# Manifestation of the Purcell Effect in Current Transport through a Dot–Cavity–QED System

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Hamiltonian of the Total System

## 3. Results

^{th}is not found anymore at ${g}_{\gamma}=0.3$ meV (see Figure 6b).

## 4. Summary

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

QD | Quantum Dot |

${T}_{L}$ | Temperature of the left lead |

${T}_{R}$ | Temperature of the right lead |

MB | Many-Body states |

0 | Ground-state energy |

1$\gamma $0 | one-photon replica of the ground-state |

2$\gamma $0 | two-photon replica of the ground-state |

1st | first-excited state |

1$\gamma $1st | one-photon replica of the first-excited state |

2nd | second-excited state |

3rd | third-excited state |

4th | fourth-excited state |

5th | fifth-excited state |

6th | sixth-excited state |

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**Figure 1.**(

**a**) Schematic diagram demonstrates the QD system (black) connected to the leads (blue) where the chemical potential of the left lead (${\mu}_{L}$) is higher than the that of the right lead (${\mu}_{R}$). The red zigzags indicate the quantized photon field in the cavity (red rectangle). (

**b**) The potential ${V}_{\mathrm{QD}}$ defining the QD embedded in a short quantum wire that is connected diametrically to the leads in the x-direction. (

**c**) The potential ${V}_{\mathrm{QD}}$ in relation to the chemical potentials of the leads and the three lowest one-electron states of the system.

**Figure 2.**Many-Body energy spectra of the cavity-QD system versus the photon energy for x- (

**a**) and y-polarized (

**b**) photon field, where brown squares refer to zero-electron states (0ES), blue circles display one-electron states (1ES), and red triangles are two-electron states (2ES). The chemical potential of the left and the right leads are ${\mu}_{L}=1.65$ meV (purple line) and ${\mu}_{R}=1.55$ meV (green line), respectively. 0 is the one-electron ground-state energy, $1\gamma $0 and $2\gamma $0 demonstrates the one- and two-photon replica of the 0, and 1st, 2nd, 3rd, 4th, 5th, 6th indicate the one-electron first-, second-, third-, fourth-, fifth- and sixth-excited state, respectively. The $1\gamma $1st indicates the one-photon replica state of the 1st. The photon number initially in the reservoir ${n}_{\mathrm{R}}=1$, ${g}_{\gamma}=0.1$ meV, and $\kappa ={10}^{-5}$ meV. The magnetic field is $B=0.1\phantom{\rule{3.33333pt}{0ex}}\mathrm{T}$, $e{V}_{\mathrm{g}}=0.651$ meV, ${T}_{\mathrm{L},\mathrm{R}}=0.5$ K and $\hslash {\Omega}_{0}=2.0\phantom{\rule{3.33333pt}{0ex}}\mathrm{meV}$.

**Figure 3.**Current versus the photon energy is plotted for the photon number ${n}_{\mathrm{R}}=0$ (purple squares), 1 (green triangles), and 2 (blue diamonds) in the case of x- (

**a**) and y-polarized (

**b**) photon field with ${g}_{\gamma}=0.1$ meV, and $\kappa ={10}^{-5}$ meV. The vertical red lines indicates the positions of the resonance states. The chemical potential of the left lead is ${\mu}_{L}=1.65$ meV and the right lead is ${\mu}_{R}=1.55$ meV. The external magnetic field is $B=0.1\phantom{\rule{3.33333pt}{0ex}}\mathrm{T}$, $e{V}_{\mathrm{g}}=0.651$ meV, ${T}_{\mathrm{L},\mathrm{R}}=0.5$ K, and $\hslash {\Omega}_{0}=2.0\phantom{\rule{3.33333pt}{0ex}}\mathrm{meV}$.

**Figure 4.**Current versus the photon energy for the cavity–reservoir coupling $\kappa ={10}^{-5}$ meV (purple squares), ${10}^{-4}$ (green triangles), and ${10}^{-3}$ (blue diamonds) in the case of x- (

**a**) and y-polarized (

**b**) photon field with ${g}_{\gamma}=0.1$ meV, and ${n}_{\mathrm{R}}=1$. The vertical red lines indicates the positions of the resonance states. The chemical potential of the leads are ${\mu}_{L}=1.65$ meV, and ${\mu}_{R}=1.55$ meV. The external weak magnetic field is $B=0.1\phantom{\rule{3.33333pt}{0ex}}\mathrm{T}$, $e{V}_{\mathrm{g}}=0.651$ meV, ${T}_{\mathrm{L},\mathrm{R}}=0.5$ K, and $\hslash {\Omega}_{0}=2.0\phantom{\rule{3.33333pt}{0ex}}\mathrm{meV}$.

**Figure 5.**The partial occupation of the spin-up component of 1st and 1$\gamma $1st for different cavity–reservoir coupling is plotted x- (

**a**) and y-polarized photon field (

**b**). Furthermore, the partial occupation of the spin-up component of 0 and 1$\gamma $0 for different cavity–reservoir coupling is presented for x- (

**c**) and y-polarized photon field (

**d**). The cavity–reservoir coupling is assumed to be $\kappa ={10}^{-5}$ (purple for 0, and brown for 1st), ${10}^{-4}$ (green for 0, and yellow for 1st), and ${10}^{-3}$ (light blue for 0, and dark blue for 1st) in the case of x- (

**a**) and y-polarized (

**b**) photon field with ${g}_{\gamma}=0.1$ meV, and ${n}_{\mathrm{R}}=1$. The vertical red lines indicates the positions of the resonance states. The chemical potential of the leads are ${\mu}_{L}=1.65$ meV and ${\mu}_{R}=1.55$ meV. The magnetic field is $B=0.1\phantom{\rule{3.33333pt}{0ex}}\mathrm{T}$, $e{V}_{\mathrm{g}}=0.651$ meV, ${T}_{\mathrm{L},\mathrm{R}}=0.5$ K, and $\hslash {\Omega}_{0}=2.0\phantom{\rule{3.33333pt}{0ex}}\mathrm{meV}$.

**Figure 6.**Many-Body energy spectra of the cavity-QD system versus the photon energy for x- (

**a**) and y-polarized (

**b**) photon field, where brown squares refer to zero-electron states (0ES), blue circles display one-electron states (1ES), and red triangles are two-electron states (2ES). The chemical potential of the left and the right leads are ${\mu}_{L}=1.65$ meV (purple line) and ${\mu}_{R}=1.55$ meV (green line), respectively. 0 is the one-electron ground-state energy, $1\gamma $0 and $2\gamma $0 demonstrates the one- and two-photon replica of the 0, and 1st, 2nd, 3rd, 4th, 5th, 6th indicate the one-electron first-, second-, third-, fourth-, fifth- and sixth-excited state, respectively. The $1\gamma $1st indicates the one-photon replica state of the 1st. The photon number initially in the reservoir ${n}_{\mathrm{R}}=1$, ${g}_{\gamma}=0.3$ meV, and $\kappa ={10}^{-5}$ meV. The magnetic field is $B=0.1\phantom{\rule{3.33333pt}{0ex}}\mathrm{T}$, $e{V}_{\mathrm{g}}=0.651$ meV, ${T}_{\mathrm{L},\mathrm{R}}=0.5$ K and $\hslash {\Omega}_{0}=2.0\phantom{\rule{3.33333pt}{0ex}}\mathrm{meV}$.

**Figure 7.**Current versus the photon energy for ${g}_{\gamma}=0.1$ (purple squares), $0.2$ (green triangles), and $0.3$ meV (blue diamonds) in the case of x- (

**a**) and y-polarized (

**b**) photon field. The cavity–reservoir coupling is $\kappa ={10}^{-5}$, and ${n}_{\mathrm{R}}=1$. The vertical red lines indicates the positions of the main resonance states. The chemical potential of the left leads are ${\mu}_{L}=1.65$ meV and ${\mu}_{R}=1.55$ meV. The magnetic field is $B=0.1\phantom{\rule{3.33333pt}{0ex}}\mathrm{T}$, $e{V}_{\mathrm{g}}=0.651$ meV, ${T}_{\mathrm{L},\mathrm{R}}=0.5$ K, and $\hslash {\Omega}_{0}=2.0\phantom{\rule{3.33333pt}{0ex}}\mathrm{meV}$.

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**MDPI and ACS Style**

Abdullah, N.R.; Tang, C.-S.; Manolescu, A.; Gudmundsson, V.
Manifestation of the Purcell Effect in Current Transport through a Dot–Cavity–QED System. *Nanomaterials* **2019**, *9*, 1023.
https://doi.org/10.3390/nano9071023

**AMA Style**

Abdullah NR, Tang C-S, Manolescu A, Gudmundsson V.
Manifestation of the Purcell Effect in Current Transport through a Dot–Cavity–QED System. *Nanomaterials*. 2019; 9(7):1023.
https://doi.org/10.3390/nano9071023

**Chicago/Turabian Style**

Abdullah, Nzar Rauf, Chi-Shung Tang, Andrei Manolescu, and Vidar Gudmundsson.
2019. "Manifestation of the Purcell Effect in Current Transport through a Dot–Cavity–QED System" *Nanomaterials* 9, no. 7: 1023.
https://doi.org/10.3390/nano9071023