Functional Methods in Quantum Dynamics: Fixed-Point Approach

Quantum theory describes our world on the microscopic level, e.g. atoms and molecules. Since most of those microscopic systems consist of many particles that interact with each other, we are in general confronted with what is called an interacting many-body quantum system. Although we know the equations that govern the dynamics of interacting many-body quantum systems in the case of atoms or molecules this is usually the Schrödinger equation we do typically not know the full solutions of these equations (their wave functions). Even with nowadays supercomputers we are in general not able to calculate these wave functions. This spawned a lot of interest into the question whether one can find (computationally) simpler equations to directly calculate a physical observable without the use of the wave function. To do so one can, for instance, reformulate time-dependent quantum theory in such a way, that no longer the wave function is the fundamental object to be determined, but a special observable. The most prominent of those theories is time-dependent density-functional theory, where one calculates only the probability to find one of the many particles of the quantum system at some point is space (the one-particle density). This theory is extensively used in different fields of physics, chemistry and material science to calculate properties of time-dependent many-particle systems. Nevertheless, the formal foundations of these time-dependent functional methods are not well understood and the approximations that one has to use in practice lead to wrong predictions in certain physical situations. In this research project we explore a new approach to these time-dependent functional methods. First, by reformulating the basic equations in terms of what is called a fixed- point equation, we will employ well-established mathematical techniques to give these theories a rigorous foundation and extend their applicability. A second topic of the project is to calculate the exact solutions of these equations for certain cases to understand their properties. The fixed-point formulation actually constitutes a constructive procedure to do so. With this we will consider those situations where current approximations give wrong results and will derive new approximations capable of dealing with these situations. Finally, the fixed-point approach can also be used as a novel quantum control technique, where the many-body quantum system is governed by a prescribed observable in time. Such control problems are very important in the fields of quantum optics and quantum information. The goal here is to establish time-dependent functional ideas as practical methods in the field of quantum optic and quantum information. In conclusion, the proposed research project will introduce a new viewpoint on time-dependent functional methods, strengthen their foundations and will open up new and exciting possibilities of application.

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