Optimum Control for Open Quantum Systems

This work deals with the theory of (optimal) control of quantum systems which, among other things, is of fundamental relevance to the implementation of quantum algorithms and the operation of nanostructured (electronic) devices based on quantum interference. Two main goals will be addressed in this theoretical work: steering and trapping of a quantum system by an external control. This work consists of several aspects: Optimum coherent control is formulated for open quantum systems. Open (i.e., real) quantum systems are governed by non-Markovian kinetic equations rather than the Schroedinger or von Neumann equation. These differential equations act as constraints to a cost functional which accounts for the physical quantity to be optimized. We use the variational approach and Pontryagin`s minimum principle to achieve this task. An account of the non-Markovian nature of quantum (sub-) systems enhances the complexity of the optimization problem, on one hand, but opens more channels for quantum interference and thus enhances the possibility for successful steering of a quantum system, on the other hand. The potential of this idea will be studied at the example of selected quantum systems in the second step. In particular, we study feasibility of enhanced stabilization and dynamical control of open quantum systems, such as coupled electron-phonon systems, by minimizing dissipative processes via quantum interference channels in the system-bath (system-environment) interaction which are opened by external electromagnetic control fields (the "control``). On sufficiently short time scales and on a microscopic level, any system-environment interaction represents a quantum process and, hence, is susceptible to quantum interference. The simplest example studied in this context consists of one spin (two-level system) in a bath of phonons (spin- boson model). It is extended subsequently to a model for two spins (two two-level systems) in a phonon system. We study this 2-qubit model of an elementary quantum gate regarding its controllability and stabilization in pure, mixed and entangled states. Kinetic equations are obtained within two main approaches: the projection operator approach and the density matrix approach. Optimization is performed within several numerical approaches, such as the conjugate gradient method and genetic algorithms. Optimization schemes utilize both general field profiles and a parameterized series of Gaussian wave packets. In parallel we pursue optimization of coherent control of complex many-body systems as an extension of earlier work. We study optimized coherent control of electron dynamics and its consequences on optical properties of semiconductor heterostructures within a microscopic model accounting for the electron-electron and electron- phonon interaction. Due to the complexity of the problem, both from the physical and numerical standpoint, iterative direct optimization will be utilized initially, whereby interaction processes are successively added between iterations, utilizing the Fermi-liquid properties of the system. In this fashion optimal pulse properties are tailored to optimize optical gain or loss using quantum interference processes.

This project has concerned itself with the theoretical investigation of coherent optimal control of open quantum systems in form of quantum bits (qubits) and quantum gates. Qubits constitute the main building blocks for the formulation and implementation of quantum information processing, i. e. computation based on quantum dynamics. Using the parallelism inherent in quantum dynamics it has been shown before that quantum-based algorithms may scale significantly more favorably than algorithms based on classical computers. Main obstacle in the realization of quantum information processing is dissipation occurring due to inevitable coupling of the quantum system to its environment. Dissipation tends to destroy quantum-coherent dynamics and hence the benefits envisioned. Formidable problems also arise when the complexity of quantum networks increases (scaling problem). The design of high fidelity qubits and quantum gates is of utmost importance in order make meaningful and to reduce the need for quantum error correction, which exponentially increases the number of qubits needed. In this project we have addressed the problem of unwanted system-bath interaction and its consequences on device performance. The goal was to steer the quantum system in such a way, that dissipative effects are minimized and the desired gate operation could be executed with highest fidelity compatible with the design. For this we have studied within microscopic models the interaction of specific quit and quantum gate realizations with its environment and have carried out performance estimates to guide concurrent and future experimental efforts. We first developed optimal control theory for the (state-independent) control of dissipative quantum systems and have come up with several solutions: a formulation based on perturbation theory for weakly dissipative systems which avoids the use of Lagrangean multipliers, a superoperator formulation and a formulation based on the process tomography matrix (used in experiment), the latter two being fully general. This has enabled us to study optimal performance of quit realizations within the Lindblad and Bloch-Redfield equation (within spin-boson models). Specifically we have studied superconducting qubits (Josephson charge qubits) and self-assembled InGaAs/GaAs quantum dot molecules. Apart from development of several complementary formulations of state-independent optimal control theory for open quantum systems our studies have shown that, contrary to some claims in the literature, Josephson charge qubits should perform equally well as other superconductor-based qubits, such as the flux qubit. We have also proposed a hole-spin qubit realization in a quantum dot molecule which combines lower decoherence rates than electron-spin based realizations with all electric control via g-tensor manipulation. The latter makes this realization an attractive device because it can be operated electrically rather than by a magnetic field.