders of the ver-tices should be taken. The photon absorption at the A vertex isdrawn only at the rightmost vertex to show the structure of thediagram. This photon absorption can take place at any of theA vertices, and all combinations should be taken.

Fig. 9. Diagrams contributing to b2. All time orders of the ver-tices should be taken. The photon absorptions at the A verticesare drawn only at the rightmost vertices to show the structure ofthe diagrams. These photon absorptions can take place at any ofthe A vertices, and all combinations should be taken.



This expression does not involve approximations yet.Now we want to make a diagrammatic expansion in whichthe time integrals are changed to energy denominators.This step can, of course, be done only approximately, sinceAE and X are unknown functions of time. We foundabove in the discussion of the linked expansion that theconcept of self-energies (E is the ground-state self-energy) is valid only in the case of slow turn-on. Thismeans that the proper approximation at this point is theassumption that AE and X are slowly varying functions oftime. Note, however, that we do not assume that AE issmall or that X is close to unity, so we are not making alow-intensity approximation. In terms of an arbitrarytime-dependent function )(t), we make the followingapproximation to get rid of the time integral:

rt tf exp[1(t')]dt' = exp[(D(t)] f exp[1D(t') - D(t)]dt'

rt= exp[cF(t)] f exp[F(t) (t' - t)]dt'

= exp['D(t)]/I(t). (46)

This approximation contains more than the assumption ofslow time dependence. First, it is assumed that the realpart of cF(t) is such that there is no contribution from thelower limit of the integral. Second, it is assumed that theintegrand is dominated by its values at the end of the inte-gration interval. In our case we have

¢D(t) = iAt + rt - i f AEdt - In X. (47)

We now assume that X is slowly varying and also neglectthe imaginary part that is due to 12 Equation (45) thenbecomes

b = Dio exp(iAt + rt) (48)A - AE

This gives b as a function of AE. When we use this ex-pression in Eq. (25), we get

AE - ID 1012 exp(2rt) (49)

A - AE 1

This is now a second-order equation for AE, and the solu-tion is

A FA2 11~~~~~~~/2AE = 2 ± [- + ID012 exp(2rt) J (50)

This is the same result that we obtained from explicitlysumming the linked diagrams [Eq. (28)]. When we usethis solution in Eq. (48) and calculate the norm from

X = 1 + bl2,

to low intensities, but it means that the best we can dowith diagrams is to find the Floquet state that is adiabati-cally connected to the ground state. Now we find that theapproximation in Eq. (46) gives the same result. There-fore a diagrammatic formalism based on this approxima-tion will also represent the best that we can do.

Our idea in this section has been to assume that theground state has an unknown self-energy AE and then tofind an equation for it. We can also use this assumptiondirectly in the diagrams. The linked expansion in theRWA was shown in Fig. 5. The first diagram gives-Do exp(iAt + t)/A, where A = E - E- co. If theground state has an energy shift AE, we therefore getA - A - AE. When we make this change in the energydenominator of the first diagram, we get Eq. (48). Note,however, that the shift should be made only in the energydenominator, not in the exponential time factor. Fur-ther, in Fig. 6 we show the diagrams for AE. The firstdiagram gives AE = - Dlol2 exp(2Ft)/A. The energyshift in the denominator now gives Eq. (49), which whensolved gives the full RWA solution. With the extra dia-grammatic rule that all energy denominators should beshifted by AE, the RWA solution is then given by the twodiagrams in Fig. 10. There is now one diagram for b andone for AE instead of an infinite set for each. The evalu-ation of the diagrams is in this case quite simple and is avast improvement over the previous analysis. It is clearthat these diagrams are not perturbative but are truehigh-intensity diagrams.

For time-independent interactions the method above isequivalent to using the Brillouin-Wigner perturbationtheory.46 It is therefore clear that the method is nonper-turbative for the two-level system just because there areonly two levels. If the basis set is infinite, however, theBrillouin-Wigner theory is still perturbative, so the prob-lem of summing the diagrams to infinite order will stillremain in the general case. Nevertheless, the discussionabove shows that for the two-level system this is the rightapproach for finding the Floquet solution.

There is another advantage of the method we use in thissection. In the case of the two-level problem we get asecond-order equation for AE, which we can solve analyti-cally. This is not true for the general discrete system, forwhich the equation for AE will be of higher order. It is,however, no problem to solve such an equation numeri-cally, which makes the approach useful in the general casealso. To generalize this method, we must, however, re-member that it applies only if AE is slowly varying, whichit is in the case of the RWA. When we include counterro-tating terms, which we discussed in Subsection 2.C above,AE will contain higher harmonics of the field and will berapidly oscillatory [see Eq. (32)]. We therefore cannot use

(51)

we again find the full Floquet solution that is adiabaticallyconnected to the ground state. That is, the approxima-tion made in Eq. (46) is equivalent to omitting the imagi-nary parts in the energy denominators. We concludedabove that in any diagrammatic expansion with energydenominators one would have to neglect the imaginaryparts of the energy denominators. This does not limit us

I AE =

Fig. 10. Diagrams for the amplitude ratio b and the ground-state energy shift AE in the RWA. If all the energy denomina-tors are shifted by AE, then these are the only diagramsnecessary for the two-level system in the RWA.


b =


fio

0

Fig. 11. Damped one-level system. The continuum states arelabeled by their energy , and it is assumed that one photon isenough to reach the continuum from the ground state.

this method directly. The problem can be solved by goingto a dressed-state picture and, instead of using ao to formthe amplitude ratios by, using only the slowly varying partof ao. Then AE will be slowly varying, too, and it is pos-sible to obtain an equation for it just as before. This gen-eralized formulation is described in part II of this paper.4 7

First, however, we must investigate what the diagramsyield when we include continuum states.

3. DAMPED SYSTEMS

The discussion so far has concerned discrete systems only.We now discuss some of the fundamental problems thatarise when the atom can ionize. In the case of a time-independent field there is no major difference betweendiscrete and continuum states as long as the ground stateis bound and nondegenerate. This is because there areonly virtual transitions to the continuum, and there areno real transitions that would ionize the system. If theexternal field is harmonic, however, the situation is differ-ent. There will always be a multiphoton transition that isenergetic enough to ionize the atom. This fact is enoughto invalidate the adiabatic switching. If the ionizationrate is different from zero, the system will become com-pletely disintegrated at finite times. The mathematics ofthe Goldstone derivation of the time-dependent diagram-matic expansion still applies, however, since it requiresonly that the switching be exponential, not adiabatic.Just as for discrete systems one is forced, in practice, toassume that the turn-on is slow. Above we showed that inthe discrete case this approximation, when the linked dia-grams are used, gives results that are valid at high inten-sities. In this section we show that when continuumstates are present this is not true. The problems, whichare fundamental, appear even for the simplest system andfor any values, no matter how small, of the ionization rate.It is therefore sufficient for us to investigate a few simplemodel systems to illustrate these points.

Let us first discuss a single state interacting with a con-tinuum and assume that one photon is enough to ionizethe system. The system is shown in Fig. 11. The groundstate has index 0, and the continuum states are labeledby their energy e, which varies continuously between 0and o. The interaction Hamiltonian is, as before,

HI = 2D exp(Ft)cos wt. (52)

We assume that there is no static dipole moment in theground state and no coupling between the continuumstates, that is, Doo = Dee' = 0 but Do = Do,* • 0. We fur-ther assume that the coupling to the continuum is weak,so that we need to calculate the ground-state energy shiftand ionization rate only to the lowest nonvanishing order.

The relevant diagrams, those for weak coupling, in theunlinked expansion for the ground-state amplitude aoare shown in Fig. 12. These are the same as the RWAdiagrams for the two-level system, but now we must inte-grate over the continuum in each bubble. The corre-sponding expansion is [cf. Eq. (9)]

ao = 1 + exp(2Ft) f dE JD2

exp(417t) C De'012 de ID o12

(i2F)(i4r)J AetO - iJ A - i3+*(53)

where Aeo = -Eo - w. As in the discrete case, it isnot possible to keep track of the numbers in front of Fin the energy denominators. We therefore make theapproximation

fdE ID fl2 I l 2 _ I, a infinitesimal,

(54)

where I is defined to be the integral evaluated with aninfinitesimal imaginary part. We can then also write

I = Pd f [ + irlD~ol -+E 2, (55)

where 9P denotes the principal part of the integral and thematrix element in the imaginary part is taken at the fixedenergy £ = O + Eo.

The series for a0 can now be summed and gives

ao = exp[ 2F )I = exp[i f I(t)dtl

(56)= exp(-if AEodt - 1 frdt)

where we have defined I(t) = I exp(2rt) and

AEo = -0 J d JDeo 1 exp(2rt)

ro = 2T-D£o0 l=-+E 0 exp(2Ft).

Equation (57b) is a reasonable expression, considering theapproximations made. The ground state is shifted by thetime-dependent energy AEo and is damped exponentiallywith the time-dependent ionization rate F0. Note, how-ever, that the transitions to the continuum occur at theunshifted energy e = w + Eo. A more consistent resultwould have been e = c + Eo + AEo, including higher-

1 + + +

Fig. 12. Unlinked diagrams for the ground-state amplitudewhen the coupling to the continuum is weak.

(57a)

(57b)



order effects. One might think that this shift would haveappeared if we had included more diagrams, but this is notthe case. The shift of the electron peak is included in theRWA diagrams. It is when we made the approximationNP -8 that we lost this shift. To show this, we start byassuming that the energy denominator in I is shifted byAEO, but we still assume that it is a small shift. We canthen expand the denominator in the following way:

!AD 1 - IDI,5+ Efde ID o I2f de A~ ~ ~ fd - (L - iS (AS a+A &Aoi) 2

_ I + AEJ, (58)

where J is by definition equal to the integral with thesquared denominator. Note that this integral is diver-gent. This kind of divergent integral always appears inperturbation theory when an energy denominator for acontinuum state has a shift that has not been includedexplicitly. Such integrals are well-defined as generalizedfunctions, and it is well known how they should behandled.48 We now go back to Eq. (53) and make a simi-lar expansion of the rightmost integral:

I den A -i3r f | den A,° ir + i2f | de(AIDE- ir)2

(59)Putting this expansion back into Eq. (53) and lettingP - in the remaining integrals, we get

exp(2rt) + [i ex2P2rt) 2

i exp(4rt)4p IJ +...

= 1 + i I(t)dt + 2 i I(t)dt] - i f %(t)J(t)dt +

=1 + i [I(t) - I(t)J(t)]dt + 2 [i I(t)dt + -

(60)

where J(t) = J exp(2rt). Comparing Eq. (60) withrelation (58), we see that Eq. (60) implies that the denomi-nator of the first integral is shifted by AE = -I(t).

Just as for the discrete systems, we conclude that withthe unlinked graphs we get only the lowest-order energyshift when we make the imaginary parts of the energydenominators infinitesimal. This limitation is in itselfsufficient for us to abandon the unlinked expansion as acandidate for high-intensity interactions. This conclu-sion is then true for any system, damped or not, and wehave shown above that when we go to the linked expansionthis problem is solved. There is, however, another prob-lem that arises for damped systems, which cannot besolved merely by going to the linked expansion.

Under the assumption that the coupling is weak, so thatit is sufficient for us to calculate the lowest-order shiftand damping, the unlinked expansion gives a qualita-tively reasonable result for the ground-state amplitude[Eq. (56)]. This is not the case if we calculate the ampli-tude for one of the continuum states. Before we show thisstatement to be true, however, we calculate the exactweak-coupling amplitudes directly from the Schridingerequation, so that it is clear what the correct result is.

In the interaction picture the Schrodinger equation forour model system gives

ibe = D, 0 exp(iAdot + t)ao,

iao = | deDo, exp(-iA ot + rt)a,.

(61a)

(61b)

Under the assumption of weak coupling, Eqs. (61) are eas-ily solved by integrating Eq. (61a) and using the result inEq. (61b). We get

iao = f deDo, exp(iAeot + rt)

tx -i J. D,0 exp(iAeot + rt)ao dt

= de IDI exp(2rt) 0 It)ao- (62)

ao is simply taken outside the time integral in the firstline, which is where the weak-coupling approximation isused. The solutions to Eqs. (61) then become

a = -i L Dfo exp(iA/ot + t)aodt

= -i | DO exP[iAeot + rt + i I(t')dt' dt, (63a)

ao = exp[i f I(t)dt' ]. (63b)

We see that Eq. (63b) is equal to Eq. (56), which is what theunlinked expansion gave for ao.

The unlinked diagrams for ae are shown in Fig. 13.When evaluated, they give

Dfo exp(iAeot + rt) _ Deo exp(iAeot + 3rt)'A6 - ir (Ae0 - i3r)(-i2P)

x Idet 1D6 ol2 _ D6 o exp(iAlot + 5t)Ae' - (A0 - i5F)(-i4r)(-i2F)

X de' D6 r f dE" l D6 i, ...A,6o- i3P d i

D6o exp(iA'ot + Pt) D6 0 exp(iAEot + 3t)A60 - ir A60 - i3r

t ~% , 'D 6o exp(iA~ot + 5t)X [ I 2() ] I A- o - i5r

X 2 [i I(t)dt! .... (64)

y E + y :+Vs

p.

Fig. 13. Unlinked diagrams for the amplitude of the continuumstate e when the coupling to the continuum is weak.



If we also make the approximation Nr - 8 in the denomi-nators that are not integrated over, we get

f I(t)dta = - A io exp[iAeot + t + i

A60 - is exp(iAeot + t)ao. (65)

This result is qualitatively incorrect, since the continuum-state amplitude is proportional to the ground-stateamplitude. Among other things, this leads to the noncon-servation of probability, since after an infinite time boththe ground state and the continuum will be empty. Thisexpression for a6 is what one gets if ao is taken outside thefinal time integral in Eq. (63a). The fact that Eq. (65) isincorrect is also shown if we try to use it to calculate thecontinuum probability, since the square of the absolutevalue of Eq. (65) is not well defined when a is infinitesi-mal. Keeping finite does not change the conclusionsabove, however. This result shows that for the continuumstates we cannot transform the last time integral intoan energy denominator. This point is emphasized bymaking a term-by-term time derivation in the series inrelation (64) and then summing it. This procedure gives

a = -iDeo exp(iAeot + rt) - iDo exp(iAeot + rt)

tx i f I(t)dt - iD6 o exp(iAfot + r

t 2X i I(t)dt] *

t

= -iDeo exp iAlot + rt + i f I(t)dt], (66)

which is equivalent to Eq. (63a). The problem arisessolely from the performance of the last time integral. Ifwe keep this integral we get the correct result. Theconclusion is that the expansion for a6 is qualitatively un-physical, but the expansion for ae gives reasonable results.This result is related to the well-known fact that adiabaticswitching can be used to find the ionization rate but notthe ionization probability or the continuum amplitudesthemselves.

Let us now turn to the linked expansion. This is theexpansion of the amplitude ratios b = a,/ao. In theweak-coupling limit we obtain the correct solution fromEqs. (63), which yields

b = -i exp[-i f I(t)dt]

x f Do exp iAeot + rt + i f I(t')dt']dt. (67)

The diagrams for be are shown in Fig. 14. Note that wehave now written the V vertices at the same horizontallevel, but we have explicitly shown the time order ofthe A vertices. This means that all time orders of theV vertices should be taken, but the time order of theA vertices should be fixed to the one shown in each dia-gram. The diagrams all have the same structure as theones for the two-level system in the RWA, and they can be

evaluated with the method discussed in Appendix A. Theonly difference for the two-level case is that for each parti-cle line there is a continuum integration. This situationleads to the fact that diagrams that are numerically equalfor the two-level system, despite being topologically dis-tinct, will give different results for the present system.With this in mind, the method in Appendix A can still beused. When the diagrams in Fig. 14 are evaluated, weget the following series:

b D.0 exp(iA~ot + t) I D,0 exp(iAeot + 3ft)A,0 - ir (A,, - i3r) (Ao - if

X f dE' JD6'o12 _ D, 0 exp(iAeot + 5t)J Ao - ir (A,, - i5r) (A6O - i3r) (Ao - i)

X dE' 1Do1 dE" D6oJ Ao - ir f Ae, - ir

_ D 0 exp(iAeot + 3rt) fde' 1D6'o12

(A.O - isr) (A60 - if) (A6.50 - i) (A6'0 - i3f)

x de 6H1 f ir + .... (68)

In the last term we have a continuum integral in whichthe integrand has two energy denominators with the samecontinuum level. In the higher-order terms in the rest ofthe series there are integrals with any number of energydenominators with the same continuum level. Just as forthe unlinked case, these integrals give energy shifts to thedenominators in the integral with only one denominator.In the weak-coupling limit we neglect such shifts, which,translated to diagrams, means that we keep only the dia-grams shown in Fig. 15. Note that in the unlinked casethe shift of the energy denominators in the continuum in-tegrals appears only when we explicitly keep track of thenumbers in the imaginary parts. Here these shifts arerepresented by new diagrams, so we do not lose theseshifts if we make the approximation Nr , 8. For thepresent argument we do not need to take these shifts intoaccount, so we omit the corresponding diagrams, but, ifwe wanted to include them, we could. In the unlinkedcase this option was not possible. This contrast clearlyillustrates the importance of working with the linkedexpansion.

+ 6 '+ 6

Fig. 14. Linked diagrams for the amplitude ratio b, = a,/ao.The A vertices should be kept fixed to the positions shown in thediagrams, but all time orders of the V vertices should be taken.

Fig. 15. Linked diagrams for the amplitude ratio b. = a,/aowhen the coupling to the continuum is weak. The A verticesshould be kept fixed to the positions shown in the diagrams, butall time orders of the V vertices should be taken.



The diagrams in Fig. 15 give

b~ Dfo exp(iAeot + t) + Dfo exp(iAeot + 3t) IA,0 A - ir (Alo - i)(Afo- ir)

Dfo exp(iAsot + 5t) I2(Ado - i ) (AO - i3F) (Afo - i)

D+ o exp(iAot + 7t) I3- (69)(Afo - i7) ... (Ao- ir) X

where we let Nr -> ( only in the continuum integrals.When we let Nr --> 3 in the denominators that are not in-tegrated over also, we obtain

b D,0 exp(iAEot + rt)Ao - iS

X - GO + 1(t)2_ (t)3 + .[1 A.E_ isj (A~o - i)2 (A.EJ i

D, 0 exp(iAfot + t) (70)A 0 + (t)

When we use relation (70) in Eq. (20) for AE, we get

AE =-|de ID o12 exp(2rt) (71)A 0 + 1() 71

This demonstration shows that, even though we neglectedthe diagrams that contain the shift, the lowest-order shiftstill appears in the final result for b and AE. The transi-tion peak in the continuum now also has a width thatcomes from the imaginary part of I(t). This interpreta-tion might seem physically reasonable, but we show belowthat it is wrong. We note, in passing, that one can antici-pate that, had we included all the linked RWA diagrams,the result for AE would have been

AE = - | de IDo 12 exp(2rt)

Afo - AE(72)

It is not hard to see that a similar derivation, such as forthe two-level system in Subsection 2.D, would give thisresult, too [cf. Eq. (49)]. Once again we therefore findthat the assumption of an unknown energy shift's alreadyoccurring at the outset of the calculation quickly leads tothe same result as taking the limit Nr - S. Even thoughthe result for AE is correct, however, the expression for b,above is just as incorrect as the one in the unlinked case.First, we have a = bao, so we still obtain a continuumamplitude that is proportional to the ground-state ampli-tude, a result that can never be correct. By performing aterm-by-term derivation in relation (69), we get

b = -iDeo exp(iAfot + R) - iI(t)be,

This approach will give the desired effect that the contin-uum amplitude increases when the ground-state ampli-tude decreases. In the case we are discussing b' issimply b' = D,0 exp(iAfot + Rt), which we get directlyfrom Eqs. (63). If we omit the final-state energy denomi-nator, this expression is equal to the first diagram inFig. 15. Thus it turns out that in this case be' could berepresented by a single diagram, if we add the rule thatthe last energy denominator should not be included. Bycombining this expression with Eq. (72), taken to lowestorder, and the definition ao = exp(-ift AEdt), we get thesolution in Eqs. (63).

Finally, let us discuss the meaning of the imaginarypart of a self-energy when it appears in an energy denomi-nator. We have the system in Fig. 16 as a starting point.It is a two-level system in which each level interactsweakly with a continuum. The two levels interacted withtwo different continua that do not couple to each other.The effect of the two continua is to shift and to broadenthe two levels. Without the continua the lowest-ordercontribution to the amplitude ratio b = a /a is

b -Do exp(iAt + ft)/A, A = E - E - to. (75)

To see how the presence of the continuum that couples tothe ground state changes this amplitude, we sum the dia-grams in Fig. 17(a). This procedure gives us

Dio exp(iAt + rt)A + Io(t)

Io = deeo - -- iJ o- Eo- - iS

(76)

The corresponding diagrams for the influence of the othercontinuum are shown in Fig. 17(b). They give us

Dio exp(iAt + t)A - (t)

I =| de1 ID 2 (77)

The total effect of the continua, in the weak-couplinglimit, is then

b = Dio exp(iAt + rt)l\,+ Mot) - Mlt)

The real parts of the integrals I and Io give the energyshifts that are due to the continua, and we see that the

C , -- - - -

. .. .. .. (73)

which is solved by Eq. (67). This result shows that it isnot the series in relation (69) that is the problem, but thestep NF - . This insight does not help us much, since ifwe use the expression for be from relation (70) in Eq. (73)we do not get the correct result for b. Keeping the finaltime integral in b will not help solve this problem. Thesolution is, instead, to make a linked expansion by takingthe ratio b = /ao, which then gives

a,= -i bodt. (74)

so

fio

0Fig. 16. Damped two-level system. The discrete excited statehas index 1, and the ground state has index 0. The continuumstates that couple to the ground state are labeled by their en-ergy eo, and the continuum states that couple to the excited stateare labeled by their energy el. is the laser angular frequency.


(78)

---- ----- - 1


)/+ A |+ W ,, (a)

+ V + +... (b)

Fig. 17. Linked diagrams for the excited-state amplitude thatcontribute to the damping and resonance shift of the two-levelsystem when the continuum coupling is weak. (a) Diagrams thatshow the effects that are due to the continuum that couples to theground state. (b) Diagrams that show the effects that are due tothe continuum that couples to the excited state. In (a) all timeorders of the V vertices should be taken, while all other verticesshould be kept fixed.

resonance energy in relation (78) is shifted by the differ-ence of the shifts of the two levels. This result is what weexpect to find. We also find, however, that, if we inter-pret the imaginary parts of I, and Io as the widths of thetwo levels, we run into trouble, since this interpretationwould imply that the width of the resonance is equal to thedifference of the widths of the individual levels. Thisconclusion would not be correct. On the other hand, it istrue that the imaginary parts of I, and Io are equal to thedamping rates of the two levels. It is just that the correctway to use this information is not to have these rates ap-pear in the energy denominators. In the Goldstonederivation there is no mechanism for treating the realand the imaginary parts of the self-energies differently.Therefore the above problem will appear for all self-energies, and, to get correct results, we have to introducesomething new.

It is an implicit assumption that, for the above formal-ism to work, we must not have any multiphoton resonancesin our system. This prohibition is needed because suchresonances will cause Rabi oscillations, which come fromthe mixing of different Floquet states. We showed above,however, that the results of our formalism will be theFloquet state that is adiabatically connected to the groundstate and is unmixed with any other Floquet state. Inthe undamped case this conclusion implies that the detun-ing from resonance must be large compared with the laserbandwidth, so that there are no resonance effects. Thisconsequence explains why neglecting the imaginary partiNF in the energy denominators leads to a pure Floquetstate. The term iNE just describes the uncertainty in theresonance energy that is due to the laser bandwidth. Fordamped systems there is an additional uncertainty in theresonance energy that is due to the width of each level,caused by its decay. To be sure that there is no mixingbetween Floquet states, we must therefore make sure thatthe detuning is large compared with the sum of all thesewidths. The conclusion is that we should disregard allimaginary parts in the energy denominators, irrespectiveof their origin. However, if the energy denominator is

followed by an integration over the continuum, we mustkeep an infinitesimal imaginary part.

For the two-level system above we get

b ~: -~ D10 exp(iAt + Vt)A + Re[Io(t)] - Re[Il(t)]

(79)

The effects of the damping come from the imaginary partof AE through multiplication by ao = exp(-if t AEdt). Ifwe include only the lowest-order contribution from eachcontinuum, the diagrams for AE shown in Fig. 18 are theones that contain the damping effects. These give

AE = -Io(t) - ADol exp(2t) Mt) (80)

and the imaginary part of this expression gives the weak-coupling damping rate that is due to both continua.

4. DISCUSSION AND SUMMARY

At low intensities diagrammatic many-body theory is auseful tool in the study of many-electron effects in pho-toionization. In this paper we have raised the question ofwhether it can also be used at high intensities. For thetheory to be used at high intensities, two conditions mustbe satisfied; it must be possible, in practice, to sum thediagrammatic expansion to infinite order, and the resultof such a summation must be physically reasonable.

We have chosen to start our discussion by investigatingthe second of these points. The reason is that it would bea waste of time to try to sum the diagrams to infiniteorder if we did not have some assurance that the final re-sult would behave at least qualitatively correctly. There-fore, in this paper we have applied the diagrammatictechniques to model systems in which the diagrams can besummed directly, even though we do not expect this tech-nique to be successful in the general case. As an exampleof a discrete system, we have studied the two-level systemin the RWA, and as examples of damped systems we havestudied one level that is weakly coupled to a continuumand a weakly damped two-level system.

The formal starting point for our discussion was theobservation that in Goldstone's diagrammatic formulationthe exponential turn-on of the interaction is the crucialrequirement. In Goldstone's original derivation the expo-nential turn-on is used as a mathematical tool for a time-independent Hamiltonian. At the end of the derivationthe adiabatic limit is taken, which gives time-independentresults. In our case we have seen the exponential turn-onof the harmonic laser field as part of the physical interac-tion. This change of view, however, does not change themathematics. The only major difference is that we can-

fo +0

Fig. 18. Energy diagrams whose imaginary part gives thedamping of the two-level system when the coupling to the con-tinua is weak.

L. Jonsson and G. Wendin


not take the adiabatic limit, and the results will, there-fore, be explicitly time dependent.

The exponential turn-on of a laser pulse is an idealiza-tion, but we have found that in the case of slow turn-onthere are ways to relax this condition in the final formula-tion. There is no problem connected with the fact thatthe exponential pulse goes to infinity, since the atomionizes and therefore will not be affected by this ever-increasing intensity. In fact, in many of today's experi-ments the neutral atom is ionized long before the intensityreaches its maximum, and the atom therefore exits onlyduring the rising of the pulse. In such a case the expo-nential turn-on is not an unreasonable idealization. Thisidealization, however, is not a crucial problem at thispoint. Mathematically the exponential turn-on is thesimplest, and if we can show that the diagrammatic tech-niques cannot handle this case then it will hardly workfor other amplitude functions. On the other hand, if wefind a way to treat this case properly, there will be cases,especially for slow turn-on, in which the physics is notdrastically changed when the amplitude happens to devi-ate from the truly exponential one.

In the case of the discrete two-level system we foundthat in the RWA the diagrammatic expansion could besummed to give the Floquet state that is adiabatically con-nected to the ground state. This calculation was bestdone in the linked expansion, in which one could neglectthe imaginary parts of the energy denominators and stillget the high-intensity result. This simplification was notpossible for the unlinked expansion. We have also arguedthat it is a necessary condition of the formulation that wecan neglect these imaginary parts. To take them into ac-count leads to calculations that are too complex. There-fore we can say that the requirement of slow turn-on isfundamental to the kind of diagrammatic expansion thatwe have discussed in this paper, an expansion in whichtime integrals are performed to yield energy denomina-tors. The conclusion for discrete systems is then that it ispossible to get high-intensity results but that one cannotrelax the condition of slow turn-on. The resultingdiagrams, when summed to infinite order, then givethe Floquet state that is adiabatically connected to theground state.

For damped systems we have shown that a straightfor-ward summation gives incorrect results. We have found,however, that in the case of the unlinked expansion theerror was due only to the performance of the last timeintegral in the expansion for the amplitudes of the contin-uum states. For the linked expansion the problemswere not solved by retaining the last time integral, how-ever. We concluded that a linked expansion of the timederivative of the wave function would avoid all the prob-lems of the original expansions.

We have also discussed the meaning of the imaginaryparts of the energy denominators. We have shown that inthe Goldstone formulation the imaginary parts of the self-energies give incorrect results when they appear in theenergy denominators, while the time-dependent energyshifts that are due to the real parts of the self-energiesappear in a qualitatively acceptable way. We concludethat the real and the imaginary parts of the self-energiescannot be treated together. The formalism must be re-derived so that the real parts appear in the denominators

but the imaginary parts do not. However, the imaginaryparts, which give the damping rates of the various levels,should appear in the overall normalization factor.

The above arguments then show that, if one makes alinked expansion of the time derivative of the wave func-tion, one will obtain a formalism that works well for thecase of a slowly turned-on harmonic field. Such a formal-ism will describe the atom in terms of a Floquet state withtime-dependent self-energies and damping rates. Wehave also shown that this result is the best that one can dowith the kinds of diagrams that we have used. This con-clusion is then our answer to the question of what the dia-grams give when summed to infinite order.

The question of whether one can actually sum the dia-grams still remains. We have shown, by trying to treatthe two-level system outside the RWA, that the direct sum-mation used in this paper cannot be used in general. Wehave also shown, however, that there is an implicit way ofsumming the diagrams that, at least for the systemstreated here, is much easier than direct summation. Infact, we have shown that with this alternative formulationthe whole RWA solution for the two-level system is repre-sented by only two diagrams, one for the amplitude andone for the energy. The idea is that, if the ground-stateenergy shift is introduced as an unknown function beforethe time integrals are performed, it will automatically ap-pear in the energy denominators. Then the diagrams forthis energy will give an equation that, if it can be solved,will give the full solution. This technique correspondsto using the Brillouin-Wigner expansion in the time-independent formalism. For the systems treated in thispaper this method was much simpler than summing thediagrams explicitly. The method can readily be general-ized to more-complex systems, since, if there is no analyticsolution, the equation can be solved numerically. Thecrucial step in this method was the transformation of thetime integrals into energy denominators. This transfor-mation must involve some approximation, since the inte-grand contains an unknown function. For a generalfunction (t) we conclude that the approximation thatgives the correct results for all the cases discussed is

i I = ej I xp[(t)] (81)

Relation (81) applies only if 1 is a slowly varying functionof time, and it also contains other assumptions. We havenot analyzed this formula in detail, but we have shownthat it gives exactly the same result as the direct summa-tion when the imaginary parts of the energy denomina-tors are neglected. In Part II of this paper" we willderive a diagrammatic formulation based on the describedmethod of performing the time integrals.

APPENDIX A

We now show how the linked diagrams for the ratiob = a/ao can be evaluated when = 0 in all energy de-nominators. The diagrams are shown in Fig. 19. Weshow only the bare diagrams without photon lines. It isunderstood that at every vertex a photon is either emittedor absorbed, and we have to include all possible combina-tions. It is also understood that all time orders shouldbe taken.

L. Jonsson and G. Wendin


t+++..Fig. 19. Linked diagrams for the amplitude ratio b. The photonlines are not shown, but at each vertex a photon is either ab-sorbed or emitted. All combinations should be taken. Further,for each such combination all the time orders of the verticesshould be taken.

We introduce the following notation: A particle-hole-pair annihilation will be called a A vertex, and a particle-hole creation will be called a V vertex. The first diagramin Fig. 19 will be called a V diagram. In any given evalu-ation we refer to the order of the diagram as the numberof V vertices in the diagram. The sum of the whole set ofdiagrams in a given order n will be denoted Dn. We cantherefore write b = Di + D2 + D3 + ... and D = V. Wefurther write

b = (-Dio/A)exp(iAt + Ft)b', (Al)

and a primed variable below will refer to the expansion ofb'. We also define

q = A/(A - 2ncu), (A2)

x = D2 exp(2rt)/A2 . (A3)

Let us now examine the structure of the diagrams in de-tail. If we take a given diagram and remove the A vertexthat has the latest time, we get two parts that are dia-grams of a lower order. Let us call their values b, and b2,respectively. There are a number of diagrams that whenbroken in this way will give exactly the same two lower-order diagrams that the original diagram did. Thesediagrams differ only in the time order of the vertices inpart 1 relative to the vertices in part 2, while the internaltime order of each part is the same. We now use the gen-eral rule that when one adds all time orders (in the abovesense) of two disconnected diagrams, the result is theproduct of the two diagrams.3 7 This rule means that,when we add all the diagrams above, the value of the sum,which we will call b3, is equal to bi, times b2, times thecontribution from the A vertex that connects the twoparts, divided by the final-state energy denominator.The A vertex contributes a factor Do, exp(-iAt + Vt) if aphoton is emitted and Do, exp[-i(A + 2G)t + rt] if a pho-ton is absorbed. Note that the sign is positive, since thereis one minus from the vertex and one from the additionalinternal hole line. When a photon is emitted at the finalvertex we therefore get

b3 =b1b2Do, exp(-iAt + t)

A - 2nwo

ton is absorbed in the final vertex, we instead get

b3' = -xqn exp(-i2wt)bj'b 2 '.

To shorten the notation further, we also define

e = exp(-iwt)

and introduce the operation Q, defined by

Q(e 2,) = qe2,,

(A6)

(A7)

(A8)

which formally describes the process of dividing by thefinal-state energy denominator.

We can now evaluate all diagrams by starting with thefirst-order diagrams, which when combined give all second-order diagrams, and so on. If we symbolize the above pro-cess of combining two diagrams by *, we get

D = V,

D2 = Di *Di,

D3 = D2 *Di + Di *D2 = 2(D2 *Di),

D4 = 2(D 3 *DI) + D2 *D2 ,

D = 2(D 4 *D1 ) + 2(D3 *D2),

(A9a)

(A9b)

(A9c)

(A9d)

(A9e)

and so on; the factor 2 comes from the fact that when wecombine two diagrams from two different orders they givedifferent diagrams, depending on which one is on the leftand which on the right, but these combinations will havethe same numerical value even if they are topologicallydistinct. Algebraically, the operator * is defined, inaccordance with Eqs. (A5)-(A6), as

D.' *Dm' = Q[-x(l + e2 )DnDm'], (A10)

which includes both emission and absorption of the finalphoton. The diagrams contributing to V' are shown inFig. 20. They give

V = 1 + q-,e 2.

For D2 ' we then get

D2' = Dl' *Dl' = Q[-x(1 + e2 )(1 + qje 2)2 ]

= Q[-x(l + 2qj + e2 + 2qle -2

+ q i2e-

2+ q i

2e-

4)]

= -x(l + 2qj + qie2 + 2q-12 e 2

+ q_ 1 3e-2 + q _2q-2e-4).

(All)

(A12)

This expression corresponds to the diagrams in Fig. 21.The diagrams are shown in the same order as the corre-sponding terms. Diagrams within parentheses are equal,and their sum gives the term in the series. We can nowcontinue to generate all the higher-order diagrams by ap-

D10 exp(iAt + t) b 'bA xqn lb2',

(A4)

which gives

(A5)

where n = (net number of absorbed photons - 1)/2 andemission is counted as negative absorption. When a pho-

Fig. 20. Definition of the V diagram, which is the lowest-ordercontribution to the amplitude ratio b.

L. JBnsson and G. Wendin

b3' = -xq.bib2',


+ + + "A +

D5 ' = (lOp 2 + 60p3 + 180p4 + 320p5 + 300p6

+ 180p 7 + 70p8+ 14p9 )x4

and, for b0,

Di' = 1,

D2' = -(1 + 2p)x,

D3' = (2 + 6p + 8p 2 + 2p 3)x2 ,

+KA+We+ +W

Fig. 21. All the second-order diagrams (D2) for the amplituderatio b. In each diagram both time orders of the V verticesshould be taken. Two diagrams within the same pair of bracketshave the same numerical value.

plying the scheme in Eqs. (A9). This will generate all thediagrams for b'. The number of diagrams increases fastwith each order, and it becomes tedious to retain allterms. It is, however, a straightforward method, and wedo not have to worry about identifying all diagrams in agiven order, since the above procedure automatically givesthe correct numerical factor for each term.

In the RWA the procedure is trivial, since then allqnw = 0 and there are no factors containing e, so theQ operator can be excluded. We get

D1' = 1,

D2' = 1 * 1= -x,

D4' = 2(2x2* 1) + (-x * -x) = 5x3,

and so on. Up to the eighth order we get

bRwA' = 1 - x + 2 2 - 5X3 + 14x 4 - 42x5

+ 132X6 - 429x7 + ....

(A13a)

(A13b)

(A13c)

(A13d)

(A14)

Equation (A14) is the result used in Section 2.For the discussion in Section 3 we need the series for

the following cases: b 1 and bo when only q is differentfrom zero; b when only q is different from zero; and b2when only q and q2 are different from zero. The coeffi-cients b are here the slowly varying Fourier coefficientsof b', which are defined through

D4' = -(5 + 2 0p + 40p2 + 46p3 + 20p4 + 4 5 )x3 ,

(A17d)

D5'= (14 + 70p + 180p2 + 300p3 + 320p 4 + 180p 5

+ 60p6 + 1Op 7 )x4 .

In the second case we get, for b1,

Di' = 0,

D2' = -qx,

D3' = (2q + 2q 2)x2 ,

D4' = -(5q + 6q2 + 4q3 )x3 ,

D,5'= (14q + 18q2+ 16q3 + 8q 4)x4

,

D6 ' = -(42q + 56q2 + 56q3 + 40q 4 + 16q 5)x5,

(A17e)

(A18a)

(A18b)

(A18c)

(A18d)

(A18e)

(A18f)

D7' = (132q + 180q2 + 192q 3 + 160q 4 + 96q5 + 32q6)X6,

(A18g)

D8' = -(429q + 594q2 + 660q3 + 600q4 + 432q5

+ 224q6 + 64q7)X7

and in the third case, for b2,

D1 '= 0,

D2' = 0,

D3'= 2qrx2,

D4 = -(6qr + 5q 2 r + 4qr2 )x3 ,

D5' = (18qr + 20q2r + 12q3 r + 16qr2

+ 10q 2r2 + 8qr3 )x4 ,

D6' = -(56qr + 70q 2r + 60q 3 r + 28q4 r + 56qr2

+ 50q2r2 + 24q3 r2 + 40qr3

+ 20q2r3 + 16qr4 )x5 ,

(A18h)

(A19a)

(A19b)

(Al9c)

(A19d)

(A19e)

= b exp(- i2nwt). (A15)

The diagrams that contribute to b for a given n are allthose that have a net absorption of 2n + 1 photons. Toshorten the expressions somewhat, we define p = q,q = q, and r = q2. The termination of each series isdetermined by the amount of labor required. In the firstcase we get, for bL,

(A16a)

(A16b)D2 = -(2p2 + p3 )x,

D3' = (2p 2 + 8p 3 + 6p4 + 2p5)X2, (A16c)

D4 = _(4p 2 + 20p3 + 46p4 + 40p5 + 20p6+ 5p7)X3 ,

(A16d)

(A19f)

D7' = (180qr + 240q 2r + 240q 3r + 168q 4r + 64q5r)x6 ,(A19g)

D8' -(594qr + 825q2r + 900q3r + 756q 4r

+ 448q 5 r + 144q6r)x7 , (A19h)

where in D7' and D8' we keep only the terms that are lin-ear in r.

ACKNOWLEDGMENTSThis research was supported by the U.S. Office of NavalResearch under contract N00014-87-K-0558, the U.S.Air Force Office of Scientific Research under con-tract FQ8671-8801536, the Strategic Defense InitiativeOrganization under contract N00014-86-C-2354, the

(A16e)

(A17a)

(A17b)

(A17c)



Lawrence Livermore National Laboratory under subcon-tract B039918, and the Swedish Natural Science ResearchCouncil.

*Present and permanent address, Institute of Theoreti-cal Physics, Chalmers University of Technology, S-412 96Goteborg, Sweden.

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Diagrammatic many-body theory for atoms in high-intensity laser fields Part I

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Transcript of Diagrammatic many-body theory for atoms in high-intensity laser fields Part I