A mathematical justification to apply the secular approximation to the Redfield equation

TL;DR

This work provides a rigorous mathematical justification for the secular approximation by deriving the Quantum Optical Master Equation (QOME) from the Redfield equation through a self-consistent truncation at the same order in the system–bath coupling. It shows that the secular-approximated dynamics emerge naturally within the Redfield framework and are in the same equivalence class as both the Redfield and Universal Lindblad Equations. Analytical arguments, complemented by numerical evidence on a two-level system, indicate that the QOME can yield more accurate results than the Universal Lindblad Equation while preserving the Lindblad form. The results clarify the validity of the secular approximation and offer guidance on choosing between Lindblad-type master equations in open quantum systems, with caveats about the necessity of system eigenstructure knowledge for QOME construction.

Abstract

Quantum master equations are widely used to describe the dynamics of open quantum systems. All these different master equations rely on specific approximations that may or may not be justified. Starting from a microscopic model, applying the standard Born and Markov approximations results in the Redfield equation that does not guarantee to preserve positivity. The latter is typically achieved by additionally applying the secular approximation resulting in a quantum master equation in Lindblad form. There are other ways to obtain an equation in Lindblad form, one of which is the recently proposed Universal Lindblad Equation. It has been shown that it is in the same equivalence class of approximations as the Redfield master equation although avoiding the heuristic secular approximation [arXiv:2004.01469]. In this work, we prove that the solutions of the master equation obtained by applying the secular approximation are also obtained by an approximation of the same order as the one performed to obtain the Redfield equation. We hereby provide a mathematical justification for the secular approximation. We show that the result of applying the secular approximation is obtained naturally by applying a self-consistency argument. This shows that the resulting master equation is also in the same equivalence class of approximations as the Redfield master equation and the Universal Lindblad Equation. We furthermore compare it to the Universal Lindblad Equation numerically and show numerical evidence that the master equation obtained through the secular approximation yields more accurate solutions.

Figures (4)

• Figure 1: Illustration of the simplification process for the matrix. Only permutations that swap $\mathcal{O}(g^2)$-elements on the diagonal are of lowest order in $g^2$. This means we only have to consider the determinant of the reduced matrix $\tilde{\mathcal{A}}$, which consists of the $\mathcal{O}(g^2)$-elements together with the squares in the figure.
• Figure 2: Comparison of the resulting superoperators of the Universal Lindblad equation (ULE) and the Quantum Optical master equation (QOME) with respect to the Redfield equation. We see that in the case of the Universal Lindblad equation, in addition to the matrix elements that can be compared to $\omega$, the other matrix elements are also changed.
• Figure 3: Comparison of the time dependent expectation values of the Pauli operators $\langle \hat{\sigma}_i\rangle$ after evolving a TLS coupled to a JC bath with the ULE, Redfield Equation (RE) and QOME respectively indicated in blue, red and green. Setting $E = 1$, so that $t_{\text{final}} = 10/E$, the parameters used to simulate the plotted evolution are as follows: $\alpha = 0.2$, $T = 0.05$, $\theta = {\pi}/{2}$, $\phi = {\pi}/{4}$, $\varphi = {\pi}/{2}$, $\Omega = 50$.
• Figure 4: Histograms showing the number of instances whose final states (at a given time $t_f=10/E_1$) have a trace distance smaller than a certain value $\mathcal{D}_{\text{thres}}=0.01$ as a function of the coupling to the bath $\alpha$ and the temperature $T$. The parameters $\alpha$ and $T$ are linearly sampled within the intervals in Table \ref{['Tab: ParamsLin']}, while the mixing angle and the Bloch sphere angles are randomly chosen within the ranges in Table \ref{['Tab: ParamsInit']}. Plots a. and c.: $\mathcal{D}_{\text{UL-RE}}(t_f)< 0.01$ for an Ohmic and a JC bath respectively. Plots b. and d.: $\mathcal{D}_{\text{QO-RE}}(t_f)< 0.01$ for an Ohmic and a JC bath respectively.