Conditions for bounded solutions of non-Markovian quantum master equations

Abstract

For the most general quantum master equations, also called Nakajima–Zwanzig or non-Markovian equations, we define suitable boundedness conditions on integral kernels and inhomogeneity terms in order to derive with mathematical rigor an upper bound on solutions, as required by the von Neumann conditions. Such equations are of importance for quantum dynamics of open systems with arbitrary couplings to environment and arbitrary entangled initial states. The derivation is based on an equivalent coherence-vector representation in finite dimension n leading to coupled Volterra integro-differential equations of second kind and convolution type in an (n 2−1)-dimensional real vector space. As examples, analytical and numerical model solutions are worked out for 2-level systems in order to test suitable trial functions for input quantities. All this is motivated by the fact that exact solutions can hardly be found but appropriate trial functions may provide a reasonable semiphenomenological description of complicated quantum dynamics.