Numerical implementation of the partial secular approximation and unified master equation in structured open quantum systems

TL;DR

This paper introduces a versatile numerical code for implementing the partial secular approximation and unified master equation in structured open quantum systems, enabling more accurate modeling of their dynamics beyond traditional approximations.

Contribution

The work presents a general computational tool for applying the partial secular approximation and unified master equation to complex multipartite quantum systems with structured environments.

Findings

The code accurately reproduces the dynamics of open quantum systems under the partial secular approximation.

It can generate both local and global master equations for the same physical setup.

Application to superconducting qubits demonstrates the code's effectiveness in studying steady-state heat flow.

Abstract

The Markovian dynamics of open quantum systems is typically described through Lindblad equations, which are derived from the Redfield equation via the full secular approximation. The latter neglects the rotating terms in the master equation corresponding to pairs of jump operators with different Bohr frequencies. However, for many physical systems this approximation breaks down, and thus a more accurate treatment of the slowly rotating terms is required. Indeed, more precise physical results can be obtained by performing the partial secular approximation, which takes into account the relevant time scale associated with each pair of jump operators and compares it with the time scale arising from the system-environment coupling. In this work, we introduce a general code for performing the partial secular approximation in the Redfield equation for structured open quantum systems. The code can be applied to a generic Hamiltonian of any multipartite system coupled to bosonic baths. Moreover, it can also reproduce the unified master equation, which captures the same physical behavior as the Redfield equation under the partial secular approximation, but is mathematically guaranteed to generate a completely positive dynamical map. Finally, the code can compute both the local and global version of the master equation for the same physical problem. We illustrate the code by studying the steady-state heat flow in a structured open quantum system composed of two superconducting qubits, each coupled to a bosonic mode, which in turn interacts with a thermal bath. The results in this work can be employed for the numerical study of a wide range of complex open quantum systems.