Selective initial state

Demirplak and Rice developed the counter-diabatic control protocol while studying control methods that efficiently transfer population between a selected initial state and a selected target state of an isolated molecule [11-13]. The protocol has been studied for manipulation of atomic and molecular states [11, 12, 19] and spin chain systems [20, 21]. Experiments with the counter-diabatic protocol have been demonstrated for the control of BECs [22] and the electron spin of a single nitrogen-vacancy center in diamond [23]. The counter-diabatic field (CDF) protocol is identical with the transitionless driving protocol, independently proposed by Berry a few years later [24]. A discussion of the relationship between these approaches and several of the other proposed shortcuts to adiabaticity can be found in the review by Torrontegui and coworkers [10]. 

The idea exploited by Demirplak and Rice is very simple Given a fleld that would generate adiabatic population transfer from a selected initial state to a selected final state if strong enough, but which does not generate complete population transfer because of lack of intensity, find another fleld, called the CDF, that when combined with the first field generates the state that would have been generated by a strictly adiabatic transformation. Demirplak and Rice show that the CDF exists, and they show how it can be calculated. 

It should be noted that the inclusion of temperature effects raises additional issues bound to the choice of the starting vibrational states in the initial electronic state. An evident way to limit the treatment is to use a threshold on the Boltzmann population of each vibrational state. In practice, this threshold is set with respect to the population of the ground state. Similar to the final states, a division in classes is performed among all selected initial states. For each class, a set is defined by the initial states sharing the same simultaneously excited modes, so that they differ only... 

When spectroscopists speak of electronic selection niles, they generally mean consideration of the integral over only the electronic coordinates for wavefiinctions calculated at the equilibrium nuclear configuration of the initial state, 2 = 0,... 

Figure B3.4.6. Reaction probabilities for the initial-state-selected process H2(v = 0,J = 0)+OH(v,y = 0) — H2O+H, for zero total angular momentum. Taken from [75] with pennission.

The methodology presented so far allows the calculations of state-to-state. S -matrix elements. However, often one is not interested in this high-level of detail but prefers instead to find more average infomiation, such as the initial-state selected reaction probability, i.e. the probability of rearrangement given an initial state Uq. In general, this probability is... 

At times, however, even the infomiation presented by "nis too detailed. If one wants to rigorously calculate the themial rate of rearrangement reactions, the initial vibrational state is not important. The relevant quantity is the sum of tire initial-state-selected probabilities... 

Under the usual conditions their ratio is kinetically controlled. Alder and Stein already discerned that there usually exists a preference for formation of the endo isomer (formulated as a tendency of maximum accumulation of unsaturation, the Alder-Stein rule). Indeed, there are only very few examples of Diels-Alder reactions where the exo isomer is the major product. The interactions underlying this behaviour have been subject of intensive research. Since the reactions leadirig to endo and exo product share the same initial state, the differences between the respective transition-state energies fully account for the observed selectivity. These differences are typically in the range of 10-15 kJ per mole. ... 

Here the sum runs over all possible initial states and the operator describes the interaction of the electrons and the radiation field with wave vector q and polarization A. In Eq. (1) it has been assumed that the detector selectively counts photo electrons with energy E, wave vector k, and spin polarization The corresponding final... 

Figures 3.58 and 3.59 show some typical space-time patterns of two-dimensional rules starting from random initial states. The figures show snapshots of runs taken at times i = 1,5,10,25,50 and a cross-sectional view of the y-axis sites taken along an arbitrarily selected x site value for all times t = 1 through t = 50.

Recognizable regularity in probability patterns, starting from randomly selected initial site-states, generally appears only when the measurement threshold is significantly less than certainty. 

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. 

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. 

Figure 9.3 Cross-section and top views of selected states along the O2 dissociation path on Pt(lll), from the initial state (t-b-t) to the dissociated product state (fee x 2). The third image is the transition state. (Reproduced with permission from Xu et al. [2004].)...

Selection of Optimal Sampling Interval and Initial State for Precise Parameter Estimation... 

Our selection of the initial state, x0, and the value of the manipulated variables vector, u(t) determine a particular experiment. Here we shall assume that the input variables u(t) are kept constant throughout an experimental run. Therefore, the operability region is defined as a closed region in the [xoj.xo, , Xo,n, U u2,...,u,]T -space. Due to physical constraints these independent variables are limited to a very narrow range, and hence, the operability region can usually be described with a small number of grid points. 

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