日本原子力研究開発機構機関リポジトリ
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Title

Quantum optimal control of the isotope-selective rovibrational
excitation of diatomic molecules

Author(s)Kurosaki Yuzuru, Yokoyama Keiichi

Citation

Chemical Physics, 493, p.183-193 (2017)

Text Version

Author Accepted Manuscript

URL

https://jopss.jaea.go.jp/search/servlet/search?5059879

DOI

https://doi.org/10.1016/j.chemphys.2017.07.003© 2017. This manuscript version is made available under the

Right

CC-BY-NC-ND 4.0 license
http://creativecommons.org/licenses/by-nc-nd/4.0/

Japan Atomic Energy Agency

Quantum optimal control of the isotope-selective rovibrational
excitation of diatomic molecules
Yuzuru Kurosakia and Keiichi YokoyamabaQuantum Beam Science Research Directorate, Takasaki Advanced Radiation Research Institute,
National Institutes for Quantum and Radiological Science and Technology, Tokai, Ibaraki 319-1195,
Japan,b
Materials Science Research Center, Japan Atomic Energy Agency, 1-1-1 Sayo-cho, Sayo-gun, Hyogo
679-5148, Japan

ABSTRACT

We carry out optimal control theory calculations for isotope-selective pure rotational andvibrational-rotational excitations of diatomic molecules. The fifty-fifty mixture ofdiatomic isotopologues, 7Li37Cl and 7Li35Cl, is considered and the molecules areirradiated with a control pulse. In the wave packet propagation we employ the method
quantum mechanically rigorous for the two-dimensional system including both the radial
and angular motions. We investigate quantum controls of the isotope-selective purerotational excitation for two total times 1280000 and 2560000 a.u. (31.0 and 61.9 ps) andthe vibrational-rotational excitation for three total times, 640000, 1280000, and 2560000
a.u. (15.5, 31.0, and 61.9 ps) The initial state is set to that both the isotopologues are in
the ground vibrational and rotational levels, v = 0 and J = 0. The target state for purerotational excitation is set to 7Li37Cl (v = 0, J = 1) and 7Li35Cl (v = 0, J = 0); that forvibrational-rotational excitation is set to 7Li37Cl (v = 1, J = 1) and 7Li35Cl (v = 0, J = 0).
The obtained final yields are quite high and those for the longest total time are calculated
to be nearly 1.0. When total time is 1280000 a.u., the final yields for the pure rotational
excitation are slightly smaller than those for the vibrational-rotational excitation. This isbecause the isotope shift (difference in transition energy between the two isotopologues)
for the pure rotational transition between low-lying levels is much smaller than that for
the vibrational-rotational transition. We thus theoretically succeed in controlling theisotope-selective excitations of diatomic molecules using the method including bothradial and angular motions quantum mechanically.

1

Keywords:
Quantum control
Optimal control theory
Isotope-selective rovibrational excitation
Laser pulse
Diatomic molecule
Wave packet
E-mail addresses: kurosaki.yuzuru@qst.go.jp (Y. Kurosaki);yokoyama.keiichi@jaea.go.jp (K. Yokoyama).

21. Introduction
Quantum control has been expected to bring a breakthrough not only in basic sciencebut also in applied research leading to novel industrial technology. Thus a great numberof studies has been devoted to quantum control for more than two decades [1-4]. One of
the most important applications of quantum control is the discrimination of a mixture ofphysically or chemically similar species, where different species are driven into different
target states at the same time with a single laser pulse. Rabitz and co-workers [5-7]presented the formulation called optimal dynamic discrimination (ODD), which is atheoretical tool to predict a control laser pulse to enable such discrimination of a mixtureof similar quantum systems. It is noteworthy that some experimental demonstrations of
ODD have been reported, such as the isotope-selective ionization of diatomic molecules[8,9] and the discrimination of competitive product channels for large biomolecules[10,11].
We [12-15] have studied, using optimal control theory (OCT) [16-18], the quantum
optimal control of isotope-selective vibrational excitations of a diatomic alkali halide. As
a result of the numerical simulations, ODD for a mixture was successfully demonstrated;
it was verified that the obtained optimal field drives the system wave packets of different
initial states into different final states. In those studies molecular rotational motion wasfrozen and only the radial (vibrational) motion was considered.
On the other hand, we [19-22] are also interested in isotope-selective rotational
excitations of diatomic alkali halides using an optical frequency comb. This is because in
the current laser shaping technology the design of the frequency comb for rotational
excitations of heavy diatomic molecules is more realistic than that of the laser pulse forvibrational excitations. Very recently, we [23] have proposed a method of the isotopeselective rovibrational excitation for a gas-phase mixture of diatomic alkali halides at a
finite temperature using a laser pulse consisting of a frequency comb for rotational
excitations and a subsequent bunch of pulse for vibrational excitations; it is expected thatthe frequency comb tuned to the rotational transition energies of one isotopologuechanges the rotational-state distributions and magnifies the difference in vibrationaltransition energies between the isotopologues, thus enabling the isotope-selective
vibrational excitation even with the single frequency pulse. The numerical simulation has
shown that the proposed scheme works well for the isotope-selective rovibrational
3

excitation for a gas-phase mixture of the diatomic 7Li37Cl and 7Li35Cl molecules at 70 K.
In the present study we perform OCT simulations to obtain optimal electric fields
that enable the isotope-selective rovibrational excitation for the isotopologues: 7Li37Cl
and 7Li35Cl. In the wave-packet calculations both the radial (vibrational) and angular(rotational) motions are treated quantum mechanically; this is more challenging than ourprevious studies [16-18] mentioned above, where only the radial motion was considered.
The ultimate goal might be the control of a mixed ensemble of molecules thermally
distributed up to high quantum levels according to the temperature of the ensemble. We
focus here, however, on the simplest cases of the isotope-selective excitations andexamine the following two excitations: (i) pure rotational excitation; (ii) vibrationalrotational excitation. In the two cases the initial state is set to the condition that both
7

Li37Cl and 7Li35Cl are in the ground vibrational and rotational levels, v = 0 and J = 0.

The target states in these cases are set as follows: (i) 7Li37Cl (v = 0, J = 1), 7Li35Cl (v = 0,
J = 0); (ii) 7Li37Cl (v = 1, J = 1), 7Li35Cl (v = 0, J = 0).
Theoretical details are described in Sec. II. The results of calculation are presented
in Sec. III and conclusions are given in Sec. IV.

2. Theoretical details

2.1. OCT for isotope-selective excitation
OCT provides an efficient tool to find an electric field (t) that causes the transition
between predetermined initial and target states with a high probability. In the present
study a gas-phase mixture of isotopologues is considered and the objective functional J
to be maximized is given asJ   p A  A (T ) O A  A (T )  A dt (t )
T

2

0

 T


 2 p A Re  dt  A (t )
 i ( H 0A   (t ))  A (t )  .
t
A
 0




(1)

The first term is the transition probability to be maximized and is referred to as final yield,
where pA is the population of isotopologue A, (t) is the wave packet of isotopologue A
propagated from time (t) = 0 to T, and A is the target operator of isotopologue A. The

4

second term is the laser fluence to be minimized, where  is a penalty factor. In order toobtain the optimal fields that start and end with the zero amplitude, a time dependence isintroduced in :   (t ) 

0

(2)

s (t )

where  is a positive constant and s(t) is a shape function having a form t 
s(t )  sin 2   .
T 

(3)

The third term is the dynamical constraint, where (t) is the Lagrange multiplier ofisotopologue A, Re denotes the real part of a complex number, H0A is the field-free
Hamiltonian of isotopologue A, and  is the dipole moment. In the present OCT
calculations we consider the 1:1 mixture of 7Li37Cl (A) and 7Li35Cl (B) and pA = pB = 0.5.
Varying the objective functional J with respect to (t), (t), and (t), we obtainthe following three coupled equations for each isotopologue:
 A (t ) ( H 0A   (t )) A (t ),  A (0)   A, 0t (t )i A  ( H 0A   (t ))  A (t ),  A (T )  O A (T )ts (t )
 (t )  
 p A Im  A (t )   A (t ) ,i

0( 4a )
(4b)
(4c)

A

with Im in Eq. (4c) denoting the imaginary part. These equations are iteratively solved
until the value of J is converged using the monotonically convergent algorithm [24], i.e.,the optimization of J is achieved by iteratively solving the following equations at the kth step: Ak (t )i
 ( H 0A   k (t )) Ak (t ),  Ak (0)   A, 0ts(t )
 k (t )  (1   )~ k 1 (t )  p A Im  Ak 1 (t )   Ak (t )

0

(5b)

A

 Ak (t )
 ( H 0A  ~ k (t ))  Ak (t ),  Ak (T )  O Ak (T )ts(t )
~ k (t )  (1   ) k (t )  p A Im  Ak (t )   Ak (t ) .
i

0

(5a)

A

2.2. Wave packet propagation
5

(5c)
(5d )

The time-dependent Schrödinger equation for the wave packet (t) of nucleus, inthe Born-Oppenheimer framework, is given byi (t )  H (t ) ,t(6)

where the total Hamiltonian H is the sum of the field-free and the light-moleculeinteraction parts:

H  H0  HI .

(7)

The field-free Hamiltonian H0 is composed of the kinetic energy operators for the radial
distance R, the polar angle , and the azimuth angle, and of the potential energy operatorV(R):H 0  TR  T  T  V ( R) .

(8)

In a previous work [25] potential energy curves for the LiCl molecule were calculatedusing high-level ab initio methods and here the calculated ground X1+ state potential
energy is employed for V(R). The kinetic energy operators are given byTR  

1
  2  R
22mR R  R (9)T  

11   
 sin 2 
2mR sin   

(10)T  

1
1 2,
2mR 2 sin 2   2

(11)

where m is the reduced mass. HI is the field-molecule interaction term and is given by

H I  ( R) (t ) cos .

(12)

The ground X1+ state dipole moment calculated in the previous work [25] is employedfor (R). In the present calculation the system is assumed to interact with the electric field
only through the permanent dipole and the higher-order terms of the electric field areneglected. This might be justified because the obtained electric fields are not so strong,
as shown later; it is of interest, however, to investigate interaction through the higherorder terms and this is a future challenge. As usual, the eigenfunction for H0 can befactorized into the three factors depending on R, , and , and thus the initial wave packet
is set to an eigenfunction:6 (0)   vJ ( R)YJM ( ,  )

(13)

J

where YJMJ(,) is the spherical harmonics and vJ(R) is the eigenfunction of thevibrational state v obtained by solving the following J-dependent radial Schrödinger
equation: 1 2

1

V
(
R
)

J
(
J

1
)

  vJ ( R)  EvJ  vJ ( R) .
2
2
2
m

R
2
mR



(14)

This equation is numerically solved using the Fourier grid Hamiltonian method [26]. In
the present calculations we only consider the MJ = 0 case and thus Eq. (13) reduces to (0)   vJ ( R) (2 J  1) / 2PJ0 (cos  ) ,(15)

where P0J is associated Legendre polynomials for MJ = 0. Eq. (3) is numerically integratedusing the split-operator method [27-29]:

 (t  t )  exp[ it (TR  T  V  H I )] (t )
 exp( i(V  H I )t / 2) exp( iT t / 2) exp( iTR t )(16)

 exp( iT t / 2) exp( i(V  H I )t / 2) (t )  O( t )3

The radial grid points are chosen to be evenly spaced and the angular grid points arechosen as the Gauss-Legendre quadrature points [30]. The action of the potential andfield-molecule interaction V + HI on the wave packet is performed directly in coordinate
space by multiplication. The kinetic-energy operator TR is acted on the wave packet in
momentum space by transforming from coordinate space with the fast-Fourier-transform(FFT) algorithm, and then the wave packet is transformed back into the coordinate spacewith the reverse FFT algorithm. The kinetic-energy operator T is acted on the wave
packet in the polynomial representation by switching from coordinate space, and then the
wave packet is switched back into coordinate space, using the discrete variable
representation (DVR) technique [31]. To eliminate the artificial reflection of the wave
packet at the edge of radial grid points, we incorporate a damping function [32] that is
operative only for the last five points from the edge. Grid and time parameters used in thepresent OCT calculations are summarized in Table I.

3. Results and discussion
In the present study we investigate two total times (T’s), 1280000 and 2560000 a.u.

7

(31.0 and 61.9 ps), for pure rotational excitation and three T’s, 640000, 1280000, and2560000 a.u. (15.5, 31.0, and 61.9 ps), for vibrational-rotational excitation. The employedinitial guess for the optimal field for pure rotational excitation is just  (t )  E0 cos 2Bt ,where B is the rotational constant for 7Li37Cl and E0 = 5.0 x 10-4 and 1.0 x 10-5 a.u. for T= 1280000 and 2560000 a.u., respectively. The employed initial guess field forvibrational-rotational excitation is  (t )  E0 cos 0 t , where 0 is the transition energy
between the v = 0 and J = 0 state and the v = 1 and J = 1 state for 7Li37Cl and E0 = 1.0 x
10-3, 5.0 x 10-4, and 1.0 x 10-5 a.u. for T = 640000, 1280000, and 2560000 a.u.,
respectively.

3.1. Rotational excitation

In this subsection the results of OCT calculations for the isotope-selective rotational
excitation, 7Li37Cl (v = 0, J = 0); 7Li35Cl (v = 0, J = 0) → 7Li37Cl (v = 0, J = 1); 7Li35Cl (v= 0, J = 0), are presented. It was found that without filtering the control field frequency
the OCT calculations often yield the optimal fields containing a zero frequency
component, which is experimentally unrealistic. Thus the low/high cutoff frequencies land h are set up and frequencies lower than l and those higher than h are filtered out
in the OCT calculation. Here we adopt two sets of cutoffs: (i) l = 6.0 x 10-6 a.u. and h= 0.1 a.u.; (ii) l = 6.0 x 10-6 a.u. and h = 1.0 x 10-3 a.u. Frequency cutoff (i) allows bothrotational and vibrational transitions but (ii) allows only rotational transition.

3.1.1. T = 1280000 a.u.

First, the OCT result obtained with frequency cutoff (i) is presented. The final yieldis 0.903, the maximum field amplitude 3.014 x 10-3 a.u., and the fluence 2.606 x 10-1 a.u.,
Table II. Figure 1a shows the optimal field as a function of t and Fig. 1b its spectrum. As
shown in Fig. 1b, the spectrum is mostly composed of three components; the fundamental,
the first overtone, and the second overtone for the vibrational transition. The first overtone
has the largest contribution, the fundamental the second, and the second overtone thesmallest. Inset in Fig. 1b shows a close-up of the spectrum in a low-frequency region;
although contribution is small, several frequency components are seen around 5 x 10-5
8

a.u., corresponding to rotational transitions. In order to make sure if those components
play a role in the control mechanism frequencies lower than 4 x 10-4 a.u. were cut out
from the optimal field and the final yield was again calculated. As a result, the yield
dropped to less than 0.5, indicating that the lower-frequency components play asignificant role. Figure 2 shows temporal changes in the populations of vibrational androtational states for the two isotopologues, caused by the obtained optimal field. The timedependent population of vibrational state v, pv(t), is given by the summation, over thenumber of angular grid point Na, of the square of the projection of the rovibrational
eigenfunction onto the system wave packet (t):Na

pv (t )    (t )  vJ ( R) (2 J  1) / 2 PJ0 (cos  )

2

(17)

J

and that of rotational state J, pJ(t), is given by the summation, over the number of radialgrid point Nr, of the same quantity:Nrp J (t )    (t )  vJ ( R) (2 J  1) / 2 PJ0 (cos  )

2

.

(18)

v

At the beginning of the control pulse the populations of the vibrational v = 0 state and therotational J = 0 state are 1.0 for both 7Li37Cl and 7Li35Cl. As shown in Figs. 2a and 2b, inboth the isotopologues vibrational states are excited up to v = 4 around the middle of thepulse and finally the population of the v = 0 state is more than 0.9 at the end. The rotational
states, Figs. 2c and 2d, show similar behaviors to the vibrational-state counterparts andthey are excited up to J = 5, although the populations change very rapidly. Finally, thepopulations of the target states, J = 1 (7Li37Cl) and J = 0 (7Li35Cl), reach more than 0.9.
Next, the OCT result obtained with frequency cutoff (ii) is discussed. The final yieldis 0.921, the maximum field amplitude 6.552 x 10-4 a.u., and the fluence 6.201 x 10-2 a.u.,
Table II. Although the final yield is comparable with that with frequency cutoff (i), thefield intensity is much smaller. Figure 3a shows the optimal field and Fig. 3b its spectrum.
The optimal field changes very slowly with small rapid oscillations. The spectrum ismainly composed of frequency components around 1.0 x 10-5 a.u. with many othercomponents being distributed over a wide frequency range. Inset in Fig. 3b shows a closeup of the spectrum around 1.0 x 10-5 a.u., where filled circles are used to indicate thediscrete points at intervals of the minimum frequency. The figure reveals that thefrequency component of the largest contribution lies around 1.3 x 10-5 a.u. and that of the9second largest contribution around 7.0 x 10-6 a.u.; the former is close to the rotational J =2←1 transition energy of 7Li37Cl equal to 4B = 1.29 x 10-5 a.u. and the latter to the J = 1←0 transition energy equal to 2B = 6.45 x 10-6 a.u. Figure 4 shows temporal changes in
the populations of vibrational and rotational states for the two isotopologues. No
vibrational excitation occurs by filtering out frequencies larger than 1.0 x 10-3 a.u., Figs.
4a and 4b. Rotational states are actively excited up to around J =12, Figs. 4c and 4d, andfinally the target states, J = 1 (7Li37Cl) and J = 0 (7Li35Cl), are achieved at a transition
probability higher than 0.9.
The minimum frequency min that can be represented by the pulse of total time T isgiven by 2/T and for the present pulse of T = 1280000 a.u. min = 4.9 x 10-6 a.u. The
rotational transition energy from J = 0 to J = 1 given by 2B are 6.45 x 10-6 and 6.51 x 106

a.u., respectively, for 7Li37Cl and 7Li35Cl and the isotope shift, i.e., the difference of 2B

between the isotopologues, is 6 x 10-8 a.u., which is much smaller than min. It is a bit
surprising that the targeted control is largely successful, despite that the frequency
spectrum is much less finer than the isotope shift. The mechanism of the isotope-selective
rotational excitation might be explained as follows in terms of difference in the velocity
of wave-packet motion between the isotopologues, leading to different quantum
interference for different isotopologue. Assuming that the rotational wave packet iscomposed of two eigenstates and the transition energy is 1 for one isotopologue and 2
for the other, the time needed for the two wave packets to become out of phase by  isestimated to be = /|1-2| [13]. If the rotational wave packet for LiCl is composed ofthe lowest two eigenstates, J = 0 and 1, then  is estimated to be 5.3 x 107 a.u., which ismuch longer than the present total time. Rotational wave packets created from higher
quantum levels, however, can move faster and difference in phase between the two wave
packets increases more quickly, resulting in a shorter . For example, if the wave packet
is created from the J = 0 and 7 states, then  is estimated to be 1.9 x 106 a.u., which isgetting closer to the present total time. It has been shown in the control processes thathigh rotational levels are utilized to create rotational wave packets; in the OCT calculation
with frequency cutoff (i) the rotational state is excited up to J = 4, Figs. 2c and 2d, and in
the calculation with frequency cutoff (ii) the rotational state is excited up to J = 10, Figs.
4c and 4d. Those results strongly suggest the control mechanism that rotational wave

10

packets created from low and high quantum levels can achieve the high final yield to thetarget.

3.1.2. T = 2560000 a.u.

First, we discuss the OCT result obtained with frequency cutoff (i). The final yield
is 0,998, the maximum field amplitude 1.019 x 10-4 a.u., and the fluence 2.384 x 10-3 a.u.,
Table II. The final yield is larger than that for T = 1280000 a.u. and is nearly perfect. The
maximum field amplitude and the field fluence are substantially smaller than those for T= 1280000 a.u. Figure 5a shows the optimal field as a function of t and Fig. 5b its spectrum.
The spectrum consists mainly of a frequency component around  = 0.003 a.u.,corresponding to the vibrational v = 1←0 transition energy, and of a component of smaller
amplitude around  = 0.006 a.u., corresponding to the first vibrational overtone. Shownin inset in Fig. 5b are the components of much smaller amplitudes in a low-frequency
region; they accord with some rotational transitions. Calculating the final yield again with
cutting out frequencies lower than 4 x 10-4 a.u. from the optimal field resulted in a final
yield less than 0.5, indicating that the lower-frequency components play a significant role.
Figure 6 shows temporal changes in the populations of vibrational and rotational states
for the two isotopologues. At the beginning of the control pulse the populations of thevibrational v = 0 state and the rotational J = 0 state are 1.0 for both 7Li37Cl and 7Li35Cl.
The vibrational v = 1 state is populated to some extent both in 7Li37Cl, Fig. 6a, and 7Li35Cl,
Fig. 6b, and almost no higher vibrational states are excited in contrast to the case for T =1280000 a.u. with frequency cutoff (i), Figs. 2a and 2b. The rotational states, Figs. 6c and6d, exhibit similar behaviors to those for T = 1280000 a.u. with frequency cutoff (i), Figs.
2c and 2d, although in the present case the amplitude of rapid oscillations is smaller.
Finally, the populations of the target states, J = 1 (7Li37Cl) and J = 0 (7Li35Cl), reachnearly 1.0.
Next, the OCT result with frequency cutoff (ii) is presented. The final yield is 0,983,
the maximum field amplitude 3.285 x 10-5 a.u., and the fluence 2.861 x 10-4 a.u., Table
II. The final yield is larger than that for T = 1280000 a.u. and is nearly perfect. The
maximum field amplitude and the field fluence are much smaller than those withfrequency cutoff (i). Figure 7a shows the optimal field and Fig. 7b its spectrum. The

11

optimal field changes very slowly and the spectrum is composed of frequency
components lower than 1.5 x 10-4 a.u. Shown in inset in Fig. 7b is a close-up of thespectrum of frequencies less than 1.0 x 10-4 a.u. and it is seen that the frequency
component of the largest contribution lies around 7.0 x 10-6 a.u. and that of the second
largest contribution around 2.0 x 10-5 a.u.; the former is close to the lowest J = 1←0transition energy for LiCl equal to 2B = 6.45 x 10-6 a.u. and the latter to the J = 3←2
transition energy equal to 6B = 1.94 x 10-5 a.u. In contrast to the spectrum for T = 1280000
a.u. with frequency cutoff (ii), the component corresponding to the lowest rotationaltransition has the largest contribution in the spectrum for T = 2560000 a.u. Figure 8 shows
temporal changes in the populations of vibrational and rotational states for the two
isotopologues. No change occurs in vibrational states by filtering, Figs. 8a and 8b, while
rapid changes in rotational states are seen in Figs. 8c and 8d, but the changes are lessrapid than those for T = 1280000 a.u., Figs 4c and 4d, and the states are excited up toaround J = 8. The target states, J = 1 (7Li37Cl) and J = 0 (7Li35Cl), are finally achieved at
a transition probability nearly 1.0.
The control mechanism when using frequency cutoff (ii) can be explained in thesame way as the previous subsection for T = 1280000 a.u., because the minimum
frequency that can be represented by this pulse, 2.5 x 10-6 a.u. is still much larger than theisotope shift in the rotational transition energy, 6 x 10-8 a.u., whereas the obtained final
yield is nearly perfect. The velocity of the wave packet created from high rotational
eigenstates makes distinct difference between the isotopologues in a shorter time than thatcreated from lower eigenstates, leading, in the last stage of the pulse, to large differencein the phase of wave packet; because of the difference in phase, the wave packet fordifferent isotopologue experiences different quantum interference in the last stage,achieving finally J = 1 for 7Li37Cl and J = 0 for 7Li35Cl

3.2. Vibrational-rotational excitation

In this subsection the OCT results for the isotope-selective vibrational-rotational
excitation, 7Li37Cl (v = 0, J = 0); 7Li35Cl (v = 0, J = 0) → 7Li37Cl (v = 1, J = 1); 7Li35Cl (v= 0, J = 0), are presented and some detailed discussion is given for each total time. In thepresent OCT calculations frequencies lower than 1.5 x 10-3 a.u. are cut out for T = 640000

12

a.u. and so are those lower than 1.0 x 10-4 a.u. for T = 1280000 a.u., i.e., the frequencies
corresponding to transitions between lower rotational levels are removed from the control
field. It has been found that this effectively smooths the complex temporal change inrotational-level populations and makes the control mechanism quite simple, while
maintaining the final yields at high levels.

3.2.1. T = 640000 a.u.

As summarized in Table II, the final yield obtained for T = 640000 is 0.765; themaximum field amplitude and fluence are 2.500 x 10-3 a.u. and 9.730 x 10-2 a.u.,
respectively. Figure 9a shows the optimized field and Fig. 9b its spectrum; the amplitudeof the field is very large around t = 300000 a.u.; in the spectrum some overtones as wellas the fundamental appear and the first overtone plays a predominant role with the others
having secondary importance. This strongly suggests that the control for the present total
time needs to make use of overtone transitions to create wave packets that can movequickly, thus achieving the highest final yield possible. The large amplitude of the fieldaround t = 300000 a.u. can thus be explained by the overtone transitions that need verystrong fields because overtone transition moments are generally very small. Figure 10
shows temporal changes in the populations of vibrational and rotational states for the two
isotopologues, caused by the obtained field. The populations of the v = 1 and J = 1 statesfor 7Li37Cl, Figs. 10a and 10c, reach about 0.85, while those of the other states decrease
to nearly zero, at the end of the pulse. The populations of the vibrational and rotational
states for 7Li35Cl, Figs. 10b and 10d, exhibit significant changes in the middle of the pulse,
but they almost return back to the initial situation at the end where only the ground states
are populated, although some vibrational and rotational excited states are slightly
populated. The state populations exhibit large oscillations around t = 300000 a.u., whichis due to the large field amplitude around this time, Fig. 9a.
The vibrational-rotational transition energies from (v = 0, J = 0) to (v = 1, J = 1) are
0.002963 a.u. and 0.002976 a.u., respectively, for 7Li37Cl and 7Li35Cl, Table III, and theisotope shift, i.e., the difference in transition energy between the isotopologues, is 1.4 x
10-5 a.u. The minimum frequency for T = 640000 a.u. is min = 2/T = 9.8 x 10-6 a.u.,which is comparable with the isotope shift, and hence the control of the present isotope-

13

selective vibrational-rotational transition is achieved rather successfully. Note that theisotope shift in the pure rotational transition J = 1←0 is 6 x 10-8 a.u., which is muchsmaller than that in the present vibrational-rotational transition, and no control of purerotational transition was obtained for the same total time.

3.2.2. T = 1280000 a.u.

As shown in Table II, the final yield for T = 1280000 a.u. is 0.988, which is nearly
perfect and substantially higher than that for T = 640000 a.u; the maximum field
amplitude and fluence are 7.289 x 10-5 a.u. and 1.206 x 10-3 a.u., respectively, which aremuch smaller than those for T = 640000 a.u. Figure 11 shows (a) the optimal field as afunction of t and (b) its spectrum. The spectrum shows that only the fundamental
frequency for the vibrational transition plays a role. The inset in Fig. 11b shows a closeup of the spectrum around  = 0.003 a.u.; it is seen that the peak is at 0.00296 a.u. The
vibrational-rotational transition energy from (v = 0, J = 0) to (v = 1, J = 1) is calculated
to be 0.002963 a.u. for 7Li37Cl and 0.002976 a.u. for 7Li35Cl, Table III. The peak in the
spectrum exactly matches the transition energy for 7Li37Cl. Figure 12 shows temporalchanges in the populations of vibrational and rotational states for the two isotopologues.
For 7Li37Cl vibrational and rotational states exhibit almost the same change and the
populations of v =1 and J = 1 reach nearly 1.0 with those of the other states decreasing tozero at the end of the pulse, Figs. 12a and 12c. On the other hand, vibrational androtational states for 7Li35Cl show almost no change, Figs. 12b and 12d.

min for T = 1280000 a.u. is 4.9 x 10-6 a.u., which is less than half the isotope shift inthe vibrational-rotational transition energies, 1.4 x 10-5 a.u. Thus the control mechanism
here is explained by what is called the conventional laser isotope separation; theseparation relies only on the applied laser frequency and does not use difference inquantum interference between wave packets moving in different velocity, which areimportant in the isotope-selective pure rotational excitation, as discussed in the previous
section.

3.2.3. T = 2560000 a.u.

14

The final yield for T = 2560000 a.u. is 0.999, which is nearly perfect and slightly
higher than that for T = 1280000 a.u; the maximum field amplitude and fluence are 4.860
x 10-5 a.u. and 8.151 x 10-4 a.u., which are slightly smaller than those for T = 1280000
a.u., Table II. Figure 13 shows (a) the optimal field as a function of t and (b) its spectrum.
The spectrum includes only the fundamental frequency and no overtone appears. A closeup of the spectrum around  = 0.003 a.u., Fig. 13b, shows again that the peak at  =0.00296 a.u. exactly matches the vibrational-rotational transition energy from (v = 0, J =0) to (v = 1, J = 1) for 7Li37Cl, 0.002963 a.u. Figure 14 shows the temporal changes in the
populations of vibrational and rotational states for the two isotopologues. Those statechanges are very similar to those for T = 1280000 a.u., Fig. 12, although in the present
case much less state excitation occurs in 7Li35Cl, Figs. 14b and 14d, compared with Figs.
12b and 12d. It is thus concluded that the control mechanism for T = 2560000 a.u. is thesame as that for T = 1280000 a.u., where only the laser frequency is the key factor for
isotope-selective excitation.

4. Conclusion

In this work we have carried out OCT calculations to obtain the electric fields that
effectively steer isotope-selective transitions for the fifty-fifty mixture of diatomic
isotopologues, 7Li37Cl and 7Li35Cl. The two-dimensional time-dependent wave packet
propagation was calculated with both the radial and angular motions of the diatomic
molecule being treated quantum mechanically. We investigated quantum controls of theisotope-selective pure rotational and vibrational-rotational excitations of the diatomic
molecules. In all the controls the initial states were set to the situation that both 7Li37Cl
and 7Li35Cl are in the v = 0 and J = 0 states. The target in the pure rotational excitation
was 7Li37Cl (v = 0, J = 1) and 7Li35Cl (v = 0, J = 0) and that in the vibrational-rotational
excitation was 7Li37Cl (v = 1, J = 1) and 7Li35Cl (v = 0, J = 0). Two total times 128000
and 2560000 a.u. (31.0 and 61.9 ps) were examined for the pure rotational excitation andthree total times, 640000, 128000, and 2560000 a.u. (15.5, 31.0, and 61.9 ps) for thevibrational-rotational excitation. The obtained final yields for the pure rotational
excitation were just over 0.9 for T = 1280000 a.u. and nearly 1.0 for T = 2560000 a.u. andthose for the vibrational-rotational excitation were just less than 0.8 for T = 640000 a.u.

15

and nearly 1.0 for T = 1280000 and 2560000 a.u. It was found that the isotope-selective
pure rotational excitation can be realized by two different control fields: One consists offrequency components corresponding to both the vibrational and rotational transitions
and the other consists of components corresponding to only the rotational transition. The
control fields for the vibrational-rotational excitations were found to include components
corresponding to the vibrational transition accompanied by the rotational transition. We
have thus theoretically succeeded in controlling the isotope-selective transitions ofdiatomic molecules using the method including both radial and angular motions quantum
mechanically. The future direction might be aiming at the control of a mixed ensemble ofdiatomic molecules having a distribution over many quantum states characterized by its
temperature, which will perhaps lead to a breakthrough in the conventional industrial
technology such as isotope separation of nuclear fuel waste.

Acknowledgements

The computations were performed using the supercomputers at Center for
Computational Science & e-Systems, JAEA, Tokai, Ibaraki, Japan and also at ResearchCenter for Computational Science, Okazaki, Aichi, Japan. This work was supported by
JSPS KAKENHI Grant No. 15H02345.

16References[1] S. A. Rice and M. Zhao, Optical Control of Molecular Dynamics, Wiley, New York,2000.
[2] V. S. Letokhov, Laser Control of Atoms and Molecules, Oxford University Press,
New York, 2007.
[3] D. D'alessandro, Introduction to Quantum Control and Dynamics, Chapman & Hall,
2007.
[4] M. Shapiro and P. Brumer, Quantum Control of Molecular Processes, Wiley-VCH,
Weinheim, 2012.
[5] B. Li, G. Turinici, V. Ramakrishna, H. Rabitz, J. Phys. Chem. B 106 (2002) 8125.
[6] G. Turinici, V. Ramakrishna, B. Li, H. Rabitz, J. Phys. A 37 (2004) 273.
[7] G. Turinici, H. Rabitz, Phys. Rev. A 70 (2004) 063412.
[8] A. Lindinger, C. Lupulescu, M. Plewicki, F. Vetter, A. Merli, S. M. Weber, L. Wöste,Phys. Rev. Lett. 93 (2004) 033001.
[9] F. Vetter, M. Plewicki, A. Lindinger, A. Merli, S. M. Weber, L. Wöste, Phys. Chem.
Chem. Phys. 7, 1151 (2005).
[10] M. Roth, L. Guyon, J. Roslund, V. Boutou, F. Courvoisier, J.-P. Wolf, H. Rabitz,
Phys. Rev. Lett. 102 (2009) 253001.
[11] J. Roslund, M. Roth, L. Guyon, V. Boutou, F. Courvoisier, J.-P. Wolf, H. Rabitz, J.
Chem. Phys. 134 (2011) 034511.
[12] Y. Kurosaki, K. Yokoyama, A. Yokoyama, J. Mol. Struct. THEOCHEM 913 (2009)38.
[13] Y. Kurosaki, K. Yokoyama, A. Yokoyama, J. Chem. Phys. 131 (2009) 144305.
[14] Y. Kurosaki, K. Yokoyama, A. Yokoyama, Comput. Theoret. Chem. 963 (2011) 245.
[15] Y. Kurosaki, A. Ichihara, K. Yokoyama, J. Chem. Phys. 135 (2011) 054103.
[16] S. Shi, H. Rabitz, J. Chem. Phys. 92 (1990) 364.
[17] W. Zhu, J. Botina, H. Rabitz, J. Chem. Phys. 108 (1998) 1953.
[18] K. Sundermann, R. de Vivie-Riedle, J. Chem. Phys. 110 (1999) 1896.
[19] K. Yokoyama, L. Matsuoka, T. Kasajima, M. Tsubouchi, A. Yokoyama in Advancesin Intense Laser Science and Photonics ed by J. Lee, Publishing House for Scienceand Technology, Hanoi, 2010, p. 113.

17

[20] L. Matsuoka, A. Ichihara, M. Hashimoto, K. Yokoyama, Proc. Int. Conf. on Towardand Over the Fukushima Daiichi Accident (GLOBAL 2011), 2011, 5017532[21] A. Ichihara, L. Matsuoka, Y. Kurosaki, K. Yokoyama, Proc. 12th Asia Pacific Physics
Conference, Chiba, 2013, JPS Conf. Proc. 1 (2014) 013093.
[22] A. Ichihara, L. Matsuoka, Y. Kurosaki, K. Yokoyama, Chinese J. Phys. 51 (2013)1230.
[23] A. Ichihara, L. Matsuoka, Y. Kurosaki, K. Yokoyama, Opt. Rev. 22 (2015) 153.
[24] Y. Maday, G. Turinici, J. Chem. Phys. 118 (2003) 8191.
[25] Y. Kurosaki, K. Yokoyama, J. Chem. Phys. 137 (2012) 064305.
[26] C. C. Marston, G. G. Balint-Kurti, J. Chem. Phys. 91 (1989) 3571.
[27] M. D. Feit, J. A. Fleck, Jr., A. Steiger, J. Comput. Phys. 47 (1982) 412.
[28] M. D. Feit, J. A. Fleck, Jr., J. Chem. Phys. 78 (1983) 301.
[29] M. D. Feit, J. A. Fleck, Jr., J. Chem. Phys. 80 (1984) 2578.
[30] W. H. Press, B. P. Flannery, S. A. Teukolsky, and . T. Vetterling, Numerical Recipes,
Cambridge University Press, Cambridge, MA, 1986.
[31] J. C. Light, I. P. Hamilton, V. J. Lill, J. Chem. Phys. 82 (1985) 1400.
[32] S. Mahapatra, N. Sathyamurthy, J. Chem. Soc., Faraday Trans. 93 (1997) 773.

18

Table 1
Grid and time parameters used in the present OCT calculations
ParameterValue
2.2677 – 5.6692

Range of the Li-Cl bond length (bohr)
Number of radial grid points

32

Number of angular grid points

16

Total time (a.u.)

6400001280000

256000

Number of time steps

13107226214452428819

Table 2
Properties of the resultant optimal electric fields
T / au







Max. field
amp. / a.u.

Fluence /
a.u.

Yield

Rotational excitation: 7Li37Cl (v = 0, J = 0); 7Li35Cl (v = 0, J = 0) → 7Li37Cl (v = 0, J = 1);
7 35
Li Cl (v = 0, J = 0)1,280,000 (~ 31.0 ps) a 10.0

0.01

0.01

3.014×10-3

2.606×10-1

0.903

1,280,000 (~ 31.0 ps) b

0.5

0.01

0.01

6.552×10-4

6.201×10-2

0.921

2,560,000 (~ 61.9 ps) a 10.0

0.01

0.01

1.019×10-4

2.384×10-3

0.998

2,560,000 (~ 61.9 ps) b 100.0

0.01

0.01

3.285×10-5

2.861×10-4

0.983

Vibrational-rotational excitation: 7Li37Cl (v = 0, J = 0); 7Li35Cl (v = 0, J = 0) → 7Li37Cl (v= 1, J = 1); 7Li35Cl (v = 0, J = 0)a640,000 (~ 15.5 ps)

1.0

0.05

0.05

2.500×10-3

9.730×10-2

0.765

1,280,000 (~ 31.0 ps)

50.0

0.01

0.01

7.289×10-5

1.206×10-3

0.988

2,560,000 (~ 61.9 ps)

50.0

0.01

0.01

4.860×10-5

8.151×10-4

0.999

Using frequency cutoff (i).
Using frequency cutoff (ii).

b20

Table 3
Rovibrational state energies for 7Li37Cl and 7Li35Cl (a.u.).
7

Li37Cl7

J

Li35Cl
J

v

0

1

2

0

1

2

0

-0.154213-0.154207-0.154194-0.154206-0.154200-0.154187

1-0.151257-0.151250-0.151237-0.151237-0.151230-0.151217

2-0.148341-0.148335-0.148322-0.148309-0.148302-0.148289

21

Figure captions
Fig. 1. (a) The optimal control field (t) for the isotope-selective rotational excitation withfrequency cutoff (i) for T = 1280000 a.u.; (b) the spectrum of (t).
Fig. 2. Temporal changes in state populations for the isotope-selective rotational
excitation with frequency cutoff (i) for T = 1280000 a.u.: (a) vibrational states for7

Li37Cl; (b) vibrational states for 7Li35Cl; (c) rotational states for 7Li37Cl; (d)rotational states for 7Li35Cl
Fig. 3. (a) The optimal control field (t) for the isotope-selective rotational excitation withfrequency cutoff (ii) for T = 1280000 a.u.; (b) the spectrum of (t).
Fig. 4. Temporal changes in state populations for the isotope-selective rotational
excitation with frequency cutoff (ii) for T = 1280000 a.u.: (a) vibrational states for7

Li37Cl; (b) vibrational states for 7Li35Cl; (c) rotational states for 7Li37Cl; (d)rotational states for 7Li35Cl
Fig. 5. (a) The optimal control field (t) for the isotope-selective rotational excitation withfrequency cutoff (i) for T = 2560000 a.u.; (b) the spectrum of (t).
Fig. 6. Temporal changes in state populations for the isotope-selective rotational
excitation with frequency cutoff (i) for T = 2560000 a.u.: (a) vibrational states for7

Li37Cl; (b) vibrational states for 7Li35Cl; (c) rotational states for 7Li37Cl; (d)rotational states for 7Li35Cl
Fig. 7. (a) The optimal control field (t) for the isotope-selective rotational excitation withfrequency cutoff (ii) for T = 2560000 a.u., obtained by filtering out the frequencies
larger than 0.001 a.u.; (b) the spectrum of (t).
Fig. 8. Temporal changes in state populations for the isotope-selective rotational
excitation with frequency cutoff (ii) for T = 2560000 a.u., obtained by filteringout the frequencies larger than 0.001 a.u.: (a) vibrational states for 7Li37Cl; (b)vibrational states for 7Li35Cl; (c) rotational states for 7Li37Cl; (d) rotational statesfor 7Li35Cl.
Fig. 9. (a) The optimal control field (t) for the isotope-selective vibrational-rotational
excitation for T = 640000 a.u.; (b) the spectrum of (t).
Fig. 10. Temporal changes in state populations for the isotope-selective vibrationalrotational excitation for T = 640000 a.u.: (a) vibrational states for 7Li37Cl; (b)22vibrational states for 7Li35Cl; (c) rotational states for 7Li37Cl; (d) rotational statesfor 7Li35Cl
Fig. 11. (a) The optimal control field (t) for the isotope-selective vibrational-rotational
excitation for T = 1280000 a.u.; (b) the spectrum of (t).
Fig. 12. Temporal changes in state populations for the isotope-selective vibrationalrotational excitation for T = 1280000 a.u.: (a) vibrational states for 7Li37Cl; (b)vibrational states for 7Li35Cl; (c) rotational states for 7Li37Cl; (d) rotational statesfor 7Li35Cl
Fig. 13. (a) The optimal control field (t) for the isotope-selective vibrational-rotational
excitation for T = 2560000 a.u.; (b) the spectrum of (t).
Fig. 14. Temporal changes in state populations for the isotope-selective vibrationalrotational excitation for T = 2560000 a.u.: (a) vibrational states for 7Li37Cl; (b)vibrational states for 7Li35Cl; (c) rotational states for 7Li37Cl; (d) rotational statesfor 7Li35Cl

23

0.004(a)ε(t) / a.u.

0.002
0.000
-0.002-0.00404000008000001200000t / a.u.
0.0025|ε(ω)| / arb. unit

(b)
6E-07

0.00204E-07

0.00152E-07

0.00100E+00
0.0000 0.0001 0.0002 0.0003

0.00050.0000
0.000

0.005

0.010

ωa.u.

Fig. 1

24

0.015

Rotational states

Vibrational states
1.0

1.0

(a)

7

v=0

0.8

0.6

0.6

0.4

0.4
v=1

0.2

7

Li37Cl(c)

Population

Population

0.8

Li37Cl

J=0

J=1

0.2

1.0

1.0

(b)
v=0

0.8

7

Li35Cl0.8

7

Li35Cl

(d)
J=0

0.6

0.6

0.4

0.4
v=1

0.2

0.2

0.0

J=1

0.0
0

40000080000012000000400000800000Time / a.u.

Time / a.u.

Fig. 225

1200000

0.0010(a)ε(t) / a.u.

0.00050.0000-0.0005-0.001004000008000001200000t / a.u.
0.00015|ε(ω)| / arb. unit

(b)
0.003

0.00010
0.002
0.001

0.00005

0.00000
0.00000.000
0.00000

0.00020.00040.00005

0.0006ωa.u.

Fig. 326

0.000100.00080.0010Rotational states

Vibrational states
1.0

1.0

(a)

(c)

v=0

0.6

0.4

0.4

Population

Population

J=1

J=0

37

Li Cl

0.6

Li37Cl

0.8

0.8
7

7

0.2

0.2

v=1
0.0
1.0

1.0

(b)

v=0

(d)

0.8

0.8

7

Li35Cl

J=0
7

0.6

35

0.6

Li Cl

0.4

0.4

0.2

0.2

v=1
0.0

0.0
0

40000080000012000000400000800000Time / a.u.

Time / a.u.

Fig. 427

1200000

0.00015(a)ε(t) / a.u.

0.000100.00005
0.00000-0.00005-0.00010-0.00015
08000001600000

2400000t / a.u.
0.000010|ε(ω)| / arb. unit(b)0.000005

1.5E-07
1.0E-07
5.0E-08
0.0E+00
0.00000

0.00005

0.000100.000000
0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007

ωa.u.

Fig.5

28

Rotational states

Vibrational states
1.0

1.0

(a)

0.8

Li37ClJ=07

Li37Cl

J=1

0.6

0.4

0.4

Population

v=1

Population

7

0.8

v=0

0.6

(c)

0.2

J=2

0.2

1.0

J=3

1.0

(b)
0.8

7

7

Li35Cl

0.8

v=0

0.6

(d)
J=0

0.6

35

Li Cl

0.4

0.4
v=1

0.2

J=1

0.2
J=2

0.0

0.0
0

8000001600000

240000008000001600000

Time / a.u.

Time / a.u.

Fig. 629

2400000

0.00004(a)ε(t) / a.u.

0.000020.00000-0.00002-0.00004
08000001600000

2400000t / a.u.

|ε(ω)| / arb. unit(b)0.000004
0.000004

0.000002

0.000002
0.000000
0.00000

0.000000
0.00000.00020.00005

0.00040.0006ωa.u.
Fig. 730

0.000100.00080.0010Rotational states

Vibrational states
1.0

1.0

(a)
7

Li37Cl0.6

7

Li37Cl

J=0

0.8

0.8

J=1

0.6

0.4

0.4

Population

Population

(c)

v=0

0.2

0.2

v=1
0.0
1.0

1.0

(b)

v=0

0.8
0.6

(d)

0.8

7

Li35Cl

J=0

0.6

7

Li35Cl

0.4

J=1

0.4

0.2

0.2

v=1
0.0

0.0
0

8000001600000

240000008000001600000

Time / a.u.

Time / a.u.

Fig. 831

2400000(a)ε(t) / a.u.

0.010
0.005
0.000
-0.005-0.0100200000400000600000t / a.u.
(b)|ε(ω)| / arb. unit

0.00100
0.000750.000500.000250.00000
0.000

0.005

0.010

0.015

0.020

ωa.u.

Fig. 932

0.025

Vibrational states

Rotational states

1.0

1.0

(a)
0.8

7

(c)

37

Li Cl

7

Li37Cl

0.8

v=0

J=0

0.6

0.6

0.4

0.4

Population

Population

v=1

0.2

J=1

0.2

v=2

1.0

J=2

1.0

(b)

0.8

7

35

Li Cl

0.8

7

Li35Cl

J=0

v=0

0.6

(d)

0.6

0.4

0.4
v=1

0.2

J=1

0.2

v=2

0.0

J=2

0.0
0

2000004000006000000Time / a.u.

200000400000Time / a.u.

Fig. 1033

6000000.00010(a)ε(t) / a.u.

0.00005
0.00000-0.00005-0.00010
1200000

8000004000000t / a.u.
0.00010|ε(ω)| / arb. unit(b)0.000100.00005

0.00005
0.00000
0.00290 0.00295 0.00300 0.00305

0.00000
0.000

0.005

0.010

0.020

0.015

ωa.u.

Fig. 1134

0.025

Rotational states

Vibrational states
1.0

1.0

(a)

(c)

0.8
0.6

7

Li37Cl0.6

7

Li37Cl

0.4

0.4
v=1

0.2

J=1

Population

Population

J=0

0.8

v=0

0.2

v=2
1.0

J=2

1.0

(b)

v=0

0.8
0.6

(d)

J=0

0.8
0.6

7

Li35Cl0.4

7

Li35Cl

0.4

0.2

0.2
v=1

J=1

0.0

0.0
0

40000080000012000000400000800000Time / a.u.

Time / a.u.

Fig. 1235

1200000

0.00010(a)ε(t) / a.u.

0.00005
0.00000-0.00005-0.00010
08000001600000

2400000t / a.u.
0.000075|ε(ω)| / arb. unit(b)0.000050

0.00006
0.00004
0.000020.00000

0.000025

0.000000
0.000

0.00290 0.00295 0.00300 0.00305

0.005

0.010

0.015

ωa.u.

Fig. 13

36

0.020

0.025

Rotational states

Vibrational states
1.0

1.0

(a)
v=0

0.8

Li37Cl7

Li37Cl

0.6

0.4

0.4
v=1

0.2

Population

Population

J=0

0.8

7

0.6

(c)

0.2

v=2
1.0

1.0

(b)

v=0

0.8

J=1
J=2

(d)

J=0

0.8
7

Li35Cl

0.6

0.6

0.4

0.4

0.2

0.2

7

Li35Cl

J=1

v=1
0.0

0.0
0

8000001600000

240000008000001600000

Time / a.u.

Time / a.u.

Fig. 1437

2400000