Dynamics in non-Hermitian systems with nonreciprocal coupling

TL;DR

This work analyzes dynamics in non-Hermitian systems with nonreciprocal (unidirectional) coupling, showing that an inherent source from directed master-slave interactions can amplify initial states without external gain. It contrasts two extremes—complex eigenenergies with orthogonal eigenstates and real eigenenergies with non-orthogonal eigenstates—under periodic versus open boundary conditions, and dissects time evolution through right-eigenstate, left-eigenstate, and biorthogonal lenses. The study reveals transient growth and long-time behavior shaped by non-normality and nonreciprocity, including a concrete $2\times 2$ PT-symmetric appendix that demonstrates left-eigenstate-driven amplification at the exceptional point. These results illuminate how non-Hermitian skin-like localization and directed coupling enable amplification and point to future nonlinear and pumped-system extensions with practical implications for wave and quantum control.

Abstract

We reveal that non-Hermitian Hamiltonians with nonreciprocal coupling can achieve amplification of initial states without external gain due to a kind of inherent source. We discuss the source and its effect on time evolution in terms of complex eigenenergies and non-orthogonal eigenstates. Demonstrating two extreme cases of Hamiltonians, namely one having complex eigenenergies with orthogonal eigenstates and one having real eigenenergies with non-orthogonal eigenstates, we elucidate the differences between the amplifications from complex eigenenergies and from non-orthogonal eigenstates.

Figures (5)

• Figure 1: Schematic figures of (a) an oscillator driven by an external driving force and (b) coupled oscillators with unidirectional coupling. The dashed boxes represent the systems under consideration.
• Figure 2: Time evolutions of the amplitudes of the elements $x_j (t)$ of $\left|\psi(t)\right>$ when the initial states are localized on the tenth site under (a) PBC and (b) OBC. Solid red lines represent the Euclidean norms of the states. The inset is a log-log scaled plot, $\left| x_1 (t) \right| \propto t^9$. (c,d) The same conditions with additional net loss. (e,f) The same conditions with different initial states localized on the first site. In (f), $x_j = 0 ~ (j=2,3,\dots,10)$.
• Figure 3: Time evolutions of the amplitudes of the elements $y_j (t)$ of $\left|\phi\right>$ when the initial states are localized on the tenth site under (a) PBC and (b) OBC. Solid red lines represent the Euclidean norms of the states. (c,d) The same conditions with additional net loss. In (b) and (d), $y_j = 0 ~ (j=1,2,\dots,9)$.
• Figure 4: Time evolutions of the amplitudes of the elements $y_{j}^{*} x_j$ when the initial states are localized on the tenth site under (a) PBC and (b) OBC. Solid red lines represent the biorthogonal norms of the states. In (b), $y_j^{*} x_j = 0 ~ (j=1,2,\dots,9)$ since $y_j = 0 ~ (j=1,2,\dots,9)$. (c,d) The same conditions with partially directed coupling, $t_l = 1.0$ and $t_r = 0.5$, instead of unidirectional coupling.
• Figure 5: Time evolutions of the amplitudes of the elements (a) $\log \sqrt{x_j^{*} x_j}$, (b) $\log \sqrt{y_j^{*} y_j}$, and (c) $\mathrm{sgn}(y_{j}^{*} x_j)\log \sqrt{|y_{j}^{*} x_j|}$ when the initial states are localized on the tenth site under PBC with partially directed coupling, $t_l = 1.0$ and $t_r = 0.5$. The brown colors when $t=0$ represent zero initial amplitudes. Time evolutions of the amplitudes of the elements (d) $\sqrt{x_j^{*} x_j}$, (e) $\sqrt{y_j^{*} y_j}$, and (f) $\sqrt{y_{j}^{*} x_j}$ when the initial states are localized on the tenth site under OBC with the same partially directed coupling.