le
Molecules
M.V. Fedorov
General Physics Institute, Russian Academy of Sciences
38 Vavilov St., Moscow, 119111, Russia
Abstract. The existing theoretical and experimental works on interference stabilization of
Rydberg atoms are overviewed. Physical origin of the phenomenon, its main features and the
conditions of existence, as well as the main theoretical approaches used for its description are
discussed. These ideas are used to describe interference stabilization of molecules with respect to
photodissociation. Specific calculations are carried out for the hydrogen molecular ion H* . The
main features of strong-field photodissociation are described. The conditions of stabilization and
destabilization of a molecule are found and related to features of the initial-state molecular
vibrational wave function.
INTRODUCTION
In the most general form, stabilization of a decaying quantum system means that its
decay can be slowed down under the influence of some external fields or forces. The
decay of atoms driven by a strong laser field is associated usually with the irreversible
process of one-photon or multiphoton ionization. In molecules the decay process to be
slowed down by a strong field can be related to either ionization or dissociation. In
this report two such phenomena will be discussed: strong-field photoionization of
Rydberg atoms and photodissociation of molecules. In both cases the mechanism of
stabilization is assumed to be related to the strong-field Raman-type transitions
between close bound levels: Rydberg levels in atoms and vibrational levels of the
ground electronic state in molecules. This mechanism is known as the interference
stabilization (IS) [1-3]. Specific manifestations of IS in atoms and molecules can differ
rather significantly because of big differences in features of atomic and molecular
spectra and transitions. Discussion of differences and similarity between the atomic
and molecular IS is very instructive and useful for understanding the physics of the
phenomena. Both rather well established and new results on IS in atoms and
molecules are described.
OF
The effect of IS in Rydberg atoms was described and discussed for the first time in
1988 [1], Many features and manifestations of this phenomenon are summarized in
Ref. [2]. As mentioned above, IS in Rydberg atoms arises owing to strong-field
CP634, Science of Superstrong Field Interactions, edited by K. Nakajima and M. Deguchi
© 2002 American Institute of Physics 0-7354-0089-X/02/$ 19.00
19
FIGURE 1. A scheme of Raman-type transitions between Rydberg levels.
Raman-type transitions between close Rydberg levels En, with the main contribution
to the corresponding two-photon matrix elements given by the atomic continuum (Atype transitions, Fig. 1). Subsequent transitions to the continuum from coherently repopulated levels En interfere with each other, and this interference suppresses the
process of photoionization and gives rise to IS. The existence criterion of IS is
equivalent to the criterion of efficient A-type transitions, and it has the form TI > En+\
- En, where F) =(V2)eo|^«£:| *s the Fermi-Golden-Rule (FGR) ionization width of
the Rydberg level En, d is the dipole moment, and 80 is the light field-strength
amplitude. Near by the ionization threshold, n » I and E « 1 (in atomic units) the
dipole matrix elements dnE can be approximated by their quasiclassical expression [3,
4] dnE ~ /T3/2co~5/3, where CO is the light frequency, CO « 1, and E ~ En + CO. This
reduces the IS criterion to the form 80 > co5/3.
In the theory of IS, the wave function *P(£) of an atom in a light field e(0=£o cos(cot)
is expanded in a series of the field-free atomic wave functions
(D
with unknown time-dependent probability amplitudes Cn(i) and CE(?). Equations for
these probability amplitudes follow from the Schrodinger equation, and these
equations can be significantly simplified owing to a series of beautiful features of
Rydberg atoms. Not dwelling upon the details, let us mention only the main steps of
such simplifications. The approximations which can be used are RWA and adiabatic
elimination of the continuum. The latter is related to the flat-continuum
approximation, which means that the dipole matrix elements dnE sufficiently slowly
depend on the energy in the continuum E. With the procedure of adiabatic elimination
performed, the set of equation for Cn(f) and C^(0 is reduced to the set of differential
equations for the discrete-level probability amplitudes Cn(f) only. Connection with
continuum in these equations is characterized by the only decay constant
F ~80/w3G)10/3 identical for all the initial and final levels En and En>.
The simplest model in which IS exists is the model of two discrete levels (denoted
as, e.g., E\ and E2) plus a continuum. In such a model, equations for C\(f) and Ci(i)
take the form
20
(2)
Both in this simple model and in a general case of a multilevel Rydberg atom one
can either solve the initial value problem or look for the stationary solution of
equations like Eqs. (2). In the last case, one gets the quasienergy solutions
Cn(t)=&xp(-~ijt)an and complex quasienrgies of the system under consideration. For the
simplest two-level system the arising complex quasienergies are easily found to be
given by
(3)
These complex quasienergies are plotted in Fig. 2 in the form of quasienergy zones,
the boundaries of which are determined as Re(y ± )+|lm(y ± )| and Re(y ± )-|lm(y ± )|.
In dependence on F °c e^ the quasienergy zones at first broaden, than they have a
branching point at T = E2-E19 and at F > E2 - El one of the two quasienergy levels
continues to broaden whereas the other one narrows. Formation of a narrow longliving qausienergy level in the strong-field limit indicates a possibility of stabilization
of the system. At T>E2~El both narrow and wide quasienergy levels are seen to be
centered at Re(j+) = (E2 -E^/2, i.e., exactly between the field-free levels E\ and E2.
In a multilevel system of Rydberg levels En decaying to the continuum, all the
quasienergy levels behave similar to the narrowing quasienergy level of the two-level
system: they narrow with a growing fild-strength amplitude 80, and they appear to be
localized between the field-fee Rydberg levels, at (En+l~En)/2 [1,2].
To see explicitly the effect of stabilization, one has to solve the initial-value
problem to get, for example, the total probability of ionization per pulse in its
dependence on the peak light intensity. Such calculations were done [6] for the pulse
envelope of the form 8o(0=£osin2(7C//T). Both multilevel structure of Rydberg levels
and degeneracy in the angular momentum / were taken into account. One of the results
0
E2-Ei
T
FIGURE 2. Quasienergy zones of the model decaying two-level system characterized by Eqs. (2), (3).
21
2.7 ps
0.6 ps
0.1
0
0.1
0.2
0
10
20
FIGURE 3. (a) Tlie calculated [6] total probability of ionization per pulse w? vs. fiuence F=Ix i (F in
arbitrary units}, and (h) experimentally measured [7] ion yield Ni(F), F in J/cm , in both cases n = 27;
tK=2n n3 is the Kepler period.
is shown in Fig. 3a, The probability of ionization is plotted in dependence on fiuence
F=/XT for two different pulse durations. The curves of Fig. 3h show the
experimentally measured (under similar conditions) ion yield vs. fiuence F [7]. In both
cases the curves corresponding to shorter pulse duration and, hemnce, higher intensity
are located lower than the curves corresponding to longer pulses and lower intensity.
This is a direct confirmation of stabilization, Le., decrease of the yield with a growing
light intensity. Theoretical and experimental results are seen to close to each other.
To conclude this Section, it's reasonable to mention an approach to the theory of IS
alternative to that discussed above. This approach involves an attempt to use
quasiclassical (WKB) description for solving directly the nonstationary Schrodinger
equation [8]. The result is given by the following very simple analytical expression for
the probability of ionization from a Rydberg level
-.2/3
—— \\-«
(4)
,5/3
CO
> K \ \
J'J
where T and Jo are the gamma- and zero-order Bessel functions. The dependence Wf(I)
calculated with the help of this formula for a trapezoidal pulse envelope is plotted in
Fugure 4 together with the results of exact numerical solution of the Schrodinger
equation [9]. The coincidence looks rather impressive.
1
10"
10"
/ [W/craf]
12
10
13
10
14
10
1015 1016
FIGURE 4. The total probability of ionization per pulse vs. the peak light intensity: numerical (dots)
and analytical (4) (solid line) solutions; n = 5 and trapezoidal envelope £ 0 (/) with the switch-on/off
and plateau periods equal to, respectively, 2 and 10 optical cycles [9]; dashed line - perturbation theory.
22
RAMAN-TYPE TRANSITIONS IN MOLECULES
A scheme of Raman-type transitions to be taken into account is shown in Fig. 5.
The frequency w is assumed to be high enough to provide one-photon transition
between the ground and first excited, unstable, electronic state. The levels to be repopulated via A-type transitions are the vivbrational levels Ev of the ground electronic
state. In analogy with Eq. (1), let us expand the wave function of a molecule in a series
of field-free wave functions
(5)
where R and r are the internuclear distance and the electron position vector, %V(R)
and %E(R) are nuclear wave functions in the electronic states f/o and f/i, v is the
vibrational quantum number and E is the energy of a molecule in the unstable state f/i,
\|/0(r,/?) and XI/^F, R) are the electronic wave functions of a molecule in the states Uo
and U\, The main difference with the case of Rydberg atoms concerns the dipole
matrix elements dVE. They were calculated for the molecular ion //* [3], and the
dependence of dVE on E was shown to have an oscillating character. This means that
the continuum of nuclear motion in the excited electronic state is not flat. As a
consequence, the procedure of adiabatic elimination of the continuum appears to be
inapplicable. However, a kind of a generalized procedure of semi-adiabatic
elimination (explained below) was shown to be valid. In the framework of this
procedure of semi-adiabatic elimination equations for the ground-state vibrational
probability amplitudes Cv(t) have the form
(6)
where Qv v,(co) are the Raman-type two-photon matrix elements
Gvy (co) =dt' Rv,, (?) exp{- i
(7)
and the function Rv v,(t) are given by
0.4
0.2
0
-0.1
1
2
3
4
5
6
7
FIGURE 5. Potential curves f/0, i(/?) (in atomic units) and Raman type transitions.
23
u.u
Re[/?Vi ,,•(?)]> a.u.
-0.4
-2,5
/
<
''*""* >*<i">t' ' "'</**\'*>>,J "i^'
0
-"-..,;l--^-^1 •^><i^3^r_.Jv^*'^ "" , ,
t
-2.0
-1.5
-0.5
-1.0
FIGURE 6. Real parts of the functions Rv>v>(f) (8); solid line - v=2 and v'=0, dashed line - v=2 and
v'=l, dash-dotted line - v=2 and v'=2, dotted line - v=2 and v'=3.
vE dEv.
(8)
In a pure form adiabatic elimination of the continuum is valid when RVtV<(t') can be
approximated by the delta- functions. However, for molecules this approximation does
not hold good. Calculated explicitly for //* , real parts of several functions Rvy(t') are
shown in Fig. 6. These functions are seen to be localized in a time interval 8f ~ 2 f s
around f=Q. For pulse duration of 70 fs 8? « T and 1/0% ~ 10 fs, where cob ~ 0.01 a.u.
is the vibrational frequency of //* . For this reason, the slow functions 8o(ff+0 and
Cv{f+f) can be taken out of the integrals over f at f=t to give Eqs. (7). On the other
hand, the function exp(— /coO is not slow compared to Rvv,(f), and must be retained
under the symbol of integral in Eq. (7). This is why such an approximation is defined
as the semi-adiabatic elimination of the continuum. The factor exp(-/<oO under the
symbol of integral in Eq. (7) determines the difference between the pure adiabatic and
semi-adiabatic elimination of the continuum, as well as the dependence of the Ramantype matrix elements Qvv, on the light frequency ox
With the help of transformation Cv(t)=&Kp(-iEvf)hv(t) Eqs. (6) can be reduced to the
form of equations with constant coefficients, which have stationary, or quasienergy,
solutions. The corresponding eigenvalues are the complex quasienergies y, to be found
from the equation
Det
= 0.
(9)
Two quasienergy zones (starting form v = 2 and 3) are shown in Fig. 7 in the
dependence on the light intensity / for co = 0.338. The boundaries of zones are
determined as Re(y) + Im(y)| and Re(y) - Im(y)|. Spacing between the boundaries
equals the width of quasienergy zones (levels) Im(y)|. It's seen that the two zones
show in Fig. 7 behave very similar to the two-level zones in Rydberg atoms (Fig. 2).
Again, these zones at first broaden, then overlap, and then a narrowing zone is formed
on the background of the broadening one. Narrowing of a quasienergy zone is a
manifestation of stabilization.
24
-0.05
-0.06
-0.07
v=2
-0.08
2
3
/ [1014W/cm2]
4
5
6
FIGURE 7. Quasienergy zones; CO = 0.338 [all in a.u.]
Alternatively, Eqs. (6) can be solved directly to determine the time-dependent
probability amplitudes Cv(0» populations of vibrational levels |CV(^)| , probability of
dissociation wD(t) = l-^\Cv(t] , and total probability of dissociation per pulse
V
w
=W
X
Dtot D( ) • To estimate the role of Raman-type transitions, the time-dependent and
total probabilities of dissociation will be compared with the FGR with saturation
probabilities
t
(
^
(10)
and WD™ =WpGR(i). The calculated dependencies wDtot(l) and w£^(/) are shown in
Fig. 8 (correspondingly, by the solid and dotted lines). The exact probability of
1.0
0.5
0
0 1 2 3 4 5 6
/ [1014 W/cm2
FIGURE 8, Total probability if dissociation per pulse vs. the peak light intensity: exact calculation
(solid line) and FGR with saturation formula (10) (dashed line); CO = 0.338 a.u., v0 = 2.
dissociation is seen to have a knee-structure, which indicates in this case a partial
stabilization of a molecule with respect to photodissociation.
An informative cvharacteristics of the degree of stabilization is the difference
between the FGR and exact probabilities, X((Q9l) = w^t-wDtot. Stabilization and
destabilization regions correspond to X(co,/)>0 and X(co,/)<0, respectively. The
calculated total dissociation probabilities and the function X(co,/) are plotted in
25
Fig. 9. in their dependence on frequency 0) at a given peak intensity /.
FOR
-1
0.5 0.6
0.1 0.2
FIGURE 9. Exact (dashed line ) and FGR with saturation (solid line) total probabilities of dissociation
per pulse (a) and their difference X (b) vs. frequency 0) (in atomic units); / = 2x 1014 W/cm2, VQ = 2.
The deep hollows of w^fr(co), in accordance with the Franck-Condon rule,
correspond to almost zero matrix elements for constant-J? transitions from such
intrenuclear distances where the initial vibrational wave function turns zero. The
strong field smoothes these deep hollows and shifts them mainly to higher-frequency
regions. For this reason the regions where wDtot((d)> w^(co) and wDtot(cb)< w^(co)
arise around every hollow of the curve w^fr(co), and these are, correspondingly,
destabilization and stabilization regions. Owing to the Franck-Condon principle this
frequency-picture can be converted to the internuclear-distance picture, as it's shown
in Fig. 10. Stabilization and destabilization regions correspond to the left- and
0.5
1.5
2.5
3.5
45
FIGURE 10. Stabilization/destabilizatioti frequencies and a structure of the initial-state vibrational
wave function.
right-hand sides of the hollows of the initial squared vibrational wave function
|(p(/0| • Tte corresponding frequencies can be found as distances along the vertical
lines from the initial vibrational level £2 to the potential curve U\(R). They are shown
by solid lines with arrows for stabilization and dashed line for destabilization regions.
26
REFERENCES
1. Fedorov M. V., and Movsesian, A. M., / Phys. B 21, LI 55-L158 (1988).
2. Fedorov, M.V., Atomic and free electrons in a strong light field, Singapore-London-New York: World
Scientific, 1997.
3. Sukharev M.E. and Fedorov M.V., Phys. Rev, A 65, 033419(12) (2002).
4.
5.
6.
7.
8.
9.
Ua. Bersos, JOSA B 7(5), 617-621 (1990).
Delone, N.B., Goreslavsky, S.P., and Krainov V.P., /. Phys. B 27, 4403-4419 (1994).
Fedorov, M.V., Tehranchi, M.-M., and Fedorov, S.M., /. Phys. B., 29, 2907-2924 (1996).
Hoogenraad, J., Vrijen, R.B., and Noordam, L.D., Phys. Rev. A 50(6), 4133-4138 (1994).
Fedorov, M.V., and Tikhonova, O.V., Phys. Rev. A, 58, 1322-1334 (1998).
Popov, A.M., Volkova, E.A., and Tikhonova, O.V., Sov. Phys, JETP 86, 328 (1998).
27