Non-Hermitian adiabatic transport in spaces of exceptional points

TL;DR

This paper investigates the adiabatic transport of non-Hermitian Hamiltonians with exceptional points, revealing that the geometric phase acquired depends only on the loop's topology and is quantized, with implications for quantum state control.

Contribution

It demonstrates that adiabatic transport around loops in non-Hermitian spaces yields a topologically quantized phase, independent of the traversal speed, extending understanding of geometric phases in non-Hermitian systems.

Findings

The geometric phase depends only on the homotopy class of the loop.

The phase is an integer power of e^{2πi/n}, quantized by the system's dimension.

The results hold when the transported state is the slowest decaying state at all points.

Abstract

We consider the space of $n \times n$ non-Hermitian Hamiltonians ($n=2$, $3$, . . .) that are equivalent to a single $n\times n$ Jordan block. We focus on adiabatic transport around a closed path (i.e. a loop) within this space, in the limit as the time-scale $T=1/\varepsilon$ taken to traverse the loop tends to infinity. We show that, for a certain class of loops and a choice of initial state, the state returns to itself and acquires a complex phase that is $\varepsilon^{-1}$ times an expansion in powers of $\varepsilon^{1/n}$. The exponential of the term of $n$th order (which is equivalent to the "geometric" or Berry phase modulo $2\pi$), is thus independent of $\varepsilon$ as $\varepsilon\to0$; it depends only on the homotopy class of the loop and is an integer power of $e^{2\pi i/n}$. One of the conditions under which these results hold is that the state being transported is, for all points on the loop, that of slowest decay.