Projective coherent states

The general theory of classical limits of algebraic models is formulated not in terms of the group coherent states of Eq. (7.17) but rather in terms of projective coherent states. The ground-state projective coherent state is... 

As in the previous case of a single U(4), we introduce projective coherent states, constructed with condensate boson operators (Leviatan and Kirson, 1988 Shao, Walet, and Amado, 1992, 1993)... 

To proceed, we note that the initial state (0)) usually represents a coherent state on the excited-state potential surface (because of the assumed broadband excitation). Moreover, as emphasized in Sec. 2 above, the strong nonadiabatic coupling effects at conical intersections may lead to a pronounced mixing of vibrational levels of the upper and lower of the intersecting surfaces. For these reasons, it is appropriate to introduce the overall electronic population Pi t) (of electronic state i) as a direct and natural measure of the internal-conversion dynamics on strongly coupled surfaces. The population of electronic state 1 ) is defined as the expectation value of the projection operator... 

Figure 2. Franck-Condon windows lVpc(Gi, r, v5) for the Na3(X) - N83(B) and for the Na3(B) Na3+ (X) + e transitions, X = 621 nm. The FC windows are evaluated as rather small areas of the lobes of vibrational wavefunctions that are transferred from one electronic state to the other. The vertical arrows indicate these regions in statu nascendi subsequently, the nascent lobes of the wavepackets move coherently to other domains of the potential-energy surfaces, yielding, e.g., the situation at t = 653 fs, which is illustrated in the figure. The snapshots of three-dimensional (3d) ab initio densities are superimposed on equicontours of the ab initio potential-energy surfaces of Na3(X), Na3(B), and Na3+ (X), adapted from Ref. 5 and projected in the pseudorotational coordinate space Qx r cos

Using this approach the +) and —) states are not coupled by the field of the ion, but are only split in energy. At high collision velocities the initial state 0) is simply projected onto the 0 + 1) state, a coherent superposition of +) and -) states, by the dipole matrix element. However, at lower velocities the change in energy of the +) and -) states during the collision allows the +) and -) states themselves to be populated rather than only a coherent superposition. The latter feature allows nondipole transitions at lower collision velocities, as observed experimentally. 

Figure 1. The creation, evolution, and detection of wave packets. The pump laser pulse pump (black) creates a coherent superposition of molecular eigenstates at t — 0 from the ground state I k,). The set of excited-state eigenstates N) in the superposition (wave packet) have different energy-phase factors, leading to nonstationary behavior (wave packet evolution). At time t = At the wave packet is projected by a probe pulse i probe (gray) onto a set of final states I kf) that act as a template for the dynamics. The time-dependent probability of being in a given final state f) is modulated by the interferences between all degenerate coherent two-photon transition amplitudes leading to that final state.

According to Section IV.A.3, for each wave vector K such that gcTK > A , there is an imaginary solution of (4.25). Thus, since localized states have a projection on every K> state, a very fast photon emission occurs (coherent emission) in the direction determined by K. In second-order perturbation theory124 126 (Fermi s golden rule"), / K takes values between 0 and x for... 

PRISM represents the first major collective effort at the European level to develop ESM supporting software in a shared and coherent way. This effort is recognised by the Joint Scientific Committee (JSC) and the Modelling Panel of the World Climate Research Programme (WCRP) that has endorsed it as a key European infrastmcture project . It is analogous to the ESME project (http //www. earthsystemmodeling.org) in the United States. 

There is a remarkable amount of excellent catalysis research being conducted in the United States. Unfortunately, at many of the national labs, there is a lack of critical mass in catalysis. Many have perhaps only one to three full-time catalysis researchers along with a few surface scientists, microscopists, and/or synthetic chemists who may include some projects related to catalysis in their portfolio. The effort to get the national laboratories to work closely together is an effort to create a virtual catalysis community where critical mass is achieved. It is also an effort to leverage and capitalize on programs technically related to catalysis, to fully utilize all DOE materials facilities (such as the synchrotron at Brookhaven and neutron diffraction at Los Alamos), and to provide a coherent program that will attract industry s interest. As a team, the labs can also be more effective in raising the profile of catalysis research. If industry finds the laboratories catalysis research useful, it will also likely become an advocate of catalysis research support. 

To complete the projection step, one has to transfer the atom in its coherent superposition back to the 60/ state this is achieved by the same pumping sequence as in the first step. The mismatch of the Clebsch-Gordan coefficient products will cause again the error probability l — rj. The information is then restored with very high probability and the system is ready to undergo a new protection cycle. 

We have considered repeated projective measurements on an ancilla as a tool for manipulating the evolution of a dynamic quantum system of interest. Due to an interaction between the dynamic system and the ancilla, the nonunitary evolution of the ancilla extends equally to the dynamic system, but close to the Zeno-limit the coherence of the dynamic system may still be preserved. Of particular interest here are systems coupled with a nondemolition interaction, since they can be described in an essentially simplified manner. Depending on the dimension Na of the ancilla, individual elements of the reduced state of the dynamic part obey master equations that are iV order differential equations in time. Equivalently, the master equations can be written in the Nakajima-Zwanzig or time convolutionless form. 

The current state of design processes can essentially not be improved by making only small steps. Instead, a new approach is necessary. Thereby, we face principal questions and nontrivial problems. We find new questions and corresponding problems by coherently and uniformly modeling the application domain and by defining new and substantial tool functionality. The layered process/product model is a scientific question which - even in a long-term project like IMPROVE - can only be answered partially. 

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