Local versus global master equation with common and separate baths: superiority of the global approach in partial secular approximation

Abstract

Open systems of coupled qubits are ubiquitous in quantum physics. Finding a suitable master equation to describe their dynamics is therefore a crucial task that must be addressed with utmost attention. In the recent past, many efforts have been made toward the possibility of employing local master equations, which compute the interaction with the environment neglecting the direct coupling between the qubits, and for this reason may be easier to solve. Here, we provide a detailed derivation of the Markovian master equation for two coupled qubits interacting with common and separate baths, considering pure dephasing as well as dissipation. Then, we explore the differences between the local and global master equation, showing that they intrinsically depend on the way we apply the secular approximation. Our results prove that the global approach with partial secular approximation always provides the most accurate choice for the master equation when Born-Markov approximations hold, even for small inter-system coupling constants. Using different master equations we compute the stationary heat current between two separate baths, the entanglement dynamics generated by a common bath, and the emergence of spontaneous synchronization, showing the importance of the accurate choice of approach.the case for most of the applications of the two-qubit problem, we will consider memory-less reservoirs, that is to say, we will study a Markovian master equation.Our detailed derivation allows us to establish the validity of the so-called local approach for the master equation in comparison with a global one in a rather general setting. The global approach arises naturally when deriving the master equation from a microscopic model considering the full system Hamiltonian, i.e. in presence of interactions between its subsystems (here the two qubits), while the local one follows from the approximation which neglects these interactions. Recently, the problem of characterizing the range of applicability of the local rather than global master equation has received much interest [25][26][27][28][29], mostly related to the consistency of this decription in quantum thermodynamics. It is our aim to show here that an accurate application of the secular approximation in the global approach always leads to a correct Markovian master equation, independently of the value of the coupling constant between the subsystems. The deep interconnection between a correct application of the secular approximation and the local versus global issue is discussed starting from the first principles of the derivation of the master equation. Deviations from the most accurate (global partial secular) approximation are illustrated by looking at the open system dynamics as well as the steady state. Moreover, we observe how the steady state heat current, the entanglement dynamics and the presence of quantum beats or quantum synchronization vary when using distinct master equations, so as to corroborate the validity (or inaccuracy) of each approach according to physical con...

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