In this thesis we consider the properties of a class of Z2 topological phases on a two-dimensional square lattice. The ground states of Z2 topological order are generally degenerate on a periodic lattice, characterized by certain global Z2 quantum numbers. This property is important for application in quantum computing as the global quantum numbers can be used as protected qubits. It is therefore instrumental to construct and study Z2 topological order from a general framework. Our results in this thesis provide such a framework. It is based on the simplest and most illustrative Z2 topological order: the Toric Code (TC), which contains static and non-interacting anyonic quasiparticles e, m and ε. Building on this interpretation, in the first part of the thesis two exact mappings are presented from the spin Hilbert space to the Hilbert space of (e,m) and (e,ε). The mappings are derived on infinite, open, cylindrical and periodic lattices respectively. Mutual anyonic statistics as well as the effect of the global Z2 quantum numbers are taken into account. Due to the mutual anyonic statistics of the elementary excitations, the mappings turn out to be highly non-local. In addition, it is shown that the mapping to e and ε anyons can be carried over directly to the honeycomb lattice, where the anyons become visons and Majorana fermions in the Kitaev honeycomb model. The mappings allow one to rewrite any spin Hamiltonians as Hamiltonians of anyons. In the second part of the thesis, we construct a series of spin models which are mapped to Hamiltonians of free anyons. In particular, a series of Z2 topological phases `enriched by lattice translation symmetry' are constructed which are also topological superconductors of ε particles. Their properties can be analyzed generally using the duality and then the theory of topological superconductivity. In particular, their ground state degeneracy on a periodic lattice may depend on lattice size. For these phases a classification scheme is proposed, which generalizes classification by the integer Chern number. Some of the conclusions are then verified directly by exact solutions on the spin lattice. The emergent anyon statistics of e-particles in these phases is also analyzed by computing numerically the Berry phase of their motion on top of the superconducting vacua. For phases with C=0 yet still topologically non-trivial, we discover examples of `weak symmetry breaking': the e-lattice splits into two inequivalent sublattices which are exchanged by lattice translations. The e-particles on the two sublattices acquire mutual anyonic statistics. In topological phases with non-zero C, the mutual braiding of e is confirmed explicitly. In addition, the Berry phase due to background flux of each square unit cell is quantized depending on the underlying topology of the phases. This quantity is related to properties of the vison band in Kitaev materials. Lastly, the ZN (N>2) extension of Z2 topological order is discussed. Constructing the duality to `parafermions' in this case is much more complex. The difficulties of deriving such a mapping are pointed out and we only present exact solutions to certain Hamiltonians on the spin lattice.

In this Bachelor thesis project, the Lindblad master equation is derived, both as the most general way of modeling interaction with an environment that lacks memory, and through microscopic derivations focused on assumptions about the way the system interacts with its environment (weak-coupling, Born-Markov and rotating wave approximations). It is then applied to a two-level system (qubit).

Direct detection of dark matter remains an important outstanding problem since abundant astrophysical evidence points towards its existence, but no experiment has succeeded in detecting it. Axions and axion-like-particles are some of the most compelling candidates for dark matter given their appearance in many theories of physics beyond the Standard Model and their relatively unexplored parameter space compared to other candidates. Recently, the Cosmic Axion Spin Precession Experiment-Electric (CASPEr-e) has used nuclear magnetic resonance (NMR) to search for effective magnetic fields created by axionic dark matter. By decreasing technical noise sources, CASPEr-e is projected to reach the standard quantum limit where spin projection noise is the dominant noise source limiting sensitivity. However, some axion models predict axion couplings to normal matter that would be too small for even a quantum limited CASPEr-e experiment to detect. This creates a need for surpassing the spin projection noise limit in NMR dark matter searches. In this thesis, I explore the prospects for surpassing the quantum limit in NMR by using spin squeezed states, entangled states with variance in one projection reduced below the standard quantum limit. First, I propose an experimental scheme for generating squeezed states by coupling the spins to an off-resonant circuit to create a One-Axis-Twist Hamiltonian. Then, using exact results and numerical simulations, I determine the amount of squeezing that can be achieved given decoherence and noise. Next, I perform modeling to show that squeezing can accelerate dark matter searches despite earlier results that argued squeezing cannot improve experimental sensitivity when subject to decoherence. Finally, I apply these results to the CASPEr-e experiment and show that at axion frequencies near 100MHz, squeezing can speed up the experiment by a factor of up to 30, corresponding to a sensitivity improvement by a factor of over 5.

A variant of RABBITT pump-probe spectroscopy in which the attosecond pulse train comprises both even and odd harmonics of the fundamental IR probe frequency is explored to measure time-resolved photoelectron emission in systems that exhibit autoionizing states. It is shown that the group delay of both one-photon and two-photon resonant transitions is directly encoded in the energy-resolved photoelectron anisotropy as a function of the pump-probe time-delay. This principle is illustrated for a 1D model with symmetric zero-range potentials that supports both bound states and shape-resonances. The model is studied using both perturbation theory and solving the time-dependent Schodinger equation on a grid. Moreover, we study the case of a realistic atomic system, helium. In both cases, we demonstrate faithful reconstruction of the phase information for resonant photoemission.

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