Targeting Eigenstates by "Simulated Measurements" using a Decoherence based Nonlinear Schr\"odinger Equation

Inspired by the idea of mimicking the measurement on a quantum system through a decoherence process to target specific eigenstates based on Born's law, i.e. the hiearchy of probabilities instead of the hierarchy of eigenvalues, we transform a Lindblad equation for the reduced density operator into a nonlinear Schr\"{o}dinger equation to obtain a computationally feasible simulation of the decoherent dynamics in the open quantum system. This gives the opportunity to target the eigenstates which have the largest $L^2$ overlap with an initial superposition state and hence more flexibility in the selection criteria. One can use this feature for instance to approximate eigenstates with certain localization or symmetry properties. As an application of the theory we discuss \textit{eigenstate towing}, which relies on the perturbation theory to follow the progression of an arbitrary subset of eigenstates along a sum of perturbation operators with the intention to explore for example the effect of interactions on these eigenstates. The easily parallelizable numerical method shows an exponential convergence and its computational costs scale linear for sparse matrix representations of the involved Hermitian operators.