Adiabatic following

Entirely nonadiabatic transitions, in which the electrons cannot adiabatically follow the change in the positions of the proton and the medium molecules. 

Note that since the profile of the lower adiabatic potential energy surface for the proton depends on the coordinates of the medium molecules, the zeroth-order states and the diabatic potential energy surfaces depend also on the coordinates of the medium molecules. The double adiabatic approximation is essentially used here the electrons adiabatically follow the motion of all nuclei, while the proton zeroth-order states adiabatically follow the change of the positions of the medium molecules. 

The normal vibrations q and q are related to the shifts of the ions Y and X . The low-frequency part of the inertial polarization of the medium, k(cok co 9 co ), cannot follow these shifts. The high-frequency part of the inertial polarization, /(a>/ co 1, co )9 adiabatically follows the shifts of the ions Y" and X-, and the equilibrium coordinates of the effective oscillators describing this part of the polarization depend on the normal coordinates of the corresponding normal vibrations, viz. /0i(gl), (iof(q )-... 

The expression for uj-j. thus obtained, assuming an adiabatic following of Ej), obviously has contributions from the populations as well as from the coherences However, the contributions of populations to involve which,... 

Although it is possible with the adiabatic following of STIRAP to produce not only complete population transfer but also, through fractional STIRAP [16], any superposition of quantum states 1 and 3, so too is it possible to design a series of rotations that will produce an arbitrary change of polarization. However, in both cases, the full change of variables (quantum state or polarization) is more robust than a partial change. 

Adiabatic passage schemes are particularly suited to control population transfer between states, since the adiabatic following condition assesses the stability of the dynamics to fluctuations in the pulse duration and intensity [3]. The time evolution of the wave function does not depend on the dynamical phase, and is therefore slow in comparison with the vibrational time scale. This fact guarantees that the time variation of the observables during the controlled dynamics will be slow. Adiabatic methods can therefore be of great utility to control dynamic observables that do not commute with the Hamiltonian. We are interested in the control of the bond length of a diatomic molecule [4]. 

It is shown in Ref. 19 that, if we assume that the bulk dynamics adiabatically follows the evolution of the membrane bound species and does so in linear response to them, (l6) gets replaced by (assuming a single relevant membrane species, B = 1, for simplicity)... 

As e " 0 the dynamics of the 1 modes is much quicker than that of the slow, = 1, modes. Hence putting (3 0 into (31) and letting 0 -t 0 we get the "adiabatic following" relation... 

Thus the fast modes follow the slow 1-modes. To obtain the equation for the slow mode amplitudes Y v> v = 0, 1, we put (3 0 into (32) for v = 1, take e " 0 and insert the adiabatic following relation (35) to obtain... 

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