Executive Summary : | The control of quantum systems through external periodic driving has become a significant field of research, known as Floquet engineering. A general formalism has been developed to design periodic driving protocols to steer simple systems to more exotic desired systems, which are based on the underlying Lie algebraic structure in their Hamiltonians. This proposal aims to extend this formalism to open quantum systems, a more realistic scenario where quantum systems interact with their surrounding environment. To do so, the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation must be cast into a form that displays an underlying Lie algebraic structure. The third-quantization formalism, introduced to solve quadratic open quantum systems of many-particle fermions, is also proposed to be extended to time-periodic systems to extract a Lie algebraic structure in Floquet systems. Preliminary investigations indicate that the third-quantization formalism can cast specific quadratic open quantum fermionic systems into a form with underlying Lie algebraic structure. In open quantum systems, the central object of interest is investigating non-equilibrium steady states (NESS), which survive under dissipation and can have one or multiple NESSs. The multiplicity of NESSs is determined by the presence and absence of symmetries in the systems. The proposed driving protocol will introduce symmetries in open quantum systems without any symmetry, indicating the presence of decoherence-free subspaces (DFS) in the system. This approach can be used for error-free quantum computation. |