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

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.

💡 Research Summary

This paper presents a comprehensive theoretical study on the wave-packet dynamics in general one-dimensional non-Hermitian (NH) lattices, uncovering a series of unexpected phenomena that challenge conventional wisdom from Hermitian physics.

The core of the investigation lies in deriving and analyzing the equations of motion for a Gaussian wave packet. The authors show that in NH systems, the wave packet’s central momentum, k_max(t), evolves in time due to the gradient of the imaginary part of the complex energy band, E_I(k), causing it to drift toward the momentum k* where E_I(k) is maximum. This momentum evolution has profound consequences. First, it leads to an “anomalous group velocity.” While the average velocity of the wave packet’s center of mass, ¯n(t), is related to the average of the real band slope ⟨dE_R/dk⟩, the instantaneous group velocity V_g(t) contains an additional term proportional to ⟨E_I v_g⟩ - ⟨E_I⟩⟨v_g⟩. This means the true propagation speed is not simply given by the band’s real dispersion but is dynamically modified by the gain/loss profile encoded in E_I(k).

A striking manifestation of this combined effect is the prediction of “self-induced Bloch oscillations.” In models where E_R(k) oscillates rapidly (e.g., with long-range couplings) and E_I(k) has a clear maximum, the drifting k_max(t) and the anomalous V_g(t) conspire to produce an oscillatory motion of the wave packet in real space—without any external electric field. Unlike standard Bloch oscillations, the envelope of these oscillations can amplify linearly in time.

Another major discovery is the phenomenon of “disorder-free non-Hermitian jumps” in systems with entirely real energy spectra under open boundary conditions (OBC). The authors demonstrate that even without disorder or complex OBC spectra, a wave packet can undergo a sudden jump to a state with momentum k*. This is explained through the lens of the complex band structure under periodic boundary conditions (PBC). In a finite-sized system, the initial wave packet is a superposition of discrete momentum states. Over time, the component with the largest E_I(k) (i.e., the largest gain) experiences exponential amplification relative to others. At a critical time t_c, this dominant component overwhelms the rest, causing an abrupt transition or “jump” of the entire wave packet to that specific momentum state. The jump time t_c depends on the system size.

Finally, the paper explores a different mechanism for NH jumps that occurs near the boundary of systems exhibiting the non-Hermitian skin effect. When a wave packet approaches the edge in such a system, the reflection process can involve a jump not to the state with maximum E_I(k), but to a state with momentum opposite to the initial one. The authors interpret this using a similarity transformation that maps the NH Hamiltonian to a Hermitian one, allowing the reflection to be understood via the creation and motion of an “auxiliary wave packet.” This reflection process is associated with a temporal analogue of the Goos-Hänchen shift (TGHS). The analysis suggests that the NH jump at the edge can lead to both positive and negative temporal shifts, corresponding to frequency blue-shifts and red-shifts, respectively.

In summary, this work establishes a general framework for understanding wave dynamics in NH lattices, revealing anomalous velocity renormalization, field-free oscillations, and novel jump phenomena linked to the complex-valued nature of the energy bands. It significantly expands the catalog of unique dynamical features possible in non-conservative, non-Hermitian systems.