Quantum–classical mechanics and the Franck–Condon principle related to quantum mechanics are discussed as two alternative theoretical approaches to molecular optical spectroscopy. The statement of the problem is connected with the singularity of quantum mechanics in describing the joint motion of electrons and nuclei in the transient state of “quantum” transitions. This singularity can be eliminated by introducing chaos into the transient state. Quantum mechanics itself, supplemented by chaos (dozy chaos), is called quantum–classical mechanics. Using the simplest example of quantum transitions, it is shown that the results of quantum–classical mechanics in the case of strong dozy chaos correspond to the physical picture based on quantum mechanics and the Franck–Condon principle. The same chaos can be strong for small molecules in standard molecular spectroscopy and simultaneously weak in the photochemistry and nanophotonics of large molecules, where quantum mechanics no longer works. To describe the chaotic dynamics of the transient state, it is necessary to apply quantum–classical mechanics. Thus, the erroneous Franck–Condon physical picture of molecular “quantum” transitions is workable from a practical point of view as long as we are dealing with sufficiently small molecules, just as the erroneous geocentric picture of the world was workable until we went out into outer space.
We contrast two sets of conditions that govern the transition in which classical dynamics emerges from the evolution of a quantum system. The first was derived by considering the trajectories seen by an observer (dubbed the "strong" transition) [Bhattacharya et al., Phys. Rev. Lett. 85, 4852 (2000)], and the second by considering phase-space densities (the "weak" transition) [Greenbaum et al., Chaos 15, 033302 (2005)]. On the face of it these conditions appear rather different. We show, however, that in the semiclassical regime, in which the action of the system is large compared to h, and the measurement noise is small, they both offer an essentially equivalent local picture. Within this regime, the weak conditions dominate while in the opposite regime where the action is not much larger than h, the strong conditions dominate.
Using the key properties of chaos, i.e. ergodicity and exponential instability, as a resource to control classical dynamics has a long and considerable history. However, in the context of controlling "chaotic" quantum unitary dynamics, the situation is far more tenuous. The classical concepts of exponential sensitivity to trajectory initial conditions and ergodicity do not directly translate into quantum unitary evolution. Nevertheless properties inherent to quantum chaos can take on those roles: i) the dynamical sensitivity to weak perturbations, measured by the fidelity decay, serves a similar purpose as the classical sensitivity to initial conditions; and ii) paired with the fact that quantum chaotic systems are conjectured to be statistically described by random matrix theory, implies a method to translate the ergodic feature into the control of quantum dynamics. With those two properties, it can be argued that quantum chaotic dynamical systems, in principle, allow for full controllability beyond a characteristic time that scales only logarithmically with system size and
ℏ−1. In the spirit of classical targeting, it implies that it is possible to fine tune the immense quantum interference with weak perturbations and steer the system from any initial state into any desired target state, subject to constraints imposed by conserved quantities. In contrast, integrable dynamics possess neither ergodicity nor exponential instability, and thus the weak perturbations apparently must break the integrability for control purposes. The main ideas are illustrated with the quantum kicked rotor. The production of revivals, cat-like entangled states, and the transition from any random state to any other random state is possible as demonstrated.