In our research we develop models for the theoretical description of the control of quantum dynamics by external fields. Typically, we investigate systems from the realm of molecular or chemical physics which we model quantum-dynamically in real time, and on a microscopic scale. The goal is to manipulate internal and/or external molecular degrees of freedom, ultimately aiming at actively controlling the course of a chemical reaction. In cases where the "curse of dimensionality" makes a full dimensional treatment computationally too expensive, we strive at developing reduced dimensional models.
Model reduction in light-induced control of quantum systems
Burkhard Schmidt with Boris-Schäfer-Bung
Cooperation with Carsten Hartmann (now at BTU Cottbus-Senftenberg)
Support by ECMath (Einstein Center for Mathematics Berlin) through project SE 11
Note also our CECAM workshop 1309 (September 2016)
While the numerical effort to solve the Schrödinger equation (isolated quantum systems) scales linearly with the number of quantum states involved, the situation changes for the worse when the influence of the environment has to be modeled, e. g., by coupling the quantum system to a heat bath (open quantum systems). Treating (reduced) density matrices instead of state vectors, the effort to solve the corresponding Liouville-von Neumann equation (typically with a dissipative part in Redfield or Lindblad form) scales quadratically with the original number of quantum states. While direct solutions are currently limited to systems with few degrees of freedom , effective strategies for dimension reduction are imperative for more complex systems.
To this end, we apply the balanced truncation method originally developed for linear systems in engineering, and extend it to the bi-linear problems in quantum dynamics. The basic principle of balanced truncation is to identify a subspace of jointly easily controllable and observable states and then to restrict the dynamics to this subspace without changing the overall response of the system beyond a controllable error . Alternatively, also methods aiming at H2-optimal model reduction, minimizing an H2-type error, are considered. The focus of our work is on the numerical implementation and a thorough comparison of the methods. Structure and stability preservation are investigated, and the competitiveness of the approaches is shown for practically relevant, large-scale examples .
Most of the above methods have been implemented within our WavePacket software package, which is freely available via the SourceForge platform. For extensive documentation, see also   .
Control of molecular wavepacket dynamics by short laser pulses
Burkhard Schmidt with group of Prof. Jörn Manz (Chemistry, FU Berlin)
Support by Deutsche Forschungsgemeinschaft through SFB 450 and SPP 470
Since the advent of lasers in the 1960s, chemists and physicists have used these light sources to manipulate the course of molecular dynamics in the context of various photo-chemical, photo-physical, and/or photo-biological processes. In modern experiments, internal and external degrees of freedom are controlled by means of shaped laser pulses, typically on timescales of pico-seconds, femto-seconds or even atto-seconds. Despite of their short duration, these pulses can permanently change the state of the systems with which they interact. Apart from the quest for high yields of molecules in the desired target states, the main goal is to achieve high selectivity with respect to different quantum states or different reaction pathways leading to different products.
In our theoretical/computational work we mainly use quantum models based on the time-dependent Schrödinger equation, where - in most cases - the coupling of the quantum systems to external electromagnetic fields is described within the semi-classical dipole approximation. For examples, see our results, mainly for small molecules, in the following fields:
photoassociation , , , , , , , [B]
molecular orientation/alignment , , , , , , , , [D]
Most of these simulations have been carried out with our WavePacket software package, which has been developed in parallel with these projects. For an extensive documentation, see also   .