PHYSICAL REVIEW A 66, 047401 ~2002!

Comment on ‘‘Scheme for multistep photoionization of atoms’’

A. A. Makarov and V. S. LetokhovInstitute of Spectroscopy, Russian Academy of Sciences, Troitsk, Moscow Region, 142190 Russia

~Received 6 February 2002; published 24 October 2002!

In a recent paper, Bo Liu and Xi-Jing Ning@Phys. Rev. A64, 013401 ~2001!# compared a multistepphotoionization scheme where the resonant laser pulses act in consecutive order with one where the same laserpulses act simultaneously. Their conclusion is that the ‘‘consecutive’’ scheme is much more~several orders ofmagnitude! efficient than the ‘‘simultaneous’’ one, which they call traditional. We show, however, that optimumconditions for efficient photoionization with the simultaneous~or traditional! scheme are very different fromthose assumed by the authors. So the difference between two schemes, in terms of efficiency, is of much lowerimportance, if any, than that believed by the authors.

DOI: 10.1103/PhysRevA.66.047401 PACS number~s!: 32.80.Rm, 33.80.Rv

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The authors of the recent work@1# drew a rather unex-pected conclusion concerning the expenditure of laser enin the two- and three-step photoionization of atoms withhigher autoionization state. Their conclusion is that effectphotoionization by means of simultaneously acting lapulses requires an energy allegedly higher by several orof magnitude than that in the case of consecutively aclaser pulses.

Of course, there can be no doubt that the scheme whea laser pulse makes theentire population of the lower levemove to the first excited level, and thereafter another lapulse makes theentire population of the first excited levemove to the next excited one, and so on, is optimal. Moover, the use of such an excitation scheme is absolutely ccal ~for more detail, see@2#! where it is necessary to achievanultrahigh selectivity, for example, with the view to detecting rare radioactive isotopes. However, as for the consution of energy, the attainable gain cannot be so significanthe authors of@1# believe. Their doubtful conclusion resulfrom what they call~but do not prove to be! ‘‘necessaryconditions for efficient photoionization.’’ In fact, for example, in the case of the two-step excitation scheme~see Fig.1 in Ref. @1# and also the diagram of levels in our Fig.below!, the authors’ conditions for efficient photoionizatiolook like

V1;V2;g, ~1!

whereV1 is the Rabi frequency for the first transition,V2that for the second transition, andg the rate of decay of thehigher~autoionization! state into the ionization continuum.is quite obvious, however, that the general condition foroptimal relation between the parameters of the problemquires only that the ‘‘rate’’ of the first transition~the quantityV1) should be approximately equal to the rate of the lasinduced photoionization from the first excited state, whichthe particular case where the desired inequality

V2!g ~2!

is satisfied~we are aspiring to make the laser fields as weas possible, after all! is V2

2/g. So the correct condition fo

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the optimal relation between the problem parameters incase of simultaneous action of two laser pulses must havefollowing form:

V1;V2

2

g. ~3!

Let us illustrate what has been said above by a numerexample.

We take our original equations in the form of the threquations~1! from Ref. @1#, wherein we assume that the eact resonance conditions are met:

iC151

2V1C2 ,

iC251

2V1C11

1

2V2C3 , ~4!

iC351

2V2C22

1

2igC3 ,

where C1 , C2 , and C3 are probability amplitudes of theground, first excited, and autoionization state, respectivWe further assume that the inequality~2! is fulfilled. Thisinequality imposes, to a very good approximation, the cdition of quasistationarity on the third of the above equatio~4!, which leads to the following relation between the proability amplitudesC2 andC3 :

V2C25 igC3 . ~5!

Substituting Eq.~5! simplifies the system of equations~4! asfollows:

iC151

2V1C2 ,

iC251

2V1C12 i

V22

2gC2 . ~6!

It is now quite simple to write down the solution for a stewise pulse subject to the initial conditionsC1(t50)51 and

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COMMENTS PHYSICAL REVIEW A 66, 047401 ~2002!

C2(t50)50 and hence find the photoionization yieNi(t)512uC1(t)u22uC2(t)u2 as a function of the timetelapsed from the beginning of the pulse. We will characterthe process by some ‘‘average decay time’’ of the total polation N(t)512Ni(t) at levels below the ionization limitWe use the natural definition of this time:

t 5*0

`tN~ t !dt

*0`N~ t !dt

. ~7!

We further assume that the Rabi frequencyV1 is fixed andplot t as a function of the rate of photoionization from thexcited stateV2

2/g ~see Fig. 1!. Hence we see that the maxmum rate of ionization~for fixed V1) occurs at some optimaphotoionizing field intensityI opt which corresponds in ordeof magnitude to the estimate~3!, or, to be more exact, afollows from the solution of Eqs.~6!, to the equality

~V2opt!25gV1 /&. ~8!

As for the condition~1! presented in@1#, it deliberatelymakes senselessly steep demands on the laser pulse inties.

FIG. 1. Average decay timet @Eq. ~7!# of the total populationN(t) as a function of the intensity of the laser field inducing ttransition from the excited to the autoionization state. This secpulse laser intensity is plotted in the dimensionless unitsV2

2/gV1 ,

and time in the unitsV1 t . The decay rateg of the autoionizationstate~see the diagram of levels in the right-hand part of the figu!is assumed to be higher than the Rabi frequencyV2 for the transi-tion from the excited to the autoionization state.

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The fact that the photoionization rate drops as the intsity of the second field increases in excess ofI opt appearsparadoxical at first glance only. There is a simple explation: the laser-induced transition from the excited level inthe ~auto!ionization continuum actually results in the broaening of this level, and as soon as this broadening stexceeding the Rabi frequency for the transition from tground state, the excitation rate at the first step drops, ana consequence, so does the rate of the entire photoionizaprocess. This is illustrated in Fig. 2.

The dynamics of~quasi!resonance excitation of multilevesystems by laser radiation was subject to wide discussiothe literature in the past~see@3,4# for review!. At the presenttime, this subject has once again gained in popularity in cnection with the so-called ‘‘quantum control’’ problem, sthat some effects revealed earlier have frequently been recovered. In particular, the effect considered here, wherstrong transition serves as a factor equivalent to the broening of the preceding, weaker transition and thus slodown the entire process, was also noted earlier in@5,6#, alongwith other not quite obvious frequency-dynamical effects

We acknowledge partial support of this work by thRussian Foundation for Basic Research~Grant No.00-15-96612!.

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FIG. 2. Photoionization yieldNi as a function of time plotted inthe dimensionless unitsV1t. The top curve corresponds to the otimal intensity~8! of the laser radiation inducing the transition frothe excited to the autoionization state~the rate of this process,s i I'V2

2/g, is schematically shown in the right-hand part of the fiure!. The bottom curve corresponds to a laser intensity five timehigh as the optimum value.

n

@1# Bo Liu and Xi-Jing Ning, Phys. Rev. A64, 013401~2001!.@2# A. A. Makarov, Sov. Phys. JETP58, 693 ~1983!.@3# C. D. Cantrell, V. S. Letokhov, and A. A. Makarov, inCoherent

and Nonlinear Optics, edited by M. S. Feld and V. S. Letokho~Springer-Verlag, Berlin, 1980!, pp. 165–269.

@4# B. W. Shore, The Theory of Coherent Atomic Excitatio~Wiley, New York, 1990!.

@5# A. A. Makarov, Sov. Phys. JETP45, 918 ~1977!.@6# V. S. Letokhov and A. A. Makarov, Appl. Phys.16, 47

~1978!.

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