A degeneracy in a non-Hermitian system where both eigenvalues and eigenvectors coalesce.An exceptional point is a special degeneracy in a non-Hermitian system where not only the eigenvalues (energies or frequencies) but also the eigenvectors (states) of the system coalesce. Unlike a standard degeneracy, where two states share the same energy but remain distinct, at an exceptional point the two states become identical, leading to a breakdown of the usual mathematical description. This concept arises in systems that are open, meaning they exchange energy or particles with their environment, and it has profound implications for how such systems behave.
Exceptional points were first studied in the context of non-Hermitian quantum mechanics in the 1960s, but they gained prominence in the 2000s with experiments in optics and photonics, where gain and loss can be precisely engineered. The term 'exceptional point' was coined by T. Kato in his 1966 book on perturbation theory.
In a non-Hermitian system, such as one with gain or loss, the eigenvalues are complex, with the imaginary part representing decay or growth. An exceptional point occurs when two eigenvalues and their corresponding eigenvectors merge, which requires both the real and imaginary parts to match. Near this point, the system's response to perturbations is highly sensitive, and the behavior of the system changes dramatically — for example, the usual rules of quantum mechanics, like the orthogonality of states, break down. This sensitivity can be exploited for enhanced sensing and control.
Exceptional points are used to design ultra-sensitive sensors, where a small perturbation near the exceptional point causes a large response, and to control light propagation in photonic devices, such as lasers and waveguides.
At an exceptional point, the system's eigenvectors become parallel, meaning the two states are indistinguishable — a phenomenon that has no analogue in closed, Hermitian systems and can lead to effects like unidirectional invisibility.








