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Exact time-evolving resonant states for open double quantum-dot systems with spin degrees of freedom

Akinori Nishino, Naomichi Hatano

TL;DR

This work extends Siegert boundary conditions to an open two-electron, spinful double quantum-dot system with on-dot and interdot Coulomb interactions, yielding exact non-Hermitian effective Hamiltonians that capture resonant transport. By diagonalizing the one- and two-body effective Hamiltonians, the authors obtain four two-body resonance energies, including an exceptional point where two resonances coalesce. They construct exact time-evolving two-body resonant states, revealing that while amplitudes on the dots decay, those on the leads grow within a causally expanding finite interval, with interference effects governed by an so(4) structure. Using these solutions, survival and transition probabilities are computed, showing interaction-insensitive lifetimes for some initial states and interaction-dependent or oscillatory behavior for others, depending on the detuning ΔU and coupling v'. The results illuminate how non-Hermitian dynamics and many-body interactions shape resonant transport in nanoscale open quantum systems and connect to broader formalisms like Feshbach projections.

Abstract

We study time-evolving resonant states in an open double quantum-dot system, taking into account spin degrees of freedom as well as both on-dot and interdot Coulomb interactions. We exactly derived a non-Hermite effective Hamiltonian acting on the subspace of two quantum dots, where the non-Hermiticity arises from an effect of infinite external leads connected to the quantum dots. By diagonalizing the effective Hamiltonian, we identify four types of two-body resonant states. For the initial states of localized two electrons with opposite spins on the quantum dots, we exactly solve the time-dependent Schroedinger equation and obtain time-evolving two-body resonant states. The time-evolving resonant states are normalizable since their wave function grows exponentially only inside a finite space interval that expands in time with electron velocity. By using the exact solution, we analyze the survival and transition probabilities of localized two electrons on the quantum dots.

Exact time-evolving resonant states for open double quantum-dot systems with spin degrees of freedom

TL;DR

This work extends Siegert boundary conditions to an open two-electron, spinful double quantum-dot system with on-dot and interdot Coulomb interactions, yielding exact non-Hermitian effective Hamiltonians that capture resonant transport. By diagonalizing the one- and two-body effective Hamiltonians, the authors obtain four two-body resonance energies, including an exceptional point where two resonances coalesce. They construct exact time-evolving two-body resonant states, revealing that while amplitudes on the dots decay, those on the leads grow within a causally expanding finite interval, with interference effects governed by an so(4) structure. Using these solutions, survival and transition probabilities are computed, showing interaction-insensitive lifetimes for some initial states and interaction-dependent or oscillatory behavior for others, depending on the detuning ΔU and coupling v'. The results illuminate how non-Hermitian dynamics and many-body interactions shape resonant transport in nanoscale open quantum systems and connect to broader formalisms like Feshbach projections.

Abstract

We study time-evolving resonant states in an open double quantum-dot system, taking into account spin degrees of freedom as well as both on-dot and interdot Coulomb interactions. We exactly derived a non-Hermite effective Hamiltonian acting on the subspace of two quantum dots, where the non-Hermiticity arises from an effect of infinite external leads connected to the quantum dots. By diagonalizing the effective Hamiltonian, we identify four types of two-body resonant states. For the initial states of localized two electrons with opposite spins on the quantum dots, we exactly solve the time-dependent Schroedinger equation and obtain time-evolving two-body resonant states. The time-evolving resonant states are normalizable since their wave function grows exponentially only inside a finite space interval that expands in time with electron velocity. By using the exact solution, we analyze the survival and transition probabilities of localized two electrons on the quantum dots.
Paper Structure (12 sections, 1 theorem, 72 equations, 8 figures, 1 table)

This paper contains 12 sections, 1 theorem, 72 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Open double quantum-dot systems
  3. Hamiltonian
  4. Time evolution of two-electron states
  5. Siegert boundary conditions and an effective Hamiltonian
  6. Time-evolving resonant states
  7. Resonance energies in a special case
  8. Time-evolving two-body resonant states
  9. Two-body resonant states at an exceptional point
  10. Survival and transition probabilities
  11. Concluding remarks
  12. Algebraic structure of the effective Hamiltonian

Key Result

Proposition 3.1

In the case $\Delta U\neq 4\Gamma$ in which there is no exceptional point, the solution of the set of time-dependent Schrödinger equations (eq:Sch-eq_2elec-g_5), (eq:Sch-eq_2elec-e_5) and (eq:Sch-eq_2elec-f_5) under the initial conditions in Eqs. (eq:initial-condition) is given by Here $Q=(Q_{1}, Q_{2})$ is a permutation of $(1, 2)$, $x_{12}=x_{1}-x_{2}$, $E^{(1)}_{{\rm R}\pm}$ are the eigenvalu

Figures (8)

  • Figure 1: A schematic diagram of the open double quantum-dot (QD) system. The thick double-headed arrows represent on-dot and interdot Coulomb interactions.
  • Figure 2: Electron scattering around the origin $x=0$ of each lead described by the wave functions $e_{m\alpha,\sigma\overline{\sigma}}(x,t)$.
  • Figure 3: Electron transport on the external leads of the open double quantum-dot system under the Siegert boundary conditions in Eqs. (\ref{['eq:Siegert-2elec-bc_1']}). Outgoing waves from the origin appear for right-moving electrons with $v_{\rm F}>0$, while incoming waves toward the origin appear for left-moving electrons with $v_{\rm F}<0$.
  • Figure 4: Arrangement of the two-body resonance energies $E^{(2)}_{{\rm R}, 1}$, $E^{(2)}_{{\rm R}, 2}$ and $E^{(2)}_{{\rm R}, 3\pm}$, which are given by Eqs. (\ref{['eq:2-resonance-energy']}), on the complex-$E$ plane in the case of $v^{\prime}=0$ and (a) $\Delta U<4\Gamma$, (b) $\Delta U>4\Gamma$.
  • Figure 5: The flow chart of the construction of the time-evolving two-body resonant states in Proposition \ref{['prop:time-evolving-resonant-state']}.
  • ...and 3 more figures

Theorems & Definitions (1)

  • Proposition 3.1