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

TL;DR

This work addresses open quantum systems in the one-body limit, showing how non-Hermiticity arises when the environment is infinite and how resonant states defined by the Siegert boundary condition relate to a non-Hermitian effective Hamiltonian $H_{ ext{eff}}(E)$ obtained via Feshbach projection. It introduces a new complete set that includes resonant and anti-resonant states in a time-reversal-symmetric framework, enabling a unified description of scattering and non-Markovian dynamics. The authors demonstrate the approach on tight-binding and continuous models, deriving a nonlinear (quadratic) eigenproblem in $\\lambda=e^{iKa}$ that connects to Tolstikhin's continuum formulation, and show how contour-integral expansions yield a natural split between decaying and growing components for $t>0$ vs $t<0$. They further reveal non-Markovian memory effects, quantum Zeno short-time behavior, and long-time power-law tails, highlighting the framework's potential for advancing strong-coupling open-system physics and guiding future extensions to many-body and Liouville-space descriptions.

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 Schrödinger 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.

Figures (25)

• Figure 1: (a) A schematic view of the system and the environment of an open quantum system. The system indicated by a small circle is embedded in the environment indicated by a big circle, with a coupling between them indicated by a solid red curve. (b) In the problem of potential scattering, the potential area is identified as the system, while the rest, flat space is identified as the environment.
• Figure 2: (a) The potential of three delta functions in Eq. \ref{['eq50']}. (b) A schematic potential we have in mind.
• Figure 3: Plots of the two curves Eq. \ref{['eq110a']} (dashed blue curves for even solutions and dot-dashed red curves for odd solutions) and Eq. \ref{['eq110b']} (solid black curves). The vertical broken gray lines indicate multiples of $\pi/2$ on the $\xi$ axis. The crossing points indicated by crosses with green squares indicate resonant states of even parity and those indicated by crosses with green circles indicate resonant states of odd parity. The example is given for $\alpha_1=1$ with $V_0=0$.
• Figure 4: The transmission coefficient $\mathcal{T}$ (blue curve) for $v_0=0$ and the locations $\{\xi_n,\eta_n\}$ of the resonance poles, which are the same as the ones in Fig. \ref{['fig3']}. For the values of the former and the latter, see the left and right vertical axes, respectively. The example is given for $\alpha_1=1$ with $V_0=0$.
• Figure 5: The density plot of the function \ref{['eq150']}. The black spots indicate negative infinity. The example is given for $\alpha_0=3$ and $\alpha_1=1$.