Gaussian coherent state

Coherent states for the translational degrees of freedom (in d dimensions) are Gaussians located at a point (q, p) in phase space. For convenience we choose them as... 

The choice of the value of the coherent state width 7 is arbitrary, since this parameter does not affect the mathematical properties of the Gaussian basis set which determine the form of the semi-classical propagator. In fact, the value of this quantity is usually chosen in practical implementations so as to facilitate the numerical convergence. In the following, we shall set 7 = 1/2 since with this choice (25) simplifies considerably and becomes... 

It is clear from (28) that g(a) is always positive, since p is a positive definite operator. For a coherent state ao), g(a) = (l/ji)exp(— a — ao 2) is a Gaussian in the phase space Re a, Im a which is centered at a0. The section of this function, which is a circle, represents isotropic noise in the coherent state (the same as for the vacuum). The anisotropy introduced by squeezed states means a deformation of the circle into an ellipse or another shape. 

In order to make the END formalism practical, some approximations must be made in the representation of the waveffinctions of the electrons and nuclei. The particulars are outlined in the next section. In the simple model employed in this work, we choose to represent the electronic wavefunction by a group theoretical coherent states parametrization of a single determinant (SD). The nuclear wave function is formulated in terms of a frozen Gaussian wave packet (FGWP) and the limit of a narrow width is taken. This latter approximation corresponds to the classical limit for describing the nuclei. The coherent state representation of a single determinant leads to the so-called Thouless parametrization (29). For the description of the nuclear wavefunction, a quantum description has also been worked out (26, 30), but is yet to be implemented. 

The Gaussian wave packet is a coherent state and can be expressed as a superposition of oscillator states. This means that... 

The Gaussian wave packet in this form is the original coherent state . Generalizations of this concept have been made, in particular the work of Perelomov [14] has introduced so-called group-related coherent states. Such a state is formed by the action of a Lie group operator exp acting on a reference state lO). The Zm are the, in general... 

A more general coherent state description of a Gaussian wave packet is required when we allow the width parameter to evolve in time. The corresponding Lie group is then Sp(2, K), which is isomorphic to SU(1,1) or SO(2,1). The generators of the Sp(2, R) Lie algebra are... 

It is instructive to compute the time correlation function in the simple case that the ground and excited state potentials are harmonic but differ in their equilibrium position and frequency. This is particularly simple if the initial vibrational state is the ground state (or, in general, a coherent state (52)) so that its wave function is a Gaussian. We shall also use the Condon approximation where the transition dipole is taken to be a constant, independent of the nuclear separation, but explicit analytical results are possible even without this approximation. A quick derivation which uses the properties of coherent states (52) is as follows. The initial state on the upper approximation is, in the Condon approximation, a coherent state, i /,(0)) = a). The value of the parameter a is determined by the initial conditions which, if we start from a stationary state, are that there is no mean momentum and that the mean displacement (x) is the difference in the equilibrium position of the two potentials. In general, using m and o> to denote the mass and the vibrational frequency... 

What this means is that the function really represents a Gaussian in phase space. The label coherent state is often reserved for the special case when x and p are uncorrelated, which corresponds to a state of minimum uncertainty, i.e., one for which the... 

Other quantum mechanical approaches based on Gaussian wavepackets or coherent-state basis sets are those by Methiu and co-workers [46] and Martinazzo and co-workers [47] as well as the multiple spawning method developed by Martinez et al. [48] by which the moving wavepacket is expanded on a variable number of frozen Gaussians. Elsewhere [49] such an approach, especially conceived to be run on the fly, has been utilized for computing the ethylene spectrum by directly coupling it with electronic structure calculations. 

Equations (16.7) and (16.8) govern the spectral width of the absorption, i.e., how many stationary states u(R) are coherently excited by the laser pulse. Let us assume, for simplicity, that Eo(t ) is a Gaussian in time centered around to,... 

In the control scheme [13,17] that we have focused on, the time evolution of the interference terms plays an important role. We have already discussed more explicit forms of Eq. (7.75). One example is the Franck-Condon wave packet considered in Section 7.2.2 another example, which we considered above, is the oscillating Gaussian wave packet created in a harmonic oscillator by an (intense) IR-pulse. Note that the interference term in Eq. (7.76) becomes independent of time when the two states are degenerate, that is, AE = 0. The magnitude of the interference term still depends, however, on the phase S. This observation is used in another important scheme for coherent control [14]. 

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