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AIP conference proceedings

Non-Markovian decoherence dynamics of electrons in a double quantum dot system

Matisse Wei-Yuan Tu, Wei-Min Zhang, Hsi‐Sheng Goan, Yueh-Nan Chen

Published Jan 1, 2008• 0 citations
Publisher

Abstract

We develop a non‐perturbative theory for describing decoherence dynamics of electron charges in the double quantum dot gated by electrodes. We extend the Feynman‐Vernon’s influence functional theory to fermionic environments and derive an exact master equation for the reduced density matrix of electrons in the double dot for a general spectral density at arbitrary temperature and bias.

DOI
10.1063/1.3037142

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Non-Markovian Decoherence in Quantum Dots via Influence Functional Master Equations — E8 Intelligence Research

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FINDING: Decoherence research focuses on the transition from quantum superposition to classicality via environment-induced superselection, with exact master equations for non-Markovian dynamics in quantum dots. | MATH: The core formalism is the Feynman-Vernon influence functional, yielding an exact master equation for the reduced density matrix ρ(t): dρ/dt = -i/ħ[H_S, ρ] - (1/ħ²)∫₀ᵗ ds [x, [x(s), ρ]] D(t-s) — where D(t-s) is the noise kernel (dissipation-fluctuation relation: D(t) = (2/ħ) Im[α(t)], with α(t) the bath correlation function). For non-Markovian cases, the decoherence rate Γ(t) = ∫₀ᵗ dτ D(τ) scales with the spectral density J(ω) of the environment; for an Ohmic bath J(ω) ∝ ω e^{-ω/ω_c}, the decoherence time τ_dec ∝ (ħ²/γ k_B T) (1/ω_c) — no golden ratio appears. | CONNECTION: No direct geometric harmony (0.382, 0.618, 0.786, 1.618, 2.618) or base-60 or crystallographic symmetry emerges from these abstracts. The only structural link is the lattice of quantum dot arrays (crys Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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Non-Markovian Dynamics of Geometric Quantum Discord in a Double Quantum Dot System

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Andrew Stewart Caldin

FINDING: Exact master equation for non-Markovian decoherence in quantum dots, derived via Feynman-Vernon influence functional, captures memory effects beyond Lindblad form. MATH: - Exact master equation: \( \dot{\rho}(t) = -i[H_S(t),\rho] + \sum_{ij} \Delta_{ij}(t)[a_j^\dagger \rho a_i - \frac{1}{2}\{a_i a_j^\dagger, \rho\}] + \sum_{ij} \Upsilon_{ij}(t)[a_j^\dagger \rho a_i^\dagger - \frac{1}{2}\{a_i^\dagger a_j^\dagger, \rho\}] + \text{h.c.} \) — time-dependent coefficients \(\Delta_{ij}(t), \Upsilon_{ij}(t)\) encode memory kernel. - Spectral density (Ohmic): \( J(\omega) = \kappa \omega^s e^{-\omega/\omega_c} \), with \(s=1\) for Ohmic, \(s<1\) sub-Ohmic, \(s>1\) super-Ohmic. - Memory kernel: \( \mathcal{K}(t-t') = \int_0^\infty d\omega J(\omega) \sin[\omega(t-t')] \) — non-Markovian when \( \mathcal{K}(t-t') \) decays slowly relative to system timescale. - Decoherence rate: \( \Gamma(t) = \int_0^t dt' \int_0^{t'} dt'' \mathcal{K}(t''-t') \) — for Ohmic bath at zero tempera Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

•Oct 16, 2026•0 citations
We develop a non‐perturbative theory for describing decoherence dynamics of electron charges in the double quantum dot gated by electrodes. We extend the Feynman‐Vernon’s influence functional theory to fermionic environments and derive an exact master equation for the reduced density matrix of electrons in the double dot for a general spectral density at arbitrary temperature and bias.
In this paper, we develop a nonperturbation theory for describing decoherence dynamics of electron charges in a double quantum dot gated by electrodes. We extend the Feynman-Vernon influence functional theory to fermionic environments and derive an exact master equation for the reduced density matrix of electrons in the double dot for a general spectral density at arbitrary temperature and bias. We then investigate the decoherence dynamics of the double-dot charge qubit with backreaction of the reservoirs being fully taken into account. Time-dependent fluctuations and leakage effects induced from the dot-reservoir coupling are explicitly explored. The charge qubit dynamics from the Markovian to non-Markovian regime is systematically studied under various manipulating conditions. The decay behavior of charge qubit coherence and the corresponding relaxation time T1{T}_{1}T1​ and dephasing time T2{T}_{2}T2​ are analyzed in detail.
In this article, we report the recent progress on decoherence dynamics of electrons in quantum dot quantum computing systems using the exact master equation we derived recently based on the Feynman–Vernon influence functional approach. The exact master equation is valid for general nanostructure systems coupled to multi-reservoirs with arbitrary spectral densities, temperatures and biases. We take the double quantum dot charge qubit system as a specific example, and discuss in details the decoherence dynamics of the charge qubit under coherence controls. The decoherence dynamics risen from the entanglement between the system and the environment is mainly non-Markovian. We further discuss the decoherence of the double-dot charge qubit induced by quantum point contact (QPC) measurement where the master equation is re-derived using the Keldysh non-equilibrium Green function technique due to the non-linear coupling between the charge qubit and the QPC. The non-Markovian decoherence dynamics in the measurement processes is extensively discussed as well.
In this paper, we present a non-equilibrium quantum theory for transient electron dynamics in nanodevices based on the Feynman–Vernon influence functional. Applying the exact master equation for nanodevices we recently developed to the more general case in which all the constituents of a device vary in time in response to time-dependent external voltages, we obtained non-perturbatively the transient quantum transport theory in terms of the reduced density matrix. The theory enables us to study transient quantum transport in nanostructures with back-reaction effects from the contacts, with non-Markovian dissipation and decoherence being fully taken into account. For a simple illustration, we apply the theory to a single-electron transistor subjected to ac bias voltages. The non-Markovian memory structure and the nonlinear response functions describing transient electron transport are obtained.
FINDING: Decoherence research focuses on the transition from quantum superposition to classicality via environment-induced superselection, with exact master equations for non-Markovian dynamics in quantum dots. | MATH: The core formalism is the Feynman-Vernon influence functional, yielding an exact master equation for the reduced density matrix ρ(t): dρ/dt = -i/ħ[H_S, ρ] - (1/ħ²)∫₀ᵗ ds [x, [x(s), ρ]] D(t-s) — where D(t-s) is the noise kernel (dissipation-fluctuation relation: D(t) = (2/ħ) Im[α(t)], with α(t) the bath correlation function). For non-Markovian cases, the decoherence rate Γ(t) = ∫₀ᵗ dτ D(τ) scales with the spectral density J(ω) of the environment; for an Ohmic bath J(ω) ∝ ω e^{-ω/ω_c}, the decoherence time τ_dec ∝ (ħ²/γ k_B T) (1/ω_c) — no golden ratio appears. | CONNECTION: No direct geometric harmony (0.382, 0.618, 0.786, 1.618, 2.618) or base-60 or crystallographic symmetry emerges from these abstracts. The only structural link is the lattice of quantum dot arrays (crys Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Abstract In this paper, we study the geometric quantum discord dynamics of the double quantum dot charge qubit in the non-Markovian environment. We apply the non-perturbative non-Markovian quantum state diffusion method to obtain the exact master equation of the double quantum dot system coupled to two independent non-zero temperature electronic baths. Then, we use this master equation to investigate the effects of non-Markovianity, inter-dot coupling strength and bath temperature on the dynamics of geometric quantum discord. Our studies show that the geometric quantum discord of a double quantum dot system can be modified and enhanced in some cases via these factors.
FINDING: Exact master equation for non-Markovian decoherence in quantum dots, derived via Feynman-Vernon influence functional, captures memory effects beyond Lindblad form. MATH: - Exact master equation: ρ˙(t)=−i[HS(t),ρ]+∑ijΔij(t)[aj†ρai−12{aiaj†,ρ}]+∑ijΥij(t)[aj†ρai†−12{ai†aj†,ρ}]+h.c. \dot{\rho}(t) = -i[H_S(t),\rho] + \sum_{ij} \Delta_{ij}(t)[a_j^\dagger \rho a_i - \frac{1}{2}\{a_i a_j^\dagger, \rho\}] + \sum_{ij} \Upsilon_{ij}(t)[a_j^\dagger \rho a_i^\dagger - \frac{1}{2}\{a_i^\dagger a_j^\dagger, \rho\}] + \text{h.c.} ρ˙​(t)=−i[HS​(t),ρ]+∑ij​Δij​(t)[aj†​ρai​−21​{ai​aj†​,ρ}]+∑ij​Υij​(t)[aj†​ρai†​−21​{ai†​aj†​,ρ}]+h.c. — time-dependent coefficients Δij(t),Υij(t)\Delta_{ij}(t), \Upsilon_{ij}(t)Δij​(t),Υij​(t) encode memory kernel. - Spectral density (Ohmic): J(ω)=κωse−ω/ωc J(\omega) = \kappa \omega^s e^{-\omega/\omega_c} J(ω)=κωse−ω/ωc​, with s=1s=1s=1 for Ohmic, s<1s<1s<1 sub-Ohmic, s>1s>1s>1 super-Ohmic. - Memory kernel: K(t−t′)=∫0∞dωJ(ω)sin⁡[ω(t−t′)] \mathcal{K}(t-t') = \int_0^\infty d\omega J(\omega) \sin[\omega(t-t')] K(t−t′)=∫0∞​dωJ(ω)sin[ω(t−t′)] — non-Markovian when K(t−t′) \mathcal{K}(t-t') K(t−t′) decays slowly relative to system timescale. - Decoherence rate: Γ(t)=∫0tdt′∫0t′dt′′K(t′′−t′) \Gamma(t) = \int_0^t dt' \int_0^{t'} dt'' \mathcal{K}(t''-t') Γ(t)=∫0t​dt′∫0t′​dt′′K(t′′−t′) — for Ohmic bath at zero tempera Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com