Non-Hermitian Quantum Systems and Time-Optimal Quantum Evolution

Abstract

Recently, Bender et al. have considered the quantum brachistochrone problem for the non-Hermitian $\mathcal{P}\mathcal{T}$-symmetric quantum system and have shown that the optimal time evolution required to transform a given initial state $|{\psi }_i⟩$ into a specific final state $|{\psi }_f⟩$ 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 $|{\psi }_i⟩$ and $|{\psi }_f⟩$. In this paper, we study a similar problem for the generic non-Hermitian Hamiltonian, focusing our attention on the geometric aspects of the problem.

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