Optimal control of driven quantum many-body systems

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Driven systems offer an exciting analytic tool to investigate clean realizations of solid state models, featuring high temperature super-conductivity, fractional quantum Hall effect, spin Hall effect and states with topological order. One of the most promising implementations is based on optical lattices in which atoms are trapped in a standing laser field. Optical lattices permit to trap atoms in lattices of different geometries, and the dynamics of the atoms can be controlled by periodically modulating (shaking) the lattice. A central limitation on current experiments is that the shaking heats up the sample, what ultimately destroys the desired effects. In practice, heating is controlled merely in terms of the choice of frequency with which the system is shaken. If this frequency is far off the resonance frequencies of the system, then the heating is not too large. One would, however, expect that heating can be substantially reduced if destructive interference prevents atoms from occupying highly excited states. For this purpose, one can drive the system poly-chromatically and identify driving profiles that minimize the occurrence of undesired processes. For reaching such a goal different techniques can be employed. On the one hand a variety of perturbative techniques have been employed to deal with periodically driven systems. It provides a path to analytically find optimal modulation of the system in order to engineer effective dynamics of interest while ensuring the suppression of detrimental side-effects. This approach is limited by an exponential increase in complexity when many body interactions come into play. Another way would be through the use of Machine Learning (ML) techniques. ML has been successfully employed in finding optimal control schemes for classical systems, however only more recently it has been successfully applied to quantum system. Research has been driven by results showing in that solving any optimization of quantum control problem should be easy. Examples of the use of ML for quantum systems include the preparation of target quantum states with high fidelities, cooling procedures of ultra-cold gazes, shaping of ultrafast light pulses, and coherent manipulation of matter waves. Among the plethora of possible techniques, (physical) model-free approaches (e.g. reinforcement learning, Bayesian inferences, evolutionary algorithms) provide interesting alternatives to the more commonly used gradient descent methods. This project is in collaboration with an experimental group at Cambridge setting up modulated optical lattices, with whom I can discuss realistic experimental constraints and the viability of the control schemes developed. In order to prepare for the project I did a placement at Microsoft Research to familiarize myself with ML (reinforcement learning) techniques.Driven systems offer an exciting analytic tool to investigate clean realizations of solid state models, featuring high temperature super-conductivity, fractional quantum Hall effect, spin Hall effect and states with topological order. One of the most promising implementations is based on optical lattices in which atoms are trapped in a standing laser field. Optical lattices permit to trap atoms in lattices of different geometries, and the dynamics of the atoms can be controlled by periodically modulating (shaking) the lattice. A central limitation on current experiments is that the shaking heats up the sample, what ultimately destroys the desired effects. In practice, heating is controlled merely in terms of the choice of frequency with which the system is shaken. If this frequency is far off the resonance frequencies of the system, then the heating is not too large. One would, however, expect that heating can be substantially reduced if destructive interference prevents atoms from occupying highly excited states.