Coherent-state theory

Let us start with a brief review of spin-coherent state theory. For simplicity we focus on a two-level (or spin 1/2) system. The coherent states for a two-level system with basis states /i), /2) can be written as [136, 139]... 

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 time dependence of the molecular wave function is carried by the wave function parameters, which assume the role of dynamical variables [19,20]. Therefore the choice of parameterization of the wave functions for electronic and nuclear degrees of freedom becomes important. Parameter sets that exhibit continuity and nonredundancy are sought and in this connection the theory of generalized coherent states has proven useful [21]. Typical parameters include molecular orbital coefficients, expansion coefficients of a multiconfigurational wave function, and average nuclear positions and momenta. We write... 

R J Glauber, in Coherent States in Quantum Theory, Mir, Moscow, 1972, p 26... 

The first case has already been considered section 2.0 the second case leads to a strong classical spin-orbit coupling, which is reflected in a Hamiltonian nature of the classical combined dynamics. In both situations the procedure is to find a suitable approximate Hamiltonian Hq( ) that propagates coherent states exactly along appropriate classical spin-orbit trajectories (x(l,),p(t),n(l,)). (For problems with only translational degrees of freedom this has been suggested in (Heller, 1975) and proven in (Combescure and Robert, 1997).) Then one treats the full Hamiltonian as a perturbation of the approximate one and calculates the full time evolution in quantum mechanical perturbation theory (via the Dyson series), i.e., one iterates the Duhamel formula... 

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... 

Within the theoretical framework of time-dependent Hartree-Fock theory, Suzuki has proposed an initial-value representation for a spin-coherent state propagator [286]. When we adopt a two-level system with quantum Hamiltonian H, this propagator reads... 

We have elucidated the nature of pulsed-shaping control of photodissociation from the viewpoint of energy-resolved coherent control theory. Clearly, when excitation is from a superposition of states, as in the vast majority of control scenarios, the role of the pulse shaping is to enhance a different set of interfering pathways for each control objective. 

Now, returning to Pfeifer s model of chirality we see that we have to make a choice of representation when selecting states to use in a Hartree variational calculation of the ground-state of the molecule-radiation field system. In Pfeifer s calculations the trial functions are chosen as coherent states, say t/N based on the photon Fock space n) in the Coulomb gauge theory an inequivalent set of trial functions is obtained by choosing coherent states, ip, based on the gauge-invariant photon Fock space n). One then has to compare the results of two minimization calculations involving, (cf. Eq. 5.4),... 

The initial photon state can be a number state (with a not well-defined phase) or a linear combination of number states, for instance a coherent state. We formulate the construction of coherent states in the Floquet theory and show that choosing one as the initial photon state allows us to recover the usual semiclassical time dependent Schrodinger equation, with a classical held of a well-defined phase (see Section II.C). 

In this Appendix we show that the coherent states are represented in Floquet theory by a generalized function "T>e0 (0), which is real and depends on 0 — 0o, where % E S1 is a fixed angle, and... 

The subscripts in the scalar product symbols (( ) r/>) indicate on which space they act. Thus we conclude that in Floquet theory the photon coherent states are represented by the square root of a 8-function, which we denote by cI>e0(0) = (27i)1/281/2(0 — 0o). Since we will be interested in expectation values, only e0 2 will appear in our calculations. The formal calculus rules for 81/2(0 — 0o) are given in Ref. 9. 

The END theory was proposed in 1988 [11] as a general approach to deal with time-dependent non-adiabatic processes in quantum chemistry. We have applied the END method to the study of time-dependent processes in energy loss [12-16]. The END method takes advantage of a coherent state representation of the molecular wave function. A quantum mechanical Lagrangian formulation is employed to approximate the Schrodinger equation, via the time-dependent variational principle, by a set of coupled first-order differential equations in time to describe the END. 

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