the effect of continuum-continuum transitions on laser-induced continuum structures

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The Effect of Continuum-continuumTransitions on Laser-inducedContinuum StructuresM.E.St J. Dutton a & B.J. Dalton a b ca Department of Physics , The University of Queensland ,Queensland, 4072, Australiab Optics Section, Blackett Laboratory , Imperial College ofScience, Technology and Medicine , Prince Consort Road,London, SW7 2BZ, Englandc Clarendon Laboratory , University of Oxford , Parks Road,Oxford, OX1 3PU, EnglandPublished online: 12 Mar 2007.

To cite this article: M.E.St J. Dutton & B.J. Dalton (1993) The Effect of Continuum-continuumTransitions on Laser-induced Continuum Structures, Journal of Modern Optics, 40:1, 123-162,DOI: 10.1080/09500349314550141

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JOURNAL OF MODERN OPTICS, 1993, VOL . 40, NO . 1, 123-162

The effect of continuum-continuum transitionson laser-induced continuum structures

Strong probe case

M. E . ST J. DUTTONDepartment of Physics, The University of Queensland,Queensland 4072, Australia

and B. J . DALTONtOptics Section, Blackett Laboratory,Imperial College of Science, Technology and Medicine,Prince Consort Road, London SW7 2BZ, Englandand Clarendon Laboratory, University of Oxford,Parks Road, Oxford OXI 3PU, England

(Received 13 July 1992; revision received 26 August 1992)

Abstract. Laser induced continuum structures (LICS) can be produced by astrong laser embedding an excited atomic bound state into a flat atomiccontinuum, leading to a tunable resonance of adjustable width that can be probedby a second laser . If this second laser is of similar intensity to the embedding laserthen the distinction between the two becomes artificial and saturation effectsbecome important. In this paper the effect on LICS of transitions caused by bothlasers between the original atomic continuum and a second atomic continuum arestudied in the Markoff approximation by means of Laplace transform methodsand allowing for the ionization of each atomic state by both laser fields. Formalresults correct to all orders are also given in terms of a T-matrix approach .Numerical calculations are presented showing the effects on the LICS resonantstructures of continuum-continuum coupling processes . Significant changes tothe Fano profiles (second laser weak) and to coherence holes (second laser strong)occur. Analytical results are also given .

1 . IntroductionSingle quantum states coupled to a flat continuum of states can result in a

structure with interesting properties . In autoionization [1] a discrete atomic statecoupled via intra atomic interactions to atomic continuum states can be probed fromthe atomic ground state using a weak laser field, and the resulting ionization rateversus laser detuning curve is the asymmetric Fano profile [1] . Less familiar are laserinduced continuum structures [2,3] [LICS] in which effectively an autoionizingstate is created by coupling a higher (embedded) bound atomic state to thecontinuum via a second strong laser . However, as in autoionization, the basic physicsinvolves a quantum mechanical interference between processes starting and endingin the same pair of states and leading, in the weak probe laser case, to an enhancement

j' Permanent address : Department of Physics, The University of Queensland, Queensland4072, Australia .

0950-0340/93 $10. 00 1993 Taylor & Francis Ltd .

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M. E. St J. Dutton and B. J. Dalton

or a diminution of the ionization rate according as this interference is constructive ordestructive . At the position of the Fano window all the population is trapped in theweak probe case . LICS effects were first observed in polarization experiments [4],later in third-harmonic generation [5, 6], and recently in ionization experiments[7, 8, 9] . LICS phenomena have been reviewed recently [10, 11] .

The effects on the basic LICS phenomenon of factors such as photo ionization ofthe higher or ground atomic state by the other laser [11], Raman processes involvingpathways between the ground and embedded states via other discrete states ratherthan the atomic continuum [12,13], phase fluctuations in the laser field [14] andpulse shape effects [11, 15] have also been considered. Generally the consequence ofthese contributions will be to partially destroy the LICS interference feature, butthey can be minimized with careful system choice . If the first probe laser becomesstrong, saturation and more complex partial population trapping effects [10, 16,17]can occur. As the distinction between probe and embedding laser disappears theFano profile curve for the ionization rate changes until for equal ionization rates forthe ground and embedded states (ignoring the complications described in theprevious paragraph) half the atomic population is trapped and the coherence holeshifts to the position of zero two photon detuning . In this situation the phenomenumcan be described in terms of two photon dressed states [10], in which the coupling ofthe ground and embedded states via the continuum is described by the two photonRabi frequency [18,19] .

The complicating effects on LICS phenomena of transitions induced by theembedding laser between the original atomic continuum and a second atomiccontinuum (for example, with a differing angular momentum quantum number)have been studied previously [20], using Laplace transform techniques andinvolving a Markoff approximation [21] . However, the analysis was restricted to theweak probe, strong embedding laser case, although the formalism allowed forcontinuum-continuum coupling due to the strong laser to become comparable withthe continuum-bound state coupling . General results for this case were given interms of a T-matrix [22,23] approach. The pure weak probe LICS case would beobtained by setting all continuum-continuum matrix elements to zero . Systemsinvolving real autoionizing states and two atomic continua have also been studied[1,24] .

The process whereby the system makes a transition to the second continuumis an above threshold ionization (ATI) and such phenomena have been studiedextensively [25, 26] in terms of calculations of the photoelectron spectrum . Here theemphasis is on the bound states and total ionization probability only .

In the present paper the treatment of continuum-continuum coupling isextended to allow for the situation where both lasers can be strong . The methodsused are the same as in the previous paper. As both lasers can now couple the boundstates to the continuum states and continuum states to each other, processesassociated with a given power of the laser intensity now involve a much greaternumber of contributions. The time development of the Fano profile (weak probecase) and the coherence hole (strong probe case) is examined numerically in the pureLICS situation, where continuum-continuum coupling can be ignored, such as withlow overall intensities for both lasers . Numerical calculations are also carried out forthe situation where the continuum-continuum coupling becomes important. Thesecalculations are presented in terms of the effects on the ionization rate and ionizationprobability versus two photon detuning curves (all suitably scaled) as the overall

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Laser-induced continuum structures

1 25

intensity of both lasers is increased by two orders of magnitude (K= 102), (alongwith a similar reduction in the time scale), thereby changing from a situation wherecontinuum-continuum coupling is negligible to a situation where second-ordercontinuum-continuum coupling is of moderate size. This study is presented in termsof a possible experimental investigation of the effects of continuum-continuumcoupling, and the results show quite distinctive departures from the usual Fanoprofile (weak probe case) and coherent hole (strong probe case) depending on thedetails of the second-order continuum--continuum coupling parameters .

In Section 2 the basic LICS model and theoretical analysis is presented . Resultscorrect to the first and second powers of the intensity are given along with numericalstudies in Sections 3 and 4 . Section 5 deals with the results correct to all orders in theintensity via a T matrix approach . Conclusions are drawn in Section 6 .

2. Theory of LICS2.1 . LICS model and approximations

The model is shown in figure 1 . An initial atomic state 10> is coupled to atomiccontinuum states i f > via lasers a, b of frequencies wa, wb and polarizations sa, sbrespectively . This continuum is also coupled to a second bound state 11 > by the samelasers and with frequencies such that

wO+Ulbti0)1 +Wa •

( 1 )

In our extension to the basic LICS situation a second atomic continuum with statesIg), differing from the first say via angular momentum quantum numbers, is alsocoupled to the first continuum via both lasers .

In the simple weak probe LICS model laser b say would be weak and laser astrong. In this situation laser b acts as a probe of the structure embedded in thecontinuum via the strong laser a . In the general case treated here the intensities of a, bcould be comparable and the simple LICS interference between the processes

10 ; nanb> bif ; nanb

1'>

1°>

Figure 1 . Simplified atomic-level scheme, showing basic LICS system of initial atomic state10) coupled by lasers a, b to atomic continuum states If) to which a second bound stateI1) is coupled due to lasers a, b. A second continuum fig) is coupled to the first via lasersa, b . The frequencies are such that mO+(Ub is approximately o) 1 +w so that some of theprocesses shown are not energy conserving .

10

(2 a)

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due to laser b causing transitions between the bound states, and processes such as

and

I0; nanb)b+If; nanb-1 a+Ig; n a-1 nb-1 °* If ; nanb - 1X (2 d)

I0;nanb>-+If ;nanb-1 > b+Ig;nanb - 2>b+If ;nanb - 1>,

(2e)

due to transitions between the continuum states caused by either laser . Here na and nbdenote the number of photons of frequencies Wa , wb respectively .

In the model the following approximations will be made :

(a) Raman processes involving virtual bound states of the same parity as if > aredisregarded;

(b) phase fluctuation effects in the laser fields are ignored ;(c) the rotating-wave (RWA) and electric-dipole approximations [21] have been

used;(d) the no back reaction approximation is used ;(e) a Markoff type approximation for treating the coupling to the continuum

states will be used;(f) the secular approximation involving the inclusion of only processes that are

approximately energy conserving overall (so called resonant processes) willbe made;

(g) spontaneous emission processes are ignored ;(h) continuum threshold effects are not treated ; and(1) ATI effects are weak .

For simplicity dipole matrix elements will be taken to be real implying that thephotons have linear polarizations . The same approximations and assumptions wereused previously [20] . However the previous assumptions of ignoring ionization ratesand level shifts of I0> due to laser a and of I1 > due to laser b will not be applied. Wenote that the interaction V (below) only allows 0-+f, 1--+f, f--+g processes viaabsorption (the reverse via emission) . The basis of this RWA approximation is that forprocesses conserving energy and involving continuum states not too far abovethreshold, other possibilities do not occur. For example f--+g via emission would putg below the continuum threshold . The neglect of such non RWA processes will ofcourse affect the frequency shift terms .

Initially the atom is in state IO>, and the two laser fields are in coherent states ofthe form

~,,, C(nanb)inanb>,nanb

with mean photon numbers k, nb>". The corresponding intensities are Ia and Ib ,which are proportional to na/V and nb/V respectively, where V is the volume for thelaser fields.

126

and

M. E. St J. Dutton and B . J. Dalton

IO;nanb> -If.,nanb - l> °~I1;na+lnb-I> a If; nanb- t> (2 b)

is affected by other processes such as

IO;nanb>b_If;nanb - 1>b+Il ;nanb>b+If;nanb - 1>, (2c)

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The unperturbed Hamiltonian takes the form

Ho=Will ><1l+hwoIO><OI+ JdWirhWIf>f <fl

+ Jdo)g hwglgXgl+atahwa+btbho)b ,

(3)

where a, at, b, and bt are the annihilation and creation operators for the laser fields aand b respectively . The continuum states are normalized as <fl f'>=6((of-w'f) and<gig> =8(wg-w'a) .

The interaction Hamiltonian has the form

V= Jdwf (aIf><1Ihi+bIfX1ih4)Vf

+ Jdw, (alf><Olh~af+blf><Oih~f)

•

JJdwi dwg (a>(I o+bXfi g)lgfOflg0f+

Hermitian conjugate,

(4)where

(i)

hXf=-i<fld •s.ll> 2 V)

(5 a)

1/2(ii)

h~`f=-i<fld-seiO> 2~V) ,

( 5 b)0

llz(iii)

h6fg = - i<gld • 8.1f> 2s ) ,

(5 c)0

with c =a, b . The dipole matrix elements will be assumed real .In terms of the essential states approach [18], the system state vector is

expanded as

Iv >= FF ,, ,,b exp( -lWO9anbt)IOnanb>aanb

•

Jdw, Clnanb exp( =lwfnanbt)Ifnonb>

•

Jdcog CS„an b exp (-iwgnaft t)gn anb>

+ Cl-.-b exp (- iw 1 n,nbt)I l nanb>J,

(6)

where wb ,,nb=wl +nawa+nbwb is the total angular frequency of the state linanb>(i=0,1,f,g) .

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M. E. St J . Dutton and B. J . Dalton

2.2 . Coupled amplitude equationsSubstituting into the time dependent Schrodinger equation we then have a set of

coupled equations for the amplitudes Cb,oab in the interaction picture .

Coa,nb = -if[e(na)1/2 exp (iaoft) Ctj,=lnb+l;I(16) 1/2 exp (iB1t) CJn.n e=1] dwI

(7 a)

+

Cj'nQ„y = - I l, J. (na +1) 1/2 (eXp-i6 ft)COea+Inb+ 4(nb+ 1) 1/2 exp ( - iSOft)ConOb+1

Af(na+ 1) 1/2 exp(-i81ft)C1 n,+1 nb+ 2f(nb+ 1) 1 /2 exp ( - i81ft)ClR,, +I

+ J[o(na)h12

jgeXp p;,t)Cg.;=.T.+ ®(nb)1/2 eXp (icfat) Cg,nb=I ] dwe1,

(7 b)

ega ,ab =_iJ[&fg(na+1) 1 /2exp (- i8fgt)Cfn,+1 +Ofg(nb+ 1) 1 /2

exp( - i8fgt)Cfibn b+1 ] dwf,

(7 c)

C1n,a b = -1f

[.1f(na) 1 J2 exp (i81ft) Cfna=1 +Af(nb ) 1 /2 exp (i81tt) Cln,nb=1] dwt,

(7 d)where the detunings are given by

bi = CO;-C0;+C0"

c=a,b,

i,j=0,1,f,g.

(8)Assuming that the laser fields are initially in coherent states laa>, Iab> with

ot,=(nn) 1/2 exp(-i4,) (c=a,b) and n,>>1 we can apply the approximation thatimportant na, n b are clustered around na , fib , with standard deviations (n.)1/2, (1b)1/2that are much smaller than the mean photon numbers . Hence it will be a reasonableapproximation to replace the factors .Jn,, (n,+1) 1 J2 by Iii,, (n'+ 1) 1/2 so that thecoefficients in the coupled amplitude equations will be essentially independent of naand nb .

In this case we can factorize the state amplitudes via the no-back reactionapproximation .

Cb,,ab = C; X C(na, nb),

(9 a)

C(na, nb) = <nnl aa><nbIab>,

(9b)

where the atomic factors C, (i = 0,1j, g) are independent of the photon numbers . Wenote that

E IC(na, nb)I 2 =1

(9 c)n,.nb

ClearlyC(na ±1,nb)xC(na,nb ) xexp(-F-iq ),

(10a)

C(na, n b ± 1) ti C(na , n b ) x exp (+ upb ),

(10b)

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and together with the initial condition of the atom being in state 10) we findCo(0)=1,

(11 a)

C,(0)=0,

(i=l,f,g) .

(11b)

We write

X1 f=(n,) 1/2Afexp (- ipc),

(12a)Xof=(nc)1JZzef exp (-i p,),

(12b)

Wf,=(nc)1~2b9te exp ( - i(pc),

(12c)

where the magnitudes are proportional to bound-continuum and continuum-continuum Rabi frequencies . Substituting the no back reaction approximation intothe coupled amplitude equations we obtain coupled equations for the atomicamplitudes.

(C is the contour from + oo to - oo just above the real axis) we obtain the followingequations for the Laplace transform of Co , . . . ,Cg from the coupled equations for theamplitudes

WO(Q) =1+ J

dwfEXafCf(Q+8of),

(15 a)C

QC1(Q)= JdoofyX`fCf(52+81J),

(15b)C

60 = -i I

JdoJf[EX 1a exp (i8oft) ]Cf},

(13 a)

6 1 = - i {J

dcof[EXa exp (i81ft)]Cf},

(13 b)

Of = - i SlCc~X of exp ( -8oft)]Co

+[>X1if

ex (-i811.t)JC1

c

+ Jdwg[~ g~ fexp (i8't)]Cg },

(13 c)

Cg = -i{JdwfCE®jg exp (-i8fgt)1C1.}c

C

(13 d)

2.3 . Bound atomic amplitudes-Laplace transform solutionsDefining the Laplace transforms via

d(Q) = dt exp (iQt) a(t),

Im S2, 0,

(14a)f w0a(t) = dQ exp (- iQt) a(Q),

t ,>O,

(14b)ti fc

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QC!(Q)=EXofCo(Q-bof)+EXi1C 1(o-bl !)+ fdwa~efaC a (S2+Sfg), (15c)

S2Ca(S2)- fdDf &qgCf(Q_~ g ) .y`f(15d)

Then solving formally for Gel , Ca we obtain

C,!(Q)=E °J Co(Q-bo!)+E ~tl!C 1(S2-bl !)+ Jd(O9 Eec9Ca(Q+o 9),~

l (16 a)

and

('`a (S2)= Jdwh OCh(Q_).

(16b)

Eliminating 0, gives

Cf(12)=> 'Of Co(Q-8o!)+Eii!C1(S2-bl!)

+EEJ Jdw,,dwaaefaa +bfged gCk(ll-b,'e +b !g) .

(17)

It is now convenient to introduce the notation

Aio= Slo! '

Xi!Ac - 62 ,

c' d

J%S2

dwaS2+b`~,

(c, d=a, b) .

(18c)!e

Alsob

c

d!B -(ShB - wl+wc - wh -wd,

so that

C!(Q)=E

Aft(Q)C(Q-bid!)+~ dw,,J`%(Q)Ch(Q+t5%),

(20)d i=0,1

cd

and which is an integral equation for the Laplace transforms of the atomicamplitudes of the first continuum states C1(Q) .

Also

C

(18a)

(18b)

d2Co(Q)-E fdw1 Xo1CI(Q+bo!)=1,

(21 a)f

QC 1 (Q)-Z Jdof X"j*fC1(Q+bl!)=0 .

(21 b)

As the factor J% is proportional to the continuum-continuum Rabi frequencysquared we can develop a series solution for Cc(Q) by iteration of equation (20) .

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With

we find that

and


131

dCf~(Q)= Af,(Q)C,(Q-bJ)=

it Cr(Q-Sf),

CIf+1)(Q)=Cf) (Q)+Ef do,

Jfh(0)Chn)(0+afh),

Xd'~Jl(S2)= If, Cj(62- ajj')'j 62

d2+

fdo)f,

X' jJ2 C/Q + 5c2u1 _ Sjlz+, s zd '

)czd'd2j

I'Iz+

f fdof2 dwl3 Jff2(f2)Jc?f3(Q + 6`'i )czd1 c3dz d3jXd3

(62+brI2 +SJ f3) ~j(Q+8fIz+SII3- ajf3)

+ y . . Y i f J . . . Jdwi2 d(,)13 . . . dcol.CA c3dz

C.d. -' dnjJc2d ' (62)J`3d 2 (Sa + tSca dj' ) • • Jc'd^- (Q+SJIz+ . . .+flJ~ hZ ' )I't2

Izt3

I' x

L,-'

Xd„

(a+al2dfz+jh+b`f °'Iq)

Cj(~+afi2+ . . .+bJ °,f. (23)

Substituting this solution for CI into the equations for CO, C1i we then obtain

1=QCo(Q)-E YX$(1)Do(1) Yj(1)Cjo(1)1 j

- Z ZXI(1)Jo(1, 2)Do(2) Yj(2)C(2)1,2 j

- E EXI(1)Jo(1, 2)Jo(2, 3)Do(3) Yj(3)C(3)1,2,3 j

Y X I(1)Jo(1, 2)Jo(2, 3) . . . Jo(n-1, n)Do(n) Yj(n)c(n) . . . , (24a)1,2 . . . .n j

(22 a)

(22 b)

o=ac 1(62)- I1~Xr(1)D1(1)Y/l)Cj(1)

- I EX?(1)J 1 (1, 2)D1(2)Yj(2)Cj(2)1,2 j

-1 3EX?(1)J1(1, 2)J1(2, 3)D1(3) Yj(3)C(3)

- E >X?(i)J1(1, 2)J 1(2, 3) . . . J1(n-1, n)D 1(n)Yj(n) (n) . . . , (24b)1,2 . . . .n j

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1 32


where

and

Xr(1)=Xij,,

i=0,1,

(24c)

Yj(n)=Xjfn

n=1,2, . . ., j=0,1, (24d)

D i(1)=(ll+ },) -1 ,

i=0,1,

(24e)

Di(n)=(LI+af,+8?df2+ . . . + bf"if")-1, n=2,3, . . ., i=0,1, (24f)Ji(1, 2)=J`fI2(Q+8i.,),

i=0,1,

(24g)Ji(n-1)=Jf.n~ fn(~+ Sih+ af1f12+ . . . +fn 1,, 1),n=3,4, . . ., i=0,1, (24h)

C(n)Z`,{st+aif,+af f2+ . . .+a1,, „-Sfp ), n=1,2, . . .,ij =0, 1, (24 :)

Jdw 1i .1

cjd1 (241)

_ Y, E . . . EJ

$ . . . $ do)f , dwf2 . . .dwfn .

(24 k)1 .2 . . . .n ~ctd,c2d2

cndnIn these expressions c 1 , . . . , cn and d1i . . . , do are either a, b with the convention thatthe cs are associated with absorptions, the ds with emissions.

Some of the frequency factors can be simplified :

1 ac, + aced, + . . bcndn -1 - sdn( ) if,

f,f2

. f.-If.

Jfn= (wi + coc , + coc2 + . . . + wcn) - (wJ + wd, +wd2 + . . . + (Odn)

=Owij(c 1 . . . cn ; d1 . . . dn ) .

(25 a)and this equals the frequency change in the overall process i-*j via the absorption ofphotons c1, C2 . . . . cn and the emission of photons d 1 , d2 , dn .

(11) 8if1+alf2+ . . .+lSf.-if.

_ (wt + wC , + . . . + (0cn) - (wd1+wd2+ . . . + cod, -,+ (Of,)=0wifn(c 1 . . . c„; d1 . . . do-1),

(25 b)

and this equals the frequency change in the overall process i-+fn via absorption ofphotons c1 , . . , cn; emission of d1i . . . , do_ 1 .

The positive frequency component of the c mode field is given byho)c

Ec-1

(2s Vac 8c ,

( 26)0

and the average value of E, in the coherent state Ja c> is1/2

ZEwV) ~n~ exp(-i4,,)ee

(27 a)0

_ - 1290c8c,

(27 b)

where 90c is the complex amplitude .

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The various matrix elements can be written in terms of the complex amplitudesand dipole matrix elements. The factors to, x dipole matrix element h areone-photon Rabi frequencies .

(i) Xif1 = 2 c' < iid - e* Ifs >,

(28 a)

(ii) X17"= 2 " <f„ Id - sd"11>,

(28 b)

(iii) Wfe = 'roc <gld - ELI f >,

(28 c)

1(iv) Jc„d"-1 (Q+acl + . . .+,5C-1d-2 -f"-J"

if1

2fn-1) -+eo)ifn-jC 1 ,

f

.91e"<f_1Id -alga-1><g_1Id-$d"-1f >'odn-1

(28d)x dcaa"-'2h [fl+Ocoi,"-1(c1 . . . c,, ; d 1 . . . d„ -2)l

2h '

where

OCOia "_ 1(c l . . .c„ ; dl . . .d.-2)

=(c)i+coc1+CO~2+ . . .coC")-(C)dl+wd2+ . . .+COd„-2+CO8"-,), (28 e)

and this equals the frequency change in the overall process i-+g„ _ 1 via the absorptionof photons c 1 , c2 , . . . , c„ and the emission of photons dl , d2 , . . . , d„_2 .

It is now useful to specify more precisely theprocesses beginning and ending withthe bound atomic states 0, 1 . In doing this we now state the intermediate continuumstatesf, g involved along with photon types a, b that are absorbed or emitted en route .The processes that overall are approximately (in view of equation (1)) energyconserving will be referred to as resonant processes, the others as non-resonantprocesses. Note that even a resonant process could involve a non-resonantintermediate state .

A process P2i (i;j) beginning with bound state i and ending with bound state j andinvolving 2n photons is written as

i

fl

gl

f2

92

gn-1

fn

i

L cl

I f cj-- if dl __ _I- I f c3-- I. itd2-.. . . ._-0 . I Ldn- f- J tdni ,

and involves i-+f1 (absorb c1);f1 -'g1 (absorb c2) ; g1 - 'f2 (emit d 1 ) ;f2-+g2 (absorb c3 ) ;

g2-*f3 (emit d2 ) ; . . . gg_ 1-if, (emit dp_ 1 ) ;f„-ij(emit d.) . Note that the first two stepsinvolve absorptions, the last two involve emissions and in the middle an alternationbetween absorption and emission occurs .

Examples of such processes are shown diagramatically in figure 2 . Figure 2 (a)shows the case of P4 (0, 1) involving photons baaa in the above order . This would be aresonant process and would be a four-photon Raman process. Figure 2 (b) shows thecase of P4(0, 1) involving photons abab in the above order. This would be a non-resonant process. Figure 2 (c) shows the case of P2(0, 1) involving photons ba in theabove order. This would be a resonant process and would involve a two-photonRaman process. It is this process that produces the fundamental contribution toLICS .

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1 34

M. E. St J . Dutton and B . J. Dalton

a(4)

a(2)

I1)

a 19>

It> -

I 1>W)

a(l)

b(1)

(b)

II)

(C)

19>

I9>

Figure 2 . Examples of processes P21(i, j) beginning with bound state i and ending withbound state j and involving the absorption of n photons, and the emission of n photons .Upward arrows indicate absorptions, downward arrows emissions . In (a) the resonantfour photon Raman processes P 4(0,1) involving photons baaa is shown. In (b) the nonresonant four photon Raman process P4(0;1) involving photons abab is pictured. In (c)the basic LICS process P 2(0;1) involving photons ba is shown .

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135

The frequency shift for the process P2n(i, j) is given by (25 a) as

Awp=Aw j(c j c2 . .j

c" ; d, d2 . . . d") .

(29)

Note that in this notation we have written absorptions first, emissions second in theorder they occur. The actual order of photons involved is c l c 2d ic 3d2c4 . . .c„d„_ 1 d" .

In terms of these processes the equations for Co, C1 may be written as :

f2Co(Q)- F, Aoo(12)C0[f2+Awoo(c1 . . . c"; d1 . . . d„)]P2"(o,0)

Ao (Q)C1[f2+Awo1(c1 . . .c";d, . . .d")]=1,P2"(o,1)

5201(0)-

A 2100(Q)Co[Q+Aw1o(c1 . . .c"; dl . . . d")]P2,,(1,0)

- E Aii(Q)C1[f2+/iw11(c1 . . .c,, ;d1 . . .d,)}=0,

(30b)P2,(1,1)

where the sums are over all processes .In these expressions the matrices A lj•" (0) are given as follows :

A, (S2)=

J f . . . fdwf , dw9i dw12 dwa2 . . . do),. - I dwf"cl . . .c" d, . . .d"

g&, < 17d •8*,Ifl> gb`~2 <f11d'8"lg1>X 2h1 [S2+Awjf,(c1)] 2h [52+Awj,,(c1c2)]

hod,<g1Id - ed,If2>1`c,<f2Id •8e3152>X

2h [52+Aw,f2(c1c2 ; d1)] 2h [S2+Aw;H2(c1c2c3; d1)]

hod,<g2ld - sd21f3>e&,<f3ld - 8c;1g3>X

2h [t2+Awif3(c1c2c3; d1d2)] 2h [S2+Awjg3(c1c2c3c4; d1d2)]

90d3<g31d-8d31f4>gsc"<fn-1Id.Bcjg -1 >X

2h [S2+Aeoif,(c1c2c3c4; d1d2d3)]

2h [f2+Awj,"_,(c1c2 . . . c"; d1 . . . d"_ z)]

X god"-,<g"-IId-8d.,If.>'god' <f"Id,ad.17>

(31 a)2h [d2+Awlf"(c1 . . .c" ;d1 . . .d"_1)] 2h

J

The matrices Aj"(S2) are the sum over terms associated with specific photons

A 91°(S2)=

A "(cjc2 . . . c" ; d, d2 . . . d" ; S2) .

(31 b)cl . . .c,. d, . . .d"

2.4 . Bound atomic amplitudes-Markoff, RWA approximations and DampingHamiltonian

A Markoff type approximation can now be made . In view of the various matrixelements varying slowly with continuum energies the continuum integrationsJ dw f, . . . J dc of. lead to an 0 dependence of A"(0) which is weak. Hence we maysubstitute the convenient value Q: is associated with the contours in the inverseLaplace transform into our expression for A ;"(S2), thus A?-"=At;•"(ie) will no longerdepend on S2 .

(30 a)

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136


Hence

QC0(12)-

AooCo(a +Awoo) - E AoiC1(Q+Aco01) =1 , (32 a)P240,0)

P2,,(0,1)

QC1(Q)-

A10C0(Q+A(o1o) -

Al1C1(Q+A(oll) =0 .

(32b)P2.0, 0)

P2.1,1)

At this stage it is now convenient to ignore non-resonant processes . Let us definethe two photon detuning as

6=0) 1 +coa-w0-cob .

(33)

This is approximately zero from equation (1) and we note that 6 is small incomparison to 0)1-(Oo and cob -(Da .

The frequency shift for the process P2n(i ; j) is given by equation (25 a) as

Aw,j(c1 . . .ce;d1 . . . dn)=(hi+cwC,+(Ace+ . . .CDC)-(()j+cods +wd2+ . . .(Od.) .

For i =j in order to obtain aAw that is small (actually zero) there must be the samenumbers of as and bsin the absorption as in the emission terms. If they differ then AO)

would be ±(cob-()a), ±2(wb-we ), . . .which are large terms . Thus for 2n=4 wehave (c l c 2 ; d1d2 ) of the form (ab; ab), (ab ; ba), (aa ; aa) etc. which are acceptable, butalso (ab ; aa) etc. which are not .

For i = 0, j =1 in order to obtain a Aco that is small (actually - 6), the absorptionsmust contain an extra b, the emissions an extra a. Thus for 2n = 4 we have (c 1c2 ; d 1d2 )

processes of the form (ba; aa), (bb ; ab), (bb; ba) etc. which are acceptable, but also(ab; ab) etc. which are not .

For i=1, j =O in order to obtain a Aco that is small (actually +6) the absorptionsmust contain an extra a and the emissions an extra b .

The processes that yield Aco ij small are all resonant processes . Note that an overallmismatch of +6 could be involved and resonance only applied overall and not atintermediate stages . For non-resonant processes these conditions are not satisfiedand hence these terms can be ignored, since their inclusion would merely correspondto adding terms in the inverse Laplace transform of equation (32) with oscillationfrequencies that are , much different to those obtained by only including the resonantterms. This situation is rather like ignoring the non-adiabatic terms in a masterequation, often referred to as the rotating wave approximation of the second kind[21] .

Finally then

QC0(Q)-( E Aoo'C0(Q) - (

AoilC1(Q - S)= 1 ,RESONANT

RESONAN

JT

0`1(Q)- ~ Aio\C0(Q+6)-(~ A„P2„(1

'\C1(Q)=0.,0)

;W1 1)(RESONANT

RANT

(34 a)

(34b)

Hence we have a 2 x 2 matrix problem for CO(O), C1(Q-6), which is of the form

(QE2-A)( 0l=\0

),

(35)

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1)RES

This is equivalent to the amplitudes CO , C1 given by a Schrodinger equation forthe non-Hermitian damping Hamiltonian hA

Co = 00 exp ( - ' (Pb),

so that co(Q)=C0(Q) and L~1(d2)=C`1(al-S) . This transformation should also bemade in the previous work [20] .

2.5. Effect of laser phases. Symmetry of Damped Hamiltonian hAWe now consider the dependence of results on the laser phases qqQ , cpb . From

equation (27) we have

90c= 19 0 ,1 exp [ - i( (p,+n/2)) .

(40)

The resonant terms Aoo, Aii involve the same numbers of as, bs in the emissionand absorption terms. Hence the phase factors exp [-i((p,+7c/2)] must cancel outand Aoo, Aii would be the same as if all So, where replaced by kfo,I .

The resonant term Aoi contains an extra b in absorption, an extra a in emission .Hence all the phase factors cancel, except for the extra b, extra a leaving an overallfactor exp [i(gob-co a)] from the result for Aoi that would apply if all tfoc wherereplaced by le0j . Similarly the resonant term Ai" contains an overall factorexp[-1(Qb-(p-)1 •

Clearly then an alteration of the amplitudes CO and 01 by phase factors of theform

(41 a)

Laser-induced continuum structures 137

where

(36 a)A00= A 2"00,

P2„ (0, 0)

A01 - -

RES

A2"01, (36 b)

P2„(0,1)

A -10-

RES

A2n10, (36 c)P241 . 0)RES

2nA11=a+ A11 . (36 d)

dcot

~= -iA(gO ), (37)JJJ

hwith

C0(0)=1, (38 a)

C1(0) = 0 . (38 b)

Here we have made the transformation

00 = C O' (39a)

C1 =C1 exp (-i6t), (39b)

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and

C1 =C1 exp(+i(p,)

would lead to equations for CO , C'1 in which cP, and (Pb did not appear . As the boundstate populations are given by IC0 1 2, IC 1 I 2 , and only these are of interest, we canwithout loss of generality set W, and Wb to zero .

Hence we will use for the resonant processes (and replacing 0 by i8)

2n

2eAi , =

Ai} (c 1 . . .c„; dl . . .d„)C, . . .cnd1 . . .d„1

. .fdw dcu d(o

do)

da)- (2h)2",~ .

h g1 I2 - •

9„-I Inl

. . .cnd, . . .d„

CIfo~,l<iid •s*Ifl>I10c21<f1Id •E*Isl>

X(C)i + coc, - cot , + 18) (COi + cool + C), 2-0)#1 + ic)

xI1od,Kg11d•td,If2>1e'03I<f2ld • 8*Ig2>(O)i + 0)c, + OJc2 -Ood, - O)I 2+ i8) (C01 + coc , + CAc2 + COc3 - COd, - COg2 + 18)

xIgOd X921d'8d2If3>(w, + (0, + 0)C2 + COc3 -wd, -COd 2- OJt 3 + is)

x

I f0 41<f3ld -8ci1153>(C)i + Cuc, + COc2 + COc3 + CO,, - COd 1- COd2 - (O g3 + 18)

X1f0c„I<f,-1Id•8,Ig,-1>

(w,+COC,+ . . .+C)c„-COdI- . . .-COdn-2-(0gn-,+18)

I9odn j<g.-lld • 8d„-,Ifn>X I'od„I <f Id ' Sd„I1 > • (42)(C)i+CAc1+ . . .+COc„-COd,- . . .-Cod- , -WIn+ 18)

For real polarization vectors 8„ sb and by choosing continuum state vectors sothat the dipole matrix elements are real we have

<Aid • sIB> = <Bid • alA>,

(43)where A,B are 0, 1, f, g and s is s„ 8b .

Under these conditions and using the condition of small two photon detuning(0 1 + coaxw0 + cub we can show that

LA10(c1c2 . . . c. ; d1d2 . . . d„)IxES=[Aoi(dnd„_ 1 . . . d1 ; cn . . . c2c1)]&ES •

(44)This results relates the matrix element for the 1-+0 process with absorptionsc1c2 . . . c, and emissions d1d2 . . . d„ to that for the 0-+1 process with the absorptionsnow d,, . . . d 1 and the emissions c,, . . . c 1 that is, absorptions and emissions are swappedaround and in reverse order .

As the result for A01, A10 involves sums over all such resonant processes and theorder of the summation is immaterial then the matrix A is symmetric

Aol =Alo,

(45)for this case of real polarizations and real dipole matrix elements . Unless otherwisestated we will from now on assume this situation applies .

(41 b)

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139

2.6. Expressions for bound amplitudes, ionization probability, ionization rateThe Damped Schrodinger equation (37) is the basic working equation for the bound

state amplitudes and hence for the ionization probabilities and rate . The sameapproach to solving these as in [20] can be used to obtain solutions and forcompleteness we include the results here . A is symmetric, as we have shown .

Expressions for Co and C1 in terms of the eigenvalues Ai and eigenvectors Y; of Acan then be found. The latter are given by [27]

AYi = 2tYi ,

i= 1, 2,

(46 a)

with

YTY,-Bij,

i,j=1,2,

(46b)

and we have the solution for C° and C1 as

(°)

C=E[YTC(0)]Yi exp (-i~.it),

(47 a)C

`1i

with

C(0)= C0) .

(47 b)

The bound state population is given by

Ps=IC0I 2 +ICu I 2

_I:[Y C(0)][YTC(0)]"[YIYj] eXP [1(2? - 2 )t],

(48)

The ionization probability is given by

P1 =1-PB,

( 49)

and the ionization rate is defined as

RdP,dt

(50 a)

=ij~[Yj C(O)][YTC(0)]*[Y7i(A-A*)Yj]exp[i(A?-Aj)t] .

(50b)

These results apply to both strong and weak continuum-continuum coupling viaboth lasers with the bound states coupled to the continuum I f > via both lasers .

As will be seen below (equation (94)) the non-Hermitian Damped Hamiltonianmatrix hA can be written

A - so+ 8o -iiYo-iiYo'

jr,°(q+1)+JA'

°+sjZrl°/q+a)+ZA 10 r~ °D)+i a+a +~~OD' -; 1

1 X00) '

(51)l

(wo

)

1

1

2 Yl -1.Y1where 8i, 8 ;°° ) are frequency shifts for level i of first and all higher orders in theintensities respectively ; yi , y;°°~ are decay rates for level i of first and all higher ordersrespectively. The quantities 4F10(q+i), JA (0)(r(°0)+i) are two-photon and multi-photon Rabi frequencies which involve the first and all higher orders in the intensitiesrespectively . q is intensity independent and the q and r (°°) factors define the ratio ofthe real and imaginary parts. Explicit formulae are given below .

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The quantities (SV . . .yl°D) can each be written as a sum of terms of order n in theintensities I„ Ib (that is, la I% with p+q=n)

00bi °°) =

60) ,

i=0,1,

(52 a)n=2

°o7400) = L Yin) ,

i=0,1,n=2(52 b)

°oA(.°)(r(°°)+t)=

A(,°(r(n)+i) .

(52c)n=2

Explicit formulae are given below . The factors r( " ) are intensity independent .We introduce the matrix B

B=A-[b0+bo°°)]E

(53 a)

__ ziYo -iiYo-F10(q+t)+zA(.)(r(°°) +t)'

(°

(53 b)il 1°(q+i)+ A )W°°) +t) S-fiy l -Iiyl°°)

where the shifted two photon detuning is

s=6+61+si°°)-60-so°°)

(54a)

=[w1+61+S;°°)+wa]-[coo+60+B(OW)+cob], (54b)

and which now allows for the bound-continuum and continuum-continuumprocesses to affect the atomic transition frequency wlo .

The eigenvectors of B are the same as for A though with related eigenvalues ~ i

BY;=U;,

i=1, 2,

(55 a)

a(55b)

The ionization rate and the ionization probability can now be written in terms ofB where the dependence on the shifted two photon detuning is more explicit . Wehave :

R= [YJC(0)][YTC(0)]*[Yti(B-B*)Yj] exp [i(l;?-~j)t] .

(56 a)

P1 =1-~ [Y; C(0)][YTC(0)]*[YtYj] exp [i(~? - ~) t] .

(56 b)

In order to obtain finite probability amplitudes C o, C 1 the eigenvalues Ai musthave non-positive imaginary parts .

Im#,j=Im ~1< 0 .

(57)

Clearly this condition will depend on the parameters y o , ya") , F'o , . . . , An) and on thetwo photon detuning S. In numerical work condition (57) is tested for each parameterset to ensure that the set corresponds .to a physical situation.

Writing

Z=EYjYj exp(-i~jt)C(0),

(58)i

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141

we see that the ionization rate (56 a) can be written as the Hermitian form

R=ZtCZ,

(59)

with

C=i(B-B*)

(60 a)

__ Yo +Y~°°)

1 ' 10 A 10 )

lo_ to

(00)

(60 b)- 1

A(G) Y1+Y1

From (47), (53), (55 b) and (57) the condition that the probability amplitudes arefinite for the special case S=0, q = 0, r(') = 0 is

[y o +Yv)][YI+Yi°°)]>.[Fl0+A'~)]2.

(61)

This condition guarantees that the eigenvalues of C are non negative . Hence theionization rate R is always positive, so that there is never a population oscillation backto the bound states and the ionization is irreversible .

2.7 . Experimental considerationsThe ionization probability P, and the ionization rate R are closely related to

experiment. The ionization probability P, is proportional to the numbers of ions orelectrons produced in the interval 0-t [11] . The observation time t can be chosen asthe laser pulse length . The rate at which these numbers are increasing will beproportional to R.

It can easily be seen from (37), (38), (51) that at very short times the ionizationrate is yo+y0), since the short time solution to (37), (38) is

C0 exp{-i[So+S~°°)- Yo-iiYo°°)]t},

(62 a)

C` 1 0 .

(62 b)

Similarly, if the atom was initially in state 11 > the short time ionization rate wouldhave been y l +y(°°) . In principle then these quantities y o +yV), Y 1 +y(°°) could beexperimentally measured and be available for scaling the ionization rate R (in termsofYo + yo°°), see below) and the measured (unshifted) two photon detuning 6 (in termsof 1 (y I+y' )), see below). As will be seen below the theoretical expressions forionization rate and shifted two photon detuning 3are conveniently given in terms ofscaled variables R/(yo+y(O°°)) and c=S/1[y l + yr] . The experimentally more conve-nient unshifted two photon detuning 6 (=co l + COs -coo -wb) cannot be easilytranslated into a shifted two photon detuning . However apart from a sideways shiftthe experimentally measured curves for P1, R versus (scaled) unshifted two photondetuning 6 would be the same shape as the corresponding theoretical curves plottedagainst the (scaled) shifted two photon detuning 8. The sideways displacementwould be itself a measure of the frequency shift [S 1 +S1°° ) -So -So) .

At a particular intensity Ia, I6 it is not possible to conclude whether the scaledexperimental curves for R, P1 versus S have been affected by continuum-continuumcoupling, since with the replacements yo-'Yo+Yo°°), Yi-'Y1+Y(°°),

4f"°(q+i)-i1-lo

x (q+i)+ZA(°)(r(°°)+1),, S1-60-•81+5,°°)-So-So°°) the curves would be the same asfor a pure LICS situation . However experiments done at various intensities Ia, Ib andat differing times t would enable the continuum-continuum effects to be seen, sincethe it) , yi") , A~°, S~") , 81") have quadratic and higher-power dependences on l a and Ib .

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3. Results correct to first power of intensity. No continuum-continuumcoupling

3.1 . Theoretical expressionsHere we consider only the lowest-order contribution (n = 1), these terms give the

usual LICS results but with the additional feature that both modes a, b contribute tothe frequency shifts 60, ai and ionization rates y o , yi of the bound states .

The Markofff RWA2 results can be written correct to the first power of theintensity as

__ 60 Ziyo

21'10(q+i)

()63 ai1 10(q+t)

6+a1 -Ziy1 '

B=A-6 0E,

(63 b)

where we assume real polarizations and dipole matrix elements

ao - ziyo =Aoo(a; a) +Aoo(b ; b),

(64 a)

61-ziY1 =Ai1(a; a) + A1 1 (b ; b),

(64 b)

II'' o(q+ i) = Ao 1(b ; a)

(64 c)

=A10(a; b) .

(64 d)

Using the general formula we have (real polarization)

Aizcc)--l

fJdaf 1 I90,!<t1d•cIf1>lo'o,l<f1ld•$~ I i>, (65 a)t,

(2h)i

((O; +wC-wf , + ic)

A21(b,a)

1)z

, lSobKOld - cblf1>I$oal<fild - sall)(2h

Jdcof '(wo + (Z)b- wfI + ic)

(65 b)

Aio(a~ b)= 1 2 Jdwfl l .9oal<1Id.$aIfi>l'fobl<fild' Ebl~~ .

(65 c)(2h)

(COI +wa-wf1 +is)

Using the condition w l +wa xw0 +wb and the result for real 5a , Eb and real dipolematrix elements it is easy to see that

A10(a; b) "Aoi(b ; a) .

(66)

The quantities Ai0(a, b) are of course the usual two photon Rabi frequencies [18, 19] .Breaking (64) up into real and imaginary parts the contributions from each of the

modes to the ionization rates, frequency shifts etc . are:

Condition (61) is satisfied since yoYi % Y1 yo so a positive ionization rate is guaranteed .

Yo = Y'O' + YO) (67 a)

8o =ag+a0 , (67 b)

Yi =Yi +A) (67 c)

a1=ai+sl, (67d)(l-io)z=v ?

. (67 e)

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In the previous case of a weak probe b and a strong embedding laser a, thecorresponding results were

b0- iYo = Aoo(b ; b),

(68 a)

6 1 -jiy j =A', (a ; a),

(68 b)ff'1 o(q + i) =A 21(b; a)

(68 c)

=A10(a ; b),

(68 d)

and the condition (f10 )2 =y oy 1 applies .The shifted two-photon detuning s and its dimensionless form E are given as

s=6+6 1 -6 0 ,

(69 a)

s2Y1

(69 b)

3.2. Numerical resultsFigures 3, 4, 5, 6 show the time dependence of the ionization probability P, or

ionization rate R versus dimensionless shifted two photon detuning e curves . Notethe important distinction between P, and R . In traditional Fano theory the totalionization rate to a structured continuum R normalized to the rate to a flat continuum

RFLAT = Yo is the quantity of interest, so R is shown normalized in units of y o . Forsimplicity these results ignore the effects of additional ionizations so that ya=yi = 0 .

For the weak probe case we choose parameters yo = 2n x 102 , yi = 2x x 106 andq = 5 . In this weak probe case we see that at short time t = 3 x 10 -9 (figures 3 (a), and4(a)) the bound state 11> has not yet become embedded . Thus R=yo for t<<y 1. 1 ,t<< yo 1 [20] . At the somewhat longer time t=3 x 10' (see figures 3 (b) and 4(b))oscillations as a function of detuning appear with a zero in the ionization rate startingto become apparent near c = - q . For moderate time t= 3 x 10 -6 (see figures 3 (c) and4 (c)) a sufficient interval has elapsed. for the embedding state to be established butnot long enough that the initial state j0) has been significantly depleted . Here thewell known Fano profile form [1] occurs

R=Yo (1 ++ p 2

E

)

2)'

t>>yi 1 ,

t<<yo 1 ,

(70)

as shown in [20] . The Fano profile has started to change somewhat for t= 3 x 10- s(figure 4 (d)) at long times (t = 3 x 10 -4 , figures 3 (e) and 4 (e)), saturation effectsbegin to become apparent, thereby changing the Fano profile . In figure 3 (f) fort=3 x 10-2 the system is essentially ionized at detunings different from E= -qwhere a small bound state population remains (the position of the so called Fanowindow).

These figures indicate the importance of choosing the time regime correctly inorder to observe the Fano profile, as the curves obtained can vary significantly .

For the strong probe case we choose yo =yi = 2x x 106 , q= 5 . In this strong probecase at short times t = 3 x 10 -9 (figures 5 (a) and 6 (a)) embedding has not yetoccurred and the flat continuum results apply. At the somewhat longer timet=3 x 10- ' (see figures 5 (b) and 6 (b)) oscillations as a function of detuning occurwith a zero in the ionization rate starting to become apparent near E = 0. At moderatetimes t=3 x 10- 6 (figures 5 (c) and 6 (c)) a coherence hole has been essentially formed

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O

0

0

0

0

a

0

-20 0

E

(a)

-20

0

(e)

M. E. St. J . Dutton and B. J. Dalton

20

-20

20

20

-20

0

20

E

0

C d

d

NO

O

-20

-20

E

(b)

E

(d )

0

E

(f )

20

1

20

Figure 3 . Total ionization probability PI versus normalized detuning s for weak probeexcitation of a strongly-embedded LICS for various times t . In all cases 70=2n x 102 .-?,=2n x 106 . 1°0=)i=0 and q=5 . In (a) t=3 x 10 -9 , ( b) t=3 x 10 -7 , (c) t=3 x 10-6 ,(d) t=3 x 10 -5 , (e) t=3 x 10-4 and (f) t=3 x 10-2 . Continuum-continuum processesare ignored .

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O ,-20

0

20

E

(a)

0


145

-20

0

20

-20

E

(c)

¢

-20

20

(e)

-20

0

f

(b)

E

(d)

20

20

Figure 4. Ionization rate R in units of RFLAr=Yo versus normalized detuning c for weakprobe excitation of a strongly embedded laser-induced resonance at various times t . In(a) t=3 x 10 -9 , (b) t=3 x 10 -', (c) t=3 x 10 -6 , (d) t=3 x 10 -5, (e) t=3 x 10 -4. Allother parameters are as in figure 3 . Continuum-continuum processes are ignored .

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00

Ga d

0

0

0

md

O0.

NO

O

-20

-20

0

20

E

(a)

0

E

(c)

0


-20

20

0

E

a

(e)

Figure 5 . Total ionization probability P, versus normalized detuning a for strong fieldexcitation of a strongly embedded laser-induced resonance for various times t . In allcases yg =y1 = 2n x 10 6 . y ; =yg =0 and q= 5. In (a) t=3 x 10-9 , (b) t=3 x 10- 7, (c) t=3x 10-6 , (d) t=3 x 10 -5,(e) t=3 x 10-4. Continuum-continuum processes are ignored .

md

0a

fO

NO

0

-20

-20

20

0

E

(b)

0

E

(d)

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20

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0-20

0

20

(a)


a

(b)

6

E

(c)

(d)

Figure 6. Normalized ionization rate R in units of RAT =yo versus normalized detuning efor strong field excitation of a strongly embedded LICS at various times t . In (a) t=3x 10 -9 , (b) t=3 x 10 -7 , (c) t=3 x 10 -6 , (d) t=3 x 10-s . All other parameters are as infigure 5 . Continuum-continuum processes are ignored .

with population trapping (approximately half the population) becoming evident fordetuning s=0 and saturation effects become important . For the strong probe casethere are only two distinct time regimes t << yl 1, yo 1 and t > 71 1 , yo 1 and thecoherence hole starts to appear as t -yi 1 , yo 1 . Figures 5 (d) and 6 (d) show theintermediate time region t=3 x 10-s , with the coherence hole becoming narrower .At long time t = 3 x 10 -4 (see figure 5 (e)) saturation effects are dominant and thecoherence hole evolves further towards zero width .

4. Results correct to second power of the intensity4.1 . Theoretical expressions

Here we consider both the lowest (n=1) and next-order contribution (n=2) tothe matrix B. Thus the usual LICS results are corrected to the lowest order bycontinuum-continuum coupling .

The MarkoffRWA2 results can now be written correct to the second power in theintensity

_ So +So2) jiyo 1iyo)

T10(q+i)+#A I (r' 2) +t)A

(71 a)10(q+i)+jA'2(r(2)+t) a+a 1 +a 2)- Y1-iiY(2)I

1

2 1

B=A - [60+8PIE,

(71 b)

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where

Obviously there are too many quantities to give explicit results . A typical case isAIO(ba ; aa) . Using the general formula, we have

Ao1(ba;aa)= 1)a

dmt,dw,dtvf2 IeobK0Id'EbIf1>(2h f f f

(COO + wb-(O t, + ie)

xISoal<fild' 8algl>I9oal<g11d'Ealf2>I'oal<f2ld • 8a11> .

(73)(CUo+CAb+U)a -O),,+21e) (coo +Cob+CUa-U)a -0)f 2 +1E)

Using the small two photon detuning condition (0 0 +cob ~(o 1 +a)a and the result forreal Sa, ab and real dipole matrix elements we see that :

A01(ba; aa) ,& A41 O (aa; ab) .

(74)

The quantities Alo(aa, ab) etc. are four photon Rabi frequencies .In the previous case of a weak probe b and a strong embedding laser a the

corresponding results were

bo2)- 2iyo2) =Aoo(ba; ab), (75 a)

bi2)- iiyI2) =A11(aa; aa), (75 b)

A~2j(r(2) +i)=Ao I(ba; aa)

(75 c)

=A1o(aa; ab) .

(75 d)

The expression for the shifted two photon detuning is given by equation (54) withoo -> 2 and its dimensionless form c given by

Ss_2 Y1 +Y(2)]

(76)

scaled in terms of the short time ionization rate [y I +y12)] . As indicated before theionization rate R will be scaled in units of the short time ionization rate (yo + y0(2)) forstate 10> .

4.2. Numerical resultsFigures 7-12 show the effect of second-order continuum-continuum coupling on

the ionization probability P I or ionization rate R versus dimensionless shifted twophoton detuning c curves . To examine such effects note first that if the intensities I a ,Ib are replaced by KIa , KIb then the first-order frequency shifts, ionization rates and(two) photon Rabi frequencies are increased by the factor K, whilst the second-orderfrequency shifts, ionization rates and (four) photon Rabi frequencies are increasedby the factor K2 . The factors q, r(2) do not change .

So2)-4iyo2)=A0 (aa; aa)+Aoo(ab; ab)+Aoo(ab; ba)

+Ao0(ba; ab) + Ao o(ba; ba) + Ao o(bb ; bb), (72 a)

5i2)- iiyi2)=A41(aa; aa)+At j(ab; ab)+A 4 1 (ab ; ba)

+ AII(ba; ab) + At 1(ba; ba) + At 1(bb ; bb), (72 b)

2Aj2o>(r~2) +1)=Ao 1 (ab;aa)+Aol(ba;aa)+Ao 1(bb;ba)+Ao 1 (bb;ab) (72 c)

=Alo(aa; ab)+Alo(aa; ba)+Alo(ab; bb)+Alo(ba; bb) . (72 d)

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a 0

0

a o0

-20

-20

F


149

0

20

E

(a)Figure 7 . Total ionization probability P1 (a) and normalized ionization rate R (b) in units of

y0 +yb2) versus normalized detuning s for weak field excitation of a strongly embeddedLICS at low overall intensity and with negligible second-order continuum-continuumprocesses. In both cases Y0 =2n x 102 , yi =2x x 10 6, yl =yo=0, 1' 10=2x x 104, q=5 .Continuum-continuum contributions are yo2) =x x 10° , yi2~=x x 104 , A~2)'=x x 102 , rt21=5 . The time is t=3 x 10" 6 . The Fano profile is seen in (b) .

c

a

0

-20

0

F

(b)

E

20

(a)

(b)Figure 8 . Total ionization probability P1 (a) and normalized ionization rate R (b) in units of

YO +yW , versus normalized detuning a for weak field excitation of a strongly embeddedLICS at higher overall intensity (x 10 2) and with non-negligible second-ordercontinuum-continuum processes . In both cases yo=2x x 104, yi=2x x 108, y', =y'0=0,f 10 =2x x 10 6 , q=5 . Continuum-continuum contributions are yo2~=x x 104 , Yi2~=xx 108, A('2)=x x 106 , r(2)=5. The time is t=2 x 10 -8 . The Fano profile is unchanged .

The main interest would be in the effects of continuum-continuum coupling onclassic pure LICS features such as the Fano profile, which occurs for the weak probecase Ib<<I. in a time regime 1/y, << t<< 1 /y 0 and the coherence hole, which occurs forthe strong probe case Ib ' I, in a time regime t -1 /y1, 1 /y ° . These pure LICS featuresoccur for the ionization rate R and ionization probabilities PI respectively and areshown in figures 4 (c), 5 (c) .

In the weak probe case, figure 7 shows the original weak probe results (see figures3 (c), 4 (c)) with a very small amount of second-order continuum-continuumcoupling included . Here we again choose yo = 2x x 10 2 , yi = 2x x 106 ,r 1 ° = 2x x 10 4,q= 5 but with yo2'= x x 10 ° , y12j =x x 104 , A~ °) =1t x 102 , r~2~ = 5 . The time t is again3 x 10- 6 . The ionization probability (figure 7 (a)) and ionization rates (figure 7 (b))are unchanged.

Figure 8 shows the effect of increasing IQ, Ib by a factor K=102 . Here we nowhave yo = 2n x 104, ya = 2x x 10 8, r,° = 2x x 106 , q = 5, but now the second-order

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Lq

0

w

aa

N

0

-20

-20

0

E

(a)

0

E

(c)


IQn

o

20

0

-20 0

E

(b)

-20

0

E

(d)

20

20

Figure 9 . Normalized ionization rate R in units of y o +yW) versus normalized detuning a forweak field excitation of a strongly embedded LICS at higher overall intensity with non-negligible second-order continuum-continuum processes . Effects of variation of AJ2 onFano profile . In (a) A'2 = - 2. 0 x 10 6 (b) A'2 = -1 . 5 x 106 (c) A'Z = - 2. 0 x 105 (d) A'2= 5 . 0 x 105 . Other parameters are as in figure 8 .

terms are much more important with yo =n x 10 4 , yi2~=n x 10 8 , A'Zj=n x 106 ,rr 2j=5 . Choosing the time as t=2.0x10 -s results in the factors ~ ;t (see (56))remaining the same. Thus as expected the ionization probability (figure 8 (a)) andionization rate (figure 8 (b)) curves are unchanged from the pure LICS case (figure7 (a, b))-even though the second-order continuum-continuum coupling para-meters are comparable to those for pure (weak probe) LICS!

However, there is no reason for the second-order continuum-continuumparameters to have been so elegantly proportional to those for first order . Forsomewhat different initial second-order parameters to that for figure 7 the second-order parameters after increasing the intensity by factor K=10 2 could have beendifferent to those chosen for figure 8 . Figures 9, 10 display the ionization rate Rversus detuning c for a different second-order continuum-continuum coupling,showing the changes to the Fano profile due to variations in A'Z and r(2) respectively .In these figures we again choose yo ) = n x 104 , y12)=n X1011 and t=2 .0 x 10-8 .Modifying t would not restore the results in figure 8 (a, b) .

Figure 9 (a) with ADZ = - 2 .0 x 106n shows a flat profile . A small peak appears forfigure 9 (b) with A = -1 .5 x 106n, and the curve becomes more Fano like in figure9 (c, d) where A~2 = - 2 .0 x IOsn, 5 .0 x 105n respectively, although the Fano windowis partly filled in and the Fano peak height (q'+ 1) is not attained .

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a

CL

a

Figure 10. Normalized ionization rate R in units of y o +yo versus normalized detuning a forweak field excitation of a strongly embedded LICS at higher overall intensity with non-negligible second-order continuum-continuum processes. Effects of variation of r(2) onFano profile . In (a) r(2)= -10, (b) r(2)= -7.5, (c) r(2) =0, (d) r (2)= 10. Other parametersare as in figure 8 .

00

0a

ad

N0

0

-20

20

-20

0

20E

E

(a)

(b)

-20

0

z0

° -2o

20E

1

-20 0

E


151

20

a

a

-20

6

0

E

20

Figure 11 . Total ionization probability PI(a) and normalized ionization rate R (b) in units ofYo+~o ) versus normalized detuning a for strong field excitation of a strongly embeddedLICS at low overall intensity and with negligible second-order continuum-continuumprocesses. In both cases yo=2a x 106 , yi=2x x 106 , 4=y"o=0, Tlo =2n x 106 , q=5 .Continuum-continuum contributions are yW'=n x 104 , yi2) =u x 10° , A~=n x 104 , r(2)=5 . The time is t=3 x 10 -6 . The coherence hole is seen in (a) .

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a

N

-20

0

20

-20

20

E

f

(a)

(b)Figure 12 . Total ionization probability p,(a) and normalized ionization rate R (b) in units of

y o +yb2) versus normalized detuning a for strong field excitation of a strongly embeddedLICS at . higher overall intensity (x 10 2) and with non-negligible second-ordercontinuum-continuum processes . In both cases yo=2a x 10 8 , yi =2n x 10 8 , yi =YO, = 0,110 =2n x 10 8 , q=5. Continuum-continuum contributions are yo)= a x 108 , yi2>=nx10 8 , A('2 =n x 10 8 , r(2) =5 . The time is t=2 x 10 -8 . The coherence hole isunchanged .

Figure 10 (a) with r(2)= -10 shows a window at c=0 in an otherwise flat profile .In figure 10(b) with r (2) =-7.5 a rather dispersion like curve occurs . For r(2) =0 acurve like the Fano profile is seen in figure 10 (c) and in figure 10 (d) with r (2) = +10the Fano like profile has a maximum greater than q2 + 1 .

For the time-scale and rate constants used in figures 9, 10, a non-saturationregime applies, so curves of the ionization probability versus detuning have similarshapes to those for the ionization rate . These are not shown .

In the strong probe case figure 11 shows the original strong probe results (seefigures 5 (c), 6(c)) with a very small amount of second-order continuum-continuumcoupling added. Again we choose yo = 2Tt x 106 , y', = 21t x 10 6 , r, ° = 2n x 106 , q = 5,but with yo =n x 104 , yi2)=n x 104 , A'2 =n x 104, r(2) =5. The time is againt=3 x 10 -6 . The ionization probability (figure 11 (a)) and ionization rate(figure 11(b)) are unchanged .

Figure 12 shows the effect of increasing la , Ib by a factor K=102 . Here we nowhave yo = 271 x 10 8 , y', = 2n x 108 , T 1 ° = 2n x 10 8 , q = 5, but now the second-orderterms are much more important with y(2)=n x 10 8 , y12~=n x 108 , A'2=n x 10 8 ,r(2) =5. Choosing the time as t=2 .0 x 10 -8 results in the factors ~,t remaining thesame. Thus as expected the ionization probability (figure 12 (a)) and ionization rate(figure 12 (b)) curves are unchanged from the pure LICS case (figure 11 (a, b))--eventhough the second-order continuum-continuum coupling parameters are compar-able to those for pure (strong probe) LICS!

However again there is no reason for the second-order continuum-continuumparameters to have been simply proportional to those for first order . For somewhatdifferent initial second-order parameters to those for figure 11 the second-orderparameters after increasing the intensity by factor K=102 could have been differentto those chosen for figure 12 . Figures 13, 14 display the ionization probability P Iversus detuning a for a different continuum-continuum coupling, showing thechanges in the coherence hole due to variations in A'2 and r(2) respectively. In thesefigures we again choose yo2~=n x 108 , y12~=n x 108 and t=2 .0 x 10 -8 . Modifying twould not restore the results of figures 12 (a, b) .

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O

0

d0

-20

0d

da

ad

N0

0 -20

20

-20

0

20


0

0d

da

0dNO

O

E

(c)Figure 13 . Total ionization probability P, versus normalized detuning a for strong field

excitation of a strongly embedded LICS at higher overall intensity and with non-negligible second-order continuum-continuum processes. Effects of variation of A (2 oncoherence hole . In (a) A('2j= -2 . 0 x 10 8 (b) A(0)= 5 . 0 x 10 7 (c) A~2j=9 . 5 x 10'. Otherparameters are as in figure 12 .

d

mda

0dNd

0

1

t

1

-20

20

0E

(b)

-20

0

20E

C

(a)

(b)

Figure 14. Total ionization probability P, versus normalized detuning a for strong fieldexcitation of a strongly embedded LICS at higher overall intensity and with non-negligible second-order continuum-continuum processes . Effect of variation of r (2) oncoherence hole . In (a) r(2)

= - 1 0 (b) r(2)=10. Other parameters are as in figure 12.

20

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Figure 13 (a) with A"= -2 .0 x 10' in shows no coherence hole and completesaturation in the ionization probability at all detunings . A shallow coherence holebegins to appear at A'Z = 5 .0 x 10 7 Tt (figure 13 (b)), becoming much deeper forA'2 =9 .5 x 10 7 it (figure 13 (c)) .

Figure 14 (a) with r(2)= - 10 shows a very narrow coherence hole which becomesquite broad in figure 14 (b), where r(2) = + 10 .

The ionization rate also shows an interesting behaviour and this is illustrated infigures 15, 16 allowing for variations in A" and r (2) respectively. Figure 15 (a) withA'2 = - 2 .0 x 1081 shows a uniform ionization rate at all detunings . A single peak inthe ionization rate occurs for A" = 5 .0 x 10' tt (figure 15 (b)) which ultimately splitsinto the more familiar two peak curve, see figure 15 (c) with A~2 =9 . 5 x 10 7 7t .

In figure 16 (a) with r (2)= - 10 the two peak ionization rate curve is very narrow,but becomes very broad for r(2) = + 10, as in figure 16 (b) .

Thus second-order continuum-continuum coupling produces significantchanges to the ionization rate and to the ionization probability versus detuningprofiles for both the weak probe and strong probe excitation of a strongly embeddedLICS. Fano profiles and coherence holes would be significantly modified inexperiments at higher intensities and shorter time-scales designed to explore theseeffects .

0

oqOXN

0-20 0

O0

0

20

0a xN

/x

-20

0

20

f

-20

0

20

f

(C)

Figure 15. Normalized ionization rate R in units of y o+yo versus normalized detuning a forstrong field excitation of a strongly embedded LICS at higher overall intensity and withnon-negligible second-order continuum-continuum processes . Effects of variation ofA('20) on coherence hole . In (a) A,',0) = -2.0 x 108 (b) A,=5 .0 x 10 7 (c) A,2,=9 . 5 x 10 7 .Other parameters are as in figure 12 .

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s0

0-20 0


155

20

-20f

f

(a)

(b)

Figure 16. Normalized ionization rate R in units of y o +yo ) versus normalized detuning E forstrong field excitation of a strongly embedded LICS at higher overall intensity and withnon-negligible second-order continuum-continuum processes . Effect of variation of r(2)on coherence hole . In (a) r(2)= - 10 (b) r(2)=10. Other parameters are as in figure 12 .

5 . Results correct to all powers of intensity via the introduction of aT operator

5 .1 . Bound-continuum, continuum-continuum interactions . T operatorThe expressions given in this Section apply to the regimes of both strong and

weak continuum-continuum coupling via the lasers, with the LICS structures due toboth lasers also . We include contributions involving all orders in the intensity .

We introduce bound-continuum, continuum-continuum interactions VB, Vc via

VB= Jda[t)<h#t+bIf><14)wf(JhfIh

+ Jdoof(aJf><0Ikf+blf><Olh~f)+Hermitian conjugate,

(77 a)

Vc=J

Jdc01 do), (alg><fJ8rg +blgX«Ofb g)+Hermitian conjugate,

(77 b)

ho =hcoaata+hwbbtb+JdwfhwflfXfI+ Jdw9 hwg(g><gJ .

(77 c)

Then the T operator [23] associated with continuum-continuum processes is

T= Vc+ Vc(zlh

/Vc + Vc(

1h0- /Vc, x

1h0Vc + . . .

(78 a)0

=T1+T2+T3+ . . .,

(78b)

with

Z = hwo + nohcoo+ nbhcvb + ihE .

(78 c)

We now reconsider the resonant terms

Aj ,=

E A "(c 1 . . . cp ; dl . . . d„),

(79)ct . . .C.

di . . .d„(absorption) (cmissions)

and write these in terms of matrix elements of T and of VB . We note that for resonantterms : (a) i =j, there are the same numbers of a, bs in absorption (cs) and emission

20

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(ds) (b) i=0, j=1, the absorptions contain an extra b (cs) and the emissions have anextra a (ds) (c) i = 1, j =0, the absorptions have an extra a (cs) and the emissions anextra b (ds) .

The overall process involves steps due to VB and steps due to Vc:

i

Ai

g1

.f2

g2

gn-1

fn

fL CI --a- It,, lid,, t t c3 _...

L cn -. I {dn(Due to VB) n- - . . .

(Due to VV)

-- -(Due to VB)

Suppose we were to include photon states in describing the process . Then if onlyresonant processes are involved, the system in initial state IOfiafib ) can only reachIOfiafib> (equal number of a, bs in absorption and emission) or 115 .+l, fib -l>(absorptions an extra b, emissions an extra a) .

Hence we may list the resonant processes via

The states in the middle will be of the form I f(fia +p), (nb+q))Ig(fia +p), (fib+q))where p, q are related to the net number of as, bs absorbed at that stage. In theAppendix the intermediate states involved for 2n = 2 and 2n = 4 are listed . We notethat only an even number of Vc terms appear (0, 2, 4. . . . ) . Also the first intermediatestate and the last intermediate state is always an f state and these states are onlyconnected by Vc interactions . On the other hand the first and last intermediate statesare connected to 0 or 1 states via VB interactions .

A useful notation for listing the possibilities is to designate states as follows :

Here e could be an absorption or an emission .

5.2. Non-Hermitian Damping Hamiltonian and T matrix elementsConsider now the total contributions to the non-Hermitian Damping Hamil-

tonian . From equations (42), (79) :

AjJR$S =Y,EAZJ(cl ; d1)

(n=1 term)Cl d,

+

A2j(c 1 . . , c,, ; d 1 . . . da)

(n > 2 terms)

(82 a)n=2 c, . . .c„ d, . . .4,.

io) = i0fiafib) (81 a)

I1)-llfia +1,fib -1), (81 b)

If[Oe))=iffia-1, fib), e=a, (81 c)

=If".,fib-1), e=b, (81 d)

if(' el)=Iffia,fib- 1), e=a, (81 e)

=lffia +l,nb -2), a=b . (81f)

A00: i0fiafib) H . . . -I0nanb), (80 a)

A01:IOfiafib) ~ . . .-1 lna+l, fib-1), (80b)

A1 0: IN. + 1, fib -1)H . . . E-•I0nanb), (80 c)

A11 : 11fia +1, fib - 1)H . . .HIlna +1, fib -1) . (80d)

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where

and


157

00p

fd(of, ( Cif,(cl)#f,Xdl)+ E E F,c, d,

n=2 c, . . .cnd, . . .d„

x Jdco fi Jdwf.OGifl(Ci)Tfif,(C2 . . .cn;dl . . .dn_1)flf,3(dn),

(82 b)

1 <ild •8*If1>aifl(cl) =2l~o~ I

+ w

+is)', (w .,

c, - w f,

ff,,/d.) = 2bISod„I<fNId - 8d„I j>,

Tf1f„(c2 . . .cn;dl . . .d,_1)

(n>.2)

1

f

('

~,,

=(2h)2n-2f do),,, dwf2. . .

Jdwo* - 1 I9Oc2I<f1Id -8c*I(51>

1X(wi+wc1+wC2-ws1+is) I90d1I<glld'8d,1f2>

x

1

Ieoe3I<f2 d • a*(g2>(wi+wcl+wc2-wdI-wf2 +1E

xI

I~Od2K 2Id .8d21 f3>(w i+wet +o 2+wC3-wd 1-wa2 +ic)

1x(cot+wcl+wc2+wc3-wd 1- wd 2-wf3 +is)

I

(83 a)

(83 b)

I9O l I<gn-lld ,8d 11f11>dn-n(wt+we, . . .w~„-wd, . . .-wdn-2-we„_1+1E)

X ((oj+('oCj+o),' . . .+wcn-wd1, -wdz .. . .-(

-w +IE)d., _ 1

jw

Then, after renumbering the ds backwards and withf1 replaced byF1, fn by F2

AiJ,1gg=Y.Yl Jdwri aiF1(Cl)I'F1J(dl)C1 d,

(83 c)

+E J d(t)F1aC1 . 1

X

TF1F2( c 2 . . . cn; dndn_ 1 . . . d2) fF2j(dl) .

(84)Ln=2c2 . . .c„d,, . . .d2

Now considering the various matrix elements of VB, Vc we have using equations(5), (27), (77), (81), (83) and with 0a=06=0:

(a) aiF1(C1)=-<=7VBIF1[ic1l> i=0, 1,

(85)h(wi + wc1 - wF1 + is)

(b) <F2[id1]iVBli>= -~hi(d1)~

i=0,1,

(86)

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(c) <fnanbl vclgna - 1, nb> = -I,90al<f Id - sq Ig>, (87)

(d) <fnanblVclgna,nb - I>_ - I11fobl<fld - srlg> . (88)

Also with x=hwo +nahwa +nbhwb +ihc

(z - ho)Ihnanb> = ht[wo +(na -na)coa+ (nb-nb)wb -wb+ iE] Ihnanb>

and we can approximate coo + cob xco t + wa as required .Hence

hCOi-hwF I [ic]=h(Coi+CUc -COFI ),

Z=O, 1,

(90 a)

hw i -hWF2[ ; d] =h(w i +cod -wF2 ),

i=0,1 .

(90 b)

However for resonant processes P2n(i, j)

1h(OJi+COcI+O)c,+ . . .+(we.)~h(COJ+Wd .+COd._,+ . . .+Wd1),

(91 a)

so that

111(o. j+ Uc,+CAc2+ . . .+CJ c. -Wdn -Wd._,-CUd2 -COF2 +i8).h(coJ-WF2Ud,]+i6) . (91 b)

Equations (78), (85), . . . (91) can then be substituted into the equations (83), (84)to enable the non-Hermitian Damped Hamiltonian matrix hA to be written as :

A

dw 1 <II VBI F1[IciI ><Fi[Jdi]I VBI J> +

f

fdw dwF,

FZil,tss- i

F, h2

(wi-CUFI[IcI1+1E)

cl dl

1 «VsIF1[Ici]><Fi[Ici]I T IF2[Jd1]><F2[Jd1]IVsIJ>X

(92)V3

(wl -OJF, [Ici1+is)(co -0F2[Jdi1+1E)

This is of the same general form as for the case of one strong field and one probe field .The only difference is the additional processes that are involved in the case of twostrong fields .

Using (89), (92) we can also write

Ai.iRes = jt [<Il VB / ziho

/VsIJ)+<i Va(z lh

0)T(z)(z

l h

0)VsIJ>], (93)

which is again in the same form as for the previous probe field case .

5.3 . n photon frequency shifts, ionization decay rates, Rabi frequenciesThe non-Hermitian Damped Hamiltonian matrix hA for the case where all orders

of the intensities are included can be written as :

a0+b~°°1-IiYo-IiYoa°1

~.r1o(q+t)+I.A)(r[o' ) +i)A Ir'O(q+t)+2l1~~1(r[00) +t)

(94)

where in terms of T matrix elements we have :

I

<01 VBIF1[Oci ]><F1[Od1]IVBIO>So - iyo =

2c,d, fdwF '

(wo - (F,[oc,] + lE)

(95 a)

_ 1

<11VsIF1[1c1]><F1[ld1]IVsll)

)S1- iylii2c,d, f

dwF,(a -wF,[ic,J+iS)

,

(95 b

h=f,g,

(89)

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I'10(q+i)-z

1dwFt<OIVslF1[0c11><Fi{ldlllVsll>

(95 c)ctdt

(w0 - (OF, [Octl + iE)

So°°)-

WY f Jdwr1&OF,cidi

x<01V1IF1[0c1]><Fi[Oci]ITIF2[0d11><F2[0d11IVsIO>

(95 d)(w0 -WFt[octl + ia)(w0 - wF2[Odt) + is)

JffdwF t dw1

Yl -3

F211 ctdl

x<1IVsIFi[lcl]><Flhci]ITIFshd1]><F2[ld1]IVHI1>

(95 e)(w 1 -CFl [ 1 cil+le)(w 1 -WF2 [1d tl+is)

10 (m)

1t)= 3 f fdwFt dWF2ct t

x <0IVsIF1(fic1l><F1(ficlllT)F2[1d11><F2[1d1 ]IV5I1>

(w0- WFI[Octl + ic)(w1 -WF2[idt] + is) (95f)

Again these results are of the same general form as for the weak probe case,though of course all the terms embody a much larger number of contributions fromthe possible processes . The intensity dependence of the quantities above is morecomplex and will not be elaborated on here .

The behaviour of the ionization rate R and ionization probability P 1 versusshifted two photon detuning S (in appropriate scaled units) for various choices of t,yo , Y1

r1o, q, y 0 ) . Yi°°) , A(~), r(0D) could be studied in a similar manner to that given inSection 4 . The higher-order continuum-continuum coupling effects would show upvia their different dependence on the intensity .

6. ConclusionsA theory of the effect of continuum-continuum coupling on strong field LICS

associated with two strong fields has been formulated . Results correct to all orders ofcontinuum-continuum coupling have been obtained in terms of a T operator, as wellas the approximate contributions correct to the second power of the intensity. Thetheory is valid for intensities high enough to induce continuum-continuumtransitions but does not consider intensities where ATI effects dominate .

The main difference with the weak probe field LICS situation is the appearanceof a much larger number of terms corresponding to the possibility of additionalprocesses involving photons from either laser field .

The results have been examined numerically for the case of zero continuum-continuum coupling and the time development of the Fano profile (weak probe case)and the coherence hole (strong probe case) has been demonstrated .

The effects of a second-order continuum-continuum coupling on the ionizationrate and ionization probability versus two photon detuning curves (all suitablyscaled) have also been calculated in the context of a possible experiment in which theintensities of both lasers are increased by two orders of magnitude along with asuitable reduction in the time-scale . Significant changes to the Fano profile (weakprobe case) and coherence hole (strong probe case) occur as the second-ordercontinuum-continuum coupling parameters are varied .

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M. E. St. J. Dutton and B . J. Dalton

AcknowledgmentsThe authors wish to thank Professor P . L. Knight, Dr S. Barnett, Dr K . Burnett,

Dr V. Reed and Mr M . Protopapas for helpful discussions in connection with thiswork, Mr I. Mortimer for computing programming work and Miss S . Saunders fortyping the manuscript . One of us (B.J.D.) acknowledges the financial assistance of theAustralian Research Council in connection with this work . Thanks are due toProfessor P . L. Knight of Imperial College of Science and Technology, London andDrs K. Burnett and J . Ryan of the Clarendon Laboratory, Oxford for theirhospitality during the visit of B .J.D. to the UK .

AppendixIntermediate states for the resonant processes with initial and final states of the

form I0nanb>, 11na + 1, nb -1 > are as follows for the various numbers 2n of photonsinvolved .

For 2n = 2 the states involved are :

(b) For 2n=4 the states involved are :

Aoo(aa;aa):10nanb>Q i Ifna - 1,nb>- Igita-2,nb)a~ve

I fna -1, nb~ a10nanb>, (A 2 a)

Aoo(ab; ab):10nanb)a~Ifna -1 , nb)6°ilgna- 1, fib -1 ~ a°0I f ianb -1) 6H0nanb>, (A 2 b)

Aoo(ab;ba):10nanb>aH'Ifa- l,nb)6° +Igna - l,nb - 1~ °-.

1fna-1, nb)Q;I0nanb>, (A 2 c)

Aoo(a,

~ (A 1 a)a): IO1*anb)

Ifna -1 , nb)Q.' I0nanb>,

Aoo(b, b):10nanb>yb

yb (A 1 b)Ifna, nb-1 )

IOnanb>,

Aol(b,a):IOnanb)b (Al c)lfna,nb-1) 'I"'a +l,nb-1X

Aio(a, b):1".+,'b-l>

1' '1fna , (A l d)a

nb -1 }b 10nanbX

A1 (a, a):11na+l,nb- 1>pB+ Ifna, fib - 1) aH+Ilna +1,nb -1~, (Ale)

(A 1f)Af 1(b,b):I1na +l,nb -l>be-~Ifna +l,nb -2>VbIlna +l,nb -1~ .

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References


161

Aoo(ba; ab) : IOnanb

-.I fna, nb-1>a° .Igna -1, fib -1)Q -Ifna nb -1)bH-I~anb), (A 2 d)

Aoo(ba; ba) : IOnon b

-'Lfna. nb-1)a°algna -1, nb -1)b iIgfl,, 1, nb)a

vH-~I0nanb),

(A 2 e)

Aoo(bb;bb):IOnanb) bBIfna,nb - 1)-

Igna, -fib -2) b° ~

Ifna,nb - 1>-Z-3 I0fianb),

(A2f)

Ao1(ab;aa):I0nanb)a lfna -1,nb} bb -. lgna -1,nb -1)ac '

Ifna,nb- 1)a~I1n`a+1,nb-1), (A2g)

AoI (ba;aa):I anb)bB fna, '1b-1) o`~Igna -l,nb -1)a'Ifna,nb - 1)--va 11na +1,nb -1),

(A2h)

Ao1(bb;ba):IOnanb)bH >Ifna,nb- 1)-bIgna,nb -2)b° ~

Ifna,nb -1)aB +Ilna +l,nb -1), (A2 :)

AoI(bb; ab) : IOnanb)bH'1A., nb-1 )b lgna , nb - 2 )-Q°-.

Viia +1,nb -2>b~Ilna +1,nb -1) . (A2j)

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the effect of continuum-continuum transitions on laser-induced continuum structures

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