State initial, field optimized

The selective flux maximization from the FOIST scheme shown in Fig. 2 is achieved by altering the spatial profile of the initial state to be subjected to the photolysis pulse and since changes in flux are due to the flow of probability density, it is useful to examine the attributes of the probability density profiles from the field optimized initial states. 

The experimental realization of the optimal initial states is however a completely uncharted area at this time. In an earlier paper (17), we have presented the formulae to obtain field parameters required to achieve these field optimized initial states and the optimal control (30) approach may also be easily and profitably employed to attain this FOIST comprising of only three... 

The experimental realization of the optimal initial states is, however, a completely uncharted area. In an earlier paper,we have presented the formulae to obtain field parameters required to achieve these FOISTs, and the optimal control approach may also feasibly and profitably be employed to attain this FOIST, which comprises only three vibrational levels. We however believe that, while the theoretical tools are useful, the central results from our investigation - - are that, instead of putting the entire onus of selective control on a theoretically designed laser pulse that may not be easy to realize in practice, the approach where different vibrational population mixes are experimentally obtained and subjected to readily attainable photolysis pulses, leading to an empirical experimental correlation between selectivity attained for diverse photolysis pulses and initial vibrational population mix used, represents a more promising and desirable alternative. Our results, we hope, will spur experimental tests, and a concerted partnership between field and initial state shaping is required to better realize the chemical dream - of using lasers as molecular scissors and tweezers to control chemical reactions. 

Easy availability of ultrafast high intensity lasers has fuelled the dream of their use as molecular scissors to cleave selected bonds (1-3). Theoretical approaches to laser assisted control of chemical reactions have kept pace and demonstrated remarkable success (4,5) with experimental results (6-9) buttressing the theoretical claims. The different tablished theoretical approaches to control have been reviewed recently (10). While the focus of these theoretical approaches has been on field design, the photodissociation yield has also been found to be extremely sensitive to the initial vibrational state from which photolysis is induced and results for (11), HI (12,13), HCl (14) and HOD (2,3,15,16) reveal a crucial role for the initial state of the system in product selectivity and enhancement. This critical dependence on initial vibrational state indicates that a suitably optimized linear superposition of the field free vibrational states may be another route to selective control of photodissociation. 

We consider, first, whether it is in principle possible to control the quantum dynamics of a many-body system. The goal of such a study is the establishment of an existence theorem, for which purpose it is necessary to distinguish between complete controllability and optimal control of a system. A system is completely controllable if an arbitrary initial state can be transformed, without loss to other states, into an arbitrary final state at some time T. A system is strongly completely controllable if an arbitrary initial state can be transformed, without loss to other states, into an arbitrary final state at a specified time T. Optimal control theory designs a field, subject to specified constraints, that guides the evolution of an initial state of the system to be as close as possible to the desired final state at time T. 

These control schemes are very effective for a certain class of processes but are not versatile and ineffective for, for example, multilevel-multilevel transitions we shall consider in this chapter. There exist several mathematical studies that investigate controllability of general quantum mechanical systems [11,12]. The theorem of controllability says that quantum mechanical systems with a discrete spectrum under certain conditions have complete controllability in the sense that an initial state can be guided to a chosen target state after some time. Although the theorem guarantees the existence of optimal fields, it does not tell us how to construct such a field for a given problem. 

This chapter is organized as follows. In Section II, we show how quantum chaos systems can be controlled under the optimal fields obtained by OCT. The examples are a random matrix system and a quantum kicked rotor. (The former is considered as a strong-chaos-limit case, and the latter is considered as mixed regular-chaotic cases.) In Section III, a coarse-grained Rabi state is introduced to analyze the controlled dynamics in quantum chaos systems. We numerically obtain a smooth transition between time-dependent states, which justifies the use of such a picture. In Section IV, we derive an analytic expression for the optimal field under the assumption of the CG Rabi state, and we numerically show that the field can really steer an initial state to a target state in random matrix systems. Finally, we summarize the chapter and discuss further aspects of controlling quantum chaos. 

We study optimal control problems of quantum chaos systems. Our goal of control is to obtain an optimal field s(t) that guides a quantum chaos system from an initial state target state (pj) at some specific time t = T. One such method is optimal control theory (OCT), which has been successfully applied to atomic and molecular systems [4]. 

Once we fix the initial state ip,) and the final state ipj), the optimal field E(t) is obtained by some numerical procedures for appropriate values of the target time T and the penalty factor a. Though there should be many situations corresponding to the choice of ip,) and ipj), we only consider the case where they are Gaussian random vectors. It is defined by... 

We show two numerical examples for a 64 x 64 random matrix Hamiltonian One is the relatively short-time case with T = 20 and a = 1 shown in Fig. 1, and the other is the case with T = 200 and ot = 10 shown in Fig. 2. In both cases, we obtain the optimal field s(f) after 100 iterations using the Zhu-Botina-Rabitz (ZBR) scheme [13] with s(f) = 0 as an initial guess of the field. The initial and the target state is chosen as Gaussian random vectors as mentioned above. The final overlaps are Jo = 0.971 and 0.982, respectively. 

The optimization of a laser field e t) driving the system from an initial state j(O) at time t = 0 to the selected target area (defined by the projection operator X) at time t = T can be reduced to maximizing the functional L given... 

Further, the Rayleigh-Ritz variational maximization of flux by generating an optimal spatial profile for the initial wave function offers a new and flexible alternative for laser-assisted selective control of chemical reactions. It is our hope that the FOIST-based approach presented here will attract requisite experimentation, and a concerted partnership between the field and the initial state shaping advocated here will assist in keeping the dream of controlling chemical reactions by modifying the... 

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