[Paper Review] Analytical expression of geometrical pumping for a quantum dot based on quantum master equation

Ryosuke Yoshii, Hisao Hayakawa|arXiv (Cornell University)|Dec 13, 2013

Quantum and electron transport phenomena3 citations

TL;DR

This paper derives an analytical expression for geometrical pumping in a single quantum dot with electron-electron interactions using the quantum master equation (QME). It demonstrates that spin-dependent Coulomb interactions induce a Berry-like phase in parameter space, leading to a finite Berry-Sinitsyn-Nemenman (BSN) curvature and a quantized pumped current under adiabatic modulation of reservoir parameters such as chemical potentials and temperatures.

ABSTRACT

We analytically investigate a non-equilibrium quantum pumping for a single quantum dot connected to external leads on the basis of the quantum master equation (QME). We show that the Coulomb interaction associated with the spin effect in the dot induces the Berry-like phase in the parameter space and this phase results in the excess charge transfer for the cyclic modulation of parameters in leads. We obtain an analytical expression of the curvature of the phase and that for the pumped currents.

Motivation & Objective

• To analytically investigate non-equilibrium quantum pumping in a single quantum dot with electron-electron interactions using the quantum master equation (QME).
• To identify the role of spin-dependent Coulomb interactions in generating a Berry-like geometric phase in parameter space.
• To derive an analytical expression for the Berry-Sinitsyn-Nemenman (BSN) curvature in the presence of many-body correlations.
• To demonstrate that pumped current arises from adiabatic modulation of reservoir parameters (e.g., chemical potential, temperature), even in the absence of external bias.
• To clarify the explicit parameter dependence of the BSN curvature and pumped current, overcoming limitations of prior numerical approaches.

Proposed method

• Formulates a quantum dot model coupled to two leads using the Anderson Hamiltonian with on-site Coulomb interaction and spin degrees of freedom.
• Applies the quantum master equation (QME) with a counting field to compute the cumulant generating function and extract the pumped current.
• Uses a perturbative expansion in the inverse temperature times on-site energy (βU) to compute the right and left eigenstates of the Liouvillian to first order.
• Derives the BSN curvature from the inner product of the left and right eigenvectors of the Liouvillian, using the formula for geometric phase in open quantum systems.
• Evaluates the curvature in two limits: βU ≪ 1 (low U) and βU ≫ 1 (high U), obtaining analytical expressions in both regimes.
• Validated the analytical results by showing vanishing curvature in the non-interacting limit (U=0), confirming the role of electron correlation.

Experimental results

Research questions

• RQ1 How does electron-electron interaction in a quantum dot induce a geometric phase in the parameter space of reservoirs?
• RQ2 What is the analytical form of the Berry-Sinitsyn-Nemenman (BSN) curvature for a quantum dot with spin-dependent Coulomb interaction?
• RQ3 Can a quantized pumped current be generated by adiabatic modulation of reservoir parameters such as chemical potential and temperature?
• RQ4 How does the BSN curvature depend explicitly on system parameters like temperature, chemical potential, and interaction strength?
• RQ5 What is the role of spin degrees of freedom in enabling geometric pumping in the absence of external bias?

Key findings

• The Coulomb interaction in the quantum dot induces a finite Berry-Sinitsyn-Nemenman (BSN) curvature in the parameter space of reservoirs.
• The BSN curvature arises from spin-dependent electron correlations and vanishes in the non-interacting limit (U=0), confirming its many-body origin.
• An analytical expression for the BSN curvature is derived in both the low- and high-U limits, providing explicit parameter dependence.
• The pumped current is generated by adiabatic modulation of reservoir parameters (e.g., chemical potential, temperature), even without external bias.
• The curvature expression is validated by showing that the left eigenvector's first-order correction in βU vanishes for χ=0, consistent with the geometric phase formalism.
• The method provides a closed-form analytical result, resolving the ambiguity in prior numerical studies regarding parameter dependence.