# Generalized Master Equation Approach to Time-Dependent Many-Body Transport

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## Abstract

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## 1. Introduction

## 2. Formalism

#### 2.1. Generalized Master Equation for Hybrid Systems

#### 2.2. ‘Hybrid’ States and Diagonalization Procedure

#### 2.3. Numerical Implementation and Observables

#### 2.4. Coupling between Leads and Central System

## 3. Many-Body Effects in the Transient Regime

#### 3.1. Transient Charging of Excited States

#### 3.2. Coulomb Switching of Transport in Parallel Quantum Dots

## 4. Thermoelectric Transport

## 5. Electron Transport through Photon Cavities

#### 5.1. The Electron-Photon Coupling

#### 5.2. Results

## 6. Steady-State

#### 6.1. The Steady-State Limit

#### 6.2. Results

## 7. Summary

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**A schematic of the coupling of the system to the leads. The transparent green areas correspond to the contact regions defined by the nonlocal overlap function ${g}_{qn}^{\mathrm{L},\mathrm{R}}$ in ${H}_{T}\left(t\right)$.

**Figure 2.**(

**a**) The chemical potentials ${\mu}_{g,N}^{\left(i\right)}$ (red crosses) and ${\mu}_{x,N}^{\left(i\right)}$ (blue crosses) for N-particle configurations, $N=1,2,3$. For a given particle number N the chemical potentials are ordered vertically according to the index $i=0,1,\dots $. The horizontal lines correspond to specific values of the chemical potentials in the leads (see the discussion in the text); (

**b**) The time-dependent currents in the left lead at different values of the bias window ${\mu}_{L}-{\mu}_{R}$.

**Figure 3.**The populations of ground (g) and excited (x) N-particle states ($N=1,2,3$) for two bias windows; (

**a**) ${\mu}_{L}=4$ meV, ${\mu}_{R}=2$ meV; (

**b**) ${\mu}_{L}=5.5$ meV, ${\mu}_{R}=4$ meV. In panel (

**a**) ${P}_{3}$ is negligible and was omitted.

**Figure 4.**(

**a**) The transient currents ${J}_{L,N}$ and ${J}_{R,N}$ associated to one and two-particle configurations for ${\mu}_{L}=5.5$ meV, ${\mu}_{R}=4$ meV; (

**b**) The same currents for a bias window ${\mu}_{L}=4$ meV, ${\mu}_{R}=2$ meV; (

**c**) The charge ${q}_{N}$ accumulated on N-particle states at ${\mu}_{L}=5.5$ meV, ${\mu}_{R}=4$ meV; (

**d**) ${q}_{N}$ for ${\mu}_{L}=4$ meV, ${\mu}_{R}=2$ meV, and ${q}_{N}$ are given in units of electron charge e.

**Figure 5.**A sketch of the parallel double-dot system. Each QD is coupled to source-drain particle reservoirs described by chemical potentials ${\mu}_{Ls}$ and ${\mu}_{Rs}$, $s=a,b$. There is no interdot electron tunneling but the systems are correlated via Coulomb interaction.

**Figure 6.**The transient currents in the two systems for different chemical potentials of the leads: (

**a**) ${\mu}_{La}={\mu}_{Lb}=4.25$ meV, ${\mu}_{Ra}={\mu}_{Rb}=3.75$ meV; (

**b**) ${\mu}_{La}={\mu}_{Lb}=4.75$ meV, ${\mu}_{Ra}={\mu}_{Rb}=4.35$ meV; (

**c**,

**d**) The charge occupations of the two systems associated to the currents in Figure 6a,b. The charges ${Q}_{a,b}$ are given in units of electron charge e.

**Figure 7.**(

**a**) The time evolution of the currents in the left and right leads, ${J}_{L,R}$, driven by a temperature bias where ${T}_{L}=5.8$ K and ${T}_{R}=0.58$ K. With red color the results for the chemical potential ${\mu}_{L}={\mu}_{R}=48$ meV, and with blue color for ${\mu}_{L}={\mu}_{R}=54$ meV. In the steady state the currents have opposite sign; (

**b**) The current in the steady state for two different temperatures of the left lead, ${T}_{L}=5.8$ K (red) and ${T}_{L}=11.5$ K (blue), for variable chemical potentials ${\mu}_{L}={\mu}_{R}$.

**Figure 8.**The expectation value $\langle {(r/{a}_{w})}^{2}\rangle $ for the first photon replica of the two-electron ground state in the closed system at $t=0$ for x- and y-polarization of the photon field. $\hslash \omega =2.0$ meV, $B=0.1$ T. Two parallel quantum dots are embedded in the central system.

**Figure 9.**The left (black) and right (gold) currents and the mean electron number (blue) for initially fully entangled Rabi-split singlet two-electron states as the interacting system discharges in the transient regime. $\hslash \omega =2.0$ meV, $B=0.1$ T. Two parallel quantum dots are embedded in the central system.

**Figure 10.**(

**upper left**) For the closed system as functions of the number of the eigenstate $\mu $, the many-body energy (squares), the mean photon ($\gamma $) and electron content (e), and the mean spin z-component (${S}_{z}$). The horizontal yellow lines represent the chemical potentials of the left (${\mu}_{L}$) and right leads (${\mu}_{R}$) when the system will be coupled to them. (

**upper right**) The mean electron (solid) and photon number (dashed) in the central system as a function of time. The mean occupation of the many-body eigenstates of the system for ${g}_{\mathrm{EM}}=1\times {10}^{-6}$ meV (

**lower left**), and ${g}_{\mathrm{EM}}=0.05$ meV (

**lower right**). ${V}_{g}=-1.6$ mV, $\hslash \omega =0.8$ meV, x-polarization, $\kappa =1\times {10}^{-5}$ meV, ${L}_{x}=150$ nm, and $B=0.1$ T. No quantum dots in the short wire.

**Figure 11.**The mean occupation of the many-body eigenstates of the system when the initial state is the ground state $|1)$ (

**left**), or the first photon replica of the ground state $|2)$ (

**right**). ${g}_{\mathrm{EM}}=0.05$ meV. ${V}_{g}=-2.0$ mV, $\hslash \omega =0.8$ meV, x-polarization, ${L}_{x}=150$ nm, and $B=0.1$ T. Two parallel quantum dots embedded in the short wire, but no photon reservoir.

**Figure 12.**The spectral density $S\left(E\right)$ of the emitted cavity radiation for the central system in a steady state (

**left**), and the spectral densities for the current–current correlations ${D}_{l{l}^{\prime}}\left(E\right)$ (

**right**). ${g}_{\mathrm{EM}}=0.1$ meV, ${V}_{g}=-2.0$ mV, $\hslash \omega =0.72$ meV, $\kappa =1\times {10}^{-3}$ meV, and ${L}_{x}=150$ nm. Two parallel quantum dots embedded in the short wire.

**Figure 13.**The mean electron (e), photon ($\gamma $), z-component of the spin (${S}_{z}$), trace of the reduced density matrix, and the Réniy-2 entropy (S) as functions of time. $\hslash \omega =0.373$ meV, x-polarization, $\kappa =1\times {10}^{-5}$ meV, ${g}_{\mathrm{EM}}=0.05$ meV, and ${L}_{x}=180$ nm. Two asymmetrically embedded quantum dots in the short wire.

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**MDPI and ACS Style**

Moldoveanu, V.; Manolescu, A.; Gudmundsson, V. Generalized Master Equation Approach to Time-Dependent Many-Body Transport. *Entropy* **2019**, *21*, 731.
https://doi.org/10.3390/e21080731

**AMA Style**

Moldoveanu V, Manolescu A, Gudmundsson V. Generalized Master Equation Approach to Time-Dependent Many-Body Transport. *Entropy*. 2019; 21(8):731.
https://doi.org/10.3390/e21080731

**Chicago/Turabian Style**

Moldoveanu, Valeriu, Andrei Manolescu, and Vidar Gudmundsson. 2019. "Generalized Master Equation Approach to Time-Dependent Many-Body Transport" *Entropy* 21, no. 8: 731.
https://doi.org/10.3390/e21080731