General theory of slow non-Hermitian evolution

TL;DR

This work develops a general theory of slow non-Hermitian evolution, separating a noiseless adiabatic regime from a noise-assisted regime. It proves a non-Hermitian adiabatic theorem and shows that, with noise, the final state is determined by the end-point spectrum rather than the entire trajectory, often aligning with the end-point fastest growing eigenstate. The framework explains a broad class of anomalous state-conversion phenomena, including decoupling of chirality from exceptional points, and provides quantitative tools for predicting outcomes without full time evolution. The results have practical implications for designing non-Hermitian devices and extend to quantum Lindbladian and hybrid Liouvillian dynamics, with clear experimental verification paths.

Abstract

Non-Hermitian systems are widespread in both classical and quantum physics. The dynamics of such systems has recently become a focal point of research, showcasing surprising behaviors that include apparent violation of the adiabatic theorem and chiral topological conversion related to encircling exceptional points (EPs). These have both fundamental interest and potential practical applications. Yet the current literature features a number of apparently irreconcilable results. Here we develop a general theory for slow evolution of non-Hermitian systems and resolve these contradictions. We prove an analog of the adiabatic theorem for non-Hermitian systems and generalize it in the presence of uncontrolled environmental fluctuations (noise). The effect of noise turns out to be crucial due to inherent exponential instabilities present in non-Hermitian systems. Disproving common wisdom, the end state of the system is determined by the final Hamiltonian only, and is insensitive to other details of the evolution trajectory in parameter space. Our quantitative theory, leading to transparent physical intuition, is amenable to experimental tests. It provides efficient tools to predict the outcome of the system's evolution, avoiding the need to follow costly time-evolution simulations. Our approach may be useful for designing devices based on non-Hermitian physics and may stimulate analyses of classical and quantum non-Hermitian-Hamiltonian dynamics, as well as that of quantum Lindbladian and hybrid-Liouvillian systems.

Figures (9)

• Figure 1: Open-trajectory evolution under Hamiltonian (\ref{['eq:nHH_examples']}). The most growing state wins, in qualitative agreement with the naïve theory of Sec. \ref{['subsec:Summary_Perfect-slow-evolution']}. This also agrees with the advanced theory of Sec. \ref{['subsec:Summary_Slow-evolution-with-fast-noise']}, as the most growing state is also the end-point fastest growing one. (a) The trajectory in the parameter plane (orange) where the trajectory direction is shown with arrows and the orange dot depicts the starting point. The trajectory corresponds to the following parameters in Eq. (\ref{['eq:trajectory_general']}): $\delta_{0}=0$, $g_{0}=1$, $R=0.3$, $T=500$, $\omega=-\pi/T$, $\phi=0.4\pi$. The black dots correspond to the EPs of the Hamiltonian, and the dashed lines correspond to the branch cuts of $\lambda_{\pm}$. (b) The trajectory of the eigenvalues $\lambda_{\pm}$ with arrows showing the trajectory direction. The eigevalues $\lambda_{\pm}$ at the start of the parameter trajectory are depicted as blue and red dots. (c) Riemann surface plot of $\mathrm{Im}\,\lambda_{\pm}(t)$ (red surface for $\lambda_{-}$, blue surface for $\lambda_{+}$). The state trajectory (black), with black dot depicting the starting point and the arrow showing the trajectory direction, corresponds to $\left|c_{+}(t)\right|^{2}\mathrm{Im}\,\lambda_{+}(t)+\left|c_{-}(t)\right|^{2}\mathrm{Im}\,\lambda_{-}(t)$. (d) The integrals $\mathrm{Im}\int_{0}^{t}dt\,\lambda_{\pm}(t)$ that govern the non-Hermitian state conversion under the naïve theory. (e) Population dynamics: the prediction of the naïve theory vs the numerical simulation. While qualitatively they agree --- the state is ultimately converted to $\left|\psi_{+}\right\rangle$ --- the location of the population switch is not the same ( $t_{1}\approx0.8T$ vs $t_{1}'\apprle0.7T$).
• Figure 2: Open-trajectory evolution under Hamiltonian (\ref{['eq:nHH_examples']}). The most growing state loses, in disagreement with the naïve theory of Sec. \ref{['subsec:Summary_Perfect-slow-evolution']}. The end-point fastest growing state wins, in agreement with the advanced theory of Sec. \ref{['subsec:Summary_Slow-evolution-with-fast-noise']}, which accounts for the errors of numerical simulation (in this example) and Hamiltonian control (in real experiments). (a) The trajectory in the parameter plane (orange) where the trajectory direction is shown with arrows and the orange dot depict the starting point. The trajectory corresponds to the following parameters in Eq. (\ref{['eq:trajectory_general']}): $\delta_{0}=0$, $g_{0}=1$, $R=0.3$, $T=500$, $\omega=\pi/T$, $\phi=-0.6\pi$. The black dots correspond to the EPs of the Hamiltonian, and the dashed lines correspond to the branch cut of $\lambda_{\pm}$. (b) The trajectory of the eigenvalues $\lambda_{\pm}$ with arrows showing the trajectory direction. The eigevalues $\lambda_{\pm}$ at the start of the parameter trajectory are depicted as blue and red dots. (c) Riemann surface plot of $\mathrm{Im}\,\lambda_{\pm}(t)$ (red surface for $\lambda_{-}$, blue surface for $\lambda_{+}$). The state trajectory (black) , with black dot depicting the starting point and the arrow showing the trajectory direction, corresponds to $\left|c_{+}(t)\right|^{2}\mathrm{Im}\,\lambda_{+}(t)+\left|c_{-}(t)\right|^{2}\mathrm{Im}\,\lambda_{-}(t)$. (d) The integrals $\mathrm{Im}\int_{0}^{t}dt\,\lambda_{\pm}(t)$ that govern the non-Hermitian state conversion under the naïve theory. (e) Population dynamics: the prediction of the naïve theory vs the numerical simulation. The naïve theory predicts conversion to the most growing state $\left|\psi_{+}\right\rangle$, whereas the numerical simulation results in conversion to $\left|\psi_{-}\right\rangle$. The latter is in agreement with the advanced theory, as $\left|\psi_{-}\right\rangle$ is the end-point fastest growing state.
• Figure 3: Non-chiral state conversion when encircling an exceptional point. The numerical result is in clear discrepancy with the predictions of the naïve theory. The advanced theory explains the numerical result, as $\left|\psi_{-}\right\rangle$ is the end-point fastest growing state for both encircling directions. (a) The trajectory for clockwise encircling in the parameter plane (orange) corresponds to the following parameters in Eq. (\ref{['eq:trajectory_general']}): $\delta_{0}=0$, $g_{0}=1$, $R=0.3$, $T=500$, $\omega=-2\pi/T$, $\phi=-3\pi/4$. The trajectory direction is shown with arrows and the orange dot depict the starting point. The black dots correspond to the EPs of the Hamiltonian, and the dashed lines correspond to the branch cuts of $\lambda_{\pm}$. (b) The trajectory of the eigenvalues $\lambda_{\pm}$ corresponding to the clockwise trajectory. The eigevalues $\lambda_{\pm}$ at the start of the parameter trajectory are depicted as blue and red dots, and the arrows show the trajectory direction. (c) The population dynamics for clockwise encircling when the system is initialized in $\left|\psi_{-}\right\rangle$. (d) The population dynamics for clockwise encircling when the system is initialized in $\left|\psi_{+}\right\rangle$. (e) The trajectory for counterclockwise encircling in the parameter plane (orange) corresponds to the following parameters in Eq. (\ref{['eq:trajectory_general']}): $\delta_{0}=0$, $g_{0}=1$, $R=0.3$, $T=500$, $\omega=2\pi/T$, $\varphi=-3\pi/4$. (f,g,h) Same as (b,c,d), but for the counterclockwise trajectory.
• Figure 4: Chiral state conversion without encircling an exceptional point. The naïve theory predicts no conversion whatsoever, while the numerics predicts conversion to $\left|\psi_{-}\right\rangle$ ($\left|\psi_{+}\right\rangle$) for clockwise (counterclockwise) trajectory. This is explained by the advanced theory as the respective states are the end-point fastest growing in the respective cases. (a) The clockwise trajectory (orange) corresponds to the following parameters in Eq. (\ref{['eq:trajectory_general']}): $\delta_{0}=0$, $g_{0}=0.5$, $R=0.3$, $T=500$, $\omega=-2\pi/T$, $\phi=0$. The trajectory direction is shown with arrows and the orange dot depict the starting point. The black dots correspond to the EPs of the Hamiltonian, and the dashed lines correspond to the branch cuts of $\lambda_{\pm}$. (b) The trajectory of the eigenvalues $\lambda_{\pm}$ corresponding to the clockwise trajectory. The eigevalues $\lambda_{\pm}$ at the start of the parameter trajectory are depicted as blue and red dots, and the arrows show the trajectory direction (c) The population dynamics for clockwise encircling when the system is initialized in $\left|\psi_{-}\right\rangle$. (d) The population dynamics for clockwise encircling when the system is initialized in $\left|\psi_{+}\right\rangle$. (e) The counterclockwise trajectory (orange) corresponds to the following parameters in Eq. (\ref{['eq:trajectory_general']}): $\delta_{0}=0$, $g_{0}=0.5$, $R=0.3$, $T=500$, $\omega=2\pi/T$, $\varphi=0$. (f,g,h) Same as (b,c,d), but for the counterclockwise trajectory.
• Figure 5: Open-trajectory evolution under the perturbed Hamiltonian $\bar{H}(t)$ (\ref{['eq:nHH_examples_perturbed']}). The parameter trajectories in panels (a) and (e) are identical to those in Fig. \ref{['fig:open_trajectory_no_surprises']}(a) and Fig. \ref{['fig:open_trajectory_qualitative_surprise']}(a) respectively: $\delta_{0}=0$, $g_{0}=1$, $R=0.3$, $T=500$, $\omega=-\pi/T$, $\phi=0.4\pi$ for panel (a) and $\delta_{0}=0$, $g_{0}=1$, $R=0.3$, $T=500$, $\omega=\pi/T$, $\phi=-0.6\pi$ for panel (e). The trajectory direction is shown with arrows and the orange dot depict the starting point. Panels (b) and (f) show the evolution of the instantaneous eigenvalues of $\bar{H}(t)$. The eigevalues $\lambda_{\pm}$ at the start of the parameter trajectory are depicted as blue and red dots, and the arrows show the trajectory direction. Panels (c) and (g) show the population dynamics along the respective trajectories when starting in $\left|\psi_{-}\right\rangle$, while the panels (d) and (h) show the same when starting in $\left|\psi_{+}\right\rangle$. The evolutions in numerical simulations and as predicted by the advanced theory of Sec. \ref{['subsec:Summary_Slow-evolution-with-fast-noise']} are in quantitative agreement.