Three-state system numerical study

The results above indicate that M averaged control is possible for configuration B, assuming that the fields significantly couple only the ten indicated levels. A generalized, albeit computational, extension of these results to include all M levels is provided in this section. Specifically, we show that the full system of states that are coupled by three perpendicular linearly polarized lasers shows control, even upon M averaging. Note, however, that numerical studies do show that the ten-level approximation is, in most cases, an excellent approximation. 

The main objective of any theory is to be able to understand and predict the results of experiments. Since the world is three dimensional one cannot limit oneself to the study of collinear systems. In section IV we show how the collinear analysis based on periodic orbits may be generalised to three dimensional systems.We provide a 3D adiabatic transition state theory which is used to analyse numerical computations as well as experimental results. A 3D analysis of quantal resonances predicts that one should hope that quantal resonances will provide a new spectroscopy of transition states. A discussion of the future role of periodic orbits and reactive scattering is given in section V. 

The dynamics of populations of the electronic states in a 4,4 -bis(dimethylamino) stilbene molecule (two-photon absorption) was studied against the frequency, intensity, and shape of the laser pulse [52]. Complete breakdown of the standard rotating wave for a two-photon absorption process was observed. An analytical solution for the interaction of a pulse with a three-level system beyond the rotating wave approximation was obtained in close agreement with the strict numerical solution of the amplitude equations. Calculations showed the strong role of the anisotropy of photoexcitation in the coherent control of populations that can affect the anisotropy of photobleaching. The two-photon absorption cross section of an ethanol solution of a trans-stilbene and its derivatives exposed to radiation of the second harmonic of a Nd YAG laser (532 nm) of nanosecond duration has been detected [53]. In experiments, the method based on the measurement of the photochemical decomposition of examined molecules was used. The quantum yield of the photoreaction (y266) of dyes under one-photon excitation (fourth harmonic Nd YAG laser 266 nm) was detected by absorption method. 

To construct a CS-QDT formulation, one shall treat the effects of system-bath coupling H [Eq. (2.1c)] to the second order exactly for not only the reduced density operator p t) evolution, but also the initial canonical state of the total composite system, Pt( o —oo) = p (T), before the external field excitation. Various CS-QDT formulations differ at their partial resummation schemes for the higher order contributions. We have recently arrived at three forms of CS-QDT in terms of differential equations of motion [38]. Two of them are in principle equivalent to the conventional second-order COP [Eq. (1.2)] and POP [Eq. (1.3)] formulations (cf. AppendixB). For the sake of clarity, we shall present here only the unconventional one that may be particularly suitable for the numerical study of non-Markovian dissipation in the presence of external time-dependent fields. 

The static bifurcation behavior and its practical implications have been investigated. We have also formulated the unsteady-state dynamic model and we have used it to study the dynamic behavior of the system by solving the associated IVP numerically. Both the controlled and the uncontrolled cases have been investigated. Two particular reactions have been studied, one with three steady states, and one with five steady states. 

Unsteady-state or dynamic simulation accounts for process transients, from an initial state to a final state. Dynamic models for complex chemical processes typically consist of large systems of ordinary differential equations and algebraic equations. Therefore, dynamic process simulation is computationally intensive. Dynamic simulators typically contain three units (i) thermodynamic and physical properties packages, (ii) unit operation models, (hi) numerical solvers. Dynamic simulation is used for batch process design and development, control strategy development, control system check-out, the optimization of plant operations, process reliability/availability/safety studies, process improvement, process start-up and shutdown. There are countless dynamic process simulators available on the market. One of them has the commercial name Hysis [2.3]. 

Abstract In a solid with orbital degree of freedom, an orbital configuration does not minimize simultaneously bond energies in equivalent directions. This is a kind of frustration effect which exists intrinsically in orbital degenerate system. We review in this paper the intrinsic orbital frustration effects in Mott insulating systems. We introduce recent our theoretical studies in three orbital models, i.e. the cubic lattice orbital model, the two-dimensional orbital compass model and the honeycomb lattice orbital model. We show numerical results obtained by the Monte-Carlo simulations in finite size systems, and introduce some non-trivial orbital states due to the orbital frustration effect. 

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