Quantum exact response theory based on the Dissipation Function

The exact response theory based on the Dissipation Function applies to general dynamical systems. In recent years, it has offered excellent results in various applications, from the field of non-equilibrium molecular dynamics to dynamical systems relevant in biological context and beyond. In this thesis, we derive its quantum translation. We first presented a brief summary of the most used quantum response theories. In many quantum dynamics, it has not yet been possible to overcome the perturbative approach, which is an approximation. Future extensions of our work could prove important in this regard. Firstly, we applied the quantization rule to the classical Dissipation Function. Then, we derived a quantum expression for calculating the expectation value of the observables over time, in a form analogous to that of the classical Dissipation Function response theory. To do this, we worked in finite-dimensional Hilbert spaces, as in the rest of the thesis. We tested the validity of the new formalism by applying it to qubit systems, which are of fundamental importance in quantum mechanics and at the basis of quantum computing. The dissipation operator used is not Hermitian. In an attempt to make it so, we derived a second expression, equivalent to the first. In addition to the Hermitian dissipation operator, the contribution of an antihermitian operator also appears in this second expression. Next, we studied the role that these dissipation operators have in the time evolution of the density operator. In conclusion, we tried to exploit the new formalism to solve Lindblad equations, which describe dissipative quantum dynamics, suggesting possible future extensions for this work.