A. Wójcik and R. Parzyński
Vol. 12, No. 3 / March 1995 / J. Opt. Soc. Am. B
369
Dark-state effect in Rydberg-atom stabilization
A. Wójcik and R. Parzyński
Quantum Electronics Laboratory, Institute of Physics, A. Mickiewicz University,
Grunwaldzka 6, 60-780 Poznań, Poland
Received May 2, 1994; revised manuscript received September 22, 1994
We show that, when an atom prepared in a high Rydberg state is one-photon ionized by a laser, a near
resonance with some lower-lying state permits the formation of a nondecaying dark state of the coupled
laser – atom system. This dark state traps population in an amount that increases to some upper limit as
the laser radiation used becomes stronger. Thus the dark state formed causes a decrease in the Rydbergatom ionization with increasing laser intensity, i.e., stabilization, which survives even for long times. This
long-time stabilization is described by a simple zero-pole formula, and the physical meaning of the term long
time is explained. We estimate, for model situations, the laser frequencies that ensure dark-state formation
and maximum ionization suppression. The laser pulse intensity and duration that are required for the
observation of the dark state are evaluated.
PACS numbers: 32.80 Rm, 42.50 Hz.
1.
INTRODUCTION
Noordam et al.1 recently showed that, as a result of
Raman coupling through a continuum, the initial population in a single Rydberg state can be redistributed over
several Rydberg neighbors when the atom is exposed to
an intense short laser pulse. There seems to be experimental evidence of a quantum interference mechanism
for Rydberg-atom stabilization.2–11 The first step in the
experiment of Noordam et al. consisted in preparing a
barium atom in a selected long-lived Rydberg state 6snd,
n ­ 26, by excitation with two nanosecond lasers (Fig. 1).
The Rydberg barium atom was then exposed to a 1.5-ps
pulse of intensity 4 3 1012 Wycm2 at focus. After the
picosecond pulse had passed, they used a field ionization
technique to determine the final Rydberg-state population. In this way they detected a few percent of the
population in the Rydberg states lying above the initially
occupied one. To identify the redistribution channels,
Noordam et al. measured the redistribution for different
wavelengths of a picosecond laser tuned around the wavelength of the atomic transition from the initial Rydberg
state to the lower state 5d6p (Fig. 1). Because they
observed no effect on the redistribution as a result of
resonance, the conclusion was that in the experiment the
L two-photon transition through the continuum shown in
Fig. 1 was the only redistribution channel.
Despite such an experimental result, we show here
that the lower state jel in Fig. 1, opening the V channel
of redistribution, is capable of drastically changing the
Rydberg-atom photoionization situation. We show that
this may happen because the lower resonant state under
some frequency conditions permits the formation of an
otherwise unattainable perfectly dark state of a strongly
coupled laser – atom system. This dark state has something in common with dark states known, e.g., from resonance fluorescence.12 In our model the dark state found
is perfect in the sense that its width is zero for all laser intensities. Thus it remains perfectly stable, and we show
that it traps population in an amount that increases with
0740-3224/95/030369-08$06.00
increasing laser intensity. With this property the dark
state gives rise to Rydberg-atom stabilization, even over
long times. This long-time stabilization is the effect of
the V-type Raman redistribution, through the lower resonant state jel, of the initial population in a given Rydberg
state over its neighbors. Recently this kind of Rydbergstate redistribution was experimentally observed in potassium atoms by Jones et al.13 In their experiment the
redistribution took place among different nf s12 , n , 18d
states owing to a resonance with the lower 3d state under
the action of a 772-nm laser pulse of 100-fs duration and
an intensity of 2 3 1011 Wycm2 .
2. RYDBERG-ATOM MODEL
WITH RESONANCE
The atomic model to be considered is sketched in Fig. 1.
It is composed of three groups of states: (i) a set hjjlj
of Rydberg states with state jil initially populated, (ii) a
flat atomic continuum hjclj to which the atom is laser ionized, and (iii) an jel state lying below the Rydberg family
that comes into resonance with the initial Rydberg state
when the laser frequency is tuned. We start by writing
the set of coupled equations for Laplace transforms b̃ of
the original Schrödinger probability amplitudes bstd. To
make the appropriate mathematics as simple as possible
we apply square-pulse and rotating-wave approximations.
With our choice of the energy origin as the energy of the
jel state plus that of the accompanying photons, this set is
sb̃e ­ 2i
X
(1)
Vj b̃j ,
j
ss 2 idj db̃j ­ dji 2 iVj b̃e 2
fs 2 isdc 1 di dgb̃c ­ 2
i X
Vcj b̃j ,
" j
i X
Vjc b̃c ,
" c
(2)
(3)
where s is the Laplace variable, dj ­ v 2 sEj 2 Ee dy" and
dc ­ v 2 sEc 2 Ei dy" are laser detuning for bound – bound
1995 Optical Society of America
370
J. Opt. Soc. Am. B / Vol. 12, No. 3 / March 1995
A. Wójcik and R. Parzyński
sb̃e 1 iVA ­ 0 ,
iVS b̃e 1 s1 1 GSdA ­
fi
,
s 2 idi
(9)
(10)
where
A­
X
fj b̃j ,
(11)
fj 2 .
s 2 idj
(12)
j
S­
X
j
Fig. 1. Model of Rydberg-atom photoionization with resonance:
hjj lj, Rydberg states with jil initially populated; hjclj, flat atomic
continuum; jel, near-resonant state.
and bound – free transitions, respectively, Vj ­ Vej y"
is the bound – bound Rabi frequency, Vab is the fielddependent transition matrix element, and dji is the
Kronecker symbol with the initial condition that at t ­ 0
only the jil Rydberg state is populated. With the use
of Eq. (3) we eliminate continuum state amplitudes from
Eq. (2), applying the standard pole approximation and replacing the sum over the continuum states by an energy
integral including the continuum state density r. Then
Eq. (2) becomes
P
(4)
ss 2 idj db̃j ­ dji 2 iVj b̃e 2 G jj 0 b̃j 0 ,
We obtained Eq. (10) in this set by dividing Eq. (4) by
ss 2 idj d, then multiplying it by fj , and finally summing
the result over all j. We solve this set with respect to
b̃e and A and then substitute this solution into Eqs. (4)
and (3). As a result we find that, under ansatz (8), the
solution of the starting set of Eqs. (1) –(3) is
b̃e ­ fi
b̃j ­
2iV
,
ss 2 idi dfss1 1 GSd 1 V 2 Sg
(13)
dji
V 2 1 sG
,
2 fj fi
s 2 idj
ss 2 idi dss 2 idj dfss1 1 GSd 1 V 2 Sg
(14)
b̃c ­ 2
sVc
i
.
fi
" fs 2 isdc 1 di dgss 2 idj dfss1 1 GSd 1 V 2 Sg
(15)
j0
where
Gjj 0
s1 1 iqjj 0 d
(5)
2
is the two-photon Raman coupling between any two
Rydberg states through the continuum, with
G jj 0 ­
P
qjj 0 ­
Z V V0
cj j c
rdsdc d
dc 1 di
"Gjj 0 y2
(6)
being the Fano parameter for this coupling and
Gjj 0 ­
2p
Vjc Vcj 0 r
"
(7)
the Fermi golden-rule ionization rate if j ­ j 0 .
The advantage of using Eqs. (1) and (4) is that they
form a set that applies to discrete-state amplitudes only.
This set can be solved straightforwardly if one uses the
factorization ansatz
Vaj ­ fj Va ,
(8)
where a ­ e or c, Ve fi Vc , and fj describes the dependence
of coupling matrix elements on Rydberg-state quantum
numbers. In the case of highly excited Rydberg states
we have roughly fj ­ n23/2 if angular-momentum effects
are neglected. This ansatz makes the Fano parameters
independent of the Rydberg-state indices, i.e., it reduces
the number of Fano parameters to only one, denoted q.
Moreover, Vj ­ fj V, with V ­ Vey", and Gjj 0 ­ fj fj 0 G,
with G ­ s2py"djVc j2 r, giving G jj 0 ­ fj fj 0 G, with G ­
s1 1 iqdGy2. Under this ansatz the set that is to be
solved is
3. DARK-STATE FORMATION AND
LONG-TIME STABILIZATION
Equations (13) and (14) point to a substantial difference
between the photoionization models without sV ­ 0d and
with sV fi 0d the resonant jel state, i.e., to the effect of
the resonant jel state on Rydberg-atom photoionization.
With no jel state and for j fi i the second term in Eq. (14)
is seen to have no purely imaginary or zero pole. For
j ­ i this term has a purely imaginary pole, s ­ idi ;
however the residuum of this term at this pole is canceled
by the residuum of the first term. All this means that the
Rydberg states in the model without the resonant jel state
decay completely to the continuum on a long-time scale,
i.e., there is no population trapping by bound states. It
is our aim to show that the situation changes drastically
when the model does include the resonant jel state and
the laser frequency is specifically chosen. That is, if the
frequency is such that
Sss ­ 0d ­ 0 ,
(16)
i.e., the one that fulfills the condition
X fj 2
­ 0,
dj
j
(17)
s­0
(18)
then
is seen to be the pole of both b̃e and the second term in
Eq. (14). Because of this zero pole the bound states of
the model trap some amount of population, even on the
long-time scale t ! `, preventing the atom from being completely ionized. The long-time asymptotics
A. Wójcik and R. Parzyński
Vol. 12, No. 3 / March 1995 / J. Opt. Soc. Am. B
of discrete-state population amplitudes, found by the
method of residua, are
be s`d ­
fi
V
,
di sp 1 1 V 2 R
(19)
bj s`d ­
fi fj
V2
,
sp
sp
di dj 1 1 V 2 R
(20)
where
√
dS
R ­ lim
s!0
ds
!
­
X
j
0
12
@ fj A
dj sp
(21)
and the detunings dk sp are those calculated at the specific
frequency defined by Eq. (17). The above amplitudes allow us to construct the long-time wave function of the
atom, which does not decay to continuum, as
2
0
1 3
X
f
fi
V
6
@ j Ajjl7
4jel 1 V
5 . (22)
jCs`dl ­ sp
di 1 1 V 2 R
dj sp
j
In the dressed-state language the s ­ 0 pole corresponds to the formation of a zero-width dark state decoupled from the continuum and thus un-ionizable in
whole intensity range. In view of what has been said
above this dark state needs for its formation (i) the presence of a resonant state below the Rydberg family of states
and (ii) the choice of a specific laser frequency in the way
determined by Eq. (17). Let us see that the condition for
the specific frequency has sense only for at least a pair
of Rydberg states. For the family of N Rydberg states,
Eq. (17) is equivalent to an algebraic equation of the order of N 2 1 with respect to frequency, and this equation
generates as many as N 2 1 different specific frequencies,
ensuring dark-state formation and population trapping.
This dark state jDl, which survived after t ! `, can be
found from the condition
jCs`dl ­ CD jDl ,
(23)
where CD is the dark-state population amplitude. From
the condition that at t ! ` the population in the dark
state jDl must be equal to that in all bare discrete states,
i.e.,
P
(24)
jCD j2 ­ jbe s`dj2 1 jbj s`dj2 ,
j
we find that
CD ­
V
fi
,
p
di sp 1 1 V 2 R
(25)
and then
2
0
1 3
X
f
1
6
@ j Ajjl7
4jel 1 V
5.
jDl ­ p
dj sp
1 1 V 2R
j
(26)
The dark state found is, of course, normalized to unity:
kD j Dl ­ 1. This dark state is a coherent superposition
of bare discrete states of our model without decay to
the continuum. Formed at specific frequencies, it traps
the population in the amount of jCD j2 . As a result, the
371
ionization after a long time is not complete and amounts
to
Pion s`d ­ 1 2 jCD j2
!2
√
V2
fi
.
­12
sp
di
1 1 V2R
(27)
It is striking that the trapping in this dark state increases with increasing laser intensity up to the asymptotic value R 21 s fiydi sp d2 , and thus the ionization decreases
to 1 2 R 21 s fiydi sp d2 when the laser intensity increases.
This decrease in the probability of ionization with increasing laser intensity is what is usually considered atomic
stabilization against intense laser ionization. In our case
it is the stabilization of the Rydberg atom on the longtime scale. As was shown above, the long-time stabilization mechanism consists in the formation under a
specific laser frequency of a dark state that is populated
with a probability increasing to some upper limit when
the laser gets stronger and stronger. This mechanism
requires near resonance between the initially populated
Rydberg state and some state lying below the Rydberg
quasi-continuum. With no jel state below the Rydberg
quasi-continuum, i.e., with V ­ 0, there is no dark state
formed sCD ­ 0d, and the long-time ionization gets completely saturated at the level of unit probability. As a result there is no long-time stabilization effect when there is
no jel state. This example seems to be the most striking
one, pointing to a fundamental difference in the behavior of the models with and without a near-resonant state
below the Rydberg quasi-continuum.
We emphasize that all equations derived in this section, along with the qualitative conclusion drawn from
them, are general in the sense that they apply to any
quasi-continuum that satisfies factorization ansatz (8) for
matrix elements. Obviously for different quasi-continua
we will have different specific frequencies dj sp and different parameters R, resulting finally in different amounts
of trapped and ionized populations on the long-time scale.
However, this quantitative difference seems to be the only
one, leaving our general qualitative conclusions concerning dark-state formation as well as long-time population
trapping and stabilization unchanged.
The formulas of this section automatically include the
effect of different angular-momentum Rydberg series and
possible two-photon coupling between them. For example, if the initial Rydberg state is of the s type, both
the resonant state and the continuum must be of the p
type. The p continuum couples not only to the original
s Rydberg states but also to d ones. Consequently the
d Rydberg series is coupled to the resonant p state. In
this case the d Rydberg series does not change our original model in Fig. 1 or, as a result, the structure of the
long-time formulas derived. We must remember only
that now the index j refers to Rydberg states of both s
and d series. Thus the only effect of the additional d
series is to change the specific frequencies dj sp and the R
parameters without introducing any qualitative changes.
Slightly more complicated is the case when the initial
Rydberg state is of the p type and the resonant state is
of the s type. Through the d continuum, the f Rydberg
series can now be populated that is electric-dipole decoupled from the resonant s state. Our original model
372
J. Opt. Soc. Am. B / Vol. 12, No. 3 / March 1995
A. Wójcik and R. Parzyński
shown in Fig. 1 does not encompass such a more-general
case. However, after algebra performed for this generalized model along a similar line one finds that for specific
frequencies the long-time asymptotics of all f-state amplitudes are equal to zero, but those of the s and the
p states are again given by Eqs. (19) and (20). This is
so because the f series modifies only our G jj 0 , given by
Eq. (5), which does not enter the long-time asymptotics,
Eqs. (19) and (20).
All results and qualitative conclusions of this section also remain unaffected when one extends our
model shown in Fig. 1 by the inclusion of the continuum– continuum transitions specific for above-threshold
ionization. This statement is based on the well-known
fact2,14 that continuum – continuum transitions, if included, lead to modification of bound – free matrix elements or, equivalently, of ionization decay rates of bound
states. This modification takes the form, in general,
of an infinite continued fraction, derived by Deng and
Eberly.14 Because our main long-time results in the form
of Eqs. (19) – (27) by no means depend on bound –free
matrix elements, they cannot be influenced by continuum– continuum transitions.
4. QUANTITATIVE ESTIMATIONS FOR
THE MODEL WITH THE BIXON–JOERTNER
QUASI-CONTINUUM
Obviously, for a rigorous Rydberg quasi-continuum one
must use numerical procedures to calculate specific
frequencies from Eq. (17) and then the value of R from
Eq. (21). One can obtain simple estimations in a completely analytical way, however, by modeling high
Rydberg series with the Bixon – Joertner quasicontinuum. This means that we replace the Rydberg
series with an infinite family of equidistant states with
the frequency spacing D, each state being coupled to the
resonant jel state with the same strength s fj ­ 1d. With
this modeling, dj ­ di 2 jD, with j ­ 0, 61, 62, . . . , 6`,
and then Eq. (17) for specific frequencies converts into
√
!
X̀
p
1
pdi
­
cotan
­ 0.
(28)
D
D
j­2` di 2 jD
Thus the specific frequencies for this model are determined by the condition
di sp ­ sk 1 1/2dD ,
(29)
where k ­ 0, 61, 62, . . . , 6`. These specific frequencies
are those that ensure the formation of the dark state if the
quasi-continuum is of the Bixon – Joertner type and only
those for which our Eqs. (19) – (27) are valid. For these
specific frequencies we have, from Eq. (21),
X̀
1
R­
sp
2 j Dd2
j­2` sdi
√ !2
1 X̀
1
p
.
­ 2
(30)
2 ­
1
/
D j ­2` sk 1 2 2 jd
D
Accordingly, the dark state obtained by combining
Eqs. (26), (29), and (30) is
1
0
p X̀
1
jjl
C
B
A,
@jel 1 u
jDl ­ p
(31)
1/
k
1
2
2
j
1 1 p 2u
j ­2`
and it traps population in the amount
jCD j2 ­
1
u
,
1/ 2 1 1 p 2 u
sk 1 2d
(32)
where u ­ sVyDd2 is a dimensionless parameter proportional to the laser intensity. This population trapping
results in suppression of the long-time ionization probability from 1 to
Pion s`d ­ 1 2
1
u
.
sk 1 1/2d2 1 1 p 2 u
(33)
According to Eqs. (19) and (20) the entire trapped population is distributed over bare atomic states in the following
way:
X
j
jbe s`dj2 ­
1
u
,
sk 1 1/2d2 s1 1 p 2 ud2
(34)
jbj s`dj2 ­
1
u2
1
,
2
1/ 2
1/
sk 1 2d sk 1 2 2 jd s1 1 p 2 ud2
(35)
jbj s`dj2 ­
p 2 u2
1
,
1/ 2 s1 1 p 2 ud2
sk 1 2d
(36)
where Eq. (35) has been written under the assumption
that the initially populated state in the quasi-continuum
was that of j ­ 0.
As is seen, the population trapping in the dark state
reaches its maximum for k ­ 0 or k ­ 21, i.e., for the
laser detunings jdi sp j ­ Dy2. For these two specific
detunings, differing from each other only by sign, the
high-intensity limit su .. 1d of the population trapped
by the dark state amounts to s2ypd2 ø 0.4. As follows
from Eqs. (34) – (36), this high-intensity trapped population is distributed mainly among the initial state
fs2ypd4 ø 0.16g and the rest of Bixon–Joertner quasicontinuum fs2ypd2 2 s2ypd4 ø 0.24g. This result points
to a substantial redistribution of the initial population
in a given Rydberg state over all neighboring states after a long intense pulse of specific frequency has passed.
Obviously this high-intensity population trapping in the
dark state manifests itself as suppression of the longtime ionization probability from the saturation to the
value 1 2 s2ypd2 ø 0.6.
Figure 2 is graphic presentation of the results obtained with the assumptions of the Bixon – Joertner quasicontinuum and for the specific detuning di sp ­ dj ­0 sp ­
Dy2. Figure 2(a) shows the long-time redistribution
of the initial population in j ­ 0 over all neighboring
quasi-continuum states and the resonant jel state versus the laser intensity parameter u. Significant population of different quasi-continuum states with j fi 0,
high-intensity vanishing of the population in the resonant state, and high-intensity saturation of population
trapped by the whole quasi-continuum are what are
clearly demonstrated in Fig. 2(a). Figure 2(b) presents
the long-term ionization probability [Eq. (33)] against
laser intensity and the suppression effect with increasing
intensity. Qualitatively the same effect is expected when
instead of the model Bixon –Joertner quasi-continuum the
true Rydberg series is considered.
A. Wójcik and R. Parzyński
Vol. 12, No. 3 / March 1995 / J. Opt. Soc. Am. B
373
V to the resonant state jel. Assuming additionally that
the Fano parameter q ­ 0, we have G ­ Gy2. By further
neglect of the resonant jel state, this simplified model
would reduce to the model considered by Parker and
Stroud.3 For our two-Rydberg-state model with resonance there is only one specific frequency that results
from Eq. (17) and ensures the existence of the zero pole,
namely, such frequency that
d1 sp ­ 2d2 sp ­ Dy2 .
(37)
For this specific frequency
S­
2s
,
s2 1 sDy2d2
(38)
resulting in
R ­ 8yD2 .
(39)
The above S leads to
√
!
G
D
1
1 V2
s s1i
2
2
,
b̃1 ­
sss 2 s1 dss 2 s2 d
(40)
G
1 V2
2
,
b̃2 ­ 2
sss 2 s1 dss 2 s2 d
(41)
D
2
,
b̃e ­ 2iV
sss 2 s1 dss 2 s2 d
(42)
s
Fig. 2. Long-time discrete-state population (a) and ionization
(b) versus laser intensity for the model with the Bixon – Joertner
quasi-continuum and the specific frequency di sp ­ dj­0 sp ­ Dy2.
The initially populated is the j ­ 0 quasi-continuum state.
5. APPLICABILITY LIMITS OF THE
ZERO-POLE (LONG-TIME) RESULTS
All formulas after Eq. (16) were obtained with the inclusion of only the zero pole that was ensured by specific frequencies resulting from Eq. (17). Thus these formulas,
particularly Eqs. (27) and (33) for ionization probability,
may be referred to as the zero-pole approximation. This
zero-pole approximation is believed to be correct for times
sufficiently long to ensure that the contributions coming
from all other nonzero poles are completely damped out.
If s ­ Ressd 1 i Imssd is a given nonzero pole, then Imssd
describes the position of a given dressed state and Ressd
its width, and the damping of this dressed state is complete, provided that tjRessdj .. 1. This is the condition
for a long time with respect to that nonzero pole. To say
what a long time is and what it depends on, we have to
calculate all nonzero poles, or at least their real parts.
Given that for our general infinite-number-state model
shown in Fig. 1 it is impossible to find all these poles analytically, we choose to simplify the model drastically by
reducing the whole Rydberg family to only a pair of states
(Fig. 3). Obviously this is a crude approximation of the
real Rydberg atom, but it is sufficient to get in an analytical way an approximate answer to the question stated (see
Ref. 15 for nonanalytical treatments of complex models).
The two Rydberg states left, labeled j1l and j2l, are energy
separated by "D, and the lower of them, j1l, is assumed
to be initially populated. For the sake of simplicity these
states are considered as being coupled with the same
strength G to the continuum and with the same strength
s1i
where
1 p 2
G
6
G 2 8V 2 2 D2
28 2
9
31/2 †
2
√ !
>
=
1
D <
1
u
v 21 6 41 2 8
2 25
­
†
2 >
v v
v
:
;
s6 ­ 2
­
D
z6
2
(43)
are the two nonzero poles, additional to the zero pole,
with u ­ sVyDd2 and v ­ GyD being intensity-dependent
parameters. Because both u and v are linear in intensity, the uyv ratio is intensity independent and completely
Fig. 3. Simplified version of the model from Fig. 1 with only a
pair of Rydberg states included. G, Fermi golden-rule ionization
rate of Rydberg states; V, resonance Rabi frequency. Condition:
the case of specific laser detuning, d1 ­ Dy2.
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J. Opt. Soc. Am. B / Vol. 12, No. 3 / March 1995
A. Wójcik and R. Parzyński
governed by the involved bound – bound and bound –free
electric-dipole matrix elements. Taking into account
all three poles, we find the time-dependent Schrödinger
population amplitudes of discrete states
b1 std ­
z1 sz1 1 i 1 vd 1 4u
4u
1
expspz1 td
1 1 8u
z1 sz1 2 z2 d
2
b2 std ­
be std ­
z2 sz2 1 i 1 vd 1 4u
expspz2 td ,
z2 sz1 2 z2 d
vz1 1 4u
24u
2
expspz1 td
1 1 8u
z1 sz1 2 z2 d
vz2 1 4u
expspz2 td ,
1
z2 sz1 2 z2 d
p
p
2 u sz1 1 id
2 u
2i
expspz1 td
1 1 8u
z1 sz1 2 z2 d
p
2 u sz2 1 id
1i
expspz2 td ,
z2 sz1 2 z2 d
(44)
(45)
(46)
where t ­ tyT , with T ­ 2pyD, has the sense of the
classical Kepler time of the orbiting Rydberg electron (T ­
2pn3 in atomic units of time, where n is the principal
quantum number). These amplitudes allow us to obtain
an ionization probability valid for all times:
Pion std ­ 1 2 fjb1 stdj2 1 jb2 stdj2 1 jbe stdj2 g .
(47)
In the mathematical limit of long times, t ! `, the exponents in the above amplitudes tend to zero, and then
the ionization probability given by Eq. (47) converts simply into
Pion s`d ­ 1 2
4u .
1 1 8u
(48)
In agreement with Eqs. (27), (37), and (39), this long-time
result coincides with the zero-pole approximation for the
model with a pair of Rydberg states plus the resonant jel
state. The second term in this zero-pole approximation
is the amount of population trapped by the dark state.
According to Eq. (26) this dark state now has the form
jDl ­ p
p
1
fjel 1 2 u sj1l 2 j2ldg .
1 1 8u
laser intensity. Thus what we shall treat as a physically long time also depends on laser intensity. As follows from Eq. (43), in weak fields sv ,, 1d the widths of
both dressed states are the same, v. However, in strong
fields sv .. 1d these dressed states have different widths,
namely, 2v (j2l state) and 4suyvd (j1l state), with the latter width tending to zero as u ! 0. This means that
the j2l state has its width continuously increasing with
increasing intensity, whereas the width of the j1l state
first increases and after reaching a maximum decreases
with increasing intensity. In the decay-time language
this means that t2 is a decreasing function of intensity,
whereas t1 first decreases with increasing intensity, then
crosses a minimum, and finally increases to the asymptotic value 1ys4puyvd. Because t1 $ t2 in the entire
intensity range, validity condition (50) of the zero-pole approximation should read as t .. t1 .
Figure 4 shows the dressed-state decay times t6
against the intensity-dependent parameter v for the fixed
uyv ­ 1 ratio. As is seen, these decay times do behave
in the way that we predicted above. We show in Fig. 5
the ionization probability versus the intensity parameter v at uyv ­ 1 for different pulse durations t. Solid
curves represent the exact results obtained with the use of
Eqs. (44) – (47), and asterisks follow Eq. (48) for the zeropole approximation in which u was replaced by v. As we
can see, the longer the pulse duration the wider the intensity range that is covered by our zero-pole approximation.
This agrees with Fig. 4 and condition (50). Figure 5 allows us to tell for what intensities a given pulse duration
can be considered long. For instance, t ­ 10 is long in
the sense of applicability of the zero-pole approximation
for all intensities that satisfy the condition v $ 0.1. It
is striking that even for this long time the height of the
exact solid curve in Fig. 5 is less than 1, meaning that
no complete ionization is possible. For t ­ 10 we show
in Fig. 6 that this ionization suppression is caused by
the presence of the near-resonant state below the Rydberg doublet. When the uyv ratio is diminished, which
means neglect of the nearly resonant state, the suppression vanishes and saturation appears, which precedes
(49)
In the limit of strong fields it traps as much population as 0.5. Obviously this 50% survival probability is
not universal, and, as was shown for the model with the
Bixon– Joertner quasi-continuum, the survival probability depends on the number of states involved.
Physically, for the zero-pole approximation given by
Eq. (48) to be valid the laser pulse duration t must be
as long, as indicated by
t .. t6 ­
1
,
pjResz6 dj
(50)
where t1 and t2 are decay times (in units of Kepler period T ­ 2pyD) of the two dressed states with complex
energies z1 and z2 , respectively. According to Eq. (43)
the dressed-state widths jResz6 dj depend through v on
Fig. 4. Decay times of dressed states with z6 complex energies
against laser intensity, for the model from Fig. 3. Conditions:
d1 ­ Dy2 and uyv ­ 1.
A. Wójcik and R. Parzyński
Fig. 5. Ionization probability versus laser intensity for the
model from Fig. 3 and different pulse durations t. Conditions:
d1 ­ Dy2 and uyv ­ 1. Asterisks mark the results of the
zero-pole approximation.
Vol. 12, No. 3 / March 1995 / J. Opt. Soc. Am. B
of the lower resonant state, and I is laser intensity expressed in watts per square centimeter. Because the
Rydberg-state separation is D ­ 2pyT , with Kepler period
T ­ 2pn3 in atomic units of time, we have the approximation u ø 6 3 10218 snyn0 d3 I . Taking into account that
the ionization rates of Rydberg states are scaled roughly
by n23 , we find v to be independent of n. Then, for a
hydrogenlike atom with an ionization cross section of
the order of a few megabarns and optical photons, we approximate v ø 10216 I , with I in watts per square centimeter. As u, unlike v, depends on the principal quantum
numbers of the resonantly coupled states, we can control
the relation between u and v by the appropriate choice of
these states. According to the case of t ­ 102 in Fig. 5,
we are interested in the condition u ­ v $ 1022 . With
the use of our estimations for u and v we find that this
condition is approximately fulfilled if I $ 1014 Wycm2
and, e.g., n ­ 10 and n0 ­ 4. For the n ­ 10 initial
Rydberg state the Kepler period is 1.5 3 10213 s and,
consequently, t ­ 102 corresponds to the absolute pulse
duration of 15 ps. Thus we expect that with a 15-ps
laser pulse of intensity $ 1014 Wycm2 interacting with
the n ­ 10 initial Rydberg state nearly resonantly coupled
to the n ­ 4 state (detuning of the order of 6Dy2), a decrease in ionization probability with increasing intensity
should be observed. This decrease should follow our
zero-pole (long-time) expression. It would confirm the
effect predicted by us of the formation of a dark state in
Rydberg-atom ionization. It would also indicate that a
15-ps pulse is sufficiently long in the context of this paper. However, to satisfy the square-pulse approximation
used in our calculations the rise time of the experimental
pulse needs to be much shorter than the entire duration
of the pulse. As Fig. 5 shows, for pulses longer than
that considered above lower intensities would be sufficient to verify our zero-pole predictions. Also, Rydberg
states with relatively high ionization cross section would
diminish the experimentally required intensities.
6.
Fig. 6. Effect on long-time ionization probability st ­ 10d of coupling strength, uyv, of the Rydberg doublet to the near-resonant
jel state, for the model from Fig. 3. Condition: d1 ­ Dy2.
the stabilization region, where the ionization probability
decreases with increasing intensity. Our curve for the
smallest uyv ratio coincides with the curve of Parker and
Stroud3 for their model without resonance. Thus Fig. 6
shows with an extremely simple model the effect of near
resonance in Rydberg-atom photoionization.
On the basis of Fig. 5 we are able to assess the
laser – atom parameters, ensuring experimental verification of our theoretical predictions. Let us focus our
attention on the descending part of the solid curve,
i.e., the exact one, for t ­ 102 . This descending part
is seen to be reproduced well by the asterisk curve of
the zero-pole approximation if the laser intensities fulfill
the condition that u ­ v $ 1022 . For typical electricdipole matrix elements
in a hydrogenlike atom we have
p
roughly V ø 108 Iysnn0 d3/2 , where n is the principal
quantum number of the initial Rydberg state, n0 is that
375
SUMMARY
We have proved analytically that a near resonance with a
state lying below the initially populated Rydberg quasicontinuum is able to change drastically the conditions
for Rydberg-atom photoionization. The necessary conditions to support this situation are that (i) the Raman
coupling between any two Rydberg states through the
resonant state must be at least comparable with that
through the continuum and (ii) laser frequency is to be
specifically chosen in a way dependent on details of the
quasi-continuum. Under these conditions the near resonance permits formation of a perfectly dark state of a
strongly coupled laser –atom system that remains nondecaying for all laser intensities. This dark state prevents the Rydberg atom from getting completely ionized,
even if the laser pulse is long and strong. The dark state
exhibits the property of trapping more and more population, up to some upper limit lower than 1, with increasing laser intensity. This property of the dark state gives
rise to strong-field Rydberg-atom stabilization on a scale
of long times. This long-time stabilization has been described by a simple zero-pole approximation formula. A
comparison between the exact and the zero-pole results
376
J. Opt. Soc. Am. B / Vol. 12, No. 3 / March 1995
for a model with a pair of Rydberg states has permitted
identification of the meaning of the long-time term. We
have also evaluated the laser – atom parameters that ensure experimental observation of the effect of dark-state
formation.
ACKNOWLEDGMENTS
We thank A. Kowalewska and K. Palacz for their assistance in the final stage of this research.
We acknowledge support from Komitet Badań
Naukowych grant 2 2338 92 03, which made this
research possible.
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