Exact time-evolving scattering states in open quantum-dot systems with an interaction: Discovery of time-evolving resonant states

TL;DR

This work provides an exact construction of time-evolving many-electron states in an open, interacting double quantum-dot system with linearly dispersed leads. By solving arbitrary-initial-condition value problems, it yields explicit one- and two-electron time-evolving scattering states that converge to stationary counterparts, and introduces time-evolving resonant states that are normalizable due to causality. The analysis reveals one- and two-body resonance poles that govern survival and existence probabilities, with exceptional points marking qualitative changes in dynamics. These results offer a rigorous framework for time-dependent transport and decoherence in quantum-dot devices, and they generalize to multi-electron sectors, informing potential qubit initialization and coherent control strategies.

Abstract

We study exact time-evolving many-electron states of an open double quantum-dot system with an interdot Coulomb interaction. A systematic construction of the time-evolving states for arbitrary initial conditions is proposed. For any initial states of one- and two-electron plane waves on the electrical leads, we obtain exact solutions of the time-evolving scattering states, which converge to known stationary scattering eigenstates in the long-time limit. For any initial states of localized electrons on the quantum dots, we find exact time-evolving states of a new type, which we refer to as time-evolving resonant states. In contrast to stationary resonant states, whose wave functions spatially diverge and not normalizable, the time-evolving resonant states are normalizable since their wave functions are restricted to a finite space interval due to causality. The exact time-evolving resonant states enable us to calculate the time-dependence of the survival probability of electrons on the quantum dots for the system with the linearized dispersions. It decays exponentially in time on one side of an exponential point of resonance energies while, on the other side, it oscillates during the decay as a result of the interference of the two resonance energies.

Figures (5)

• Figure 1: A schematic diagram of the double quantum-dot system.
• Figure 2: The flow chart of the construction of the time-evolving one-electron states.
• Figure 3: The flow chart of the construction of time-evolving two-electron states.
• Figure 4: Existence probability distributions of the time-evolving one- and two-body resonant states in the case of the parallel-coupled DQD system with $v_{\ell\alpha}=v\in\mathbb{R}$, $v^{\prime}_{\alpha}=0$ for (a) $\epsilon_{{\rm d}1}=-\epsilon_{{\rm d}2}=0.8\Gamma$ and (b) $\epsilon_{{\rm d}1}=-\epsilon_{{\rm d}2}=1.2\Gamma$. The solid lines indicate the existence probability $P^{(1)}_{m}(x,t)/\Gamma$ of an electron on the lead $m$ in the one-electron case, and the dashed lines indicate the existence probability $P^{(2)}_{m{\rm d}}(x,t)/\Gamma$ of an electron on the lead $m$ and another electron on the quantum dots in the two-electron case.
• Figure 5: Survival probabilities of the time-evolving one- and two-body resonant states in the case of the parallel-coupled double quantum-dot system with $v_{\ell\alpha}=v\in\mathbb{R}$, $v^{\prime}_{\alpha}=0$ and $\epsilon_{{\rm d}1}=-\epsilon_{{\rm d}2}$. The solid thin line indicates the survival probabilities $Q^{(1)}(t)$ of the time-evolving one-body resonant states at $\epsilon_{{\rm d}1}=0.5\Gamma$, the dotted line indicates that at $\epsilon_{{\rm d}1}=1.0\Gamma$, the dashed line indicates that at $\epsilon_{{\rm d}1}=1.5\Gamma$, the chain line indicates that at $\epsilon_{{\rm d}1}=2.0\Gamma$ and the solid thick line indicates the survival probability $Q^{(2)}(t)$ of the time-evolving two-body resonant states.

Theorems & Definitions (4)

• Proposition 3.1
• Proposition 3.2
• Proposition 4.1
• Proposition 4.2