Non-Hermitian Quantum Mechanics of Open Quantum Systems: Revisiting The One-Body Problem

TL;DR

This paper reviews the non-Hermitian quantum mechanics of open systems, focusing on the one-body problem, and introduces a new complete set of eigenstates that include resonant states, facilitating better understanding of open quantum dynamics.

Contribution

It presents a new complete set of eigenstates for the one-body open quantum system, unifying different definitions of resonant states and enabling exact solutions at arbitrary coupling strengths.

Findings

Resonant states can have complex energies despite Hermitian Hamiltonians.

The effective Hamiltonian becomes explicitly non-Hermitian through Feshbach formalism.

The new complete set captures non-Markovian and time-reversal symmetric dynamics.

Abstract

We review analyses of open quantum systems. We show how non-Hermiticity arises in an open quantum system with an infinite environment, focusing on the one-body problem. One of the reasons for taking the present approach is that we can solve the problem completely, making it easier to see the structures of problems involving open quantum systems. We show that this results in the discovery of a new complete set, which is one of the main topics of the present article. Another reason for focusing on the one-body problem is that the theory permits the strong coupling between the system and the environment. In the current research landscape, it is valuable to revisit the one-body problem for open quantum systems, which can be solved accurately for arbitrary strengths of the system-environment couplings. A rigorous understanding of the problem structures in the present approach will be helpful when we tackle problems with many-body interactions. First, we consider potential scattering and directly define the resonant state as an eigenstate of the Schr\"{o}dinger equation under the Siegert outgoing boundary condition. We show that the resonant eigenstate can have a complex energy eigenvalue, even though the Hamiltonian is seemingly Hermitian. Second, we introduce the Feshbach formalism, which eliminates the infinite degrees of freedom of the environment and represents its effect as a complex potential. The resulting effective Hamiltonian is explicitly non-Hermitian. By unifying these two ways of defining resonant states, we obtain a new complete set of bases for the scattering problem that contains all discrete eigenstates, including resonant states. We finally mention the non-Markovian dynamics of open quantum systems. We emphasize the time-reversal symmetry of the dynamics that continuously connects the past and the future. We can capture it using the new complete set that we develop here.