Effect of continuum-continuum and quasibound-continuum above-threshold transitionson „111…-photon resonance-enhanced multiphoton autoionization

Krishna Rai Dastidar*Department of Spectroscopy, Indian Association for the Cultivation of Science, Calcutta 700032, India

~Received 27 October 1995!

We have studied the effect of above-threshold transitions on the~111!-photon resonance-enhanced ioniza-tion and autoionization processes. We have shown that the effect of the photoionization decay of the autoion-izing ~AI ! state to the higher continuum becomes dominant when the photoionization width of the AI state iscomparable to or larger than the autoionization width to within an order of magnitude, i.e., for the intermediaterange of laser intensity. The continuum-continuum coupling and the higher-order transitions become effectiveonly for much higher values of the above-threshold transition strengths, i.e., for much higher values of the laserintensity.

PACS number~s!: 32.80.Rm, 33.80.Rv

To obtain amplification of light without population inver-sion, the engineering on the absorption profile is done bychoosing different transition schemes to minimize the ab-sorption in a certain spectral region. In the case of single-photon autoionization, if the autoionization channel and thedirect photoionization channel are of comparable strength,they interfere to give an asymmetric autoionization profile~Fano profile! with a maximum and a minimum@1#. Re-cently, we have shown that by adding one more resonant stepto the process of autoionization and ionization, i.e., for~111!-photon resonant transition, the absorption profile~Adhya–Rai Dastidar profile! is completely different@2#from the Fano profile. Furthermore, by considering the effectof one more autoionizing~AI ! state embedded into the sameand different continuum, the absorption profile has beenfound to be further modified@3–4#. In the present work weshow that both the single-photon transitions above the ion-ization threshold, i.e., the continuum-continuum~C-C! andthe autoionizing-state–higher-continuum~AIS-HC! radiationcoupling, can lead to a significant change in the absorptionprofile. Studies on the above-threshold ionization@5–7# haverevealed that the C-C coupling becomes effective only athigh laser intensities. But we will show here that the AIS-HCcoupling affects the absorption process at much lower inten-sity than that required for the C-C coupling to be effective.

We have chosen a~111!-photon transition scheme~asshown in Fig. 1! for the excitation from the ground stateug&to the AI stateua& and the ionization continuumuc1& close tothe ionization threshold, via an intermediate resonant stateu i &. The AI state and the continuumuc1& are further coupledto the higher continuumuc2& by the single-photon radiationcoupling. The AI stateua& is coupled to the continuumuc1&by the configuration-interaction coupling, and this interactionis responsible for the autoionization decay. We have doneparametric calculations for this three-level, two-continuummodel system and have shown that the absorption profilechanges significantly when the above-threshold transitions~ATT! to the higher continuum become effective.

To obtain the absorption rate for~111!-photon autoioniz-ation and ionization in the presence of single-photon transi-tions above the threshold ionization continuum, we haveused the resolvent operator technique. A set of equations forthe matrix elements of resolvent operators derived from theequation defining the resolvent operator, i.e., (Z2H)G(Z)51, is given as

~Z2Eg!Ggg2DgiGig51, ~1a!

~Z2Ei !Gig2DigGgg2E Dic1Gc1g

dEc12DiaGag50, ~1b!

~Z2Ea!Gag2DaiGig2E Vac1Gc1g

dEc1

2E Dac2Gc2g

dEc250, ~1c!

*Electronic address: spkrd@iacs.ernet.in FIG. 1. Schematic diagram of the model system.

PHYSICAL REVIEW A APRIL 1996VOLUME 53, NUMBER 4

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~Z2Ec1!Gc1g

2Dc1iGig2Vc1a

Gag2E Dc1c2Gc2g

dEc250,

~1d!

~Z2Ec2!Gc2g

2E Dc2c1Gc1g

dEc12Dc2aGag50, ~1e!

whereGpq is the matrix element of the resolvent operatorbetween product statesup& anduq&. The energiesEg , Ei , Ea ,Ec1

, and Ec2are the total energies~photon1atom! of the

product statesug&un&, u i &un21&, ua&un22&, uc1&un22&, anduc2&un23&, respectively.Dpq is the dipole transition momentbetween the product statesup& and uq&. Vac1

is theconfiguration-coupling term between the autoionizing~AI !state and the continuumuc1&.

Solving Eqs.~1!, one can obtain formally the matrix ele-ments of resolvent operators as

Gpq~Z!5f p~Z!

Z2Z0,

whereZ0 is given as

Z05uDgiu2

Z2Ei8

andZ2Ei8 is given as

Z2Ei85Z2Ei2six

1i

2

g i

x2

FDia11

x E Dic1S Vc1a1*

Dc1c2Dc2a

Z2Ec2

dEc2D Y ~Z2Ec1!dEc1G 2

Z2Ea2s21i

2g22

1

x*

Q2

Z2Ec1

dEc1

;

Q5Vc1a1E Dc1c2

Dc2a

Z2Ec1

dEc1,

x511p2uDc1c2u2,

and let us define

R5Dia11

x E Dic1

3S Vc1a1E Dc1c2

Dc2a

Z2Ec2

dEc2D Y ~Z2Ec1!dEc1.

g i52puDic1u2 and si5P * uDic1

u2/(Z2Ec1)dEc1 are the

photoionization width and the ac Stark shift of the stateu i &for the coupling to the continuumuc1&, respectively. Simi-larly, g2 and s2 are the photoionization width and the shiftdue to the radiation coupling of the AI state with the highercontinuum uc2&, respectively. The autoionization width andthe corresponding shift of the AI stateua& due to the configu-ration couplingVac1

with the continuumuc1& are given asgaand sa , respectively. It is to be mentioned here that by thesubstitutionZ2Eg5Z8, one can write the above expressionsin terms of the detuningsdi and da from the statesu i & andua&, respectively.

From the expression ofQ it is clear that the two channelsof ionization of the AI state are present:~i! autoionizationdecay to the adjacent continuumuc1& due to configuration-interaction coupling and~ii ! the decay to the adjacent con-tinuum uc1& by the two-photon transition via the higher con-tinuum uc2&. The first channel is independent of laserintensity, but the second one depends on the laser intensity.

These two channels may compete with each other dependingon the laser intensity and the relative strength of the systemparametersVac1

, Dac2, andDc1c2

. Moreover, from the ex-pression ofR we find that the transition betweenu i & andua&levels can occur via three different channels:~i! single-photon transition between these two levels;~ii ! single-photontransition between the statesu i & and the continuumuc1&, fol-lowed by the configuration-interaction coupling between theAI state and the continuum; and~iii ! single-photon couplingbetween the stateu i & and the stateuc1&, followed by the two-photon transition betweenuc1& and ua& via the continuumuc2&.

In the lower intensity regime, the two-photon and thethree-photon terms involving continuum-continuum couplingcan be neglected, and the rate of ionization in the weak-fieldlimit is given as

dP1dt

54xuDgiu2

g i

~e1q!21y21~y/x!1q2xy

~e12q!21~y1q2x!2,

where y5g2/ga , the relative strength of photoionizationcompared to autoionization of the AI state;e5da/(ga/2) isthe dimensionless detuning from the AI state in units of auto-ionization half width;q5Dia /pDic1

Vc1ais the Fano asym-

metry parameter. In the limiting case, whereuDc1c2u→0 and

uDac2u→0, i.e., the above-threshold transitions are not im-

portant, the ionization rate reduces to the form@2#

2882 53BRIEF REPORTS

dP0dt

54uDigu2

g i

~e1q!2

~e12q!21q4.

However, in the higher intensity regime, where the higher-order terms are retained in the expression ofZ0, the ioniza-tion rate is obtained as

dP2dt

54xuDigu2

g i

~e1q!21y21q2xy1~y/x!~121/q22!1

2

xq2~e12q!1

q

q2~2/q2

2x2q/q222y!

~e12q!21@q2x1y2~2q/q2!#2 ,

where

q25Vc1a

1P*Dc1c2Dc2a

/~Z2Ec2!dEc2

pDc1c2Dc2a

.

The magnitude ofq2 depends on the value ofy anduDc1c2

u2.To demonstrate the effect of above-threshold transitions

on the~111!-photon threshold ionization and autoionization,we have studied the ionization process under four differentconditions: ~i! in the absence of any above-threshold tran-sitions (dP0/dt); ~ii ! in the presence of a single-photon C-Ctransition only, the effect of AIS-HC dipole transition beingnegligible ~dP1/dt, y50!; ~iii ! in the presence of both thesingle-photon above-threshold transitions, neglecting thehigher-order coupling terms (dP1/dt), and ~iv! consideringthe effect of higher-order coupling terms (dP2/dt). We havecalculated the absorption profile, i.e., the absorption rate, as afunction of e for different values ofx, y, and q. But wepresent here the results forq51 only. We have found that thepresence of two first-order decay channels of the AI state—~i! autoionization decay to the continuumuc1& ~AID ! and~ii !photoionization decay to the higher continuumuc2& ~PHD!—causes a spectacular change in the absorption line shape~de-pending on their relative strength! for both the smaller and

the higher values of continuum-continuum coupling. Withthe increase in the strength of PHD compared to AID, theusual interference pattern of the absorption profile is smearedout, leading to a flat absorption profile.

Figures 2 and 3 show the dependence of the absorptionprofile ony, i.e., the relative strength of photoionization andthe autoionization of the AI state, for small and large values,respectively, of the C-C coupling. It is found that for thesmaller values of C-C coupling, the interference pattern ofthe absorption profile is completely damped when the photo-ionization decay width of the AI state is ten times strongerthan the autoionization decay width. This means that the la-ser intensity at which this effect will be prominent will de-pend on the magnitude of autoionization width and the mag-nitude of the AIS-HC dipole transition parameter. If we takea typical value ofga of the order of 10

26 a.u., the value ofg2will be less than or of the order of 1025 a.u. For the highervalues of C-C coupling~Fig. 3!, this type of damping be-comes effective for higher values ofy. Figure 4 shows thedependence of the absorption profile on the C-C coupling forthe smaller values ofy. It is found that with the increase inthe value ofx, the interference pattern of the absorption pro-file is magnified, but there is no sign of damping. Moreover,the appreciable modification of the absorption profile is in-voked only at large values of C-C coupling, i.e.,p2uDc1c2

u2'0.5. Therefore, comparison of Figs. 2 and 4

FIG. 2. Absorption profile~i.e., plot ofdP1/dt vs e! for differ-ent values ofy: L, y50.1; ---.y51.0;!, y55.0,••• , y510.0. x51.000 001 for all four curves. —, plot ofdP0/dt vs e. See text.

FIG. 3. Absorption profile~i.e., plot ofdP1/dt vs e! for differ-ent values ofy: ••• , y50.1; ---,y51.0;!, y55.0;L, y510.0. x51.5 for all four curves. —, plot ofdP0/dt vs e.

53 2883BRIEF REPORTS

shows that the AIS-HC coupling becomes effective at muchlower laser intensity than that required to invoke the C-Ccoupling. Furthermore, the comparison of these three figuresdemonstrates that the photoionization decay of the AI stateplays a dominant role on the absorption profile, regardless ofthe strength of C-C coupling. Figure 5 shows the effect ofhigher-order terms involvingq2 on the absorption profile. Itis found that for higher values ofx andy this effect becomesspectacular, completely inverting the curvature of the absorp-tion profile.

In conclusion, we have shown here that the above-threshold photoionization decay of the AI state plays a domi-nant role in the absorption profile for~111!-photon

resonance-enhanced ionization and autoionization processeswhen the photoionization decay width of the AI state is com-parable to or larger than the autoionization decay width towithin an order of magnitude~i.e., for the intermediate rangeof laser intensity for a typical system!. The C-C coupling andthe higher-order transitions become effective only for highervalues of laser intensity.

This work has been sponsored by the Department of Sci-ence and Technology, Government of India, under ProjectNo. SP/S2/L-20/90.

@1# U. Fano, Phys. Rev. A124, 1866~1961!.@2# L. Adhya and K. Rai Dastidar, Phys. Rev. A50, 3537~1994!.@3# S. Sanyal, L. Adhya, and K. Rai Dastidar, Phys. Rev. A49,

5135 ~1994!; L. Adhya, S. Sanyal, and K. Rai Dastidar,ibid.52, 4078~1995!.

@4# L. Adhya and K. Rai Dastidar, Nuovo Cimento14D, 643~1992!.

@5# J. H. Eberly, J. Javanainen, and K. Rzazewski, Phys. Rep.204,331 ~1991!, and references therein.

@6# K. Burnett, V. C. Reed, and P. L. Knight, J. Phys. B26, 561~1993!, and references therein.

@7# S. Cavalieri, R. Eramo, and R. Buffa, Phys. Rev. A51, 2974~1995!.

FIG. 4. Absorption profile~i.e., plot ofdP1/dt vs e! for differ-ent values ofx: !, x51.05; ---,x51.1; ••• , x51.5. y50.01 forall three curves. —, plot ofdP0/dt vs e.

FIG. 5. Absorption profile with~plot of dP2/dt vs e! and with-out ~plot of dP1/dt vs e! higher-order coupling terms~HOCT! fordifferent values ofx: —, without HOCT; and!, with HOCT forx51.05; ---, without HOCT; and1, with HOCT for x51.1; ••• ,without HOCT; andL, with HOCT for x51.5. For all curvesy510.0.

2884 53BRIEF REPORTS