Research topics - Frank Grossmann
Quantum mechanical tunneling
The project is divided into four topics:
1. Semiclassical tunneling in the time-domain
Can real classical trajectories describe tunneling?
There has been a longstanding debate as to whether tunneling through a realistic (i.e. non parabolic) barrier can be described by using only classically allowed real trajectories. We have shown that although real trajectories can account for a transport across the barrier the results for the transmission probability are not converged with respect to certain wavepacket parameters entering the calculation /2/. The inclusion of standard semiclassical tunneling contributions in the time-domain to converge the results has not been achieved yet. Therefore we started using semiclassical coherent state path integrals. It turns out, however, that the inclusion of only a single trajectory at a time is not sufficient to cure the nonconvergence /3/. Further research using multiple complex trajectories is needed.
Recently, we put our focus back on the question if real-time real-valued trajectories can describe tunneling quantitatively. Although, in /2/, we had shown that standard semiclassics (Van Vleck-type and also Herman-Kluk type propagators) have problems, an extension of the method of Herman and Kluk, which enables a classical trajectory to spawn daughter trajectories (with different energy), can account for tunneling in a way which is comparable to the well-known uniformized WKB method /5/.
2. Thermal reaction rate constants
Can coherent states shine a light on common approximations to Miller's rate formula?
The standard way to calculate thermal reaction rate nowadays starts by using Miller's rate formula in one of its variants. However, many researchers are eager to approximate this expression by simplifying the dynamical part in analogy to classical transition state theory. For sufficiently high temperatures these procedures may be justified. For lower temperatures, where tunneling becomes important, it has long been known that the results can differ from the exact ones by orders of magnitude. By using a coherent state representation of the rate expression the shortcomings of the approximations can be explained in detail /4/. Furthermore, better approximation methods can possibly be explored.
3. The tunneling effect in periodically driven quantum systems
my PhD thesis (1989-1992) under the supervision of Prof. Dr. P. Hänggi
How is tunneling in bi- or metastable systems affected by an external sinusoidal field?
In the Floquet formalism we have studied the question how an external periodic field can influence the archetypal tunneling dynamics in a bistable system. Surprisingly it turns out that the Floquet energies corresponding to the tunneling doublet in a double well can be tuned into degeneracy by a suitably chosen light field meaning that the tunneling effect can be switched off by a laser field /1/. This effect does occur along a one dimensional manifold in the two dimensional parameter space spanned by the driving force and frequency. It has just recently been verified experimentally!
4. The influence of a thermal environment on the tunneling effect
How is tunneling through a barrier affected by couling to an environment?
There has been a long debate in the literature about the effect of an environment on probabilities for tunneling in a quantum system. With the help of the BOMCA technology developped in the group of David Tannor and the stochastic Liouville-von Neumann equation we have been able to show that for energies below the barrier, tunneling is enhanced, whereas above the barrier it is suppressed /6/.