One-Photon Vs. Two-Photon Processes

universe

Article Quantum Optimal Control of Rovibrational Excitations of a Diatomic Alkali Halide: One-Photon vs. Two-Photon Processes

Yuzuru Kurosaki 1,* and Keiichi Yokoyama 2 1 Tokai Quantum Beam Science Center, Takasaki Advanced Radiation Research Institute, National Institutes for Quantum and Radiological Science and Technology, Tokai, Ibaraki 319-1195, Japan 2 Materials Science Research Center, Japan Atomic Energy Agency, Kouto, Sayo, Hyogo 679-5148, Japan; [email protected] * Correspondence: [email protected]

Received: 29 March 2019; Accepted: 3 May 2019; Published: 8 May 2019

Abstract: We investigated the roles of one-photon and two-photon processes in the laser-controlled rovibrational transitions of the diatomic alkali halide, 7Li37Cl. Optimal control theory calculations were carried out using the Hamiltonian, including both the one-photon and two-photon field-molecule interaction terms. Time-dependent wave packet propagation was performed with both the radial and angular motions being treated quantum mechanically. The targeted processes were pure rotational and vibrational–rotational excitations: (v = 0, J = 0) (v = 0, J = 2); (v = 0, J = 0) (v = 1, J = 2). → → Total time of the control pulse was set to 2,000,000 atomic units (48.4 ps). In each control excitation process, weak and strong optimal fields were obtained by means of giving weak and strong field amplitudes, respectively, to the initial guess for the optimal field. It was found that when the field is weak, the control mechanism is dominated exclusively by a one-photon process, as expected, in both the targeted processes. When the field is strong, we obtained two kinds of optimal fields, one causing two-photon absorption and the other causing a Raman process. It was revealed, however, that the mechanisms for strong fields are not simply characterized by one process but rather by multiple one- and two-photon processes. It was also found that in the rotational excitation, (v = 0, J = 0) (v = 0, J = 2), the roles of one- and two-photon processes are relatively distinct but in the → vibrational–rotational excitation, (v = 0, J = 0) (v = 1, J = 2), these roles are ambiguous and the → cooperative effect associated with these two processes is quite large.

Keywords: quantum control; optimal control theory; diatomic molecule; two-photon absorption; Raman process

1. Introduction Recent rapid developments in laser technology have prompted a huge number of studies aiming at a better understanding of very fast dynamical events behind physical, chemical, and biological phenomena. It is not until the developments of ultrashort pulse generation and the pulse shaping technique that one can use the quantum control idea [1–4] to achieve the goal of relevant physical processes. In recent years, quantum control has attracted great interest because of its potential for opening up a new frontier in fundamental science and has found vast application in chemical reaction [5,6], molecular alignment [7], quantum computation [8], laser cooling [9,10], etc. In our view, the underlying mechanisms of quantum control may fall into three categories. The ﬁrst one is the pump–dump scheme [5] often employed for chemical reactions hindered by a high energy barrier. The molecular system in the electronic ground state at ﬁrst is electronically excited by a pump pulse and the nuclear wave packet forms and moves around on the excited potential surface.

Universe 2019, 5, 109; doi:10.3390/universe5050109 www.mdpi.com/journal/universe Universe 2019, 5, 109 2 of 15

When large portions of the wave packet are localized in the region that is geometrically close to the target, the second dump pulse is applied to deexcite the wave packet to the ground state potential. The wave packet created again on the ground state potential may reach the target with high efficiency due to the geometrical vicinity between the wave packet and the target. The second category is the control scheme relying on interference arising via multiple pathways to the target [6]. One of the scenarios is the control of the product branching ratio in the photodissociation from a single state via two pathways: the N-photon and M-photon optical routes. It was shown that the control parameters are the magnitudes of the N-photon and M-photon pulses and their phases and thus changing those values can make one dissociation channel predominate over the other. The last category is the scheme of stimulated Raman adiabatic passage (STIRAP) [11–13]. The basics of STIRAP are often described using the simplest three-state Λ model; let states 1, 2, and 3 be the initial, intermediate, and target states, respectively, and the objective is to transfer the population from state 1 to state 3 completely, induced by two coherent fields that couple states 2 to 1 and 3 (pump and Stokes pulses, respectively). When applying an appropriate electric field, the degeneracy of three eigenstates of the system Hamiltonian, which coincide with state 1 at first, is lifted and the three eigenstates temporally evolve adiabatically, and they finally become degenerate again and converge to state 3. It was shown that applying the Stokes pulse first and then the pump pulse, which is the so-called counterintuitive ordering, is needed to allow for the desired adiabatic evolution. Of particular note is that one of the eigenstates of the system Hamiltonian consists of states 1 and 3 and not of state 2, with the eigenvalue remaining at zero throughout the control process, as long as the two-photon resonance condition is met. Because of these basics, the STIRAP technique has been considered to realize efficient population transfer while minimizing loss due to spontaneous emission from state 2. One of the most significant advances in STIRAP may be the technique referred to as “shortcut to adiabaticity” [14], which enables speeding up of the control time, and the derived protocol from it called the superadiabatic technique [15,16] that suppresses the spurious nonadiabatic processes. The idea of quantum optimal control can contain all the essences of the three categories described above. As a consequence, the control mechanisms obtained by optimal control theory (OCT) [17–19] and optimal control experiment (OCE) [20,21] are sometimes hard to understand and the optimal control fields predicted by OCT are often hard to generate even with state-of-the-art pulse shaping technology; optimal control is, however, a very powerful and intriguing tool that can address various control problems for quantum systems. We have been interested in the study of quantum control problems, focusing recently on applying OCT to isotope separation, which is an important subject to explore not only in basic science but in industrial applications. Using OCT, we [22–25] studied the isotope-selective vibrational excitations of diatomic molecules; the theoretical calculations predicted the control electric fields that can efficiently realize the isotope-selective excitations. We [26–29] also theoretically investigated pure rotational excitations of diatomic molecules using an optical frequency comb; note that in the current pulse shaping technology, the design of the frequency comb for rotational excitations of heavy diatomic molecules is more realistic than that of the excitation pulse for vibrational transitions. We [30] recently proposed a control scenario for the isotope-selective rovibrational excitation of diatomic molecules at a finite temperature; first a frequency comb for rotational excitations is applied to change the rotational-state distributions and to magnify the difference in vibrational transition energies between the isotopologues, and second, a bunch of pulse is irradiated for isotope-selective vibrational excitations. Thus it was shown in the numerical simulation that the proposed scheme works well for the isotope-selective rovibrational excitation for a gas-phase mixture of the diatomic alkali-halide 7Li37Cl and 7Li35Cl molecules at 70 K. In a previous study, we [31] performed OCT simulations for the isotope-selective rovibrational excitation of the diatomic alkali-halide isotopologues: 7Li37Cl and 7Li35Cl, where both the radial (vibrational) and angular (rotational) motions were treated quantum mechanically in the wave-packet calculations. As a result, the calculated final yields were nearly 1.0, indicating that the optimal fields Universe 2019, 5, 109 3 of 15 to achieve the goal were successfully obtained with the two-dimensional (2D) quantum mechanical method. In the previous study [31] however, only linear field effects were considered, i.e., the field-molecule Hamiltonian includes only the term linearly dependent on the applied field, which is responsible for one-photon processes. In the present study we added the term proportional to the square of the electric field, which is responsible for two-photon processes. To our knowledge, this is the first attempt to carry out 2D OCT simulations for diatomic molecules with the Hamiltonian including both the linear and nonlinear interaction terms. In this paper we consider only a single isotopic species, 7Li37Cl, because we intend to examine and compare the roles of one-photon and two-photon processes in rovibrational transitions, rather than the isotopic selectivity. Two control processes were investigated, i.e., the rotational excitation and vibrational–rotational excitation of the 7Li37Cl molecule. The initial state is set to the vibrational and rotational ground levels, v = 0 and J = 0, and the target states are the v = 0 and J = 2 state for the rotational excitation and the v = 1 and J = 2 state for the vibrational–rotational excitation, i.e., we tried to theoretically control the following excitations: (i) (v = 0, J = 0) (v = 0, J = 2); (ii) (v = 0, J = 0) (v = 1, J = 2). → → Theoretical details are described in Section2. The results of calculations are presented in Section3 and conclusions are given in Section4.

2. Theoretical Details

2.1. OCT In OCT calculations [17–19], an optimal electric ﬁeld is found such that the objective functional J is maximized, which is written as

Z 2 "Z * +# T ε(t) T ∂ J = ψ(T) Oˆ ψ(T) dt 2Re dt χ(t) + iHˆ (r, θ, ε(t)) ψ(t) . (1) h | | i − 0 λ − 0 |∂t |

The first term is the transition probability to be maximized and is referred to as the final yield, where ψ(T) is the system wave packet propagated from time (t) 0 to total time T and Oˆ is the target operator. The second term is the laser fluence to be minimized, where ε(t) is the electric field and λ is a penalty factor. In order to obtain the optimal fields that start and end with zero amplitude, a time dependence is introduced in λ: λ = λ(t) = λ0s(t), (2) where λ0 is a positive constant and s(t) is a shape function having a form

πt s(t) = sin2 . (3) T

The third term is the dynamical constraint, where χ(t) is the Lagrange multiplier, Re denotes the real part of a complex number, Hˆ (r,θ,ε(t)) is the Hamiltonian depending on the internuclear distance r, the angle θ between the molecular axis and the laser polarization vector, and ε(t). Applying the variational approach to the functional J, coupled equations for ψ(t), χ(t), and ε(t) are obtained and an optimal field ε(t) is finally found by solving these equations iteratively until the value of J is converged. We adopt a monotonically convergent algorithm [32,33] that is capable of solving OCT problems in which a system interacts nonlinearly with an applied electric field. As, in this study, the nonlinear interaction is considered up to the term proportional to the square of the electric field ε(t), it is artificially divided into two components ε1(t) and ε2(t) in the algorithm. Starting the iteration by solving ψ(0)(t) with a set of initial trial fields, ε (0)(t) and ε (0)(t), the kth iteration step (k 1) according 1 2 ≥ to the monotonically convergent algorithm is presented as follows:

" ˆ # ∂ (k) H1 (k 1) (k 1) (k 1) (k 1) (k) χ (t) = Hˆ + ε − (t) + ε − (t) + Hˆ ε − (t)ε − (t) χ (t) (4) ∂t − 0 2 1 2 2 1 2 Universe 2019, 5, 109 4 of 15

(k) (k) (k) with the ﬁnal condition, χ (T) = Oˆ ψ (T), and then ε1 (t) is updated as (k) (k 1) (k) (k 1) (k 1) ε (t) = ε − (t) λ(t)Im χ (t) Hˆ 1 + 2Hˆ 2ε − (t) ψ − (t) (5) 1 1 − 2 and " ˆ # ∂ (k) H1 (k) (k 1) (k) (k 1) (k) ψ (t) = Hˆ + ε (t) + ε − (t) + Hˆ ε (t)ε − (t) ψ (t) (6) ∂t − 0 2 1 2 2 1 2

(k) (k) with the initial condition, ψ (0) = ψ0, and then ε2 (t) is updated as (k) (k 1) (k) (k) (k) ε (t) = ε − (t) λ(t)Im χ (t) Hˆ 1 + 2Hˆ 2ε (t) ψ (t) . (7) 2 2 − 1

Here Hˆ 0 is the field-free Hamiltonian and Hˆ 1 and Hˆ 2 are the molecular dipole moment and the polarizability, respectively, which are explicitly given below. The doubly divided electric-field (k) (k) components finally converge to the same value: lim ε (t) = lim ε (t) = ε(t) because all the k 1 k 2 →∞ →∞ equations have a symmetric form with respect to ε1(t) and ε2(t).

2.2. Wave-Packet Propagation We numerically solve the time-dependent Schrödinger equation:

∂ i ψ(t) = Hˆ ψ(t), (8) ∂t which describes the nuclear wave packet evolving with time. The total Hamiltonian Hˆ is the sum of the field-free and the field-molecule interaction parts, Hˆ 0 + Hˆ I. The field-free Hamiltonian Hˆ 0 is composed of the kinetic energy operators for the radial distance r, the polar angle θ, and the azimuth angle ϕ, and of the potential energy operator V(r):

" ! ! 2 # 1 ∂ 2 ∂ 1 ∂ ∂ 1 ∂ Hˆ 0 = r + sin θ + + V(r), (9) −2mr2 ∂r ∂r sin θ ∂θ ∂θ sin2 θ ∂φ2 where m is the reduced mass. The calculated values for the LiCl ground X1Σ+ state potential [34] are used for V(r). In the present study, the ﬁeld-molecule interaction was considered non-perturbatively 2 up to two-photon transitions and thus Hˆ I is the sum of two terms ε(t)Hˆ 1 and ε (t)Hˆ 2 pertaining to one- and two-photon processes, respectively. Explicitly, Hˆ 1 and Hˆ 2 represent the molecular dipole moment and the polarizability, respectively: Hˆ = µ(r) cos θ, (10) 1 − 1h 2 i Hˆ 2 = α + α α cos θ , (11) −2 ⊥ k − ⊥ where α and α are the components of polarizability perpendicular and parallel, respectively, to the ⊥ k molecular axis. We employed the values of the dipole moment obtained in the previous work [34] and calculated the polarizability here using the Gaussian09 program [35] at the RMP2/aug-cc-pVTZ level of theory, Figure S1 in Supplementary Materials. It is known that the eigenfunction for Hˆ 0 can be obtained by variable separation and the initial wave packet is set to an eigenfunction:

ψ(0) = RvJ(r)YJMJ (θ, φ), (12)

where YJMJ (θ, φ) is the spherical harmonics and RvJ(r) is the J-dependent eigenfunction obtained by solving " # 1 ∂2 1 + V(r) + J(J + 1) RvJ(r) = EvJRvJ(r). (13) −2m ∂r2 2mr2 Universe 2019, 5, 109 5 of 15

This is solved using the Fourier grid Hamiltonian method [36]. In the present calculations, only the MJ = 0 case is considered; this is justified because linearly polarized electric fields are used. Thus Equation p ( ) = ( ) ( + ) 0( ) 0 (12) reduces to ψ 0 RvJ r 2J 1 /2PJ cos θ , where PJ is the associated Legendre polynomials for MJ = 0. The system wave packet is represented as the complex amplitude at two-dimensional radial and angular grid points; radial grid points are chosen to be evenly spaced [36] and the angular ones as the Gauss–Legendre (GL) quadrature points [37]. The wave packet is evolved with time according to Equation (8), which is numerically integrated by using the split-operator method [38–40]: h i ψ(t + ∆t) = exp i Tˆ R + Tˆ + V + Hˆ I ∆t ψ(t) − h θ i exp i V + Hˆ I ∆t/2 exp iTˆ ∆t/2 exp iTˆ R∆t (14) ≈ − − θ − h i 3 exp iTˆ ∆t/2 exp i V + Hˆ I ∆t/2 ψ(t) + O ∆t , × − θ − where Tˆ R and Tˆ θ are the radial and angular kinetic-energy operators, corresponding to the first and second terms, respectively, in Equation (9). A common trick to perform numerical actions of kinetic and potential-energy operators is to calculate each operator locally. The potential energy operator V + Hˆ I is represented locally in coordinate space and its action on the wave packet is simply a multiplication. Since the kinetic-energy operators Tˆ R and Tˆ θ are non-local in coordinate space, their operations on the wave packet are performed using the discrete variable representation (DVR) technique [41,42], as described below. When operating Tˆ R on the wave packet, it is transformed from radial coordinate space into momentum counterpart with the forward fast-Fourier-transform (FFT) algorithm, being multiplied by the kinetic energy spectrum k2/2m, and again it is transformed back into coordinate space with the reverse FFT algorithm. When operating Tˆ θ, the wave packet in the point representation is transformed = ( + )1/2 1/2 0( ) into the polynomial counterpart using the transformation matrix, Tji j 1/2 wi Pj σi , where σi = cosθi and wi are the points and weights for the GL quadrature; the transformed wave packet in the polynomial representation is multiplied by the spectrum j(j + 1)/2mr2, and again it is transformed back 1 into the point representation with the reverse transformation matrix (Tji)− . To eliminate the artificial reflection of the wave packet at the edge of radial grid points, we incorporate a damping function [43] that is operative only for the last five points from the edge. Grid and time parameters used in the present OCT calculations are summarized in Table1.

Table 1. Grid and time parameters used in the present OCT calculations.

Parameter Value Range of the Li-Cl bond length (bohr) 2.2677–5.6692 Number of radial grid points 32 Number of angular grid points 20 Total time (a.u.) 2,000,000 Number of time steps 131,072

3. Results and Discussion In the present OCT calculations, total time T of the control pulse was set to 2,000,000 atomic units (a.u.) (48.4 ps), which is slightly larger than twice the rotational period of the LiCl molecule defined by the reciprocal of the J = 1 0 transition energy, 2π/2B ~23 ps, where B is the rotational constant and ← is 3.2 10 6 a.u. for the 7Li37Cl molecule. In each control excitation process we obtained weak and × − strong optimal fields by means of giving weak and strong field amplitudes, respectively, to the initial guess for the optimal field. Universe 2019, 5, 109 6 of 15

3.1. Rotational Excitation: (v = 0, J = 0) (v = 0, J = 2) → 3.1.1. Weak Field The initial guess for the optimal ﬁeld in the weak-ﬁeld regime is

ε(t) = E0s(t)[cos(2Bt) + cos(4Bt)] (15)

with E = 1.0 10 6 a.u. and s(t) defined in Equation (3). This initial guess is employed assuming the 0 × − one-photon resonant transition J = 1 0 followed by the transition J = 2 1. As summarized in Table2, ← ← the perfect control was achieved, i.e., the yield obtained by the optimal field is 1.000. The optimal field is very weak, as intended; the maximum field amplitude is 5.669 10 6 a.u., and the fluence is 5.463 Universe 2019, 5, x FOR PEER REVIEW − 6 of 15 6 × × 10− a.u. Figure1a shows the optimal field as a function of t and Figure1b its spectrum. The spectrum 6 5 is mostly composed of two frequency components centered around ω = 6.0 10ω and 1.3−6 10 a.u., spectrum is mostly composed of two frequency components centered around× −= 6.0 × 10 × and− 1.3 × 10which−5 a.u., correspond which correspond to the initially to the guessedinitially guessed transition transition energies, energies, 2B and 4B, 2B respectively.and 4B, respectively.

TableTable 2. Properties 2. Properties of of the the resultant resultant optimal electric electric fields. fields. Max. Field Init.Init. Guess Guess Yield Yield Max. Field amp./a.u. FluenceFluence / a.u./a.u. Filter/a.u. Filter/a.u. amp./a.u. Rotational excitation: (v = 0, J = 0) → (v = 0, J = 2) Rotational excitation: (v = 0, J = 0) (v = 0, J = 2) weak Equation (15) 1.000 5.669 × 10−6 → 5.463 × 10−6 no filter weak Equation (15) 1.000 5.669 10 6 5.463 10 6 no filter −3 − − ω −4 strong Equation (18) 0.950 7.505 × 10× 3 20.98× 0 < < 1.5 × 10 4 strong Equation (18) 0.950 7.505 10− 20.98 0 <ω< 1.5 10− ×−2 ω ×−4 strongstrong Equation Equation (19) (19) 0.959 0.959 1.9911.991 × 10 10 2 74.7874.78 0 <0 < ω1.5< 1.5× 10 10 4 × − × − Vibrational–rotationalVibrational–rotational excitation: excitation: (v (=v =0, 0,J =J =0)0) → ((vv = 1,1, J = 2) 5 → 3 weakweak Equation Equation (20) (20) 0.997 0.997 7.2007.200 × 10−510 − 1.2411.241 × 1010−3− no filterno filter × 2 × strong Equation (21) 0.998 1.014 −210 37.53 no filter strong Equation (21) 0.998 1.014 × 10× − 37.53 no filter strong Equation (22) 0.975 5.016 10 2 469.06 no filter strong Equation (22) 0.975 5.016 × 10×−2 − 469.06 no filter

1.0E-05 6.0E-08 (a) (b)

4.0E-08 5.0E-06

| / arb. unit | / arb. ) 2.0E-08 ω ( ε(t) / a.u. ε(t) / 0.0E+00 |ε

0.0E+00 -5.0E-06 0 500000 1000000 1500000 2000000 0.00000 0.00005 0.00010 0.00015 t / a.u. / ω a.u. Figure 1. (a) The optimal ﬁeld ε(t) for the rotational excitation, (v = 0, J = 0) (v = 0, J = 2) in the → Figureweak-ﬁeld 1. (a regime;) The optimal (b) the spectrum field ε(t) of forε(t ).the rotational excitation, (v = 0, J = 0) → (v = 0, J = 2) in the weak-field regime; (b) the spectrum of ε(t). Figure2 shows temporal changes in the populations of rotational states. The time-dependent populationFigure 2 of shows rotational temporal state Jchanges, pJ(t), is givenin the herepopulati by theons summation, of rotational over states. the numberThe time-dependent of radial grid point N , of the square of the projection of the rovibrational eigenfunction onto the system wave packet populationr of rotational state J, pJ(t), is given here by the summation, over the number of radial grid ψ(t): point Nr, of the square of the projection of the rovibrational eigenfunction onto the system wave Nr q 2 packet ψ(t): X p (t) = ψ(t) R (r) (2J + 1)/2P0(cos θ) . (16) J vJ J v 𝑝(𝑡) =𝜓(𝑡)𝑅(𝑟)(2𝐽 +1)⁄2 𝑃 (cos 𝜃) . (16)

Similarly, that of vibrational state v, pv(t), is given by the summation, over the number of angular grid point Na, of the same quantity:

𝑝(𝑡) =𝜓(𝑡)𝑅(𝑟)(2𝐽 +1)⁄2 𝑃 (cos 𝜃) . (17) It was found that no vibrational excitation occurs in the present control process. As shown in Figure 2, the J = 1 state is first excited, which is then followed by the J = 2 state, and finally the J = 2 state is totally populated at the end of the control pulse. It is seen that the J = 3 state is slightly populated around the middle of the pulse. It is thus concluded that the present rotational transition is controlled by only one-photon processes; the reasons are (i) the field intensity is very small; (ii) the field spectrum is composed of transition frequencies corresponding to the J = 1←0 and J = 2←1 transitions; (iii) the yield obtained for the partial Hamiltonian, Ĥ0 + Ĥ1, with the optimal field was found to be the same as that for the full Hamiltonian, as summarized in Table 3.

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Similarly, that of vibrational state v, pv(t), is given by the summation, over the number of angular grid point Na, of the same quantity:

XNa q 2 p (t) = ψ(t) R (r) (2J + 1)/2P0(cos θ) . (17) v vJ J J

It was found that no vibrational excitation occurs in the present control process. As shown in Figure2, the J = 1 state is ﬁrst excited, which is then followed by the J = 2 state, and ﬁnally the J = 2 state is Universe 2019, 5, x FOR PEER REVIEW 7 of 15 totally populated at the end of the control pulse. It is seen that the J = 3 state is slightly populated around the middle of the pulse.

1.0

J = 0 0.8

J = 2 0.6

Population 0.4

0.2 J = 1 J = 3

0.0 0 500000 1000000 1500000 2000000 t / a.u.

Figure 2. Temporal changes in state populations for the rotational excitation, (v = 0, J = 0) (v = 0, J = Figure 2. Temporal changes in state populations for the rotational excitation, (→v = 0, J = 0) 2), in the weak-field regime. → (v = 0, J = 2), in the weak-field regime: (a) vibrational states; (b) rotational states. It is thus concluded that the present rotational transition is controlled by only one-photon processes; the reasonsTable 3. are Comparison (i) the field of intensity the yields is verywith small;the full (ii) (Ĥ the0 + Ĥ field1 + Ĥ spectrum2) and partial is composed Hamiltonians of transition (Ĥ0 frequencies+ Ĥ1 and corresponding Ĥ0 + Ĥ2), obtained to the Jusing= 1 the0 and optimalJ = 2 fields1 transitions; for the full (iii) Hamiltonian. the yield obtained for the partial ← ← Hamiltonian, Hˆ + Hˆ , with the optimal field was found to be the same as that for the full Hamiltonian, 0 1 Init. Guess Ĥ0 + Ĥ1 + Ĥ2 Ĥ0 + Ĥ1 Ĥ0 + Ĥ2 as summarized in Table3. Rotational excitation: (v = 0, J = 0) → (v = 0, J = 2)

Table 3. Comparisonweak of the yieldsEquation with the(15) full (Hˆ 0 + 1.000Hˆ 1 + Hˆ 2) and 1.000 partial Hamiltonians 0.000 (Hˆ 0 + Hˆ 1 and Hˆ 0 + Hˆ 2), obtainedstrong using theEquation optimal ﬁelds (18) for the full 0.950 Hamiltonian. 0.184 0.338 strong Equation (19) 0.959 0.009 0.633 ˆ ˆ ˆ ˆ ˆ ˆ ˆ Vibrational–rotationalInit. Guess excitation:H0 + H1 +(vH =2 0, J = 0) →H (0v+ =H 1,1 J = 2) H0 + H2 Rotational excitation: (v = 0, J = 0) (v = 0, J = 2) weak Equation (20) 0.997 → 0.997 0.000 weakstrong EquationEquation (15) (21) 1.000 0.998 0.009 1.000 0.000 0.000 strong Equation (18) 0.950 0.184 0.338 strongstrong EquationEquation (19) (22) 0.959 0.975 0.003 0.009 0.083 0.633 Vibrational–rotational excitation: (v = 0, J = 0) (v = 1, J = 2) → 3.1.2. Strongweak Field Equation (20) 0.997 0.997 0.000 strong Equation (21) 0.998 0.009 0.000 In thestrong strong-field regime Equation we (22)prepared two initial 0.975 guesses for the 0.003 optimal fields: 0.083

𝜀(𝑡) =𝐸𝑠(𝑡) cos(3𝐵𝑡) (18) 3.1.2. Strong Field 𝜀(𝑡) =𝐸𝑠(𝑡)cos(6𝐵𝑡) + cos(12𝐵𝑡) (19) In the strong-ﬁeld regime we prepared two initial guesses for the optimal ﬁelds: with E0 = 7.5 × 10−3 a.u. for both equations. Equations (18) and (19) are used assuming the two-photon absorption and Raman process, respectively.ε(t) = InE 0thess(t)ecos calculations(3Bt) frequency components higher than(18) 1.5 × 10−4 a.u. are filtered out in order to eliminate undesired high frequencies from the control field. ε(t) = E0s(t)[cos(6Bt) + cos(12Bt)] (19)

3 with E0 =1.0E-027.5 10− a.u. for both equations. Equations5.0E-01 (18) and (19) are used assuming the two-photon × (a) (b) absorption and Raman process, respectively. In these calculations1.0E-04 frequency components higher than 1.5 10 4 a.u. are ﬁltered out in order to eliminate undesired high frequencies from the control ﬁeld. × −5.0E-03 5.0E-05

0.0E+00 2.5E-01 | / arb. unit )

ω 0.0E+00 ( ε(t) / a.u.

|ε 0.0000 0.0001 -5.0E-03

-1.0E-02 0.0E+00 0 500000 1000000 1500000 2000000 0.00000 0.00005 0.00010 0.00015 / t / a.u. ω a.u. Figure 3. (a) The optimal field ε(t) obtained with the initial guess, Equation (18), for the rotational excitation, (v = 0, J = 0) → (v = 0, J = 2), in the strong-field regime; (b) the spectrum of ε(t).

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1.0

J = 0 0.8

J = 2 0.6

Population 0.4

Universe 2019, 5, 109 0.2 J = 1 8 of 15 J = 3

0.0 0 500000 1000000 1500000 2000000 As summarized in Table2, the yield for the optimalt / a.u. ﬁeld obtained with the initial guess, Equation

(18), is 0.950, the maximum field amplitude 7.505 10 3 a.u., and the fluence 20.98 a.u. Figure3a shows × − the optimalFigure field as2. Temporal a function changes of t and in Figure state population3b its spectrum.s for the The rotational optimal excitation, field oscillates (v = 0, Jmostly = 0) with → (v = 0, J = 2), in the weak-field regime: (a) vibrational states; (b) rotational states. a single frequency, which is confirmed by the spectrum composed of a frequency component centered around ω = 1.0 10 5 a.u., corresponding to the initially guessed transition energy, 3B. As revealed in Table× 3. Comparison− of the yields with the full (Ĥ0 + Ĥ1 + Ĥ2) and partial Hamiltonians (Ĥ0 the inset in+ FigureĤ1 and Ĥ3b,0 + however,Ĥ2), obtained frequency using the components optimal fields with for the very full smallHamiltonian. amplitudes are seen in the 5 4 region 2.5 10− < ω < 1.5 10− a.u. When filtering out these components from the spectrum, we × × Init. Guess Ĥ0 + Ĥ1 + Ĥ2 Ĥ0 + Ĥ1 Ĥ0 + Ĥ2 found it very difficult to achieve a high yield and they therefore contribute to the control mechanism Rotational excitation: (v = 0, J = 0) → (v = 0, J = 2) significantly. Figure4 shows temporal changes in the populations of (a) vibrational states and (b) weak Equation (15) 1.000 1.000 0.000 rotational states. The populationsstrong Equation of rotational (18) states, 0.950 Figure 4b, 0.184 are highly 0.338oscillatory due to the high intensity of the control fieldstrong and onlyEquation the J (19)= 2 state 0.959 is significantly 0.009 populated 0.633 at the end of the pulse. The vibrational states, FigureVibrational–rotational4a, other than v excitation:= 0 are slightly (v = 0, populatedJ = 0) → (v and= 1, J they= 2) are highly oscillatory also due to the high fieldweak intensity Equation but the v(20)= 0 state 0.997 is totally populated 0.997 at 0.000 the end of the pulse. Thus the intended control isstrong achieved Equation and the (21) control mechanism 0.998 is0.009 attributed 0.000 mainly to a two-photon absorption because thestrong field spectrumEquation is (22) mostly composed 0.975 of 0.003 a single frequency 0.083 component, ω = 3B, corresponding to half the J = 2 0 transition energy, 6B. However, given the importance of other 3.1.2. Strong Field ← frequency components with small amplitudes, inset in Figure3b, and the chaotic behaviors of the rotational stateIn the populations, strong-field regime Figure we4b, prepared the control two mechanisminitial guesses may for bethe betteroptimal characterized fields: by a series of one- and two-photon processes, including𝜀(𝑡) not=𝐸 only𝑠(𝑡) cos two-photon(3𝐵𝑡) absorption, but also the(18) Raman process. As summarized in Table3, the yields obtained for the partial Hamiltonians, Hˆ 0 + Hˆ 1 and ( ) ( ) ( ) ( ) Hˆ 0 + Hˆ 2, with the optimal field for the𝜀 𝑡 full=𝐸 Hamiltonian,𝑠 𝑡 cos 6𝐵𝑡 are+ cos 0.18412𝐵𝑡 and 0.338, respectively, indicating(19) that thewith one-photon E0 = 7.5 × 10 process−3 a.u. for also both contributes equations. Equations to the control (18) and mechanism. (19) are used Theassuming yields the obtained two-photon for the partialabsorption Hamiltonians and Raman suggest process, that respectively. the mechanism In thes ise nearlycalculations the simple frequency addition components of one-photon higher than and two-photon1.5 × 10 processes.−4 a.u. are filtered out in order to eliminate undesired high frequencies from the control field.

1.0E-02 5.0E-01 (a) (b) 1.0E-04 5.0E-03 5.0E-05

0.0E+00 2.5E-01 | / arb. unit )

ω 0.0E+00 ( ε(t) / a.u.

|ε 0.0000 0.0001 -5.0E-03

-1.0E-02 0.0E+00 0 500000 1000000 1500000 2000000 0.00000 0.00005 0.00010 0.00015 / t / a.u. ω a.u. Universe 2019, 5, x FOR PEER REVIEW 8 of 15 FigureFigure 3. (a) The3. (a optimal) The optimal ﬁeld ε field(t) obtained ε(t) obtained with the with initial the guess,initial guess, Equation Equation (18), for (18), the for rotational the excitation,rotational (v = 0, excitation,J = 0) ( v(v= = 0,0, JJ = 0)2), → in the(v = strong-ﬁeld0, J = 2), in the regime; strong-field (b) the regime; spectrum (b) of theε( spectrumt). → of ε(t). 1.0 (a) (b) 0.8 v = 0 J = 0 J = 2 0.6

0.4 Population

v = 1 0.2

0.0 0 500000 1000000 1500000 0 500000 1000000 1500000 2000000 t / a.u. t / a.u.

Figure 4. Temporal changes in state populations for the rotational excitation, (v = 0, J = 0) (v = 0, Figure 4. Temporal changes in state populations for the rotational excitation, (v = 0, J →= 0) J = 2),→ caused (v = 0, by J = the 2), optimal caused ﬁeldby the shown optimal in Figure field shown3, in the in strong-ﬁeld Figure 3, in regime: the strong-field ( a) vibrational regime: states; (b) rotational(a) vibrational states. states; (b) rotational states.

As summarized in Table 2, the yield for the optimal field obtained with the initial guess, Equation (18), is 0.950, the maximum field amplitude 7.505 × 10−3 a.u., and the fluence 20.98 a.u. Figure 3a shows the optimal field as a function of t and Figure 3b its spectrum. The optimal field oscillates mostly with a single frequency, which is confirmed by the spectrum composed of a frequency component centered around ω = 1.0 × 10−5 a.u., corresponding to the initially guessed transition energy, 3B. As revealed in the inset in Figure 3b, however, frequency components with very small amplitudes are seen in the region 2.5 × 10−5 < ω < 1.5 × 10−4 a.u. When filtering out these components from the spectrum, we found it very difficult to achieve a high yield and they therefore contribute to the control mechanism significantly. Figure 4 shows temporal changes in the populations of (a) vibrational states and (b) rotational states. The populations of rotational states, Figure 4b, are highly oscillatory due to the high intensity of the control field and only the J = 2 state is significantly populated at the end of the pulse. The vibrational states, Figure 4a, other than v = 0 are slightly populated and they are highly oscillatory also due to the high field intensity but the v = 0 state is totally populated at the end of the pulse. Thus the intended control is achieved and the control mechanism is attributed mainly to a two-photon absorption because the field spectrum is mostly composed of a single frequency component, ω = 3B, corresponding to half the J = 2←0 transition energy, 6B. However, given the importance of other frequency components with small amplitudes, inset in Figure 3b, and the chaotic behaviors of the rotational state populations, Figure 4b, the control mechanism may be better characterized by a series of one- and two-photon processes, including not only two-photon absorption, but also the Raman process. As summarized in Table 3, the yields obtained for the partial Hamiltonians, Ĥ0 + Ĥ1 and Ĥ0 + Ĥ2, with the optimal field for the full Hamiltonian, are 0.184 and 0.338, respectively, indicating that the one-photon process also contributes to the control mechanism. The yields obtained for the partial Hamiltonians suggest that the mechanism is nearly the simple addition of one-photon and two-photon processes.

UniverseUniverse2019 ,20195, 109, 5, x FOR PEER REVIEW 9 of9 15 of 15

The yield for the optimal field obtained with the initial guess, Equation (19), is 0.959, the maximumThe yield forfield the amplitude optimal field is 1.991 obtained × 10− with2 a.u., the and initial the fluence guess, Equation is 74.78 a.u. (19), (Table is 0.959, 2). the Figure maximum 5a shows 2 fieldthe amplitude optimal isfield 1.991 as a 10function− a.u., of and t and the fluenceFigure 5b is 74.78its spectrum. a.u. (Table The2). spectrum Figure5a showsis mostly the composed optimal of × fieldtwo asa frequency function of componentst and Figure centered5b its spectrum. around Theω = spectrum1.9 × 10−5 isand mostly 3.8 × composed 10−5 a.u., corresponding of two frequency to the 5 5 componentsinitially guessed centered transition around ω energies,= 1.9 10 6B− andand 12B, 3.8 respectively.10− a.u., corresponding The inset in Figure to the initially5b shows guessed frequency × × transitioncomponents energies, with 6B very and small 12B, respectively.amplitudes in The the insetregion in 5.0 Figure x 10−55b < showsω < 1.5 frequencyx 10−4 a.u. We components found it very 5 4 withdifficult very small again amplitudes to achieve ina high the region yield without 5.0 10 −thes< eω components,< 1.5 x 10− a.u.suggesting We found that it they very contribute difficult to × againthe to control achieve mechanism a high yield withoutsignificantly. these components,It was found suggesting that temporal that theychanges contribute in the to populations the control of mechanismvibrational significantly. and rotational It was states, found Figure that temporalS2 in Supplementary changes in the Materials, populations are very of vibrational similar to andthe last rotationalcase with states, the Figure initial S2guess, in Supplementary Equation (18), Materials, Figure 4; arethe verybehaviors similar of to the the rotational-state last case with the populations initial guess,are Equationchaotic and (18), the Figure vibrational4; the behaviors states other of thethan rotational-state v = 0 are slightly populations populated are with chaotic high and oscillation, the vibrationalthus the statesintended other control than beingv = 0 achieved are slightly at the populated end of the with pulse. high The oscillation, control mechanism thus the intended is attributed controlmainly being to achieved a Raman at theprocess end ofbecause the pulse. the The field control spectrum mechanism is mostly is attributed composed mainly of two to a Ramanfrequency processcomponents, because the ω = field 6B and spectrum 12B, the is mostly difference composed of which of two corresponds frequency to components, the J = 2←0ω transition= 6B and 12B,energy, the di6B.ff erenceSimilarly of whichto the correspondsabove case, however, to the J = the2 control0 transition mechanism energy, may 6B. Similarly be characterized to the above by a case,series of ← however,one- and the two-photon control mechanism processes may including be characterized both two-photon by a series absorption of one- and and two-photon the Raman processes process. The includingyields bothobtained two-photon for the partial absorption Hamiltonians, and the Raman Ĥ0 + Ĥ process.1 and Ĥ0 + The Ĥ2, yields with the obtained optimal for field the for partial the full Hamiltonians,Hamiltonian,Hˆ 0 are+ H ˆ0.0091 and andHˆ 0 + 0.633,Hˆ 2, with respectively, the optimal Table field 3, for suggesting the full Hamiltonian, that the two-photon are 0.009 andprocess 0.633,contributes respectively, more Table to the3, suggestingcontrol mechanism that the two-photonthan the other. process contributes more to the control mechanism than the other.

1.0E+00 2.0E-02 (a) (b) 1.0E-04

1.0E-02

5.0E-01 | / arb. unit | / arb. )

0.0E+00 ω ( ε(t) / a.u. ε(t) /

|ε 0.0000 0.0001

-1.0E-02 0.0E+00 0 500000 1000000 1500000 2000000 0.00000 0.00005 0.00010 0.00015 t / a.u. / ω a.u. FigureFigure 5. (a )5. The (a) optimal The optimal ﬁeld ε( tfield) obtained ε(t) obtained with the initialwith the guess, initial Equation guess, (19), Equation for the rotational(19), for the excitation, (v = 0, J = 0) (v = 0, J = 2), in→ the strong-ﬁeld regime; (b) the spectrum of ε(t). rotational excitation,→ (v = 0, J = 0) (v = 0, J = 2), in the strong-field regime; (b) the spectrum of ε(t). 3.2. Vibrational–Rotational Excitation: (v = 0, J = 0) (v = 1, J = 2) → 3.2.1.3.2. Weak Vibrational–rotational Field Excitation: (v = 0, J = 0) → (v = 1, J = 2)

3.2.1.In the Weak weak-ﬁeld Field regime the initial guess for the optimal ﬁeld is

In the weak-field regime the initialε(t) = guessE0s( tfor) cos the(ω optimal01t) field is (20) 𝜀(𝑡) =𝐸𝑠(𝑡) cos(𝜔 𝑡) (20) with E = 1.0 10 6 a.u. and ω being the v = 1 0 transition energy. As summarized in Table2, the 0 × − 01 ← achievedwith E control0 = 1.0 × is 10 nearly−6 a.u. perfect, and ω01 i.e., being the the yield v = is 1 0.997.←0 transition As intended, energy. the As optimal summarized field is veryin Table weak; 2, the the maximumachieved control field amplitude is nearly isperfect, 7.200 i.e.,10 the5 a.u. yield and is the0.997. fluence As intended, is 1.241 the10 optimal3 a.u. Figure field 6isa very shows weak; × − × − the optimalthe maximum field as field a function amplitude of tisand 7.200 Figure × 106−5b a.u. its spectrum.and the fluence The optimalis 1.241 × field 10−3 highlya.u. Figure oscillates, 6a shows causingthe optimal the vibrational field as transition. a function The of spectrumt and Figure is mostly 6b its composedspectrum. ofThe a single optimal frequency field highly component oscillates, centeredcausing around the vibrationalω = 3.0 10 transition.3 a.u.; closer The inspectionspectrum is reveals mostly that composed this exactly of a matchessingle frequency the energy component of the × − (v =centered0, J = 0) around(v = 1, ωJ == 1)3.0 transition, × 10−3 a.u.; 2.962 closer10 inspection3 a.u., which reveals is an that ordinary this exactly one-photon matches rovibrational the energy of → × − transition.the (v = In 0, order J = 0) to → achieve (v = 1, the J = targeted 1) transition, control, 2.962 the additional× 10−3 a.u., rotational which is transitionan ordinaryJ = one-photon2 1 is ← needed;rovibrational the inset intransition. Figure6b In shows orderthe to achieve close-up the of targeted the spectrum control, in the the low additional frequency rotational region, transition where J there= are2←1 some is needed; frequency the componentsinset in Figure corresponding 6b shows the to close-up rotational of transitions,the spectrum the in largest the low of which,frequency region, where there are some frequency components corresponding to rotational transitions, the largest of which, centered around 1.0 × 10−5 a.u., almost matches the J = 2←1 transition energy, 4B.

Universe 2019, 5, 109 10 of 15

Universe 2019, 5, x FOR PEER REVIEW 10 of 15 Universe 2019, 5, x FOR PEER REVIEW 10 of 15 centered around 1.0 10 5 a.u., almost matches the J = 2 1 transition energy, 4B. Figure7 shows Figure 7 shows temporal× − changes in the populations of (a) vibrational← states and (b) rotational states. temporalTheFigure populations changes 7 shows in thetemporalof vibrational populations changes states, ofin (a)theFigure vibrationalpopulations 7a, exhibit statesof modest (a) vibrational and changes; (b) rotational states the vand = states.0 (b)and rotational 1 Thepopulations populations states. of vibrationalchangeThe populations slowly states, from Figureof 1 vibrational and7 0a, to exhibit 0 and states, 1, modestrespectively. Figure 7a, changes; exhibit The rotational themodestv = changes;states,0 and Figure 1 the populations v 7b, = 0 also and show 1 changepopulations modest slowly fromchanges 1change and 0 but toslowly 0 include and from 1, high respectively.1 and oscillations 0 to 0 and The 1, with respectively. rotational small amplitudes states,The rotational Figure and finally 7states,b, also the Figure showJ = 2 7b, population modest also show changes reaches modest but changes but include high oscillations with small amplitudes and finally the J = 2 population reaches include1 at high the end. oscillations with small amplitudes and ﬁnally the J = 2 population reaches 1 at the end. 1 at the end. 1.0E-04 2.0E-05 (a) (b) 1.0E-04 2.0E-05 1.0E-07 (a) (b) 1.0E-07 5.0E-05 1.5E-05 5.0E-05 1.5E-05 5.0E-08 5.0E-08 0.0E+00 1.0E-05 | / arb. unit ) 0.0E+00 1.0E-05 | / arb. unit ω 0.0E+00 ) ( ε(t) / a.u. ω |ε 0.0E+000.00000 0.00005 0.00010 ( -5.0E-05ε(t) / a.u. 5.0E-06

|ε 0.00000 0.00005 0.00010 -5.0E-05 5.0E-06

-1.0E-04 0.0E+00 -1.0E-040 500000 1000000 1500000 2000000 0.0E+000.000 0.005 0.010 0.015 0.020 0 500000t 1000000/ a.u. 1500000 2000000 0.000 0.005ω / 0.010 a.u. 0.015 0.020 t / a.u. / ω a.u. Figure 6. (a) The optimal field ε(t) for the vibrational–rotational excitation, (v = 0, J = 0) → FigureFigure 6. (a) The 6. ( optimala) The optimal ﬁeld ε( tfield) for theε(t) vibrational–rotationalfor the vibrational–rotational excitation, excitation, (v = 0, J = (v0) = 0, (Jv == 0)1, →J = 2), (v = 1, J = 2), in the weak-field regime; (b) the spectrum of ε(t). → in the weak-ﬁeld(v = 1, J = 2), regime; in the (weak-fieldb) the spectrum regime; of (εb(t)). the spectrum of ε(t).

1.0 1.0 (a) (b) v = 0 0.8 (a) (b) v = 0 J = 0 0.8 J = 0 0.6 0.6 0.4 Population 0.4 Population v = 1 0.2 J = 1 v = 1 0.2 J = 1 J = 2 J = 3 J = 2 0.0 J = 3 0.00 500000 1000000 1500000 0 500000 1000000 1500000 2000000 0 500000t 1000000/ a.u. 1500000 0 500000t 1000000/ a.u. 1500000 2000000 t / a.u. t / a.u. Figure 7. Temporal changes in state populations for the vibrational–rotational excitation, (v = 0, J = 0) Figure 7. Temporal changes in state populations for the vibrational–rotational excitation, (v = (v =Figure1, J = 2), 7. Temporal in the weak-field changes regime: in state (populationsa) vibrational for states; the vibrational–rotational (b) rotational states. excitation, (v = → 0, J = 0) → (v = 1, J = 2), in the weak-field regime: (a) vibrational states; (b) rotational states. 0, J = 0) → (v = 1, J = 2), in the weak-field regime: (a) vibrational states; (b) rotational states. To conclude,To conclude, the the present present vibrational–rotational vibrational–rotational transition is is controlled controlled by byonly only one-photon one-photon processes;processes; theTo conclude, reasonsthe reasons arethe are (i)present (i) the the field vibrational–rotationalfield intensity intensity isis veryvery small; transition (ii) (ii) the is the controlledfield field spectrum spectrum by onlyis composed is one-photon composed of of the frequenciestheprocesses; frequencies ofthe the ofreasons (thev = (v0, are= J0, =(i) J 0)= the 0) →field(v (=v intensity=1, 1,J J= = 1)1) is rovibrationalrovibrational very small; (ii)transition transition the field and andspectrum the the J = J2 is←= composed12 rotational1 rotational of the frequencies of the (v = 0, J =→ 0) → (v = 1, J = 1) rovibrational transition and the J = 2←1← rotational transition;transition; (iii) the(iii) yieldthe yield obtained obtained for for the the partial partial Hamiltonian,Hamiltonian, ĤHˆ0 0+ +ĤH1,ˆ 1with, with the the optimal optimal field field was was transition; (iii) the yield obtained for the partial Hamiltonian, Ĥ0 + Ĥ1, with the optimal field was foundfound to be to the be samethe same as that as that for for the the full full Hamiltonian, Hamiltonian, asas summarized in in Table Table 3.3 . found to be the same as that for the full Hamiltonian, as summarized in Table 3. 3.2.2.3.2.2. Strong Strong Field Field 3.2.2. Strong Field In the strong-field regime, we prepared two initial guesses again for the optimal fields: In theIn strong-field the strong-field regime, regime, we preparedwe prepared two two initial initial guesses guesses againagain for the the optimal optimal fields: fields: 𝜀(𝑡) =𝐸𝑠(𝑡) cos(0.5𝜔𝑡) (21) 𝜀(𝑡) =𝐸𝑠(𝑡) cos(0.5𝜔 𝑡) (21) ε(t) = E0s(t) cos(0.5ω01t) (21) 𝜀(𝑡) =𝐸𝑠(𝑡)cos(0.5𝜔𝑡) + cos(1.5𝜔𝑡) (22) ( ) ( ) ( ) ( ) ε(t) =𝜀 𝑡E=𝐸s(t)[𝑠cos𝑡 (cos0.5ω0.5𝜔t)+𝑡 cos+ cos(1.51.5𝜔ω t)]𝑡 (22)(22) with E0 = 1.0 × 10−2 a.u. for both equations.0 Equations01 (21) and (22)01 were used assuming the two- with E0 = 1.0 × 10−2 a.u. for both equations. Equations (21) and (22) were used assuming the two- with photonE = 1.0 absorption10 2 a.u.and Rama for bothn process, equations. respectively. Equations (21) and (22) were used assuming the photon0 absorption× − and Raman process, respectively. two-photon absorption and Raman process, respectively. As summarized in Table2, the yield for the optimal field obtained with the initial guess, Equation (21), is 0.998, the maximum field amplitude is 1.014 10 2 a.u., and the fluence is 37.53 a.u. Figure8a × − shows the optimal field as a function of t and Figure8b its spectrum. The optimal field oscillates at a high frequency corresponding to the vibrational transition. The spectrum is mainly composed of a single frequency component centered around ω = 1.5 10 3 a.u., corresponding to the initially × − Universe 2019, 5, 109 11 of 15

guessed transition energy, 0.5ω01. It thus follows that a two-photon absorption may play a key role in the control mechanism, as intended. The inset in Figure8b shows the close-up of the spectrum; in addition to the main frequency component around ω = 1.5 10 3 a.u., there are components of × − very small frequencies and the overtone of the main component. The small frequency components are associated with rotational transitions including very high rotational states, which is described below in temporal population analysis, and the overtone may contribute to the one-photon vibrational v = 1 0 transition. Figure9 shows temporal changes in the populations of (a) vibrational states and (b) ← rotational states. The vibrational states are excited up to around v = 6 and the populations show very rapid changes with fast and slow oscillations. The v = 0 and 1 populations, being 1 and 0 at t = 0, finally reach 0 and 1, respectively, at the end of the pulse. The rotational state populations show extremely fast and sharp oscillations due to the high intensity of the control field and the states up to nearly J = 20 are populated, but only the J = 2 state is highly populated at the end of the pulse. Thus the targeted control is achieved and the mechanism is likely to be dominated by a two-photon absorption because the field spectrum is composed of a single frequency component, 0.5ω01, corresponding to half the v = 1 0 transition energy. However, many frequencies other than the main component, inset in Figure8b, ← and the chaotic behaviors of the vibrational and rotational state populations, Figure9, suggest that the control mechanism is characterized by multiple one- and two-photon processes including both two-photon absorption and the Raman process. Table3 shows that the yields obtained for the partial Hamiltonians, Hˆ 0 + Hˆ 1 and Hˆ 0 + Hˆ 2, with the optimal field, are 0.009 and 0.000, respectively, indicating that neither the one- and two-photon processes work independently. Rather, they have a kind of cooperative effect of one- and two-photon processes, which is in contrast to the control mechanisms for theUniverse rotational 2019, 5, x excitation, FOR PEER REVIEW (v = 0, J = 0) (v = 0, J = 2), with the strong fields, where11 one-of 15 and → two-photonUniverse processes 2019, 5, x FOR are PEER distinct REVIEW to some extent and the cooperative effect is small, Table3.11 of 15

7.5E-01 (a) (b) 7.5E-01 6.0E-06 1.0E-02 (a) (b) 6.0E-06 1.0E-02 4.0E-06 5.0E-01 4.0E-06 5.0E-01 2.0E-06 0.0E+00

| / arb. unit| 2.0E-06

0.0E+00 ) 0.0E+00 ω | / arb. unit| ε(t) / a.u.ε(t) / ( 2.5E-01) 0.0E+000.000 0.005 0.010 |ε ω ε(t) / a.u.ε(t) / ( 2.5E-01 0.000 0.005 0.010

-1.0E-02 |ε -1.0E-02 0.0E+00 0 500000 1000000 1500000 2000000 0.0E+000.000 0.002 0.004 0.006 0.008 0.010 0 500000t 1000000/ a.u. 1500000 2000000 0.000 0.002 0.004 / 0.006 0.008 0.010 ω a.u. t / a.u. / ω a.u. Figure 8. (a) The optimal field ε(t) obtained with the initial guess, Equation (21), for the Figure 8. a tε vibrational–rotationalFigure( 8.) The(a) The optimal optimal excitation, ﬁeld fieldε( ()v( t= obtained) 0,obtained J = 0) → with with (v = the the1, J =initial 2), in guess, the guess, strong-field Equation Equation regime;(21), (21), for ( for bthe) the vibrational–rotationalvibrational–rotational ε excitation, excitation, (v = 0, (vJ == 0,0) J = 0)( v→= (1,v =J 1,= J2), = 2), in in the the strong-ﬁeld strong-field regime; regime;( (bb)) the the spectrum of (t). → spectrumthe ofspectrumε(t). of ε(t).

1.0 (a) (b) 1.0 J = 2 (a) J = (b)0 0.8 v = 0 J = 2 J = 0 0.8 v = 0 J = 1 0.6 J = 1 0.6 0.4 Population 0.4 Population v = 1 0.2 v = 1 0.2 0.0 0.00 500000 1000000 1500000 0 500000 1000000 1500000 2000000 0 500000t 1000000/ a.u. 1500000 0 500000t 1000000/ a.u. 1500000 2000000 t / a.u. t / a.u. FigureFigure 9. Temporal 9. Temporal changes changes in state in populationsstate populations for the for vibrational–rotational the vibrational–rotational excitation, excitation, (v = 0, (vJ = 0) (v =Figure1, J = 2), 9.→ caused Temporal by the changes optimal in ﬁeldstate shownpopulations in Figure for8 the, in vibrational–rotational the strong-ﬁeld regime: excitation, ( a) vibrational (v → = 0, J = 0) (v = 1, J = 2), caused by the optimal field shown in Figure 8, in the strong-field states;regime: (=b 0,) rotationalJ = (0)a) →vibrational ( states.v = 1, J =states; 2), caused (b) rotational by the optimal states. field shown in Figure 8, in the strong-field regime: (a) vibrational states; (b) rotational states. As summarized in Table 2, the yield for the optimal field obtained with the initial guess, EquationAs (21),summarized is 0.998, thein Tablemaximum 2, the field yield amplitude for the isoptimal 1.014 × field10−2 a.u.,obtained and the with fluence the initialis 37.53 guess, a.u. FigureEquation 8a shows (21), is the 0.998, optimal the maximum field as a field function amplitude of t and is 1.014Figure × 108b− 2its a.u., spectrum. and the fluenceThe optimal is 37.53 field a.u. oscillatesFigure 8a at shows a high the frequency optimal correspondingfield as a function to th ofe vibrationalt and Figure transition. 8b its spectrum. The spectrum The optimal is mainly field composedoscillates ofat aa single high frequencyfrequency correspondingcomponent centered to the around vibrational ω = 1.5 transition. × 10−3 a.u., The corresponding spectrum is tomainly the ω −3 initiallycomposed guessed of a singletransition frequency energy, component 0.5ω01. It thuscentered follows around that a two-= 1.5photon × 10 a.u., absorption corresponding may play to thea ω keyinitially role inguessed the control transition mechanism, energy, as0.5 intended.01. It thus The follows inset thatin Figure a two- 8bphoton shows absorption the close-up may of play the a spectrum;key role inin theaddition control to mechanism, the main frequencyas intended. component The inset aroundin Figure ω 8b= 1.5shows × 10 the−3 a.u., close-up there of are the ω −3 componentsspectrum; inof veryaddition small to frequencies the main andfrequency the over componenttone of the mainaround component. = 1.5 The× 10 small a.u., frequency there are componentscomponents are of associatedvery small withfrequencies rotational and transition the overtones including of the main very component. high rotational The states, small frequencywhich is describedcomponents below are in associated temporal populawith rotationaltion analysis, transition and thes including overtone very may high contribute rotational to the states, one-photon which is vibrationaldescribed vbelow = 1← in0 transition. temporal populaFigure tion9 shows analysis, temporal and the chan overtoneges in the may populations contribute ofto (a)the vibrational one-photon statesvibrational and (b) v =rotational 1←0 transition. states. Figure The vibrational 9 shows temporal states arechan excitedges in the up populations to around ofv (a)= 6 vibrational and the populationsstates and show(b) rotational very rapid states. changes The withvibrational fast and states slow oscillations.are excited Theup tov =around 0 and 1v populations, = 6 and the beingpopulations 1 and 0 showat t = 0,very finally rapid reach changes 0 and with 1, respectively, fast and slow at theoscillations. end of the The pulse. v = The0 and rotational 1 populations, state populationsbeing 1 and show 0 at textremely = 0, finally fast reach and 0 sharpand 1, oscillations respectively, due at tothe the end high of theintensity pulse. ofThe the rotational control field state andpopulations the states showup to nearlyextremely J = 20 fast are an populated,d sharp oscillations but only the due J =to 2 thestate high is highly intensity populated of the control at the end field ofand the thepulse. states Thus up theto nearly targeted J = control20 are populated, is achieved but and only the themechanism J = 2 state is is likely highly to populated be dominated at the by end a two-photonof the pulse. absorption Thus the targetedbecause controlthe field is spectrumachieved andis composed the mechanism of a single is likely frequency to be dominated component, by a 0.5two-photonω01, corresponding absorption to halfbecause the v the = 1 ←field0 transition spectrum energy. is composed However, of a many single frequencies frequency othercomponent, than ω the0.5 main01, corresponding component, inset to half in Figurethe v = 8b, 1← and0 transition the chaotic energy. behaviors However, of the many vibrational frequencies and rotational other than the main component, inset in Figure 8b, and the chaotic behaviors of the vibrational and rotational

Universe 2019, 5, x FOR PEER REVIEW 12 of 15

state populations, Figure 9, suggest that the control mechanism is characterized by multiple one- and two-photon processes including both two-photon absorption and the Raman process. Table 3 shows that the yields obtained for the partial Hamiltonians, Ĥ0 + Ĥ1 and Ĥ0 + Ĥ2, with the optimal field, are 0.009 and 0.000, respectively, indicating that neither the one- and two-photon processes work independently. Rather, they have a kind of cooperative effect of one- and two-photon processes, Universewhich2019 is ,in5, contrast 109 to the control mechanisms for the rotational excitation, (v = 0, J = 0) → (v12 =of 0, 15 J = 2), with the strong fields, where one- and two-photon processes are distinct to some extent and the cooperative effect is small, Table 3. The optimal field obtained using the initial guess, Equation (22), is found to give the final yield, The optimal field obtained using the initial guess, Equation (22), is found to give the final yield, 0.975, with its maximum field amplitude being 5.016 10 2 a.u. and its fluence being 469.06 a.u., 0.975, with its maximum field amplitude being 5.016 ×× 10−2 −a.u. and its fluence being 469.06 a.u., Table Table2. Figure 10a shows the optimal field as a function of t and Figure 10b its spectrum. The 2. Figure 10a shows the optimal field as a function of t and Figure 10b its spectrum. The optimal field optimal field oscillates at a high frequency corresponding to the vibrational transition. The spectrum oscillates at a high frequency corresponding to the vibrational transition. The spectrum is mainly is mainly composed of two frequency components centered around ω = 1.5 10 3 and 3.0 10 3 composed of two frequency components centered around ω = 1.5 × 10×−3 and− 3.0 × 10×−3 a.u.,− a.u., corresponding to the initially guessed transition energy. Hence a Raman process may play a corresponding to the initially guessed transition energy. Hence a Raman process may play a key role key role in the control mechanism, as intended. The close-up of the spectrum is shown in the inset in the control mechanism, as intended. The close-up of the spectrum is shown in the inset in Figure in Figure 10b; other than the two main frequency components, there are components with small 10b; other than the two main frequency components, there are components with small amplitudes amplitudes corresponding to rotational transitions and the v = 1 0 vibrational transition. Temporal corresponding to rotational transitions and the v = 1←0 vibrational← transition. Temporal changes in changes in the populations of vibrational and rotational states, Figure S3 in Supplementary Materials, the populations of vibrational and rotational states, Figure S3 in Supplementary materials, are found are found to show similar behavior to those obtained with the initial guess, Equation (21), Figure9; to show similar behavior to those obtained with the initial guess, Equation (21), Figure 9; the the vibrational state populations oscillate quite rapidly and so do the rotational state populations, vibrational state populations oscillate quite rapidly and so do the rotational state populations, the v the v = 1 and J =2 states being highly populated at the end of the pulse. Thus the mechanism is = 1 and J =2 states being highly populated at the end of the pulse. Thus the mechanism is likely due likely due to a Raman process because the field spectrum is composed of two frequency components, to a Raman process because the field spectrum is composed of two frequency components, 0.5ω01 and 0.5ω01 and 1.5ω01, corresponding to absorption and emission via a virtual state located at 1.5ω01. As 1.5ω01, corresponding to absorption and emission via a virtual state located at 1.5ω01. As with the with the above case, however, the control mechanism may be better characterized by multiple one- above case, however, the control mechanism may be better characterized by multiple one- and two- and two-photon processes rather than a simple Raman process. The yields obtained for the partial photon processes rather than a simple Raman process. The yields obtained for the partial Hamiltonians, Hˆ 0 + Hˆ 1 and Hˆ 0 + Hˆ 2, with the optimal field, are 0.003 and 0.083, respectively, as given Hamiltonians, Ĥ0 + Ĥ1 and Ĥ0 + Ĥ2, with the optimal field, are 0.003 and 0.083, respectively, as given in Table3, showing that the cooperative e ffect is very large like the last result. in Table 3, showing that the cooperative effect is very large like the last result.

6.0E+00 (a) (b) 5.0E-05 5.0E-02

4.0E+00 | / arb. unit

0.0E+00 ) ω ( |ε ε(t) / a.u. ε(t) / 2.0E+00 0.000 0.005 0.010

-5.0E-02

0.0E+00 0 500000 1000000 1500000 2000000 0.000 0.005 0.010 0.015 t / a.u. / ω a.u. FigureFigure 10. 10. ((aa)) TheThe optimaloptimal fieldfieldε ε(t()t) obtainedobtained with with the the initial initial guess, guess, Equation Equation (22), (22), for for the the vibrational–rotational excitation, (v = 0, J = 0) (v = 1, J = 2), in the strong-field regime; (b) the vibrational–rotational excitation, (v = 0, J =→ 0) → (v = 1, J = 2), in the strong-field regime; (b) spectrumthe spectrum of ε(t). of ε(t). 4. Conclusions 4. Conclusions In this work we investigated the roles of one-photon and two-photon processes in the laser-controlledIn this work rovibrational we investigated transitions the roles of theof on diatomice-photon alkali and two-photon halide, 7Li37 processesCl. We carriedin the laser- out 7 37 OCTcontrolled calculations rovibrational to obtain thetransitions electric fieldsof the that diatomic best yield alkali the transition halide, probabilityLi Cl. We tocarried the target. out TheOCT Hamiltoniancalculations includingto obtain boththe electric the one-photon fields that and best two-photon yield the transition field-molecule probability interaction to the terms target. was The usedHamiltonian and the 2D including time-dependent both the wave one-photon packet propagationand two-photon was performedfield-molecule with interaction both the radial terms and was angularused and motions the 2D being time-dependent treated quantum wave mechanically. packet propagation The targeted was performed processes were with pure both rotational the radial and and vibrational–rotationalangular motions being excitations: treated quantum (v = 0, J = mechanically0) (v = 0, J. =The2); targeted (v = 0, J = processes0) (v = were1, J = pure2). Total rotational time → → → → ofand the vibrational–rotational control pulse was set to excitations: 2,000,000 a.u. (v =(48.4 0, J = ps), 0) which (v = is0, slightlyJ = 2); (v larger = 0, J than= 0) twice (v = the 1, J rotational = 2). Total periodtime of the 7controlLi37Cl moleculepulse was (23 set ps). to In2,000,000 each control a.u. (48.4 excitation ps), which process is we slightly obtained larger weak than and twice strong the optimal fields by means of giving weak and strong field amplitudes, respectively, to the initial guess for the optimal field. In the weak-field regime it was found that the control mechanism is dominated exclusively by a one-photon process, as expected, in both the rotational and vibrational–rotational transitions. In the strong-field regime we obtained two kinds of optimal fields: one that mainly causes two-photon absorption and the other a Raman process. It was found, however, that the mechanisms are not simply Universe 2019, 5, 109 13 of 15 characterized by one process but rather by multiple one- and two-photon processes including both two-photon absorption and the Raman process. In the rotational excitation, (v = 0, J = 0) (v = → 0, J = 2), it was found that the roles of one- and two-photon processes are relatively distinct; in the vibrational–rotational excitation, (v = 0, J = 0) (v = 1, J = 2), however, the roles of the two processes → are ambiguous and the cooperative effect is quite large. We thus obtained the various optimal fields to efficiently achieve the target controlled by different mechanisms depending on the field strength and the type of quantum transition.

Supplementary Materials: Supplementary data to this article can be found online at http://www.mdpi.com/2218- 1997/5/5/109/s1. Author Contributions: Both the authors conceived of the simulations; Y.K. designed and performed the OCT calculations and K.Y. reviewed the data; Y.K. prepared the original paper draft and K.Y. reviewed and edited. Funding: This work was supported by JSPS KAKENHI Grant No. 15H02345. Acknowledgments: The authors thank Yukiyoshi Ohtsuki at Tohoku University for helpful suggestions. The computations were performed using the supercomputers at Center for Computational Science & e-Systems, JAEA, Tokai, Ibaraki, Japan and also at Research Center for Computational Science, Okazaki, Aichi, Japan. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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