What is the resonant state in open quantum systems?

Abstract

The article reviews the theory of open quantum system from a perspective of the non-Hermiticity that emerges from the environment with an infinite number of degrees of freedom. The non-Hermiticity produces resonant states with complex eigenvalues, resulting in peak structures in scattering amplitudes and transport coefficients. After introducing the definition of resonant states with complex eigenvalues, we answer typical questions regarding the non-Hermiticity of open quantum systems. What is the physical meaning of the complex eigenmomenta and eigenenergies? How and why do the resonant states break the time-reversal symmetry that the system observes? Can we make the probabilistic interpretation of the resonant states with diverging wave functions? What is the physical meaning of the divergence of the wave functions? We also present an alternative way of finding resonant states, namely the Feshbach formalism, in which we eliminate the infinite number of the environmental degrees of freedom. In this formalism, we attribute the non-Hermiticity to the introduction of the retarded and advanced Green's functions.

Figures (4)

• Figure 1: (a) Two electrodes are attached to a quantum scatterer by way of quantum leads without impurities. (b) The potential scattering problem of the quantum scatterer.
• Figure 2: The point spectra for the scattering problem under the boundary (a) in the complex $k$ plain and (b) in the complex $E$ plain. In the complex $k$ plain (a), the bound states are located on the positive imaginary axis, the anti-bound states are on the negative imaginary axis, the resonant states are in the fourth quadrant, and the anti-resonant states are in the third quadrant. In the complex $E$ plain (b), the bound states are on the negative real axis in the first Riemann sheet, the anti-bound states are on the negative real axis in the second Riemann sheet, the resonant states are in the lower half of the second Riemann sheet, and the anti-resonant states are in the upper half of the second Riemann sheet.
• Figure 3: Time-dependence of the survival probability of the potential site (denoted by $\ket{0}$ here) of the tight-binding model Hamiltonian $H$. The black curve with the highest central peak is the total survival probability, while the red curve with the peak on the right is the contribution of the resonant state and the blue curve on the left is the contribution of the anti-resonant state. Taken from Ref. Hatano19a. The contributions switch from the latter to the former within the quantum Zeno time, which is denoted by $t_Z$ here.
• Figure 4: A schematic view of the non-Markovian time evolution of the system through the propagation in the environment. The system state $P\ket{\Psi(x,\tau)}$ is transferred to the environment by the coupling Hamiltonian $QHP$, evolves in time during the period $t-\tau$ in the environment, and is transferred back to the system by the coupling Hamiltonian $PHQ$ to be $P\ket{\Psi(x,t)}$, which thus depends on the state in the past indirectly.