Non-Markovian dynamics of open quantum systems and its application

In this thesis, we investigate non-Markovian dynamics across various physical scenarios, categorized into bosonic and finite spin baths, and apply these findings to the metrological task of quantum thermometry and the fundamental problem of quantum memory detection.

To begin, we explore non-Markovian dynamics of few-level giant atoms coupled to surface acoustic waves (SAWs), focusing on two-level and $\Lambda$-type three-level systems. With single-phonon excitation at zero temperature, we derive analytical solutions showing strong memory effects due to phonon-mediated self-feedback. Extending to finite temperatures, we study numerically thermal effects on non-Markovian behavior in two-level atoms. Comparing zero-temperature results with analytical solutions, we identify the most suitable numerical approach for simulating the dynamics at finite temperatureand show that non-Markovianity can persist at finite temperatures.

Next, we apply the solutions for non-Markovian dynamics in two- and three-level giant atom systems to quantum memory detection. We introduce a criterion for the quantumness of memory approach based on the classical memory concept, reformulating it as a convex optimization problem. Additionally, we propose a metric to quantify the difference between the quantum memory required for a process and its classical memory counterpart. This framework can be extended to high-dimensional systems beyond qubits or two-level atom systems. By using the proposed method, we demonstrate that classical memory fails to describe spontaneous emission in these systems at zero temperature. Finally, we outline an experimental approach for detecting quantum memory in giant atoms.

Moving forward, we shift the focus on non-Markovian pure dephasing of a single or few central spins embedded in a finite spin bath, introducing a spin-lattice model to efficiently simulate the reduced system dynamics.
We begin by investigating a single spin's behavior in uncorrelated and correlated spin baths, examining non-Markovian dephasing effects. The model is also applied to two case studies: entangled spins in a 2D lattice interacting with an evolving environment, and two nuclear spins in an organic molecule interacting with a thermal spin bath at finite temperatures for NMR applications.

Lastly, we employed the spin-lattice model to investigate non-Markovian phase thermometry, a non-invasive method in which a quantum spin probe interacts with a system, like a 2D Ising spin lattice, to estimate its temperature by observing the dephasing dynamics. As the system under consideration undergoes a phase transition, we evaluated the thermometric performance in the vicinity of the phase transition, utilizing the quantum Fisher information (QFI) to quantify the precision.