Dynamical Tunneling and Control

Abstract

Tunneling is a supreme quantum e ect. Every introductory text[1] on quantum mechanics gives the paradigmexample of a particle tunneling through a one-dimensional potential barrier despite having a total energy less thanthe barrier height. Indeed, the reader typically works through a number of excercises, all involving one-dimensionalpotential barriers of one form or another modelling several key physical phenomena ranging from atom transfer re-actions to the decay of alpha particles[2]. However, one seldom encounters coupled multidimenisonal tunneling insuch texts since an analytical solution of the Schrodinger equation in such cases is not possible. Interestingly, therichness and complexity of the tunneling phenomenon manifest themselves in full glory in the case of multidimen-sional systems[3]. Thus, for instance, the usual one-dimensional expectation of increasing tunneling splittings as oneapproaches the barrier top from below is not necessarily true as soon as one couples another bound degree of freedomto the tunneling coordinate. In the context of molecular reaction dynamics, multidimensional tunneling can result instrong mode-speci city and uctuations in the reaction rates[4]. In fact, a proper description of tunneling of electronsand hydrogen atoms is absolutely essential[5, 6] even in molecular systems as large as enzymes and proteins. Althoughone usually assumes tunneling e ects to be signi cant in molecules involving light atom transfers it is worth pointingout that neglecting the tunneling of even a heavy atom like carbon is the di erence between a reaction occuring ornot occuring. In particular, one can underestimate rates by nearly hundred orders of magnitude[7]. Interestingly,and perhaps paradoxically, several penetrating insights into the nature and mechanism of multidimensional barriertunneling have been obtained from a phase space perspective[8, 9]. The contributions by Creagh, Shudo and Ikeda,and Takahashi in the present volume provide a detailed account of the latest advances in the phase space basedunderstanding of multidimensional barrier tunneling.What happens if there are no coordinate space barriers? In other words, in situations wherein there are no staticenergetic barriers separating \reactants from the \products does one still have to be concerned about quantumtunneling? One such model potential is shown in Fig. 1 which will be discussed in the next section. Here we have thenotion of reactants and products in a very general sense. So, for instance, in the context of a conformational reactionthey might correspond to the several near-degenerate conformations of a speci c molecule. Naively one might expectthat tunneling has no consequences in such cases. However, studies over last several decades[10{20] have revealedthat things are not so straightforward. Despite the lack of static barriers, the dynamics of the system can generatebarriers and quantum tunneling can occur through such dynamical barriers[21]. This, of course, immediately impliesthat dynamical tunneling is a very rich and subtle phenomenon since the nature and number of barriers can varyappreciably with changes in the nature of the dynamics over the timescales of interest. This would also seem toimply that deciphering the mechanism of dynamical tunneling is a hopeless task as opposed to the static potentialbarrier case wherein elegant approximations to the tunneling rate and splittings can be written down. However,recent studies have clearly established that even in the case of dynamical tunneling it is possible to obtain veryaccurate approximations to the splittings and rate. In particular, it is now clear that unambiguous identi cation ofthe local dynamical barriers is possible only by a detailed study of the structure of the underlying classical phasespace. The general picture that has emerged is that dynamical tunneling connects two or more classically disconnectedregions in the phase space. More importantly, and perhaps ironically, the dynamical tunneling splittings and ratesare extremely sensitive to the various phase space structures like nonlinear resonances[22{29], chaos[16{19, 30, 31]and partial barriers[32, 33]. It is crucial to note that although purely quantum approaches can be formulated forobtaining the tunneling splittings, any mechanistic understanding requires a detailed understanding of the phasespace topology. In this sense, the phenomenon of dynamical tunneling gets intimately linked to issues related to phasespace transport. Thus, one now has the concept of resonance-assisted tunneling (RAT) and chaos-assisted tunneling(CAT) and realistic systems typically involve both the mechanisms.Since the appearance of the rst book[21] on the topic of interest more than a decade ago, there have beenseveral beautiful experimental studies[34{39] that have revealed various aspects of the phenomenon of dynamicaltunneling. The most recent one by Chaudhury et al. realizes[40] the paradigmatic kicked top model using cold