Levels system mapping

In obvious analogy to Schwinger s theory of angular momentum, this N-level system can be represented by N oscillators, whereby the mapping relations for the operator and the basis states read [99]... 

The image of the A-level Hilbert space is the subspace of the A-oscUlator Hilbert space with a single quantum excitation. Again, this (physical) subspace is invariant under the action of any operator which results by the mapping (80a) from an arbitrary A-level system operator. As a consequence we obtain the propagator identity... 

For a two-level system the mapping (80b) obviously coincides with Eq. (75d) for... 

As discussed in Section VI.A for the case of spin systems, the formalism described above is not the only way to construct a mapping of a A -level system. First of all, it is clear that one may again eliminate one boson DoF by exploiting the operator (which corresponds to the identity operator in the physical... 

There is, however, an important conceptional difference between the two approaches. On the quasi-classical level, this difference simply manifests itself in the initial conditions chosen for the electronic DoF. Let us consider an electronic two-level system that is initially assumed to be in the electronic state vl/]). In the mean-field formulation, the initial conditions are action-angle variables [cf. Eq. (18)], the electronic initial distribution in (90), is given by pgj = 6 Ni — 1)6 N2). In the mapping formalism, on the other hand, the initial electronic state vl/]) is represented by the first oscillator being in its first excited state and second oscillator being in its ground state [cf. Eq. (91)]. This corresponds to the... 

Another ambiguity in defining the classical mapping Hamiltonian is related to the fact that different bosonic quantum Hamiltonians may correspond to the same original quantum Hamiltonian H. This problem was already discussed in Section VI.A.2 for A-level systems. In the context of nonadiabatic dynamics, a different version of the mapping Hamiltonian is given by... 

Another possibility to introduce a semiclasscial initial value representation for the spin-coherent state propagator is to exploit the close relation between Schwinger s representation of a spin system and the spin-coherent state theory [100, 133-135]. To illustrate this approach, we consider an electronic two-level system coupled to Vvib nuclear DoF. Within the mapping approach the semiclassical propagator for this system is given by... 

The physicochemical properties of LCP systems may be varied by changing the number of lipoamino acids, the length of the lipoamino acid alkyl side chains, the level of MAP system branching, and incorporation of different spacer molecules. The LCP systems are stable to a wide range of pH conditions in solution, as well as to peptidase enzymes, and do not need to be stored under refrigeration. 

Here we apply the LAND-map approach to compute of the time dependent average population difference, A t) = az t)), between the spin states of a spin-boson model. Here az = [ 1)(1 — 2)(2 ]. Within the limits of linear response theory, this model describes the dissipative dynamics of a two level system coupled to an environment [59,63-65]. The environment is represented by an infinite set of harmonic oscillators, linearly coupled to the quantum subsystem. The characteristics of the system-bath coupling are completely described by the spectral density J(w). In the following, we shall restrict ourselves to the case of an Ohmic spectral density... 

Function The purpose of a system. Some systems map to a single primary function (e.g., process visual information). Others (e.g., the human arm) map to multiple functions, although at any given time multifunction systems are likely to be executing a single function (e.g., polishing a car). Functions can be described and inventoried, whereas level of performance of a given function can be measured. 

The rehabdity modeling of fault-tolerant aircraft systems using SyRelAn can be divided into two modeling levels, one mapping the system architecture, the other defining the redundancy management. Therefore the SyRelAn tool uses ReUabdity Block Diagrams for the definition of the nominal system architecture. To map the multi-state behavior of different components Concurrent Finite State Machines are implemented. 

This is the usual magnetic resonance lineshape for transitions in a two-level system without damping. At resonance the population oscillates sinusoidally between the two states (this is known as Rabi oscillation). A n-pulse is an on-resonance pulse with 2bx = Tt, which transfers all the population from state (0) to state (c). In Section 15.4.3 we will discuss how this can be used in a molecular beam to map out the fields along the beamline. An on-resonance 7r/2-pulse 2bz = n/2) drives the transition only half-way, creating an equal superposition of states (0) and (c) with a definite relative phase. The density matrix element describing this coherence at the end of... 

The ultrafast initial decay of the population of the diabatic S2 state is illustrated in Fig. 16 for the first 30 fs. Since the norm of the semiclassical wave function is only approximately conserved, the semiclassical results are displayed as rough data (dashed line) and normalized data (dotted line) [i.e. pnorm P2/ Pi + P2)]. The normalized results for the population are seen to match the quantum reference data quantitatively. It should be emphasized that the deviation of the norm shown in Fig. 16 is not a numerical problem, but rather confirms the common wisdom that a two-level system as well as its bosonic representation is a prime example of a quantum system and therefore difficult to describe within a semiclassical theory. Nevertheless, besides the well-known problem of norm conservation, the semiclassical mapping approach clearly reproduces the nonadiabatic quantum dynamics of the system. It is noted that the semiclassical results displayed in Fig. 16 have been obtained without using filtering techniques. Due to the highly chaotic classical dynamics of the system, therefore, a very large number of trajectories ( 2 x 10 ) is needed to achieve convergence, even over... 

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