Ultra cold lattice gases, neutral bosonic and fermionic atoms confined in highly controlled optical potentials, are already one of the most successful platforms for analog quantum simulation, capable of outperforming classical computers on a range of tasks related e.g. to the theory of correlated quantum matter. This PhD-project will further deepen the power of this quantum simulation platform. The project will do so by designing novel experiments using tools from quantum many-body theory. This theory will build on recent beyond-state-of-the-art numerical many-body algorithms originating from the supervisor’s group, matrix product states plus mean-field theory (MPS+MF) [1,2,3], as well as massively parallelized density matrix renormalization group numerics (pDMRG) [4]. Utilizing these tools, the successful PhD-applicant will pursue three interlinked cutting-edge objectives: Design experiments for ultra cold lattice gases to establish these as simulators of non-equilibrium dynamics leading to ordered many-body states above critical temperature (Tc). Dynamically-induced ordering is a growing field, especially in high-Tc superconducting materials [5]. The prospect of obtaining even short-lived superconducting states well above Tc – and possibly close to room temperature – from quenching these materials is driving much experimental activity on high-Tc materials. However, the theoretical understanding of how ordered states above Tc might actually be generated from non-equilibrium dynamics remains rudimentary. This part of the PhD-project will deploy the MPS+MF framework to show in detail as to how ultra cold lattice gases can be turned into quantum simulators for dynamically generated ordering above Tc for three types of many-body states: Bose-Einstein condensates, Mott insulators and superconductors. Design experiments using ultra cold lattice gases on how to generate analogue states of high-Tc superconductivity in a metastable, mixed-dimensional equilibrium state. With the basic mechanism for high-Tc superconductivity still not understood even in the most simplified single-band models such as the 2D Hubbard- and tJ-models, ultra cold lattice gases have long been pursued as quantum simulators for this phenomenon. However, the magnetic superexchange energies achievable in ultra cold lattice gases were too low to realize such states. The advent of mixed-dimensional metastable states has radically changed this, boosting the possible magnetic superexchange energies to exceed the experimentally feasible temperatures [6]. This part of the PhD-project will utilize the MPS+MF framework to design experiments capable of finally realizing the analogue of a high-Tc superconducting states in ultra cold lattice gases. Lowering the experimentally achievable entropy in ultra cold lattice gas-based quantum simulators. Quantum simulators based on ultra cold lattice gases work at a fixed entropy per particle. The lower this crucial quantity is, the more useful will this platform be for quantum simulation, such as e.g. for quantum chemistry via variational energy minimization. This part of the PhD-project builds on proposals by the supervisor to lower the entropy per particle by dynamically disentangling two layers in such a way as to shift entropy out of one of the layers, resulting in a low-entropy layer [7]. This part of the project will be about modelling concrete experiments using pDMRG, finding schemes that perform especially well.