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Exact Master Equation and Non-Markovian Decoherence for Quantum Dot Quantum Computing

Ming‐Tsung Lee, Wei-Min Zhang

Published Aug 17, 2016• 11 citations
OpenAlex

Abstract

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.

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Exact Non-Markovian Master Equation for Quantum Dot Decoherence via Influence Functional — E8 Intelligence Research

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
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.
FINDING: The exact master equation for quantum dot decoherence, derived via Feynman-Vernon influence functional, provides a non-Markovian generalization of Lindblad form with time-dependent coefficients — the only mathematically rigorous exact solution in this domain. | MATH: Exact master equation: ρ˙(t)=−i[HS(t),ρ]+∑ijΔij(t)[2fjρfi†−{fi†fj,ρ}] \dot{\rho}(t) = -i[H_S(t), \rho] + \sum_{ij} \Delta_{ij}(t) [2 f_j \rho f_i^\dagger - \{f_i^\dagger f_j, \rho\}] ρ˙​(t)=−i[HS​(t),ρ]+∑ij​Δij​(t)[2fj​ρfi†​−{fi†​fj​,ρ}] where Δij(t)\Delta_{ij}(t)Δij​(t) are time-dependent dissipation coefficients (non-Markovian memory kernels), fif_ifi​ are fermionic annihilation operators. The Feynman-Vernon influence functional: F[j,j′]=exp⁡[−∫0tds∫0sdu (j(s)−j′(s))(Δ(s−u)j(u)−Δ∗(s−u)j′(u))] \mathcal{F}[j,j'] = \exp\left[ -\int_0^t ds \int_0^s du \, (j(s) - j'(s)) ( \Delta(s-u) j(u) - \Delta^*(s-u) j'(u) ) \right] F[j,j′]=exp[−∫0t​ds∫0s​du(j(s)−j′(s))(Δ(s−u)j(u)−Δ∗(s−u)j′(u))] with spectral density J(ω)=∑k∣gk∣2δ(ω−ωk) J(\omega) = \sum_k |g_k|^2 \delta(\omega - \omega_k) J(ω)=∑k​∣gk​∣2δ(ω−ωk​). Decoherence rate: Γ(t)=∫0tdτ Re[Δ(τ)] \Gamma(t) = \int_0^t d\tau \, \text{Re}[\Delta(\tau)] Γ(t)=∫0t​dτRe[Δ(τ)] — no golden ratio appears in the exact solution. | CONNECTION: None found. The exact master equation y Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Decoherence is the process by which quantum superpositions decay into classical mixtures due to environmental entanglement, with recent advances in exact master equations for quantum dot systems enabling precise modeling of non-Markovian decoherence. | MATH: Exact master equation derived from Feynman-Vernon influence functional: ρ˙(t)=−i[HS,ρ(t)]+∫0tds K(t,s)L[ρ(s)] \dot{\rho}(t) = -i[H_S, \rho(t)] + \int_0^t ds \, K(t,s) \mathcal{L}[\rho(s)] ρ˙​(t)=−i[HS​,ρ(t)]+∫0t​dsK(t,s)L[ρ(s)] where K(t,s) K(t,s) K(t,s) is the memory kernel capturing non-Markovian effects. Decoherence time τd∝1/γ \tau_d \propto 1/\gamma τd​∝1/γ where γ \gamma γ is the system-environment coupling strength. | CONNECTION: No direct geometric ratios (0.382, 0.618, 0.786, 1.618, 2.618) or base-60 or crystallographic symmetries appear in the provided findings. The exact master equation formalism involves integral kernels and convolution structures, which are analytic rather than geometric. | DEPTH: 4 — The exact master equation is a significant mathematical tool for quantum dot decoherence, but t Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Decoherence is the physical process that suppresses quantum interference, effectively splitting the quantum multiverse into classical branches, with exact master equations for quantum dot systems revealing non-Markovian dynamics. | MATH: Exact master equation derived from Feynman-Vernon influence functional: ρ˙(t)=−i[HS,ρ(t)]+∫0tds K(t,s)L[ρ(s)] \dot{\rho}(t) = -i[H_S, \rho(t)] + \int_0^t ds \, K(t,s) \mathcal{L}[\rho(s)] ρ˙​(t)=−i[HS​,ρ(t)]+∫0t​dsK(t,s)L[ρ(s)], where K(t,s) K(t,s) K(t,s) is the memory kernel. Decoherence time τd∝1/γ \tau_d \propto 1/\gamma τd​∝1/γ, with γ \gamma γ the system-environment coupling strength. No specific numerical constants or ratios reported. | CONNECTION: No explicit geometric ratios (0.382, 0.618, 0.786, 1.618, 2.618) or base-60, crystallographic symmetries, root systems, or lattice structures appear in these findings. The exact master equation formalism is general and does not inherently link to harmonic ratios. | DEPTH: 6 — The exact master equation for quantum dot decoherence is a significant mathematical advance in non-Marko Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Decoherence as a universal mechanism for classical emergence and quantum-to-classical transition, with exact master equation for non-Markovian decoherence in quantum dot systems. | MATH: Exact master equation derived from Feynman-Vernon influence functional: ρ˙(t)=−i[HS,ρ(t)]+∫0tds K(t,s)L[ρ(s)] \dot{\rho}(t) = -i[H_S, \rho(t)] + \int_0^t ds \, K(t,s) \mathcal{L}[\rho(s)] ρ˙​(t)=−i[HS​,ρ(t)]+∫0t​dsK(t,s)L[ρ(s)], where K(t,s) K(t,s) K(t,s) is the memory kernel encoding non-Markovian effects. Decoherence time scale τd∝1/γ \tau_d \propto 1/\gamma τd​∝1/γ where γ \gamma γ is the system-environment coupling strength. No specific constants or ratios reported. | CONNECTION: No explicit geometric ratios (0.382, 0.618, etc.) or crystallographic symmetries found in the provided abstracts. However, the Feynman-Vernon influence functional is rooted in path integrals over continuous spacetime, which can be mapped to lattice discretizations (e.g., Euclidean lattice field theory) — a weak link to lattice structures. No base-60 or golden ratio connections. | DEPTH: 4 — The exac Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
In this review article, we present a non-equilibrium quantum transport theory for transient electron dynamics in nanodevices based on exact Master equation derived with the path integral method in the fermion coherent-state representation. Applying the exact Master equation to nanodevices, we also establish the connection of the reduced density matrix and the transient quantum transport current with the Keldysh nonequilibrium Green functions. 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. In applications, we utilize the theory to specific quantum transport systems, a variety of quantum decoherence and quantum transport phenomena involving the non-Markovian memory effect are investigated in both transient and stationary scenarios at arbitrary initial temperatures of the contacts.
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