Interplay of quantum stochastic and dynamical maps to discern Markovian and non-Markovian transitions

TL;DR

This paper explores the relationship between quantum stochastic maps and dynamical maps to distinguish Markovian from non-Markovian quantum processes without using master equations, supported by four diverse physical examples.

Contribution

It introduces a framework using quantum stochastic and dynamical maps to identify Markovian and non-Markovian dynamics without master equations, demonstrated through four different models.

Findings

Eigenvalues of the intermediate dynamical B map indicate Markovian or non-Markovian behavior.

Positivity of the B map eigenvalues correlates with Markovian dynamics.

The approach applies to various physical systems, including qubits and photon polarization.

Abstract

It is known that the dynamical evolution of a system, from an initial tensor product state of system and environment, to any two later times, t1,t2 (t2>t1), are both completely positive (CP) but in the intermediate times between t1 and t2 it need not be CP. This reveals the key to the Markov (if CP) and nonMarkov (if it is not CP) avataras of the intermediate dynamics. This is brought out here in terms of the quantum stochastic map A and the associated dynamical map B -- without resorting to master equation approaches. We investigate these features with four examples which have entirely different physical origins (i) a two qubit Werner state map with time dependent noise parameter (ii) Phenomenological model of a recent optical experiment (Nature Physics, 7, 931 (2011)) on the open system evolution of photon polarization. (iii) Hamiltonian dynamics of a qubit coupled to a bath of $N$ qubits and (iv) two qubit unitary dynamics of Jordan et. al. (Phys. Rev. A 70, 052110 (2004)) with initial product states of qubits. In all these models, it is shown that the positivity/negativity of the eigenvalues of intermediate time dynamical B map determines the Markov/non-Markov nature of the dynamics.