Non-Hermitian Systems Reveal Novel Bound States and Real-Time Dynamics

Research demonstrates that a single emitter interacting with a structured, non-Hermitian environment exhibits unique scattering states, diverging from behaviour in Hermitian systems. Wave functions display finite localisation proportional to lattice size, with analytical solutions found for specific non-Hermitian bath configurations like Hatano-Nelson models.

The behaviour of quantum systems interacting with their surroundings is central to many areas of physics, from condensed matter to quantum optics.

Recent theoretical work demonstrates anomalous bound states and dynamics emerge when a single emitter couples to a specifically engineered, non-Hermitian environment, a departure from traditional quantum mechanical descriptions. Jimin Li from the University of Cambridge, Yuwen E. Zhang from University College London, Franco Nori from RIKEN, and Zongping Gong from the University of Tokyo, detail a general approach to understanding the scattering states arising from such interactions in their article, ‘Scattering States in One-Dimensional Non-Hermitian Baths’. They formally solve the eigenvalue equation governing these scattering states on finite lattices, revealing that, unlike Hermitian systems, the resulting wave functions exhibit a finite localisation length proportional to the lattice size, and explore the limits of established theoretical frameworks like the Lippmann-Schwinger equation in these non-Hermitian scenarios.
Recent research presents a rigorous theoretical framework for analysing how a single quantum emitter interacts with one-dimensional, non-Hermitian environments, formally solving the eigenvalue equation that governs all scattering states within finite periodic lattices. This allows for solutions beyond the limitations of perturbative approximations, particularly crucial in strongly non-Hermitian regimes where conventional methods prove inadequate. Perturbative approximations simplify complex problems by treating interactions as small deviations from a simpler, solvable system, but these fail when interactions become strong. The formal solution presented converges to the well-established Lippmann-Schwinger equation, a fundamental equation in scattering theory, when applied to more generic environments, thereby validating the approach and anchoring it within established quantum mechanical principles.

Scattering states, typically described as superpositions of plane waves representing freely propagating particles, now exhibit substantial, though finite, localisation lengths that scale proportionally with the lattice size. This fundamentally alters the propagation characteristics of the emitted radiation, confining it to a degree dependent on the structure of the surrounding environment. The research identifies specific conditions under which the Lippmann-Schwinger equation, despite its general applicability, breaks down, necessitating the development of alternative analytical methods to accurately describe the scattering process. This breakdown occurs when the non-Hermitian nature of the environment significantly distorts the wave function, rendering the assumptions underlying the Lippmann-Schwinger equation invalid.

Analytical solutions are presented for two specific non-Hermitian bath models. The Hatano-Nelson model describes a system where the probability of a particle hopping to the right differs from that of hopping to the left, creating asymmetry. The second model considers a unidirectional next-nearest-neighbour interaction, where particles interact only with particles two lattice sites away in a single direction. These solutions provide concrete examples of the predicted behaviour and offer insights into the physical mechanisms driving the observed localisation. They also enable exploration of novel quantum phenomena in non-Hermitian systems and open avenues for designing advanced photonic devices with tailored scattering properties, potentially leading to more efficient light manipulation and control.

This approach establishes a general method for studying the scattering states of a single emitter coupled to one-dimensional, non-Hermitian single-band baths. The development of alternative analytical techniques ensures accurate solutions for the scattering states even in strongly non-Hermitian regimes, employing sophisticated mathematical tools and approximations tailored to the specific non-Hermitian environment under consideration. Future research will focus on extending these analytical techniques to more complex bath configurations and investigating the implications of these findings for quantum information processing and sensing applications.