Anomalous Wave-Packet Dynamics in One-Dimensional Non-Hermitian Lattices

Abstract

Non-Hermitian (NH) systems have attracted great attention due to their exotic phenomena beyond Hermitian domains. Here we study the wave-packet dynamics in general one-dimensional NH lattices and uncover several unexpected phenomena. The group velocity of a wave packet during the time evolution in such NH lattices is not only governed by the real part of the band structure but also by its imaginary part. The momentum also evolves due to the imaginary part of the band structure, which can lead to a self-induced Bloch oscillation in the absence of external fields. Furthermore, we discover the wave-packet dynamics can exhibit disorder-free NH jumps even when the energy spectra are entirely real. Finally, we show that the NH jumps can lead to both positive and negative temporal Goos--Hänchen shifts at the edge.

Figures (26)

• Figure 1: (a1, a2) Band structures, $E_R(k)$ (red lines) and $E_I(k)$ (blue lines). (b1, b2) The evolution of a wave packet in NH lattices, where the white solid lines denote the predicted center of mass $\bar{n}$ according to Eq. (\ref{['n']}). The horizontal white dashed lines are at $n_0$. (c1, c2) The evolution of group velocities $\bar{v}_g(t)$ (red lines), $v_g[k_\mathrm{max}(t)]$ (black dotted lines), the numerical $V_g(t)$ (blue solid lines), and the predicted $V_g(t)$ (green dashed lines) according to Eq. (\ref{['v_g']}). The vertical gray dash-dotted line in (b, c) denotes $V_g=0$, while the red dotted lines highlight the agreement of zero points of $\bar{n}(t)=n_0$ with $\bar{v}_g(t)=v_g[k_\mathrm{max}(t)]=0$. (d1, d2) The evolution of the wave packet in momentum space with the theoretical $k_{\mathrm{max}}(t)$ denoted by the black dashed lines. The parameters are $J_1^L=0.9,~J_1^R=0.4$, and $J_{m>1}^{L,R}=0$ in (a1-d1), while $J_1^L=0.2,~J_1^R=-0.2$, and $J_{10}^{L}=J_{10}^{R}=0.08$ in (a2-d2). $k_0=-0.4\pi$ and $\sigma=5$.
• Figure 2: (a1, a2) Initial wave packet in momentum space with a finite (blue circles) and infinite size (red lines). The blue dots indicate the values for $k=k^*=\pi/2$. (b1, b2) The evolution of $c_k(t)$, where $c_{k=k^*}(t)$ is highlighted by the blue dashed lines. The evolution of the wave packet in (c1, c2) real and (d1, d2) momentum spaces under OBCs, where the horizontal white dashed lines denote $k=k^*=\pi/2$. The vertical red dotted lines in (b, c, d) highlight the time $t_c$ when the NH jumps occurs. The sizes in (a1-d1) and (a2-d2) are $N=60$ and $N=100$, respectively. Other parameters are $J_1^L=0.9,~J_1^R=0.1,~J_{m>1}^{L,R}=0,~k_0=0$, and $\sigma=5$.
• Figure 3: (a) Energy spectra under the PBC (colored line) and OBC (black dots) for a finite lattice of size $N=500$. The red (blue) circle denote the initial state of the initial (auxiliary) wave packet, which moves along the spectra denoted by the black dashed arrow. (b) Relative position of different wavefunctions. (c) The initial wave packet $\tilde{\psi}(0)+\tilde{\psi}'(0)$ in momentum space, where the blue dots (red line) denote the result under the finite lattice of size $2N$ (infinite case). (d) The evolution of $c_k(t)$, where the red dotted (blue dashed) line denotes the case for $k_0=-0.2\pi$ ($-k_0=0.2\pi$). (e) The evolution of the wave packet in real space. (f) The evolution of the wave packet in momentum space. The black dotted (dashed) line in (e) and (f) are theoretical center of mass $\bar{n}$ and $k_{\mathrm{max}}(t)$ for the initial (auxiliary) wave packet using Eq. (\ref{['n']}) and Eq. (\ref{['k']}), respectively. The parameters are $J_L=1,~J_R=0.92,~k_0=-0.2\pi$, and $\sigma=5$.
• Figure 4: (a, d) Typical band structures, $E_R(k)$ (red solid lines) and $E_I(k)$ (blue dashed lines) with (a) $J_L>J_R$ and (d) $J_L<J_R$. (b, c, e, f, g-i) The evolution of the wave packet in real space under the OBC. The parameters are $J_L=1,~J_R=0.96$ (b), $J_L=1,~J_R=0.98$ (c), $J_L=0.7,~J_R=1$ (e), $J_L=0.8,~J_R=1$ (f), $J_L=J_R=1$ (g), $J_L=1,~J_R=0.95$ (h), and $J_L=0.95,~J_R=1$ (i). $k_0=-0.2\pi,~\sigma=5$ for (b, c), $k_0=-0.2\pi,~\sigma=15$ for (e, f), and $k_0=-0.5\pi,~\sigma=35$ for (g, h, i). The red (blue) circles denote $n_0'$ ($n_2$). The red dashed lines in (g-i) denote the Hermitian cases.
• Figure 5: (a1, a2) Band structures, where the solid (dashed) lines denote the $\mathrm{Re}(E_{1,2})$ [$\mathrm{Im}(E_{1,2})$] and red (blue) lines represent the upper (lower) bands. $\mathrm{Im}(E_{1,2})$ is enlarged by $4$ times in (a2). The gray dashed line in (a1) denotes the $k=k^*$. The black solid arrow in (a2) denotes the jump from the state in the upper band to the state on the lower band. (b1, b2) The evolution of the normalized $\psi_{n,a}(t)$ in real space. (c1) The evolution of group velocities. (c2) The evolution of normalized participation ratio of the upper (lower) band $\tilde{c}_1$ ($\tilde{c}_2$). (d1, d2) The evolution of the wave packet in momentum space $\psi_{k,a}(t)$. The parameters in (a1-d1) are $v=1,~w=1.6,~\gamma=0.6,~k_0=0$, and $\sigma=2$, while the parameters in (a2-d2) are $v=1,~w=1.6,~\gamma=0.1,~k_0=-0.5\pi$, and $\sigma=5$.