J. Phys. B: At. Mol. Phys. 17 (1984) 719-728. Printed in Great Britain
Laser-induced autoionisation in the presence of additional
atomic continua
Jakub Zakrzewski
Jagellonian University, Institute of Physics, Reymonta 4, Krakbw, Poland
Received 21 March 1983, in final form 3 October 1983
Abstract. Laser-induced autoionisation is discussed in the simple model in which the
autoionising level is embedded in several atomic continua. The coupling to the second
continuum can drastically diminish the characteristic narrowing in the photoelectron spectra
near the Confluence of coherences point. The results seem to be important for possible
experimental verification of this narrowing, as in the real atom such coupling is hardly
avoidable. The similarity of the effects caused by the spontaneous emission and by the
coupling to the second continuum is pointed out.
1. Introduction
Laser-induced autoionisation has been the subject of considerable interest recently
(e.g. Lambropoulos and Zoller 1981, Rzqzewski and Eberly 1981). The characteristic
asymmetric profile in the excitation cross section in the weak-field limit is well understood on the basis of interference between the direct transition to the continuum and
ionisation through an unstable discrete state embedded in that continuum (Fano 1961).
The strong laser field introduces new features into the problem. The interplay between
strong radiative coupling and the interaction which produces the structure in the
continuum (in other words makes the mentioned discrete state unstable) gives rise to
new phenomena such as ‘population trapping’ (Coleman and Knight 1982) and the
existence of the sharp maximum in the long-time photoelectron spectra (Rzpiewski
and Eberly 1981, to be referred to as RE).
Much attention has been paid to the possible mechanisms leading to the broadening
of the very sharp maximum near the point of ‘confluence of coherences’. The role
played by spontaneous emission has been discussed by Agarwal et a1 (1982a) and
Haus et a1 (1983a). Rzqzewski and Eberly (1983) have included the laser incoherence
in the phase diffusion model; Haus et a1 (1983b) have discussed the effects of the
inhomogeneous (e.g. Doppler) broadening.
The purpose of this paper is to discuss yet another possible source of that broadening-coupling to the additional atomic continua. Such coupling is hardly avoidable in
the real atom (usually not only two but several configurations contribute to the real
atomic stationary state) thus the atomic model discussed here (defined in § 2) is more
realistic than the model that is usually used. The weak-field absorption profile in the
model consisting of an autoionising state coupled simultaneously to different atomic
continua (by an atomic interaction) and to a discrete state different from the initial
state (by a strong electromagnetic field) has been discussed recently by Andryushin et
0022-3700/84/050719+ 10$02.25 @ 1984 The Institute of Physics
719
720
J Zakrzewski
a1 (1982). Greenland (1982) investigated the influence of the realistic energy dependence of the dipole coupling strength in the context of 'population trapping'. I assume
the simplest possible dipole strength w dependence and concentrate on the effects of
the additional coupling only.
I define the model in 9 2, where I also recollect the perturbation results (Fano
1961), as simultaneous coupling to two continua is far less known. In 9 3 I present
the analytic solution of the problem for arbitrary laser strength. Finally, in 9 4 the
discussion of the results and conclusions are presented.
2. Description of model
The assumed system of atomic energy levels is shown in figure 1. It consists of two
discrete levels 10) and 11) and two continua I w ; I) and ( w ; 11). Both the state 11) and
the continuum I w ; I) are coupled with the initial state 10) by radiative interaction Qr.
State 11)is embedded in both continua Iw ; I) and / U ;11) due to non-zero matrix elements
V ( w )= ( U ; IIQcll) and W ( w )= ( U ; I I ~ ~ c ~Inl )the
. case of autoionisation Qc may be
the Coulomb, spin-spin or spin-orbit coupling. However, it is also possible to induce
autoionising-like resonances in the continuum using a second laser (Heller and Popov
1980, Heller et a1 1981) or by placing an atom in a static electric field (Feneuille et
a1 1979).
Figurel. Scheme of electron energy levels. State 11) mixes with atomic continua via
interaction V,; corresponding matrix elements are denot!d V(u)and W ( w ) . Both 11)
and / U ; I) are coupled to 10) via the radiative interaction V,.
I assume that the continua are prediagonalised i.e. there is no coupling between
them. It is not a restriction. Both continua are of atomic origin and therefore it is
possible to take the most convenient atomic basis. I assume that there is no coupling
between the additional (as compared with the usual RE model) continuum I w ; 11) and
the initial (e.g. ground) state 10) induced by the laser light. Such a coupling could be
easily incorporated into the model, it would, however, complicate the picture (introducing one more parameter) and therefore for the sake of simplicity is omitted.
The total Hamiltonian may be expressed in the basis of states IO), Il), / w ; I) and
I w ; 11) ( h = 1):
I+=&+ Qc+ Qr
(1)
7 21
Laser-induced autoionisation
where
I
I
fio=Eo~O)(O~+El~l)(l~+
dw w l w ; I)(w; I / + dw w l w ; II)(w; 111
er= (01~ r ~ l ) 1~1+o ) (
HC+
I
do((O/~ r l wI)lo)(w;
;
11+ HC)
(la)
(IC)
cc
In (1) f i 0 is a ‘bare energies’ Hamiltonian,
is the part of interaction which mixes
is the radiative coupling of the ground state to the
state 11) with the continua and
excited states. The requirement of states normalisation is fulfilled provided:
( w ; Ilw’; I) = ( U ; IIlw’; 11) = 6 ( w , U ’ ) , ( w ; $ 0 ’ ; 11) =o.
Following Fano (1961) we begin by diagonalising the fro+
part of the Hamiltonian (1). As every state with energy w is doubly degenerate the new continuum
states will be characterised by the energy and the additional parameter. Expanding
the new states in the old basis:
cr
cc
IW;
IW;
A ) = UA(OJ)ll)+
I
I
dw’(bA(U, W ’ ) / O ; I)+cA(w,W’)lW; 11))
(2)
B ) = U ~ ( o ) l l ) + dw’(bB(o, W ’ ) l W ; I)+cB(w,W ’ ) ( W ; 11))
Fano has found the coefficients uA,B, bA,B,c * , ~of (2) which can be expressed as:
c*( w , w ’ ) =
W ( w ’ )exp(-icp)
(.rry(w>)”’(E(w)-i)
P-
1
+.rrs(o)6(w, U ’ )
wherecp isanarbitraryphase,Bdenotestheprincipalpart, y ( w ) = .rr(I V(w)I2+l W ( w ) ( ’ )
and E ( W ) is a dimensionless Fano energy:
J Zakrzewski
722
10;
vr
The matrix elements of
between the state IO) and the new continua Iw;A),
B) are easily obtained using ( 2 ) and (3):
where q is a Fano asymmetry parameter:
and
The probability for ionisation given by first-order perturbation theory is proportional to:
I(w;
AI
QrIO)12 + /(U; BI QrI0)I2
where the Fano energy is given by the requirement of energy conservation (Fermi
golden rule). Equation ( 5 ) is convenient for discussing the influence of the additional
coupling W ( w ) in the weak-field limit. y(w) is interpreted (Fano 1961) as the
autoionising rate. Defining yo = T I V ( w ) I 2 I have y = y o ( l + 1 W(w)\’/I V(w)l’). Thus
the additional coupling increases the autoionisation rate (broadens the Fano resonance
in ( 5 ) ) . The characteristic minimum at E = - q is no longer equal to zero but has a
V ( w ) I 2 .The strength of the coupling is reduced by the factor
definite value 1 W(w)I2/\
I ~ ~ ~ ~ 1 2 / ~ l ~ ~ ~ ~ 1 2 + l ~ ~ ~ ~ 1 2 ~ .
In order to perform :alculations for arbitrary laser strength it is necessary to assume
the explicit form of (01Vrlw; I) as well as V ( w ) and W ( w ) .It is reasonable to assume
that V ( w ) and W ( w ) are proportional W ( w )= v V ( w ) . I will be interested in nearresonant transitions far above the ionisation edge (threshold effects have been discussed
by Rzazewski et a1 (1982) and by J,3iatynicka-Birula (1983)). In this region the w
dependence of matrix elements (01Vrlw; I) and V ( w ) is usually weak and smooth.
Therefore these matrix elements may be assumed w independent, which also makes
q and y independent of w .
Writing out the time dependence in (01Qrlw; I) (which oscillates as exp(iwLt),where
wL is the light frequency) (4) can be expressed as:
where Cl, = 2 ~ ( +qi ) V ( w ) exp(-icp)[(Ol Qr1w; I) exp(-iw,t)]
is made real by an
Laser-induced autoionisation
723
appropriate choice of the arbitrary phase cp. a,is in frequency units and as shown in
plays the role of the generalised Rabi frequency.
Finally, I introduce the large width (T >> 1 into the flat background terms in (6):
RE
1
-+q+i
1
ia
q+iE(w)+iu'
This is done to ensure the convergence of the integrals, which appear in th,e calculations
(§ 3, equation (10)). This is the price paid for the w independence of (01V,lw; I), which
should vanish for very large 0. At the end of calculations we take the limit (T + Q: thus
recovering the Fano shape (6).
The Hamiltonian (1) in the new basis may be expressed as:
Hamiltonian (7) will be the starting point in the next section's calculations.
3. The long-time photoelectron spectra
The dynamics of the model discussed is fully governed by the Hamiltonian (7) together
with the matrix elements (6), as the relaxation mechanisms (discussed elsewhere-see
introduction) are not included. Thus it is sufficient to work with the Schrodinger wave
equation instead of introducing the density matrix and its evolution. I expand the
Schrodinger wavevector I+(t)) in the basis of states {IO), / w ; A), I w ; B)}
I
I
k ( t ) ) = a(t)lO)+
+
dw P A ( w t ) exp(-iwd)(w; A)
dwPB(w, t) exp(-iwLt)Iw; B).
By such definition of PA,B(w, t ) the fast oscillations are eliminated from the timedependent coefficients in the expansion (8). Taking for convenience Eo = 0 and making
the rotating-wave approximation I obtain from (7) and (8) the equations:
dW(aA(W)PA(W,~ ) + ~ B ( ~ ) P B t() U) ,
724
J Zakrzewski
Equations (9) should be solved with the initial conditions ( ~ ( 0=)1, PA,B(w,0) = 0. The
set of equations (9) is solved by the Laplace transform method. The Laplace transforms
of (Y ( t ) - a (2) and of P A , B ( U , t ) - P A , B ( w, 2) are easily found from the set of algebraic
equations corresponding to (9):
The integral in the definition of K ( z ) is calculated with the ansatz ( 6 ) - ( 6 a ) , (for
excitation far above the ionisation edge the lower limit in that integral may be taken
equal to -CO) yielding:
K ( 2 )= z +
where A = wL- wl, the limit w +. CO is already taken and f = 1+ /qI2carries information
about the strength of the coupling to the second continuum. For q = 0 ( f = 1) the
result found in RE is obtained, as expected.
The probability for ionisation is given by P( t ) = 1- / a( t)I2, where
rO-+im
a ( t )=- I
2n-i
J
exp(zt)K(z)dx.
O+-ico
Thus the time evolution of P ( t ) is governed by the zeros of K ( z ) . It is easy to verify
that for f > 1 both roots of K ( z ) have negative real parts regardless of the values of
the other parameters. Therefore the atom is always ionised (in the t + CO limit), there
is no permanent population trapping (in accordance with Greenland’s (1982) remark).
The spectrum of the outgoing electrons is given by IPa(w, t)I2+IPB(w, t)I2. In the
long-time limit ( t -$ m) the spectrum can be expressed as:
W ( 0 )=
1
K ( -i( w - w L) )
12+1
aB(w)
K ( -i( w - w L) )
l2
which, taking (11) into account reads:
W ( E )=
fi2
4n-y(q2+ 1)
( E +q)2+
(f- 1 ) ( E 2 + 1)
M(E)
where
+
a4 2
16(1+q
2[f2(E2+
1
1)+ 2f(q2- 1+ 2qE) + ( q 2 +1)21.
In (12) E is the Fano energy, 6 = A/ y is the detuning and a=SZ,/fy is the effective
dimensionless Rabi frequency. From (12) it follows that the real zero in the
denominator M ( E ) for E = -4, f = 1 and a2= 4( S + q ) ( q+ 4-l) for f > 1 moves into
the complex plane, thus the spectra are broadened and the singularity at E = -q (related
to the population trapping) disappears. Note that this broadening differs from that
725
Laser-induced autoionisation
obtained in the weak-field limit in 8 2, which corresponded to change from yo to y.
The latter is fully incorporated in the definition of & ( U ) .
The photoelectron spectra are presented for small (figure 2) and large (figure 3)
values of q. The spectra show that even a small coupling to the second continuum
i
= 1 01
C
0
-1
1
f
Figure2. Electron spectra W ( E for
) q = 1, 8 = 1. Values of i I ; / y : are specified in the
figure. Higher and narrower peaks correspond to f = 1, lower ones to the f = 1.01 case as
indicated in the figure for i I ; / y ; = 12. For further explanation see text.
1.2
f
Figure 3. Spectra W ( E ) for large q(=10), 6 = 0, f = 1. Confluence occurs for i12 = 404.
f 2 given on the curves. The crosses give the peak heights for
Values of i I Z = i I ~ / y ~are
f = 1.1. For further explanation see text.
726
J Zakrzewski
gives rise to considerable changes in the heights and widths of the spectral curves near
the confluence of coherences point i.e. when one of the peaks of the Autler-Townes
doublet coincides with the Fano minimum. The positions of the maxima for f = 1 and
f = 1.01 in figure 2 slightly differ because of the decrease in the effective dipole coupling
strength (or in the corresponding effective Rabi frequency) discussed in 0 2. This
decrease has been compensated in figure 3 to show that the difference in the peak
heights is due to the additional (as compared with only change in y ) strong-field
broadening (the area under the curves has to be equal to one).
-6
0
7
9
W W L
Figure 4. Spectra showing the influence of the strong, additional coupling. The f = 1 case
corresponds to the R E result. CL,/ yo = 8, q = -3, S = -4, w - wL is in units of y = fro.
In figure 4 the spectra are presented for different values of the 'coupling constant'
f. Note the difference between the f = 1 case (no coupling to the second continuum)
and the f = 2 case (equal coupling strengths). For large f the spectrum consists of the
only one peak centred at the laser frequency-the elastic peak. It is easily understood
as large values o f f correspond to strong coupling with the flat B continuum, which
cannot saturate. This result should be compared with the effects of strong transverse
relaxation (Eberly et a1 1982, Rqzewski and Eberly 1983). In the latter case the
redistribution of radiation (Burnett et a1 1982) occurs and the corresponding spectra
are centred at w,-tlr.e energy of the discrete state 11) shifted a little from El by the
interaction with continua.
4. Discussion of results
It turns out that the coupling of the autoionising state to the additional, second
continuum can affect the photoelectron spectra considerably, especially near the point
of confluence of coherences. The results obtained seem important for proper interpretation of possible experiments in the field.
Laser-induced autoionisation
727
Experimental verification of the ‘confluence of coherences’ phenomenon by measuring the photoelectron spectra directly will be rather difficult due to the insufficient
resolution of electron energy measurements. The properties of these spectra can be
traced in double-resonance-type experiments (Lambropoulos and Zoller 1981) or in
the light spontaneously emitted during ionisation (Agarwal et a1 1982b, Haus et a1
1983a, Lewenstein et al 1983). However, in any case, attention should be paid to the
choice of the convenient atomic system. Otherwise the strong mixing of the autoionising
state even with continua which are inaccessible by direct transition from the initial
state may greatly diminish or even destroy the characteristic narrowing. The same
argument holds for autoionising-like externally induced resonances. In this case the
discrete state, which is coupled to the continuum and produces the autoionising-like
structure in that continuum, should be chosen with care to ensure that it mixes with
one continuum only.
It is interesting to note the close resemblance of effects produced by the additional
coupling and by the spontaneous emission. It manifests itself in the lack of redistribution
of radiation in both cases. It is an indirect proof of the similar role played by both
mechanisms. Both of them produce a second channel (of the photon of electron type)
by which the system can escape from being trapped by the confluence of coherences.
On the other hand they off er additional experimental opportunities: resonance fluorescence-type experiments in the case of spontaneous emission and spin-polarisation or
angular distribution of photoelectron experiments in the case of additional coupling.
Calculations on this latter topic are being performed now.
Finally a further explanation why only two continua are considered. Fano has
shown that the generalisation to an n-continua model is straightforward. It produces
after Fano diagonalisation ( n - 1) continua of the / w ; B) type, with no resonant
structure. The results of § 3 depend on /TI*, where as defined in § 2 7 describes the
strength of the coupling to the second continuum. The generalisation to n continua
would just result in a change from 1712to Z:=, 17112where v,,i = 2, . . . , n is a coupling
constant for the ith continuum.
Acknowledgments
I am indebted to Kazimierz Rzqzewski for bringing the problem of strong-field
autoionisation to my attention and for valuable remarks concerning the manuscript.
Work supported in part by the Polish Ministry of Sciences under contract MRI/S.
References
Agarwal G S, Haan S L, Burnett K and Cooper J 1982a Phys. Rev. Left. 48 1164
-1982b Phys. Rev. A 26 2277
Andryushin A I, Kazakov A E and Fedorov M V 1982 Zh. Eksp. Teor. Fiz.82 91 (Sou. Phys.-JETP 55 53)
Biaiynicka-Birula Z 1983 Phys. Rev. A 28 836
Burnett K, Cooper J, Kleiber P D and Ben Reuven A 1982 Phys. Rev. A 25 1345
Coleman P E and Knight P L 1982 J. Phys. B: At. Mol. Phys. 15 L235 (corrigendum 15 1957)
Eberly J H, Rzgiewski K and Agassi D 1982 Phys. Rev. Lett. 49 693
Fano U 1961 Phys. Rev. 124 1866
Feneuille S, Liberman S, Pinard J and Taleb A 1979 Phys. Rev. Lett. 42 1404
Greenland P T 1982 J. Phys. B: At. Mol. Phys. 15 3191
728
J Zakrzewski
Haus J, Lewenstein M and Rzaiewski K 1983a Phys. Rev. A 28 2269
Haus J, Rzpiewski K and Eberly J H 1983b Opt. Commun. 46 191
Heller Yu I, Lukinykh V K, Popov A K and Slabko V V 1981 Phys. Lett. 82A 4
Heller Yu I and Popov A K Sou. Phys.-JETP 51 255
Lambropoulos P and Zoller P 1981 Phys. Reu. A 24 379
Lewenstein M, Haus J and Rzpzewski K 1983 Phys. Rev. Lett. 50 417
Rzazewski K and Eberly J H 1981 Phys. Rev. Lett. 47 408
-1983 Phys. Rev. A 27 2026
Rzpzewski K, Lewenstein M and Eberly J H 1982 J. Phys. B:At. Mol. Phys. 15 L661