Emergent Liouvillian exceptional points from exact principles

TL;DR

The paper demonstrates that Liouvillian exceptional points (EPs) in open quantum systems can arise from exact principles, not just master-equation approximations. Using an exact Heisenberg-equation treatment of a dissipative double quantum dot, it identifies a second-order EP with $\eta^{HE}=0$ that exactly mirrors the master-equation EP with $\eta^{ME}=0$, revealing a shared square-root structure $\eta$ and a deep HE–ME correspondence. It shows that critical damping and a Mpemba-like faster relaxation to steady state persist beyond the Markovian/weak-coupling regime, with HE exhibiting a second-order EP and ME a third-order EP, related by $n$-th order HE EPs mapping to $(2n-1)$-th order ME EPs. The results extend to chain-like generalizations, suggesting EPs are rooted in fundamental open-system dynamics and warrant further exploration beyond the wide-band limit and simple models.

Abstract

Recent years have seen a surge of interest in exceptional points in open quantum systems. The natural approach in this area has been the use of Markovian master equations. While the resulting Liouvillian EPs have been seen in a variety of systems and have been associated to numerous exotic effects, it is an open question whether such degeneracies and their peculiarities can persist beyond the validity of master equations. In this work, taking the example of a dissipative double-quantum-dot system, we show that exact Heisenberg equations governing system and bath dynamics exhibit the same EPs as the corresponding master equations. To highlight the importance of this finding, we prove that the paradigmatic property associated to EPs - critical damping, persists well beyond the validity of master equations. Our results demonstrate that Liouvillian EPs can arise from underlying fundamental exact principles, rather than merely as a consequence of approximations involved in deriving master equations.

Figures (3)

• Figure 1: (a) A two-terminal double quantum dot setup, with dot energies $\epsilon_d^{(j)}$, tunnel-coupling strength $g$ and reservoir couplings $\Gamma_j$ ($j=1,2$). (b) Riemann sheets corresponding to the eigenvalues of the Heisenberg evolution matrix $A$, in the space of the detuning ($\epsilon_d^{(2)}-\epsilon_d^{(1)}$) and $g$. The EP (depicted as a red dot) lies at zero detuning. We therefore consider resonant dots ($\epsilon_d^{(j)}\equiv\epsilon_d$) throughout this work.
• Figure 2: The population of dot 1 normalized by its steady state value, $\expval{\hat{N}_1(t)}/\expval{\hat{N}_1}_{\text{ss}}$, as function of time for (a) strong and (b) weak coupling, obtained with HE. The insets show the long-time behaviour. The dashed curves in (b) show master equation predictions. Common parameters: $T_1=1$, $T_2=0.1T_1$$\epsilon_d= T_1$, $\mu_1=\mu_2=0$. Specific parameters: (a) $\Gamma_1 =0.5T_1$, $\Gamma_2=0.1T_1$, $g=3T_1$ (underdamping), $5\times 10^{-2}T_1$ (overdamping), $0.1T_1$ (EP) (b) $\Gamma_1 =10^{-2}T_1$, $\Gamma_2=10^{-3}T_1$, $g=5\times10^{-2}T_1$ (underdamping), $10^{-3}T_1$ (overdamping), $2.25\times10^{-3}T_1$ (EP).
• Figure 3: $\mathcal{R}_1$ as a function of time for strong and weak coupling, obtained with HE. $\mathcal{R}_1=1$ is marked with the dashed-gray line. The initial populations are chosen such that $\mathcal{R}_1>1$ at $t=0$. The parameters are taken from Fig. \ref{['fig:regimes']} (a) and (b), respectively. Similar plots can be obtained for $\mathcal{R}_2$.