Adiabatic passage by light-induced potentials in molecules
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Garraway, B M and Suominen, K-A (1998) Adiabatic passage by light-induced potentials in
molecules. Physical Review Letters, 80 (5). pp. 932-935. ISSN 00319007
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VOLUME 80, NUMBER 5 PHYSICAL REVIEW LETTERS 2FEBRUARY 1998
Adiabatic Passage by Light-Induced Potentials in Molecules
B.M. Garraway*
Optics Section, Blackett Laboratory, Imperial College, Prince Consort Road, London, SW7 2BZ, United Kingdom
K.-A. Suominen
Helsinki Institute of Physics, PL 9, FIN-00014 Helsingin yliopisto, Finland
and Theoretical Physics Division, Department of Physics, University of Helsinki, PL 9, FIN-00014 Helsingin yliopisto, Finland
(
Received 3 September 1997)
We present the process of adiabatic passage by light-induced potentials for transferring a wave packet
from one molecular potential to the displaced ground vibrational state of another. The process uses a
counterintuitive sequence of light pulses to couple three molecular states. It shares many features with
the process of stimulated Raman adiabatic passage, such as high efficiency and insensitivity to pulse
parameters. However, in the former there is no two-photon resonance, and the main mechanism for the
transport of the wave packet is a light-induced potential. [S0031-9007(97)05254-X]
PACS numbers: 33.80.–b, 42.50.–p
Femtosecond pulses have recently opened the possibil-
ity to create, observe, and control the internal dynamics
of molecules [1,2]. Typically, one has studied the pump-
probe situation, where a molecular wave packet, i.e., a co-
herent superposition of vibrational states, has been created
by the first pulse, and a second pulse probes the subse-
quent evolution of the wave packet. Alternatively, one
can observe the dissociation process directly. In more
subtle cases, such as the pump and dump schemes, the
final result can be some special molecular bound state in-
stead of dissociation. Such manipulations provide new
understanding of molecular dynamics and chemical re-
actions. Also, they present intriguing demonstrations of
wave packet dynamics and time-dependent quantum me-
chanics in general.
The purpose of this Letter is to demonstrate (theoreti-
cally) a mechanism for the transfer of a stationary ground
state vibrational wave packet to a stationary and dis-
placed excited state wave packet through an intermedi-
ate state which is not significantly populated during the
process. The overall effect is symbolically represented by
the diagonal arrow in Fig. 1. Of course, we can change
from a wave packet picture of the process to a picture in
terms of the vibrational levels, in which case Fig. 1 illus-
trates a process where the population of the n 0 vibra-
tional level of the ground state is directly transferred to
the n
00
0 vibrational level of the second excited state.
The overall effect appears as a violation of the Franck-
Condon principle, which can be simply stated as saying
that there should only be vertical transitions between vi-
brational states in a molecule, at least over short times.
Thus the diagonal transition seen in Fig. 1 should not be
allowed. It is further inhibited by the fact that the overlap
between the initial and final wave functions is very small
(the Franck-Condon overlap). This is because the initial
wave packet is displaced over a distance of about seven
times its width for the example of Fig. 1. Of course, there
is no real violation of the Franck-Condon principle; we
manipulate the molecular states on time scales close to,
but longer than the vibrational time scales.
To illustrate the process we have chosen the sodium
dimer, a molecule which has already been subjected to
much study in the field of wave packet dynamics, and
which opens the prospect of an experiment to test the
ideas in this Letter. Following Refs. [3] we have chosen
our three states so that the ground state is the X
1
S
1
g
of Na
2
, the first excited state is the A
1
S
1
u
state, and
the second excited state is the 2
1
P
g
state. Data for the
molecular potentials have been gathered from Refs. [3,4]
and these data are used in a numerical calculation of the
dynamics of the wave packet during the interaction of the
system with two laser pulses.
FIG. 1. The three Na
2
potential energy surfaces used in our
calculations: the X
1
S
1
g
ground state, the A
1
S
1
u
state as the first
excited state, and the 2
1
P
g
as the third state. The diagonal
sloping arrow indicates the overall effect of the two laser
pulses used.
932 0031-9007y98y80(5)y932(4)$15.00 © 1998 The American Physical Society
VOLUME 80, NUMBER 5 PHYSICAL REVIEW LETTERS 2FEBRUARY 1998
In terms of the electronic potentials the Hamiltonian for
the vibrational motion of the molecule is
H 2
¯h
2
2m
≠
2
≠R
2
I 1 U sR, td , (1)
where R is the internuclear separation, m is the reduced
mass of the molecule, and the electronic potentials and
couplings are given by
U sR, td
2
4
U
X
sRd 1 ¯hd
1
¯hV
1
std 0
¯hV
1
std U
A
sRd ¯hV
2
std
0¯hV
2
stdU
P
sRd1¯hd
2
3
5
.(2)
Here U
X
sRd, U
A
sRd, and U
P
sRd are the three potentials,
d
1
and d
2
are the detunings of the two pulses from the
lowest points of the potentials, and V
1
std m
XA
E
1
stdy ¯h,
V
2
std m
AP
E
2
stdy ¯h are the two Rabi frequencies. We
have assumed for simplicity that the two dipole moments
are independent of R and we solve the time-dependent
Schrödinger equation with Hamiltonian (1) by a numerical
method (see, e.g., Ref. [2]).
Figure 2 shows an example of the wave packet dynam-
ics following the coupling of two pulses between each
FIG. 2. The wave packet dynamics on (a) the X
1
S
1
g
ground
state and (b) the target state 2
1
P
g
. The wave packet
motion has been determined from a fully quantum mechanical
calculation. In (a) the ground state wave packet disappears
when the two pulses arrive at t 10.8 ps and 16.3 ps. In
(b) we see the slow and steady appearance of the wave packet
in the 2
1
P
g
state. Note the steady displacement of the wave
packet as it arrives adiabatically to the bottom of the lowest
vibrational state of the 2
1
P
g
state. Both pulses are red detuned
from the bottom of the potential energy surface by 2200 cm
21
and have a width of 5.42 ps.
pair of molecular states. In Fig. 2(a) we see at t 0 the
initial ground state wave packet, located at the equilib-
rium position of 3.08 Å, in the X
1
S
1
g
potential. When the
two Gaussian laser pulses (with peak intensities of 3 and
6 TWycm
2
) act on the molecule we see the disappearance
of the wave packet from the X
1
S
1
g
state, while it is also
displaced to the right (to larger distances). As the wave
packet disappears from the X
1
S
1
g
state it appears on the
excited 2
1
P
g
state [see Fig. 2(b)], still being displaced to
longer bond lengths as it appears. When the laser pulses
have been completed, the wave packet is left in the 2
1
P
g
state without any vibrational excitation (there is no mo-
tion of the wave packet).
During the process seen in Fig. 2 the X
1
S
1
g
and 2
1
P
g
states exchange their population while the population of
the A
1
S
1
u
state remains very low at all times. The process
is nearly 100% efficient in transferring population from
the X
1
S
1
g
state to the 2
1
P
g
state. This efficiency remains
high over a wide range of pulse parameters.
The process we have described uses a counterintui-
tive pulse sequence: the pulse nearly resonant with the
X
1
S
1
g
! A
1
S
1
u
transition is applied after the pulse nearly
resonant with the A
1
S
1
u
! 2
1
P
g
transition. However,
the process is not the same as the conventional STIRAP
[5–7] (stimulated Raman adiabatic passage) process,
already seen in molecular systems [6], for both trivial
and fundamental reasons. The most trivial difference with
the existing experiments is the linkage pattern; the ladder
system we consider (Fig. 1) has a different linkage pattern
from the Raman type L system after which STIRAP is
named. In the absence of spontaneous emission [8], these
different linkage patterns do not affect the dynamics in the
case of atomic systems [7].
Because we utilize a ladder system it makes sense
to consider a transition from X
1
S
1
g
sn 0d to 2
1
P
g
sn
00
0d, i.e., from the ground vibrational state of the
lowest electronic state to the ground vibrational state of
the highest electronic state in our three-level system. In
a L-type Raman scheme this would not make sense as
STIRAP is then used to create an excited vibrational state
(nfi0) from the ground state (n 0) within the same
electronic state of the molecule.
Conventional STIRAP utilizes a two-photon reso-
nance condition. For example, a suitable Hamiltonian for
STIRAP in an atomic ladder system is
H
a
2
4
0¯hV
1
std0
¯hV
1
std¯hD¯hV
2
std
0¯hV
2
std0
3
5
, (3)
where D is the two-photon resonant laser-atom detuning,
and V
1
and V
2
are the Rabi frequencies of the pump
pulse and Stokes pulse. If the pump and Stokes pulses
change sufficiently slowly, we can consider the process
to be adiabatic. Then we can utilize the instantaneous
933
VOLUME 80, NUMBER 5 PHYSICAL REVIEW LETTERS 2FEBRUARY 1998
eigenstate
c
z
std
1
q
V
2
1
std 1V
2
2
std
0
B
@
V
2
s
t
d
0
2V
1
s
t
d
1
C
A
(4)
to achieve the transfer directly from state 1 to state 3. We
note that the state c
z
is for all t an exact eigenstate of H
a
,
Eq. (3), with eigenvalue zero (sometimes called a “dark
state” [9]). If the pulses are in the counterintuitive order,
i.e., V
2
std reaches its peak before V
1
std, the eigenstate
(4) matches the initial state of the system (state 1). Since,
for long pulses, the system state adiabatically follows the
state c
z
, the occupation probability is transported from
state 1 to state 3. Because state 2 is not involved in
the eigenstate c
z
, it is not populated during the pulse
sequence.
The situation for the molecule is rather different be-
cause we have an extra degree of freedom: the molecular
coordinate which we denote by R. With the Hamiltonian
now given by Eq. (1), and the spatially varying potentials
(2), it is clear that it is impossible in this molecular case
to have the two-photon resonance condition used in the
atomic case (except at isolated points). Thus there is no
zero eigenstate in the molecular situation.
At this point it could be argued that rather than viewing
the Hamiltonian (1) in the position basis, we should utilize
a vibrational basis so that we could recover a version of
the atomic STIRAP process seen with the Hamiltonian
(3). However, while the vibrational picture and the spatial
picture are entirely equivalent, we believe that the key
to understanding the phenomenon in Fig. 2 is not the
vibrational basis but a spatial picture. This brings us
to a fundamental difference between STIRAP and the
phenomena in this Letter. If we had a STIRAP process
taking the system from the X
1
S
1
g
sn 0d vibrational
state to the 2
1
P
g
sn
00
0d vibrational state we would see
only the disappearance of the wave packet in Fig. 2(a)
and its reappearance in Fig. 2(b) without the smooth
positional shifting of the wave packet. The reason is that
the X
1
S
1
g
sn 0d vibrational state wave function will
be approximately the ground vibrational wave function of
a harmonic oscillator, and any positional movement of
the wave packet must be due to the excitation of other
vibrational states. We can say the same thing about the
2
1
P
g
sn
00
0d vibrational state, i.e., that if only the
sn
00
0d were involved there would be no shifting of
the wave packet as seen in Fig. 2(b). So the process
of Fig. 2 is not direct STIRAP transfer between X
1
S
1
g
and 2
1
P
g
.
Our explanation for the transfer of the wave packet
in the manner seen in Fig. 2 relies on light-induced
potentials [10]. For wave packets that travel sufficiently
slowly through systems of coupled potential surfaces
the nature of the field-dressed potential energy surfaces
becomes more important than the bare (i.e., not coupled
by light) energy surfaces. This means, for example, that
a laser-induced crossing becomes an avoided crossing
with an energy gap which increases with the intensity
of the laser. The energy gap can become large enough
to allow the passage of a wave packet which would
not otherwise penetrate the crossing; this leads to bond
softening [11]. The eigenvalues of (2) determine the
light-induced potentials, and in Fig. 3 we show the most
relevant one as a function of space and time. The most
striking feature is the kinked channel which is responsible
for guiding the wave packet from one position to another.
At t 0 the light-induced potential of Fig. 3 is com-
posed of the X
1
S
1
g
potential on the left hand side of the
picture, and the 2
1
P
g
potential on the right hand side
of the picture. In effect there is a very small avoided
crossing of these two potentials near R 3.4 Å. The
crossing is small because at t 0 the two laser fields
are very weak. As a result, there are two wells in the
field-dressed state at t 0, one belonging to the X
1
S
1
g
state, where the initial wave packet resides, and the other
belonging to the 2
1
P
g
state, which is where we aim to
transfer the wave packet. Because of the red detuning
of the two pulses, the field-dressed state corresponding to
the intermediate A
1
S
1
u
state (at t 0) lies well below the
field-dressed potential of Fig. 3. However, as the pulse
resonant with the A
1
S
1
u
! 2
1
P
g
transition turns on there
is a repulsion between the A-type (A
1
S
1
u
at t 0) field-
dressed state and the rhs of the field-dressed state in Fig. 3
(which is P-like). This repulsion results in the disap-
pearance of the rhs channel in Fig. 3. The repulsion also
moves the A-P avoided crossing to larger internuclear
separations.
FIG. 3. Space and time dependence of the light-induced
potential responsible for the transportation of the wave packet
in Fig. 2.
934
VOLUME 80, NUMBER 5 PHYSICAL REVIEW LETTERS 2FEBRUARY 1998
When the second pulse is turning on, the repulsion
between the X part of the state in Fig. 3 and the lower
A state pushes the main channel upwards in energy and
also displaces it to the right because the right hand part
of the state in Fig. 3 has more of the P character, and
is not repelled from the A state by the second pulse. As
the first pulse dies away the transfer of the wave packet
completes with the approach of an avoided crossing from
small internuclear separations. Eventually this avoided
crossing becomes the same X-P avoided crossing seen
near t 0, so we again have two wells in the potential.
It can be clearly seen from Fig. 3 that we need to have
a counterintuitive sequence of pulses. Having an intuitive
sequence would be like starting from the rear of Fig. 3 in
the left hand well. But the left hand well is a dead end.
Only the right hand well lies in the channel connecting
the wave packet through the pulse sequence. Because the
process is carried out slowly (adiabatically), the original
wave packet can not only be transported from one space
position to another, but can also change its shape at the
same time. For example, if the P state had a much
narrower potential (high vibrational frequency) the same
sequence of pulses could be carried out and the wave
packet would be adiabatically squeezed as it moves along
the light-induced channel into its final state.
The treatment presented here has been restricted to
only three levels in the sodium dimer. There is always
the possibility that other neighboring energy levels could
disturb the counterintuitive process by complicating the
dressed state potential seen in Fig. 3. This will be
subject to further investigation. Fortunately, the scheme
presented in this Letter is extremely insensitive to the
specific parameters (Rabi frequency and detuning) once
the appropriate regime has been discovered. Thus we
expect that there is enough freedom in the controlling
parameters to avoid any detrimental effects from other
levels.
We have described a process for the efficient transfer of
a wave packet from one molecular potential to another by
means of light-induced potentials. We have demonstrated
it with the sodium dimer using realistic pulse parameters
and potentials. A suitable name for the process is
adiabatic passage by light induced potentials (APLIP).
The process is not only efficient, but usable over a wide
range of counterintuitive pulse parameters. The range of
parameters is even wider than in a corresponding STIRAP
excitation because we do not maintain a precise two-
photon resonance, and do not have the possibility of
excitation of neighboring vibrational levels. The process
can also be quite fast, almost on vibrational time scales.
This work was supported by the United Kingdom
Engineering and Physical Sciences Research Council and
the Academy of Finland. The authors wish to thank Stig
Stenholm and Nikolay Vitanov for helpful discussions.
*Present address: SCOAP & Centre for Theoretical
Physics, CPES, University of Sussex, Falmer, Brighton,
BN1 9QJ, United Kingdom.
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