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

TL;DR

This work presents a numerical framework to implement the partial secular approximation (PSA) of the Redfield equation and the unified master equation (UME) for structured open quantum systems. It supports generic multipartite Hamiltonians coupled to bosonic baths and facilitates both local and global master equations, with optional symmetry-based block-diagonalization to reduce computational cost. The authors demonstrate the approach on a qubit–resonator chain, showing that PSA and the UME reproduce physically meaningful dynamics in regimes where the full secular approximation fails and that the unified method offers substantial speed advantages. Overall, the code enables accurate, scalable simulations of complex open quantum systems, with practical benefits for studying quantum transport and thermodynamics in structured environments.

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.

Figures (11)

• Figure 1: The system with Hamiltonian $H_\text{S}$ interacts with an environment with Hamiltonian $H_\text{B}$, making the system open. The interaction strength is given by the dimensionless coupling coefficient $\alpha$, assumed to be weak such that $\alpha\ll1$, and the interaction is mediated by the interaction Hamiltonian $H_\text{I}$. Figure adapted from breuer2002theory.
• Figure 2: Left: structure of energy levels of the system Hamiltonian for which the full secular approximation can be safely applied. The different Bohr frequencies are always well-separated, in such a way that their inverse is much smaller than the relaxation time scale. Right: in the structure of the energy levels two jump frequencies are very close to each other. In this case, their difference can be small, which means that the full secular approximation cannot be directly applied.
• Figure 3: Pictorial representation of the Bohr frequency grouping. Each cluster $\mathcal{F}_{\bar{\omega}}$ is drawn with different color and they are clearly separated from each other. A cluster frequency $\bar{\omega}$ is selected to be the average Bohr frequency of each cluster. The jump operator of a cluster $A(\bar{\omega})$ is given by the sum of all jump operators $A(\omega)$ whose frequencies $\omega$ belong to that cluster.
• Figure 4: Block structure of the Liouvillian superoperator arising from the symmetry between the $\mathcal{L}$ and the total number of particles superoperator $\mathcal{N}$. The Liouvillian is block diagonalized such that each block can be labeled by an eigenvalue $d$ of $\mathcal{N}$. Here, $d$ is equal to the difference in the number of particles between the left and right sides of the tensor product structure of the basis states $\ket{n,m}\otimes\ket{n', m'}$ in Liouvillian space. Figure modified from Vaaranta2022.
• Figure 5: Circuit diagram of the quantum system composed of two qubits and two resonators with coupling two resistors on either side.