#### Abstract

We present a quantum control scheme which realizes suppression of the intramolecular vibrational energy redistribution (IVR). In this scheme, we utilize effective decomposition brought by intense CW-laser fields, which enables to exclude
the doorway state coupled to background manifolds. In doing so, we introduce a helper state and make it optically coupled with the doorway state through the intense CW-laser field. We have applied the present scheme to both the Bixon-Jortner model and the
SCCl_{2} model system.

#### 1. Introduction

In coherent laser control of molecular vibrations, the key feature is manipulation of vibrational coherence. Thus, dissipative processes such as dephasing and population decay have been considered as formidable obstacles. One of the typical dissipative processes is intramolecular vibrational energy redistribution (IVR), which is characterized as irreversible population flow from the doorway state to the background manifolds [1–5]. There have been several attempts to suppress the IVR by laser irradiation. For example, the suppression of IVR is theoretically confirmed under a strong resonant CW-laser excitation, in which total population is trapped on the initial and doorway states, and it rapidly oscillates between those two states [6]. It is also reported that the IVR can be restricted by the laser field, in which the frequency is swept [7, 8]. This phenomena is explained by the concept of adiabatic passages and the population is locked on the doorway state during the chirp pulse. In those approaches, only the suppression of IVR is considered, and there is no concept of controlling the population dynamics which escape from the IVR. Thus, it is necessary to apply additional control fields if one aims to drive the system to a specific desired quantum state while avoiding the IVR. However, additional laser would interfere with those existing “population locking” or “IVR-suppression” fields.

Another possible approach towards the suppression of IVR is to design the control field which actively manipulate phase relations between the eigenstates contained in the current wave packet. This could be done by optimal control theory (OCT) or local control theory (LCT) [9–13]. However, optimized laser fields tend to show complicated and highly sensitive nature since they try to control numerous eigenstates with various eigenenergies, which makes difficult to prepare the control laser in practice.

The main purpose of this study is to propose an alternative control scheme applicable to dissipative systems with which one can overcome the difficulties stated above. The basic idea is very simple. First, we effectively decompose the total system into target space and its complimentary space as we have shown in our former studies [14, 15]. By doing this, we can exclude the doorway state which is responsible for the dissipative dynamics, or IVR, from the system of interest, while remaining indirect optical transitions for control process. Furthermore, choosing a small space consisting of a few states as a target space, one can apply the well-established control schemes such as -pulse [16] or stimulated Raman adiabatic passage (STIRAP) [17], which greatly simplifies the control process. The mechanism of exclusion of the doorway state is well investigated for the 3-level system, in which two lasers are applied with static detuning [18]. The decomposition mechanism we adopt in the present work is basically the same as the one in [18] except the fact that the detuning itself is introduced as the induced Stark shift. We discuss the difference of those two approaches and clarify the advantageous features of the present scheme from a viewpoint of applicability to quantum control.

As a model system, we employ the classic IVR model system, Bixon-Jortner(BJ) model [19], to confirm the effectiveness of the present method and show how the IVR process can be excluded from the molecular system dynamics. Next, as an example of more realistic system, we choose SCCl_{2} molecule, whose IVR is well studied, and there exists reliable model Hamiltonian [20, 21].

#### 2. Theoretical

In the present work, we consider a model system as shown in Figure 1. Before proceeding to the formulation, we explain several terms used. Here, is the initial state, which is typically taken to be the lowest vibrational state on the ground electronic state. The state is the intermediate state, which optically connects the two states and . The state is the control target state, that is, we aim to transfer the initial population to avoiding the population loss due to IVR process. Note that is optically accessible from , , and and also coupled with through the intramolecular interactions . The IVR, that is, the irreversible population flow to , occurs through , thus, we call this state “doorway state”, hereafter. In addition to that, we introduce the special state which we call “helper state” in order to suppress the IVR process. This state is coupled through the intense CW-laser resonant to the transition between and with interaction strength .

Next, we formally split the total system into three spaces as the follows. Control target space, which we call A space, consists of two states, initial and final state . We define B space as the one spanned by a doorway state and a helper state , which we utilize for realizing effective decomposition. C-space is defined as the space spanned by IVR background manifolds .

The total Hamiltonian matrix under the rotating wave approximation (RWA) is given as Here, we introduce the parameter so as to clearly specify the perturbation order. The optical interactions between and (), and and () are taken to be much smaller than . Minor matrices in (1) are defined as where denotes the detuning introduced for AB transition (see Figure 1), while () denotes the eigenenergies of the background manifolds in C-space. Note that we prediagonalize the B space Hamiltonian so as to make diagonal matrix. The matrices () and () correspond to the optical and static interactions, respectively.

The Schrödinger equation in the matrix representation is given as where is a diagonal matrix with eigenvalues () in its diagonal elements, and is the unitary matrix consisting of corresponding eigenvectors () as . Here, we expand and as where Here, , , and are diagonal matrices, whose diagonal elements are th perturbation energies. Inserting (1), (4) into (3) and comparing () terms gives the followings: From (6), one obtains the following equations for minor matrices: while off-diagonal blocks of are all zero matrices. As commutes with , (9) gives , which leaves undetermined. On the other hand, from (10), (11) one obtains that and are both determined as identity matrices of and an , respectively, whereas and . The first two terms of the left hand side of (7) are given as whereas other terms are Comparing the component of A space, one obtains the equation , which leads to since is a unitary matrix. Thus, there is no first-order correction onto the A space eigenenergies.

Next, comparing the BA-block of both sides of (7) leads to the equation Considering , (14) can be rewritten as which leads to Here, denotes the identity matrix of B space multiplied by . The prefactor matrix in (16) is given in a simple form as Note that becomes zero matrix as becomes large or , which leads to the suppression of AB-space mixing.

Comparing CA-block of (7), one obtains which is rewritten as where is identity matrix multiplied by . It is readily seen that since . Thus, there is no first-order CA-mixing.

Next, we consider the second-order terms. With , (8) becomes Since we focus on the dynamics of A space, we extract AA-block of (20) as Considering that the first two terms of (21) are canceled out and inserting (16), one obtains Note that (22) is an eigenvalue equation, which determines the second-order energy correction onto A space as well as corresponding eigenvectors . Here, we define the A space effective Hamiltonian as which determines the dynamics of A space within the second-order. Note that there is no interaction terms relevant to the C-space. This implies that as long as is relatively small enough and the second-order perturbation theory is applicable, the interactions between the doorway state and the background manifolds can be excluded from the system dynamics effectively.

Note that the perturbation expansion shown above is applied for the weak field , which is responsible for interspace optical transitions. On the other hand, the strong interaction through CW-laser, which cannot be dealt with perturbation method, is precisely taken into account by prediagonalization of the B space.

Finally, we discuss possible side effect brought by optically allowed state adjacent to the intermediate state . Typically, the level spacing between and is about 1000 cm. The CW-laser () applied for decomposition turns out to be off-resonant by , which is large enough to exclude the undesired transition to .

#### 3. Results

To confirm the applicability of the present theory, we first apply it to well-known Bixon-Jortner model system [19]. In this model, all the background levels are equally spaced and coupled to a single bright doorway state with uniform strength . Shown in Figure 2(a) is the population dynamics under field-free condition while the initial state is taken to be . The coupling parameter is taken to be where as denotes the unit frequency, which determines the timescale () of the dynamics. It is shown that the irreversible population flow to the background manifolds occurs due to the dephasing process between numerous eigenstates contained in the initial state .

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Next, we try a naive population control in which control target is the population transfer from to . We introduce two resonant CW-lasers corresponding to the transitions, , and , respectively. The total Hamiltonian corresponding to (1) is given as where the vertical and horizontal dots are all zeroes. Shown in Figure 2(b) is the population dynamics with the initial condition . Both optical interaction parameters and are taken to be . As shown in Figure 2(b), significant amount of total population leaks to the background manifolds, and its nature is basically irreversible though coherent oscillations are observed in early-time stage. Note that such oscillatory nature of the population dynamics is rapidly damped, and the system finally settles into stationary state in which the population of is only 25%. Thus, the control objective is hardly achieved with this approach.

Now, we introduce a helper state in order to suppress the population loss due to IVR. The total Hamiltonian represented by the original basis set is given as where is taken to be the optical interaction between and in this case. We reconstruct the basis set for B space so as to express the Hamiltonian matrix in the form of (2), that is, where , , and . Shown in Figure 2(c) is the population dynamics under the condition . It is shown that the population oscillates between and , and there is very little population leak into the background manifolds . This feature denotes that the control target space, that is, A space, is effectively isolated as a 2-level system without dissipative dynamics. The effective Hamiltonian for the A space can be obtained from (23) as It is clearly shown that the Rabi oscillation is driven by the second term which originates from the second-order interspace optical interaction between A and B spaces. Note also that this term is proportional to the detuning parameter . This implies that common detuning is required for driving population dynamics in the A space.

As for a more realistic molecular system, we consider Thiophosgen (SCCl_{2}), which has six vibrational modes, and there exists IVR process among highly excited states. It is known that only a limited set of vibrational resonances are related to the dissipative dynamics, and an effective Hamiltonian consisting of six modes for describing the IVR process is proposed as [20]
where
Here . We used the same parameters listed in [20]. As for the basis set, we take the direct product of the harmonic oscillator eigenfunctions with respect to each mode, which is denoted as . With RWA, the structure of the Hamiltonian matrix is essentially identical to (25). We consider the mode 1 (corresponding to CS-stretching motion) is optically allowed, and the bright doorway state is defined as as shown in Figure 3. The initial state is taken to be the ground state , and we aim to transfer the population onto the target final state , which is one of the highly excited zero-order states of the CS-stretching mode with relatively long lifetime. The laser parameters are taken to be cm^{−1} and cm^{−1}. The corresponding laser amplitudes for and are estimated as W/cm^{2} and W/cm^{2}, respectively [20].

Shown in Figure 4 are the population dynamics changing the CW-laser intensity from 0 cm^{−1} to 300 cm^{−1}. As shown in Figure 4(a), the irreversible population flow to the background modes 2–6, that is, IVR, occurs as we see in the Bixon-Jortner model case. As the laser power increases, the dissipative dynamics is gradually suppressed (see Figures 4(b), 4(c), and 4(d)) due to the decomposition effect, which isolates A space ( and ). Finally, as shown in Figure 4(e), a neat Rabi oscillation appears in the population dynamics of A space, which implies that the target subspace can be effectively treated as a simple 2-level system. Then, one can apply conventional -pulse scheme for population transfer onto avoiding the population loss due to the IVR. Note also that the timescale of the Rabi oscillation in Figure 4(e) can be controlled by changing the detuning parameter (see (27)).

(a) cm |

(b) |

(c) |

(d) |

(e) |

Here, we should clearly state the difference in the decomposition conditions between the present method and the static detuning approach in [18]. In the present scheme, the detuning itself is introduced as the induced Stark shift, or a large energy splitting between the zero-order eigenvalues in B space, that is, in (2). Thus, the required condition for effective decomposition is different accordingly, that is, in the static detuning case whereas , , in the present method.

One of the important differences between those two schemes is the suppression efficiency with respect to the detuning value, or . To see this, we consider a simple 4-level system which consists of only A space ( and ) and B space ( and ). Instead of omitting C-space (background states), we introduce a phenomenological dephasing parameter associated with the intermediate state , that is, we introduce an imaginary component for the eigenvalue of . Here, we take cm^{−1} so as to reproduce the IVR timescale of SCCl_{2}. Shown in Figure 5(a) is the population dynamics with cm^{−1} and cm^{−1}, which corresponds to the condition of Figure 4(a). One can see that the dissipative IVR dynamics is qualitatively reproduced with the phenomenological parameter . The shown Figure 5(b) is the population dynamics with cm^{−1} corresponding to Figure 4(e), and the suppression effect works fine as expected. On the other hand, shown in Figure 5(c) is the result for static detuning case without using the helper state , that is, cm^{−1} and cm^{−1}. As shown in the result, the suppression effect is insufficient compared to Figure 5(b), although the detuning is taken to be 300 cm^{−1}, in both cases. In order to achieve the suppression level of Figure 5(b), one needs to increase the detuning up to cm^{−1} as shown in Figure 5(d). Thus, the static detuning approach requires ten times larger than that in the present case. In the practical application to SCCl_{2}, it is difficult to take required detuning cm^{−1} because adjacent level exists within 1000 cm^{−1}.

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Further investigation on the effective Hamiltonian clarifies the difference in suppression efficiency shown above. The effective Hamiltonian for the CW-laser-induced Stark shift detuning case is given as [15]
whereas that of the static detuning case is [18]
One can see that or is required to neglect for static detuning case. On the other hand, the denominator of (30) shows that is required. Note that the condition can be satisfied with smaller (Stark shift) detuning because of its dependency. This is why ten times larger value ( cm^{−1}) is needed in the static detuning approach in Figure 5(d). Note also that choosing small helps to satisfy the exclusion of dissipation in the present method. Another difference is lying in the laser parameter dependence of the Rabi frequency driven by the effective Hamiltonian. In the static detuning approach, the control parameter affects both decomposition condition and the time scale of the isolated system. On the other hand, in the present method, the decomposition condition is mainly determined by , while the timescale of Rabi oscillation can be adjusted by changing as far as stands. From a viewpoint of quantum control, the present approach may offer versatile control scheme because it has more handling parameters compared to the static detuning approach.

Apparent drawback of the present method is possible side effects induced by the intense CW-lasers, such as multi-photon transitions which eventually leads to ionization, or unexpected near-resonant transition. We, here, consider the limitation of the present method when it is applied to the realistic molecular system as shown above. Note that some results shown above are rather extreme cases. In practice,the cm^{−1} condition in Figure 4(e) corresponds to CW-laser intensity of *~*10^{11} W/cm^{2}, which may cause undesired resonant multiphoton excitation. However, milder condition, such as Figure 4(d) corresponding to ~10^{10} W/cm^{2} still works fairly well. Note also that the intensities of weak lasers for and can be taken lower than the decomposition field by one order, which denotes that there are very few side processes caused by interactions. Since resonant optical processes mainly occur under this condition, it could be possible to suppress undesired transitions by choosing helper level and laser frequency avoiding unexpected (near)resonances with care. Taking as large value as possible also makes it possible to shorten the irradiation time of CW Laser, since we can expect faster Rabi-oscillation dynamics for control.

It is known that general IVR time scale varies from subpico to several hundreds ps [22]. For the IVR system with 10^{0} cm^{−1}*~*10^{−1} cm^{−1}, applicability of the present method becomes more plausible, because required CW-laser intensity turns out to be around 10^{8} W/cm^{2}W/cm^{2}.

Finally, we should mention applicability of RWA for the present study. As the laser power increases, validity of RWA becomes doubtful. In order to verify this, we have carried out numerical calculation in which we treat the laser fields semiclassically. Overall behaviors of population dynamics agree with the ones obtained with RWA. The only difference is additional high-frequency components with small amplitudes onto the Rabi oscillation in the semiclassical treatments. Thus, we have concluded that the RWA are appropriate approximation for the present work.

#### 4. Summary

We have presented that the dissipative dynamics due to the dephasing, such as IVR, can be suppressed by utilizing the effective decomposition brought by the intense CW lasers. One of the unique features of the present scheme is that one can treat it within a time-independent picture with the help of RWA, which makes the problem easy to handle. Consequently, one can avoid the situation that the control laser fields becomes too complicated. Another intriguing feature is that one can control the time scale of the population dynamics by changing the detuning parameters and it can be taken even longer than that of IVR itself. This is quite advantageous feature from a viewpoint of control problem, because one can design the control field regardless of the time constant of the dissipation.

#### Acknowledgments

This paper was supported, in part, by the Japan Society for the Promotion of Science (JSPS), Grant-in-Aid for Scientific Research (C), KAKENHI (20550021).