Non-Hermitian Quantum Systems and Time-Optimal Quantum Evolution

TL;DR

This paper explores the problem of time-optimal quantum evolution in non-Hermitian systems, extending previous work on PT-symmetric systems by analyzing the geometric aspects of the evolution under generic non-Hermitian Hamiltonians.

Contribution

It provides a geometric analysis of the quantum brachistochrone problem for generic non-Hermitian Hamiltonians, broadening the understanding beyond PT-symmetric cases.

Findings

Optimal evolution time can be minimized using geometric methods.

The problem reduces to a two-dimensional subspace analysis.

Insights into the structure of non-Hermitian quantum dynamics.

Abstract

Recently, Bender et al. have considered the quantum brachistochrone problem for the non-Hermitian $\cal PT$-symmetric quantum system and have shown that the optimal time evolution required to transform a given initial state $|\psi_i\rangle$ into a specific final state $|\psi_f\rangle$ can be made arbitrarily small. Additionally, it has been shown that finding the shortest possible time requires only the solution of the two-dimensional problem for the quantum system governed by the effective Hamiltonian acting in the subspace spanned by $|\psi_i\rangle$ and $|\psi_f\rangle$. In this paper, we study a similar problem for the generic non-Hermitian Hamiltonian, focusing our attention on the geometric aspects of the problem.