Coherent states expectation values

The expectation value of H in the coherent state (7.17) can be evaluated explicitly for any Hamiltonian. However, an even simpler construction of Hd (valid to leading order in N) can be done (Cooper and Levine, 1989) by introducing intensive boson operators (Gilmore, 1981). In view of its simplicity, we report here this construction. If one divides the individual creation and annihilation operators by the square root of the total number of bosons, the relevant commutation relations become... 

The classical Hamiltonian, Hch is then the expectation value of the algebraic Hamiltonian, H, in the coherent state (7.66),... 

This general feature of the stochastic scheme may cause convergence problems. For example, consider a situation in which the molecular system is predominately in a single state, say pu. Although the expectation values of the population P2 = trp22 and the corresponding coherences are zero, there are the same number of random walkers in these states which need to cancel... 

The phase factor is unimportant here because we shall generally take the expectation value of any operator in the coherent state. 

B. A. Hess The reason that macroscopic motions display coherence is that they are in most cases at the classical limit of quantum dynamics. In this case, a suitable occupation of quantum states ensures that quantum mechanical expectation values equal the classical value of an observable. In particular, the classical state of an electromagnetic field (the coherent state) is one in which the expectation value of the operator of the electromagnetic field equals the classical field strengths. 

We now have a formula for constructing the density matrix for any system in terms of a set of basis functions, and from Eq. 11.6 we can determine the expectation value of any dynamical variable. However, the real value of the density matrix approach lies in its ability to describe coherent time-dependent processes, something that we could not do with steady-state quantum mechanics. We thus need an expression for the time evolution of the density matrix in terms of the Hamiltonian applicable to the spin system. 

Expectation Values on Coherent States Relation with the Semiclassical Model... 

The last expression is the expectation value calculated with the semiclassical model with initial phase 0O. We thus conclude that, if one considers only observables of the molecule, the Floquet evolution with a coherent state in the initial condition is equivalent to the semiclassical model. We remark that a somewhat related construction, linking the evolution from cavity dressed states directly to the semiclassical model (i.e., without the intermediate level of Floquet states as we do here), was established in Ref. 16. 

Furthermore, using the well-known properties of the expectation values of Nm on coherent states, we obtain... 

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. 

Now let us be more quantitative. The interaction (16a) mapping the atomic operators PAi out on fight is very useful for a strong n and useless if k coherent states of the atomic spins and performing the first measurement pulse with outcomes A and B1, we may deduce the statistical properties of the measurement outcomes. Theoretically we expect from Eq. (16a)... 

It should be emphasized that the system under consideration completely describes the process of transmission of the phase information between the field and the atom. Initially the atom is in the ground state with the angular momentum 0 and has no SU(2) atomic phase at all. Absorption of photons induces an atomic phase that coincides with the phase difference between the two coherent components of the field. This can be concluded from a direct calculation of the expectation value of the atomic cosine operator (43) over the state (96) ... 

Such a coherent state can, for example, be generated by the interaction of the atomic ensemble with a sufficiently strong EM field that induces atomic dipole moments, which add up to a macroscopic oscillating dipole moment if all atomic dipoles oscillate in phase. The expectation value D of such an atomic dipole moment is... 

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... 

The multidimensional generalization of these equations is straightforward. Expectation values are readily obtained fi-om Eq. (15) and these expressions. Similarly, if the operator A is given by a low-order polynomial, the coherent state transform of p A can be evaluated analytically. 

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