Multi-block exceptional points in open quantum systems

Abstract

Open quantum systems can be approximately described by non-Hermitian Hamiltonians (NHHs) and Liouvillian superoperators. The two approaches differ by quantum jump terms corresponding to a measurement of the system by its environment. We analyze the emergence of exceptional points (EPs) in NHHs and Liouvillian superoperators. In particular, we show how EPs in NHHs relate to a novel type of EPs -- multi-block EPs -- in the no-jump Liouvillian, i.e. the Liouvillian superoperator in absence of quantum jump terms. We further analyze how quantum jump terms modify the multi-block structure. To illustrate our general findings, we present two prime examples: qubits and qutrits coupled to additional ground state levels that serve as sinks of the population. In those examples, we can navigate through the EP block structure by a variation of physical parameters. We analyze how the dynamics of the population of the states is affected by the order of the EPs. Additionally, we demonstrate that the quantum geometric tensor serves as a sensitive indicator of EPs of different kinds.

Figures (13)

• Figure 1: Illustration of the $N - 1$ excited levels undergoing non-Hermitian evolution. The $\Omega_{ij}$ values represent coherent drives between pairs of excited states and the $\gamma_{j}$ values denote decay rates from the excited states to the ground state. $\Gamma_k$ parametrize the dissipation terms affecting exclusively the $(N-1)$ excited levels.
• Figure 2: Realization of a non-Hermitian qubit formed by the two upper excited states in a dissipative three-level system.
• Figure 3: Formation of a non-Hermitian qutrit via a dissipative four-level system. We consider two equal drives ($\Omega_{hi} = \Omega_{ie}$). For simplicity, we assume that all parameters in the system are real.
• Figure 4: Evolution of the population of the $\ket{i}$-level of the effective qubit, factoring out the main exponential decay. The time dependence of $e^{\tau} \rho_{ii}(\tau)$ is shown for the parameters $\Omega_{ie} = \frac{\gamma_{e} - \gamma_{i}}{4}$, $\gamma_{e} = 0.9$, $\gamma_{i} = 0.2$, and for various values of $\Gamma$. Here, $\tau = t(\gamma_i+\gamma_e)/2$.
• Figure 5: Evolution of the population of the $\ket{e}$-level of the effective qubit, factoring out the main exponential decay. The time dependence of $e^{\tau} \rho_{ee}(\tau)$ is shown for the parameters $\Omega_{ie} = \frac{\gamma_{e} - \gamma_{i}}{4}$, $\gamma_{e} = 0.9$, $\gamma_{i} = 0.2$, and for various values of $\Gamma$. Here, $\tau = t(\gamma_i+\gamma_e)/2$.