In this work, we investigate a two-state system coupled to a generic dissipative environment. Conventionally, such systems are described using a non-Hermitian Hamiltonian within the complexified Pauli matrix algebra, and their dynamics are governed by the non-Hermitian Liouville-von Neumann equation, which is traditionally solved via matrix methods to find the density operator
ρ^(t). By leveraging the isomorphism between the Pauli matrix algebra
Mat(2,C) and the Clifford algebra of three-dimensional space
Cl3, we completely translate the non-Hermitian problem into the geometric framework of
Cl3. This approach natively decouples the dynamics into distinct geometric grades, successfully bypassing the need for cumbersome matrix techniques, such as finding the eigenvalues and eigenvectors of a
4×4 matrix. We demonstrate that the non-unitary time evolution is driven by a generalized geometric spinor that seamlessly unifies the dissipative decay and the Hamiltonian rotation. Finally, we test our general algebraic solution in the Hermitian regime, showing perfect agreement with well-known unitary results.
A mathematical and numerical framework has been worked out to represent the density operator in phase space and to propagate it in time under dissipative conditions. The representation of the density operator is based on the Fourier pseudospectral method which allows a description both in configuration as well as in momentum space. A new propagation scheme which treats the complex eigenvalue structure of the dissipative Liouville superoperator has been developed. The framework has been designed to incorporate modern computer architecture such as parallelism and vectorization. Comparing the results to closed-form solutions exponentially fast convergence characteristics in phase space as well as in the time propagation is demonstrated. As an example of its usefulness, the new method has been successfully applied to dissipation under the constraint of selection rules. More specifically, a harmonic oscillator which relaxes to equilibrium under the constraint of second-order coupling to the bath was studied.
Evolution equations of a laser-excited direct gap two-band semiconductor in the transient stage have been derived by using the first principle Liouville-von Neumann equation and the statistical projection operator method. To be more general, the photon field is described in a quantum mechanical manner. Quantum effects of the electron-photon coupoling as well as charge carrier Coulomb screening and dissipation are involved. The evolution of the time dependent coherent photon state is investigated. It results in additionally evolution equations. Numerical solutions of these systems are cast out for realistic parameters and are compared with previous results.