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Laser-induced resonances in ionizing radiative collisions
1 July 1996
OPTICS
COMMUNICATIONS ELSEVIER
Optics Communications 128 (1996) 30-34
Laser-induced resonances in ionizing radiative collisions Roberto Buffa Dipartimento
di Fisica. Uniuersith di Firenze, Largo Enrico Fermi 2, 50125 Firenze, Italy
Received 19 September 1995; accepted 19 January 1996
Abstract T ^^^_:-J..__-I _^^^^ I^^^ :_,..-I :~e:..,.C”lllbl”llJ ^^I,:“:^..”Calcj “..A:,.*^“r:-n+^rl n:F&.r*..r ..,.rL..*n.r^ n rl.a “^..r:.....,- :..,l..“^rl LiibC‘-,,,u”CCU ,izs”IIiiIILCT~ 111:--:_:..” 10111‘111g IdUldLIVc; Irl”r;arrplLc;u. YlllGilGillL purwaya L” UK L”I‘LIIIUUIII, IIIUULG” l
by laser-gssisted collisions, give rise to quantum interferences in the ionization cross section. The process can be viewed as a laser-induced continuum structure of transient molecules formed during atomic collisions. PACT? 32.80.Rm; 32.8O.Wr; 34.50.Rk
It has been recently argued that quantum interference effects, occurring as a result of the coherent excitation of a structured continuum of ionization of an atomic system, have to be expected also in processes induced by laser-assisted atomic collisions [ll. The present paper deals with one of these processes, discussing the effect of a laser-induced continuum structure (LICS) on laser-induced collisional ionization (LICI). LICI [2,3] belongs to a class of processes, called radiative collisions [4], which can be represented by the general equation A(i) +B(i) +nhw+(A, B) +nhw +A(f)
+B(f)
+(n+
l)hw, (1)
describing the fact that a photon of energy hw can hn -ahcnAd [A* mnler~nle .x”cJ”I_U \“A samittdl eII‘Itc”u, h-r “J the “XU trnnrbnt “c.LxAYIW.II lll”lrrL.ll (A, B), which is formed during a thermal collision between atoms A and B, leading to a change of internal states of both atoms. Examples of radiative collisions include, besides LICI, laser-induced collisional energy transfer (LICET) [51 and pair absorption and emission [6]. In the LICI process, the final
state of atom B is a free ionized state (B+ + e-j. Experimental evidence of LICI of cesium in a strontium-cesium mixture has been reported by Brkchignac et al. [2]. LICS [7,8] occurs when a featureless continuum of ionization is structured by an intense laser field, embedding a low-lying bound state into the continuum. The dressing of continuum states by the radiative coupling to the bound state gives rise to a pseudo-autoionizing resonance, which can be probed by a weak radiation field. The peculiarity of LICS, as compared to an autoionizing resonance, is that the energy position and the width of the structure can be modified by changing the wavelength and the intensity of the dressing field, respectively. The accuracy obtained in recent measurements of LICS in sodium [8] has demonstrated the feasibility of experiments nrnvirlinn naanntitcativo tnrtr fnr mnrlelr Y’“‘XU”‘6 ~1uuU”CUC’.~ &UUCU I”1 thmwetiral %11”“l”&lrU‘ IIIVUIIU. Fig. 1 shows a schematic energy-level diagram of the atomic model. While keeping the formulation general, I have in mind the case of thermal collisions between an alkaline-earth atom (A) and an alkalimetal atom (B). Atom A is assumed to be described by two bound states I ai> (i = 1, 2) and atom B by
003~4018/%/$12.00 Copyright 0 1996 Elsevier Science B.V. All rights reserved. PII SOO30-4018(96)00069-7
R. Buffa/ Optics Communications
128 (1996) 30-34
31
collision; (iv) multiphoton transitions and ionization are neglected. The process can be conveniently studied in the following product-state basi,;: d
‘P,> Atom A Fig.
Atom B
;ntnrota-.m;~
LvT”-yu”L”u =
[EC
13) = I a,> I
IE) = ta,>l
p3L
P&,
(2)
i6, = Vu, exp(iw,,r),
1.Schematic energy-level diagram of the model.
ici, = Vu, exp( -iW,2r)
three bound states 1 pj> ( j = 1 to 3) and a continuum of free states I &). States I cxy,>and I az), and I p, ) and I &) are coupled by a dipole-allowed transition, as well as states I &) and I BE), and I &> and I PE). State I &) is quasi-resonant with state I a,) and the ionization potential of atom A is quite larger than the ionization potential of atom B. Atom A, prepared in the excited state I a,), collides with atom B in the ground state in the presence of a strong radiation field of frequency wd, embedding the discrete state I &) into the continuum I & >. If a probe radiation field of frequency oP is tuned to the U1b
12)= I a,> I PA
eigenstates of an unperturbed Hamiltonian H, with eigenvalues Ej = fi wj (j = 1 to 3) and E = h wE. respectively. The following equations of motion for the probability amplitudes of states (2) are obtained in the rotating-wave approximation and in the interaction picture:
1P,>
ial>
II>= I a*)1 p,>,
t+~nritir\n Uculi)lLI”II
I p3)) - ECI a,))l/h),
,
N
\
1 R 1 tJ3/
during the
collision the excitation energy of atom A can be transferred to the dressed continuum of atom B by absorption of a photon of energy fi wP, leading to a frequency dependent ionization of atom B. To simplify the discussion, while illustrating the relevant physics, following the approach used in laser-assisted collisions to deal with the interatomic collisional interaction [5], and the approach used in LICS to describe the radiative embedding of the discrete state I p3) into the continuum I &) [7], the following approximations are made: 6) neglecting the magnetic degeneracy of the states involved in the process, the collisional interaction is described by a long-range, dipole-dipole scalar potential; (ii> all states not shown in Fig. 1 are supposed sufficiently off resonance to ensure that they are negligibly populated by radiative or collisional transitions; (iii> the laser fields are assumed constant during the
- [+x
dE
xpaE
exp(i+)~
'0 +m idi,= /0
i(i,=
-~~a,
dExd",
exp(iA,t),
exp( -iA,r)
- xdu, exp( -iA,r), (3)
with w,*=w,-w2;
AP=wP+02-wE;
Ad=~d+~3-~WE,
(4) and where V and ,yj (J = p; dj describe the collisional and radiative interactions, respectively. The initial conditions for Eqs. (3) are given by a,( - x) = 1 and a,(-~)=a~(--)=uJ--)=O. In the following it is assumed that w,~ > 0. The cross section u of the process is given by
CI
+mdE
U=
Iu,(+w)12
0
=
)
dt [ ,~~a, exp( -iA,t)
+xd"3 exd-iAdr)l
_A I)
(5)
where the brakets denote the average over the collisional parameters (impact parameters and velocity distribution). Eq. (3d) can formally be integrated in time and the result inserted in Eqs. (3b) and (3~) to eliminate the continuum by using the Markov approximation
32
R. Buffa / Optics Communications 128 (1996) 30-34
[7]. The following differential equations are obtained: ci, = -iVu,
exp(iw,,r),
ci, = -iVa, exp( -iW,2t) ci, = --(~a~ exp( -iAt)
- (YOUR - LY+ exp(iAt),
(6)
- cz3u3,
with A=A,-A,,
(7a)
(Ye= y,/2(1
+ik,),
(J-J)
LT~= y,/2(1
+ik,),
(7c) (74
(Y= Y/2(1 +iq), Y2= 27rfi I x, I 2,
(7e)
yj = 2,7fi 1 x,, 12,
(70 (w
P k,=
dw, I x, 1‘/(up + 02
/
’ I x,
7r
-
importance in the evaluation of the effective atomic parameters entering the equations of motion of the LICS process. In this case, the expression of the kj and q parameters given in Eqs. (7h)-(71) should be extended to include the contribution of bound states. However, the structure of Eqs. (6) remains unchanged. The solution for u3 can be written as ’ u2( I’) exp[ cz3(t’ - r) - i A/]
u3= -ff
i --r
= -_(y exp( -iAr)/_r[s( /(
a3
Ia,(+=)12 S(X) exp[i( x- d,)t]
(7h) - %/_T[
P k,=
do,
/
I x,, I ‘/(
Od +
w3
-
/
n
do,
X,
xd/(
mp
+
W2
I(
-
=
WE)
T( xpxd)E=E,+fro, P z
/
0
doE
id + ix)] dx
b7+4,)121
)I
xp-~Xd/(~3+idd)12,
o=2~Y2~+m
xpxd/(“d+03-oE)
&xd)E=E,+fio,
ff3
-
(9)
and the cross section (5) as a convolution integral:
(7’) T(
s( x, exp[i( x - A,$] 2
(79
’
Y-
dx
%)
r I xd i;=E,+iiW, P
(8)
where the Fourier transform S(x) of u2 has been introduced. Then the quantity I a,( +m> I ’ can be written as
WE)
L,+bo,
x) exp(ixr)
id + ix)] dx,
-
dt’
’
and where P denotes the principal part of the integrals in (7h)-(71). In (61, the kjyj terms represent energy shifts of states I j) ( j = 1, 2), while the q Fano parameter (71) describes the relative weight of the two interfering pathways leading to ionization of atom B. For I q I x 1 LICI is the dominant process, while for I q I * 1, two-photon radiative-collisional excitation of state I &), followed by photoioinization, is predominant. Dai and Lambropoulos [9] have pointed out that, in many cases, non-resonant Raman processes involving bound states can be of crucial
dw,,
(10)
where F( A,) is the Fano profile of the LICS process involving states I p2 > and I &) and the continuum I &) of atom B:
F( ‘d>
( A, + by,/2 + 9Ys/2)2 (11)
= ( Ad
+
k,y,/212
+
(
Y3/2)2 ’
For a weak probe laser field, the temporal evolution of u2 is governed only by the collisional interaction V, and the solution of Eqs. (6a), (6b) is easily obtained in the framework of the adiabatic collisional dressed states introduced in the study of the LICET process [5,10]. In fact, when the condition
33
R. Buffa/ Optics Communications 128 (19%) 30-34
IWl
holds (adiabatic collision) then the solution of (6a), (6b) can be written as a,=cos
a2
*I=w,*
8exp
i bn-WI (y
d$
=sindexp(-;:2(e)di),
(12)
with tan(28) = 2V/q2,
and where
n=Y[*+\ll+]
(13)
is the adiabatic time-dependent eigenvalue of the adiabatic collisional dressed state whose population remains constant during the collision. The modulus square of the Fourier transform of averaged over the collisional parameters, a29 ( I S( Ap> I*) reproduces the LICET line shape in the weak-field regime [5]. Very accurate theoretical and experimental studies [5,11] show that the LICET cross section shows an asymmetric shape, with a more rapid fall-off on one side, and a tail extending for several tens of cm- ’ , reflecting the van der Waals shift of the atomic levels. A very good analytical expression for ( I $A,) I ‘> is provided by ( I S( A,,) I * > = Soexp[(A, + ~,z)/A,],
I
0.9 -40
-30
-20
-10
0
10
20
30
S/AC
301
:
:
:
:
:
:
:
:
:
:
:
:
:
F
(14.a)
for (Ap + o,*) G 0, and
(l~(A,)12~=~,
w;;A,o.‘(( A, + co,*) + A, xev[
-(A,
x(A,+20,2j-!~5,
-30
+ ~12)/A,]]-o’s
-20
-10
0
10
20
30
40
h/AC
(14.b)
for (A, + o ,2) > 0, where A, (typically of the order of 1 cm- ’ ) corresponds roughly to the inverse of the average collisional time, and w,JA, =*- 1 satisfying the condition of adiabatic collisions. The cross section u has been calculated by using I_..\ I__\ . I_ a\ ^ ..^^ \luJ, (11) and (14) ror different vaiues of the atomic parameters. Fig, 2 reports some of the results obtained for w ,* = 604, and three typical values of the Fano parameter (q = 0, 1 and 10) setting k, = k, = 0. Here S = (wd + w,) - (w, + 0,) is the interatomic two-photon detuning and a0 is the LICI cross section, i.e., the ionization cross section in the absence of the dressing laser field. The curves (a), (b) and (c) refer to three different values of the dressing laser
Fig. 2. Ionization cross section u versus interatomic two-photon detuning 6 =(w,, + wJ-Cop+ 0,). calculated for o,~ =6OA,, three different values of the q Fano parameter (0, 1 and IO), and three different values of the dressing laser intensity, corresponding to a photoionization decay rate of state 1 & > y3 = A, (a), y, = 2 A, (b) and y3 = 46, (c). v0 is the ionization cross section in the absence of a dressing laser field.
intensity, ranging from a few to several tens of and corresponding to a photoionization decay rate of state I &> y3 = A, (a>, y3 = 24, (b) and y3 = 44, (cl. The effects of quantum interferences are evident in all the reported line shapes with a moderate
GW/cm’,
34
R. Buffa/
Optics Communications
ionization suppression for q = 0 and a remarkable ionization enhancement for q = 10. It is worth to notice that the predicted enhancement is not due to a two-photon radiative-collisional excitation of state 1 &) followed by photoioinization of the isolated atom B. In fact, the photoionization lifetime of state ) &) is of the order of the average collisional time (y3 z A,) and then the ionization process of atom B occurs during the collisional interaction. Indeed, the process can be viewed as a laser-induced continuum structure of a transient molecule formed during the atomic collision. In conclusion, a study of laser-induced resonances in ionizing radiative collisions has been reported. Interfering pathways to the continuum, induced by laser-assisted collisions (radiative-collisional ionization and two-photon radiative-collisional excitation followed by photoionization), give rise to Fano-type line shapes for the ionization cross section. The process can be viewed as a laser-induced continuum structure of transient molecules formed during atomic collisions. I wish to thank Dr. Manlio Matera for a critical reading of the manuscript.
128 (1996) 30-34
References [I] R. Buffa, Phys. Rev. A 46 (1992) Rll7l;
Phys. Rev. A 48 (1993) 4797; Optics Lett. 20 (1995) 204; Phys. Rev. A 53
(1996) 607. [2] C. Brechignac, Ph. Cahuzac and A. JXbarre, J. Phys. B 13 (1980) L383. [3] M. Crance and S. Feneuille, J. Phys. B 13 (1980) 3165. [4] N.K. Rahman and C. Guidotti, eds., Photon-assisted collisions and related topics (Hatwood Academic, London, 1982). (51 A. Bambini, P.R. Berman, R. Buffa, E.J. Robinson and M. Matera, Phys. Rep. 238 (1994) 245, and references therein. (61 J.C. White, G.A. Zdasiuk, J.F. Young and S.E. Harris, Optics Lett. 4 (1979) 137; J.C. White, Optics Lett. 5 (1980) 239. [7] P.L. Knight, M.A. Lauder and B.J. Dalton, Phys. Rep. 190 (1990) 1, and references therein. [8] Y.L. Shao, D. Charalambidis, C. Fotakis, Jian Zhang and P. Lambropoulos, Phys. Rev. Len. 67 (1991) 3669; S. Cavalieri, F. Pavone and M. Matera. Phys. Rev. Lett. 67 (1991) 3673; S. Cavalieri, R. Eramo and L. Fini, J. Phys. B 28 (1995) 1793; S. Cavalieri, R. Eramo, L. Fini and R. Buffa, I. Phys. B 28 ( 1995) 2637. [9] Bo-nian Dai and P. Lambropoulos, Phys. Rev. A 36 (1987) 5205; 39 (1989) 3704. [IO] A. Bambini and P.R. Berman, Phys. Rev. A 35 (1987) 3753. [l 11 L. Fini, R. Buffa, R. Pratesi, A. Bambini, M. Matera and M. Mazzoni, Europhys. Lett. 18 (1992) 23.
OPTICS
COMMUNICATIONS ELSEVIER
Optics Communications 128 (1996) 30-34
Laser-induced resonances in ionizing radiative collisions Roberto Buffa Dipartimento
di Fisica. Uniuersith di Firenze, Largo Enrico Fermi 2, 50125 Firenze, Italy
Received 19 September 1995; accepted 19 January 1996
Abstract T ^^^_:-J..__-I _^^^^ I^^^ :_,..-I :~e:..,.C”lllbl”llJ ^^I,:“:^..”Calcj “..A:,.*^“r:-n+^rl n:F&.r*..r ..,.rL..*n.r^ n rl.a “^..r:.....,- :..,l..“^rl LiibC‘-,,,u”CCU ,izs”IIiiIILCT~ 111:--:_:..” 10111‘111g IdUldLIVc; Irl”r;arrplLc;u. YlllGilGillL purwaya L” UK L”I‘LIIIUUIII, IIIUULG” l
by laser-gssisted collisions, give rise to quantum interferences in the ionization cross section. The process can be viewed as a laser-induced continuum structure of transient molecules formed during atomic collisions. PACT? 32.80.Rm; 32.8O.Wr; 34.50.Rk
It has been recently argued that quantum interference effects, occurring as a result of the coherent excitation of a structured continuum of ionization of an atomic system, have to be expected also in processes induced by laser-assisted atomic collisions [ll. The present paper deals with one of these processes, discussing the effect of a laser-induced continuum structure (LICS) on laser-induced collisional ionization (LICI). LICI [2,3] belongs to a class of processes, called radiative collisions [4], which can be represented by the general equation A(i) +B(i) +nhw+(A, B) +nhw +A(f)
+B(f)
+(n+
l)hw, (1)
describing the fact that a photon of energy hw can hn -ahcnAd [A* mnler~nle .x”cJ”I_U \“A samittdl eII‘Itc”u, h-r “J the “XU trnnrbnt “c.LxAYIW.II lll”lrrL.ll (A, B), which is formed during a thermal collision between atoms A and B, leading to a change of internal states of both atoms. Examples of radiative collisions include, besides LICI, laser-induced collisional energy transfer (LICET) [51 and pair absorption and emission [6]. In the LICI process, the final
state of atom B is a free ionized state (B+ + e-j. Experimental evidence of LICI of cesium in a strontium-cesium mixture has been reported by Brkchignac et al. [2]. LICS [7,8] occurs when a featureless continuum of ionization is structured by an intense laser field, embedding a low-lying bound state into the continuum. The dressing of continuum states by the radiative coupling to the bound state gives rise to a pseudo-autoionizing resonance, which can be probed by a weak radiation field. The peculiarity of LICS, as compared to an autoionizing resonance, is that the energy position and the width of the structure can be modified by changing the wavelength and the intensity of the dressing field, respectively. The accuracy obtained in recent measurements of LICS in sodium [8] has demonstrated the feasibility of experiments nrnvirlinn naanntitcativo tnrtr fnr mnrlelr Y’“‘XU”‘6 ~1uuU”CUC’.~ &UUCU I”1 thmwetiral %11”“l”&lrU‘ IIIVUIIU. Fig. 1 shows a schematic energy-level diagram of the atomic model. While keeping the formulation general, I have in mind the case of thermal collisions between an alkaline-earth atom (A) and an alkalimetal atom (B). Atom A is assumed to be described by two bound states I ai> (i = 1, 2) and atom B by
003~4018/%/$12.00 Copyright 0 1996 Elsevier Science B.V. All rights reserved. PII SOO30-4018(96)00069-7
R. Buffa/ Optics Communications
128 (1996) 30-34
31
collision; (iv) multiphoton transitions and ionization are neglected. The process can be conveniently studied in the following product-state basi,;: d
‘P,> Atom A Fig.
Atom B
;ntnrota-.m;~
LvT”-yu”L”u =
[EC
13) = I a,> I
IE) = ta,>l
p3L
P&,
(2)
i6, = Vu, exp(iw,,r),
1.Schematic energy-level diagram of the model.
ici, = Vu, exp( -iW,2r)
three bound states 1 pj> ( j = 1 to 3) and a continuum of free states I &). States I cxy,>and I az), and I p, ) and I &) are coupled by a dipole-allowed transition, as well as states I &) and I BE), and I &> and I PE). State I &) is quasi-resonant with state I a,) and the ionization potential of atom A is quite larger than the ionization potential of atom B. Atom A, prepared in the excited state I a,), collides with atom B in the ground state in the presence of a strong radiation field of frequency wd, embedding the discrete state I &) into the continuum I & >. If a probe radiation field of frequency oP is tuned to the U1b
12)= I a,> I PA
eigenstates of an unperturbed Hamiltonian H, with eigenvalues Ej = fi wj (j = 1 to 3) and E = h wE. respectively. The following equations of motion for the probability amplitudes of states (2) are obtained in the rotating-wave approximation and in the interaction picture:
1P,>
ial>
II>= I a*)1 p,>,
t+~nritir\n Uculi)lLI”II
I p3)) - ECI a,))l/h),
,
N
\
1 R 1 tJ3/
during the
collision the excitation energy of atom A can be transferred to the dressed continuum of atom B by absorption of a photon of energy fi wP, leading to a frequency dependent ionization of atom B. To simplify the discussion, while illustrating the relevant physics, following the approach used in laser-assisted collisions to deal with the interatomic collisional interaction [5], and the approach used in LICS to describe the radiative embedding of the discrete state I p3) into the continuum I &) [7], the following approximations are made: 6) neglecting the magnetic degeneracy of the states involved in the process, the collisional interaction is described by a long-range, dipole-dipole scalar potential; (ii> all states not shown in Fig. 1 are supposed sufficiently off resonance to ensure that they are negligibly populated by radiative or collisional transitions; (iii> the laser fields are assumed constant during the
- [+x
dE
xpaE
exp(i+)~
'0 +m idi,= /0
i(i,=
-~~a,
dExd",
exp(iA,t),
exp( -iA,r)
- xdu, exp( -iA,r), (3)
with w,*=w,-w2;
AP=wP+02-wE;
Ad=~d+~3-~WE,
(4) and where V and ,yj (J = p; dj describe the collisional and radiative interactions, respectively. The initial conditions for Eqs. (3) are given by a,( - x) = 1 and a,(-~)=a~(--)=uJ--)=O. In the following it is assumed that w,~ > 0. The cross section u of the process is given by
CI
+mdE
U=
Iu,(+w)12
0
=
)
dt [ ,~~a, exp( -iA,t)
+xd"3 exd-iAdr)l
_A I)
(5)
where the brakets denote the average over the collisional parameters (impact parameters and velocity distribution). Eq. (3d) can formally be integrated in time and the result inserted in Eqs. (3b) and (3~) to eliminate the continuum by using the Markov approximation
32
R. Buffa / Optics Communications 128 (1996) 30-34
[7]. The following differential equations are obtained: ci, = -iVu,
exp(iw,,r),
ci, = -iVa, exp( -iW,2t) ci, = --(~a~ exp( -iAt)
- (YOUR - LY+ exp(iAt),
(6)
- cz3u3,
with A=A,-A,,
(7a)
(Ye= y,/2(1
+ik,),
(J-J)
LT~= y,/2(1
+ik,),
(7c) (74
(Y= Y/2(1 +iq), Y2= 27rfi I x, I 2,
(7e)
yj = 2,7fi 1 x,, 12,
(70 (w
P k,=
dw, I x, 1‘/(up + 02
/
’ I x,
7r
-
importance in the evaluation of the effective atomic parameters entering the equations of motion of the LICS process. In this case, the expression of the kj and q parameters given in Eqs. (7h)-(71) should be extended to include the contribution of bound states. However, the structure of Eqs. (6) remains unchanged. The solution for u3 can be written as ’ u2( I’) exp[ cz3(t’ - r) - i A/]
u3= -ff
i --r
= -_(y exp( -iAr)/_r[s( /(
a3
Ia,(+=)12 S(X) exp[i( x- d,)t]
(7h) - %/_T[
P k,=
do,
/
I x,, I ‘/(
Od +
w3
-
/
n
do,
X,
xd/(
mp
+
W2
I(
-
=
WE)
T( xpxd)E=E,+fro, P z
/
0
doE
id + ix)] dx
b7+4,)121
)I
xp-~Xd/(~3+idd)12,
o=2~Y2~+m
xpxd/(“d+03-oE)
&xd)E=E,+fio,
ff3
-
(9)
and the cross section (5) as a convolution integral:
(7’) T(
s( x, exp[i( x - A,$] 2
(79
’
Y-
dx
%)
r I xd i;=E,+iiW, P
(8)
where the Fourier transform S(x) of u2 has been introduced. Then the quantity I a,( +m> I ’ can be written as
WE)
L,+bo,
x) exp(ixr)
id + ix)] dx,
-
dt’
’
and where P denotes the principal part of the integrals in (7h)-(71). In (61, the kjyj terms represent energy shifts of states I j) ( j = 1, 2), while the q Fano parameter (71) describes the relative weight of the two interfering pathways leading to ionization of atom B. For I q I x 1 LICI is the dominant process, while for I q I * 1, two-photon radiative-collisional excitation of state I &), followed by photoioinization, is predominant. Dai and Lambropoulos [9] have pointed out that, in many cases, non-resonant Raman processes involving bound states can be of crucial
dw,,
(10)
where F( A,) is the Fano profile of the LICS process involving states I p2 > and I &) and the continuum I &) of atom B:
F( ‘d>
( A, + by,/2 + 9Ys/2)2 (11)
= ( Ad
+
k,y,/212
+
(
Y3/2)2 ’
For a weak probe laser field, the temporal evolution of u2 is governed only by the collisional interaction V, and the solution of Eqs. (6a), (6b) is easily obtained in the framework of the adiabatic collisional dressed states introduced in the study of the LICET process [5,10]. In fact, when the condition
33
R. Buffa/ Optics Communications 128 (19%) 30-34
IWl
holds (adiabatic collision) then the solution of (6a), (6b) can be written as a,=cos
a2
*I=w,*
8exp
i bn-WI (y
d$
=sindexp(-;:2(e)di),
(12)
with tan(28) = 2V/q2,
and where
n=Y[*+\ll+]
(13)
is the adiabatic time-dependent eigenvalue of the adiabatic collisional dressed state whose population remains constant during the collision. The modulus square of the Fourier transform of averaged over the collisional parameters, a29 ( I S( Ap> I*) reproduces the LICET line shape in the weak-field regime [5]. Very accurate theoretical and experimental studies [5,11] show that the LICET cross section shows an asymmetric shape, with a more rapid fall-off on one side, and a tail extending for several tens of cm- ’ , reflecting the van der Waals shift of the atomic levels. A very good analytical expression for ( I $A,) I ‘> is provided by ( I S( A,,) I * > = Soexp[(A, + ~,z)/A,],
I
0.9 -40
-30
-20
-10
0
10
20
30
S/AC
301
:
:
:
:
:
:
:
:
:
:
:
:
:
F
(14.a)
for (Ap + o,*) G 0, and
(l~(A,)12~=~,
w;;A,o.‘(( A, + co,*) + A, xev[
-(A,
x(A,+20,2j-!~5,
-30
+ ~12)/A,]]-o’s
-20
-10
0
10
20
30
40
h/AC
(14.b)
for (A, + o ,2) > 0, where A, (typically of the order of 1 cm- ’ ) corresponds roughly to the inverse of the average collisional time, and w,JA, =*- 1 satisfying the condition of adiabatic collisions. The cross section u has been calculated by using I_..\ I__\ . I_ a\ ^ ..^^ \luJ, (11) and (14) ror different vaiues of the atomic parameters. Fig, 2 reports some of the results obtained for w ,* = 604, and three typical values of the Fano parameter (q = 0, 1 and 10) setting k, = k, = 0. Here S = (wd + w,) - (w, + 0,) is the interatomic two-photon detuning and a0 is the LICI cross section, i.e., the ionization cross section in the absence of the dressing laser field. The curves (a), (b) and (c) refer to three different values of the dressing laser
Fig. 2. Ionization cross section u versus interatomic two-photon detuning 6 =(w,, + wJ-Cop+ 0,). calculated for o,~ =6OA,, three different values of the q Fano parameter (0, 1 and IO), and three different values of the dressing laser intensity, corresponding to a photoionization decay rate of state 1 & > y3 = A, (a), y, = 2 A, (b) and y3 = 46, (c). v0 is the ionization cross section in the absence of a dressing laser field.
intensity, ranging from a few to several tens of and corresponding to a photoionization decay rate of state I &> y3 = A, (a>, y3 = 24, (b) and y3 = 44, (cl. The effects of quantum interferences are evident in all the reported line shapes with a moderate
GW/cm’,
34
R. Buffa/
Optics Communications
ionization suppression for q = 0 and a remarkable ionization enhancement for q = 10. It is worth to notice that the predicted enhancement is not due to a two-photon radiative-collisional excitation of state 1 &) followed by photoioinization of the isolated atom B. In fact, the photoionization lifetime of state ) &) is of the order of the average collisional time (y3 z A,) and then the ionization process of atom B occurs during the collisional interaction. Indeed, the process can be viewed as a laser-induced continuum structure of a transient molecule formed during the atomic collision. In conclusion, a study of laser-induced resonances in ionizing radiative collisions has been reported. Interfering pathways to the continuum, induced by laser-assisted collisions (radiative-collisional ionization and two-photon radiative-collisional excitation followed by photoionization), give rise to Fano-type line shapes for the ionization cross section. The process can be viewed as a laser-induced continuum structure of transient molecules formed during atomic collisions. I wish to thank Dr. Manlio Matera for a critical reading of the manuscript.
128 (1996) 30-34
References [I] R. Buffa, Phys. Rev. A 46 (1992) Rll7l;
Phys. Rev. A 48 (1993) 4797; Optics Lett. 20 (1995) 204; Phys. Rev. A 53
(1996) 607. [2] C. Brechignac, Ph. Cahuzac and A. JXbarre, J. Phys. B 13 (1980) L383. [3] M. Crance and S. Feneuille, J. Phys. B 13 (1980) 3165. [4] N.K. Rahman and C. Guidotti, eds., Photon-assisted collisions and related topics (Hatwood Academic, London, 1982). (51 A. Bambini, P.R. Berman, R. Buffa, E.J. Robinson and M. Matera, Phys. Rep. 238 (1994) 245, and references therein. (61 J.C. White, G.A. Zdasiuk, J.F. Young and S.E. Harris, Optics Lett. 4 (1979) 137; J.C. White, Optics Lett. 5 (1980) 239. [7] P.L. Knight, M.A. Lauder and B.J. Dalton, Phys. Rep. 190 (1990) 1, and references therein. [8] Y.L. Shao, D. Charalambidis, C. Fotakis, Jian Zhang and P. Lambropoulos, Phys. Rev. Len. 67 (1991) 3669; S. Cavalieri, F. Pavone and M. Matera. Phys. Rev. Lett. 67 (1991) 3673; S. Cavalieri, R. Eramo and L. Fini, J. Phys. B 28 (1995) 1793; S. Cavalieri, R. Eramo, L. Fini and R. Buffa, I. Phys. B 28 ( 1995) 2637. [9] Bo-nian Dai and P. Lambropoulos, Phys. Rev. A 36 (1987) 5205; 39 (1989) 3704. [IO] A. Bambini and P.R. Berman, Phys. Rev. A 35 (1987) 3753. [l 11 L. Fini, R. Buffa, R. Pratesi, A. Bambini, M. Matera and M. Mazzoni, Europhys. Lett. 18 (1992) 23.