Adiabatic eigenenergies

Figure 1. Adiabatic eigenenergies (r) (dashed lines) and effective potentials U (r) (full lines) for the collinear H+H2 reaction (,21) on the Porter-Karplus II surface (49) for n 0,l,2,3. The energies are relative to the dissociation energy into three free atoms.

A time-independent adiabatic approximation, based on the local separability of symmetric and anti-symmetric stretch motion in the region of the saddle point, provides a complementary picture (Pack 1976). Within the adiabatic limit the eigenenergies of the symmetric stretch motion on top of the potential ridge are defined through the one-dimensional... 

Diabatic Versus Adiabatic Dynamics Around Eigenenergy Crossings and... 

We distinguish the adiabatic evolution for nonresonant processes, for resonant processes at zero held, and for processes with a dynamical resonance. For nonresonant processes the adiabatic transport of the dressed states is simple The dynamics follows, up to a phase, the instantaneous dressed state whose eigenenergy is continuously connected to the one associated to the initial dressed state. This adiabatic transport will be generalized if more than one dressed state is involved in the dynamics. 

An alternative way to explore the zero-held resonance is to chirp a laser pulse that is switched on and off adiabatically, sufficiently far from the resonance. The chirp is such that the frequency is swept through the resonance when the held is on. The resonance appears in a dressed eigenenergy diagram (as a function of time or as a function of the held parameters) as an avoided crossing. 

Adiabatic passage can result in a robust population transfer if one uses adiabatic variations of at least two effective parameters of the total laser fields. They can be the amplitude and the detuning of a single laser (chirping) or the amplitudes of two delayed pulses [stimulated Raman adiabatic passage (STIRAP) see Ref. 69 for a review]. The different eigenenergy surfaces are connected to each other by conical intersections, which are associated with resonances (which can be either zero field resonances or dynamical resonances appearing beyond a threshold of the the field intensities). The positions of these intersections determine the possible sets of paths that link an initial state and the... 

The adiabatic passage induced by two delayed laser pulses, the well-known process of STIRAP [69], produces a population transfer in A systems (see Fig. 7a). The pump field couples the transition 1-2, and the Stokes field couples the transition 2-3. It is known that, with the initial population in state 11), a complete population transfer is achieved with delayed pulses, either (i) with a so-called counterintuitive temporal sequence (Stokes pulse before pump) for various detunings as identified in Refs. 73 and 74 or (ii) with two-photon resonant (or quasi-resonant) pulses but far from the one-photon resonance with the intermediate state 2), for any pulse sequence (demonstrated in the approximation of adiabatic elimination of the intermediate state [75]). Here we analyze the STIRAP process through the topology of the associated surfaces of eigenenergies as functions of the two field amplitudes. Our results are also valid for ladder and V systems. 

The analysis of the dynamics consists of (i) the calculation of the dressed eigenenergy surfaces of the effective quasienergy operator as a function of the two Rabi frequencies flj and if, (ii) the analysis of their topology, and (iii) the application of adiabatic principles to determine the dynamics of processes in view of the topology of the surfaces. 

Dressed eigenenergy curves (in units of S), corresponding to path (b) of Fig. 14 (f2max — 1.58) from formula (313) (dotted lines) and exact numerical result (full line). The arrow indicates the adiabatic path (big line). 

For computing the wavefunction of a localized exciton the adiabatic approximation (see (20), 28,29) can be used. The first step in this approximation consists of establishing the wavefunction x and the corresponding eigenenergy U for the electronic subsystem assuming that the positions of atomic nuclei are fixed. Thus, denoting by r the set of electronic coordinates and by R the set of nuclear coordinates, we have x = x(R), U = U(R), i.e. the wavefunction x and the energy U depend on R treated as parameters in this approximation. 

In the adiabatic gas-phase basis, the number of independent parameters drops to four Xad, AEad, AE12, and ai2, where the superscript ad refers to the adiabatic representation in which AE12 is the gas-phase gap between the eigenenergies, Eq. [29]. The equation for the free energy surfaces can then be rewritten in the basis-invariant form... 

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