Role of inefficient measurement in realizing post-selection-based non-Hermitian qubits

TL;DR

This work investigates how inefficient post-selection impacts the realization of a non-Hermitian qubit in a three-level system. By developing a hybrid-Liouvillian framework and comparing monitored quantum trajectories with Lindblad dynamics, it shows that imperfect detection can split a third-order EP into second-order EPs and induce decoherence in the excited manifolds. The analysis covers both no-jump (post-selected both dips) and jump cases (one jump not post-selected), highlighting how measurement backaction and inefficiency reshape the Liouvillian spectrum and EP structure. The findings offer fundamental insights into non-Hermitian behavior in open quantum systems under realistic measurement conditions and provide guidance for designing experiments with inefficient post-selection.

Abstract

Post-selecting against quantum jumps into the ground state confines the evolution of the three-level system to the excited states manifold, effectively realizing a PT-symmetric non-Hermitian qubit. In this work, by introducing post-selection efficiencies for both decay channels, the second-excited to first-excited and the first-excited to ground-state transitions, we formulate a hybrid-Liouvillian framework that captures the unmonitored dynamics of the non-Hermitian qubit. We find that the decoherence effects arising from quantum jumps within the second-excited and first-excited manifold also manifest under inefficient post-selection of the second-excited to first-excited transitions, thereby modifying the spectral properties of the Liouvillian and leading to a splitting of the exceptional points. A comparative analysis shows that the trajectory-based approach, obtained by ensemble-averaging stochastic measurement trajectories generated via the Bayesian state update rule, and the Lindblad evolution remain consistent. Our results highlight the fundamental role of measurement inefficiency in realizing post-selection-based non-Hermitian qubits and in shaping the structure of Liouvillian exceptional points. These findings provide new insights into how inefficient measurement processes influence non-Hermitian behavior in open quantum systems.

Figures (8)

• Figure 1: Schematic of the three-level system with monitored decay channels. The excited states $\ket{f}$ and $\ket{e}$ decay via $\ket{f} \rightarrow \ket{e}$ and $\ket{e} \rightarrow \ket{g}$ transitions with rates $\Gamma_e$ and $\Gamma_g$, respectively. These decays are monitored by photodetectors $\mathcal{D}_e$ and $\mathcal{D}_g$. The stochastic processes $dN_1^e$ and $dN_1^g$ register quantum jumps at these detectors with detection efficiencies $\eta_e$ and $\eta_g$. The processes $dN_e^L$ and $dN_g^L$ correspond to undetected jumps due to imperfect efficiency ($1-\eta_{e,g}$). The direct transition $\ket{f} \rightarrow \ket{g}$ is forbidden. The state of the system is updated based on the measurement outcome of detector $\mathcal{D}_{eg}$
• Figure 2: Quantum trajectories and their ensemble average of a three-level system undergoing spontaneous emission. (a)–(c) Quantum jump trajectories of the system represented in terms of the populations of the states $\rho_{g,e,f}(t)$ as it undergoes spontaneous emission, or quantum jumps, from the excited states to the ground state, $\ket{f} \to \ket{e} \to \ket{g}$. There are 50 trajectories in the plot. Each trajectory evolves as follows: the system initially occupies the state $\ket{f}$, such that $\rho_f(t) = 1$ and $\rho_e(t) = \rho_g(t) = 0$, until a quantum jump from $\ket{f}$ to $\ket{e}$ occurs. After the jump, the system remains in the $\ket{e}$ state ($\rho_e(t) = 1$, $\rho_f(t) = \rho_g(t) = 0$) until a second quantum jump from $\ket{e}$ to $\ket{g}$ takes place. Once the system reaches the ground state $\ket{g}$, it stays there for the remainder of the evolution ($\rho_g(t) = 1$, $\rho_f(t) = \rho_e(t) = 0$). (d) The ensemble average of all the trajectories. The solid lines represent the average of $10^4$ trajectories, which matches perfectly with the Lindblad dynamics represented by the dashed lines. The blue, red, and green lines correspond to the average populations of the $\ket{f}$, $\ket{e}$, and $\ket{g}$ states, respectively. Parameters are $\Gamma_g = 4~\text{MHz}$ and $\Gamma_e = 0.2~\text{MHz}$.
• Figure 3: Real and imaginary parts of the Liouvillian eigenvalue spectrum as a function of $\Omega~\text{(MHz)}$ are shown for the no-jump (a)–(b) and jump (c)–(d) cases. In the no-jump case, solid lines correspond to $\eta_e = 1$, while dotted lines correspond to $\eta_e = 0.6$. For the jump case, the spectrum is independent of $\eta_e$. In both cases, $\eta_g = 1$. Other common parameters are $\Gamma_g = 4~\text{MHz}$ and $\Gamma_e = 0.2~\text{MHz}$.
• Figure 4: Evolution of the state populations for the no-jump case where post-selection is applied to both the $\ket{f} \rightarrow \ket{e}$ and $\ket{e} \rightarrow \ket{g}$ jumps. Dashed lines show the normalized populations derived from Liouvillian dynamics. Solid lines represent the population derived from average trajectory dynamics, with blue, red, and green corresponding to the populations of $\ket{f}$, $\ket{e}$, and $\ket{g}$ states, respectively. (a) corresponds to the post-selection efficiencies $\eta_g = \eta_e = 1$. For inefficient detection, $\eta_e = 0.75$ with $\eta_g = 1$ is shown in (b). Panels (c) and (d) correspond to $\eta_g = 0.75, \eta_e = 1$ and $\eta_g = 0.75, \eta_e = 0.75$, respectively. The black dotted line indicates the time period of coherent oscillation of the population dynamics, $t_c = \pi / \mathrm{Re}[\Phi_0] = 1.61~\mu\mathrm{s}$, where $\Phi_0$ is the first eigenvalue of the non-Hermitian Hamiltonian in Eq. (\ref{['eq:eff_hamiltonian']}). Simulations were performed with $10^5$ trajectories. Other parameters are $\Gamma_g = 4~\mathrm{MHz}$ and $\Gamma_e = 0.2~\mathrm{MHz}$, $\Omega = 2~\mathrm{MHz}$ with.
• Figure 5: Evolution of the state populations for the no-jump case for $\Omega = 6~\mathrm{MHz}$. Dashed lines show the normalized populations derived from Liouvillian dynamics. Solid lines represent the population derived from average trajectory dynamics, with blue, red, and green corresponding to the populations of $\ket{f}$, $\ket{e}$, and $\ket{g}$ states, respectively. (a) corresponds to the post-selection efficiencies $\eta_g = \eta_e = 1$. For inefficient detection, $\eta_e = 0.75$ with $\eta_g = 1$ is shown in (b). Panels (c) and (d) correspond to $\eta_g = 0.75, \eta_e = 1$ and $\eta_g = 0.75, \eta_e = 0.75$, respectively. The black dotted line indicates the time period of coherent oscillation of the population dynamics, $t_c = 0.52\mu\mathrm{s}$. Simulations were performed with $10^5$ trajectories. Other parameters are $\Gamma_g = 4~\mathrm{MHz}$ and $\Gamma_e = 0.2~\mathrm{MHz}$.