# Quantum Thermal Amplifiers with Engineered Dissipation

## Abstract

**:**

## 1. Introduction

## 2. The Model

## 3. Non-Equilibrium Dynamics

#### Definition of Heat Currents

## 4. Quantum Thermal Transistor

#### 4.1. Amplification of Heat Currents

#### 4.2. Amplification Factor

## 5. Insights into the Transistor Effect via Entropic Measures of Correlations

## 6. Outro and Future Perspectives

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

GKSL | Gorini Kossakowski Sudarshan Lindblad |

MME | Markovian Master Equation |

NESS | Non-Equilibrium Steady State |

QTT | Quantum Thermal Transistor |

## Appendix A

**Figure A1.**The fidelity (as in Equation (A1)) between the various NESS obtained when varying the modulator temperature ${T}_{M}$ and the two states representing two very different kinds of entanglement for three particles. In (

**a**) the fidelity with the |W〉 state and in (

**b**) with the $|GHZ\rangle $ state. The common legend explains the settings we have considered in the paper.

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**Figure 2.**The three thermal currents defined in Equation (11) exchanged by the system with the three Ohmic reservoirs ($s=1$) as a function of the modulator temperature ${T}_{M}$. Upon fixing the frequency of the source qubit as reference, namely ${\omega}_{S}=\omega $, the parameters are $\omega =10{\omega}_{M}=3{\omega}_{D}={\zeta}_{SM}=6{\zeta}_{MD}={\zeta}_{SD}.$ The source and drain temperature are set to ${T}_{S}=10\omega $ and ${T}_{D}=0.01\omega $, respectively. The coupling strength of the system with the three reservoirs are ${10}^{6}{\lambda}_{S}={10}^{6}{\lambda}_{M}={10}^{4}{\lambda}_{D}=\omega $. Note that to highlight that the energy is conserved we plot $|{\mathcal{I}}_{D}|$. In the inset the current ${I}_{M}$ exchanged by the modulator reservoir with the system. The black solid line is a linear fit ${\mathcal{I}}_{M}=m{T}_{M}+q$, with $m\simeq 3.1\times {10}^{-4}$ and $q\simeq 3.4\times {10}^{-4}$.

**Figure 3.**We plot the amplification factors for different bath configurations, assuming as a reference the value giving the heat currents in Figure 2. (

**a**) The configurations are given by the temperature difference between the hot and the cold thermostat. We fix ${T}_{D}$ and set ${T}_{S}=5\omega $ (orange dashed line) and ${T}_{S}=25\omega $ (cyan dotted line). (

**b**) The configurations are given by a different spectral density for the reservoirs. We assume for all the three baths a subOhmic $s=0.5$ (orange dashed line) and superOhmic $s=1.5$ (cyan dotted line).

**Figure 4.**The tripartite mutual information defined in Equation (16) as a function of the modulator temperature ${T}_{M}$. All the configurations are labelled by the temperature of the source reservoir and by the type of the spectral density considered.

**Figure 5.**The bipartite mutual information defined in Equation (15) as a function of the modulator temperature ${T}_{M}$, when one of the three qubit is traced out. The plots refers to tracing out the qubit directly coupled with the source (

**a**), the modulator (

**b**), the drain (

**c**). In the common legend the configurations are labelled by the temperature of the source reservoir and by the type of the spectral density considered.

**Figure 6.**As measure of bipartite quantum correlation we consider the negativity defined in Equation (17) as a function of ${T}_{M}$. The negativity is defined for two-qubit systems so one of the three qubits has to be traced out. The plots refers to the two-qubit system obtained when the qubit directly coupled with, (

**a**) the source, (

**b**) the modulator, (

**c**) the drain is traced. The common legend explains the various considered settings.

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Mandarino, A.
Quantum Thermal Amplifiers with Engineered Dissipation. *Entropy* **2022**, *24*, 1031.
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**AMA Style**

Mandarino A.
Quantum Thermal Amplifiers with Engineered Dissipation. *Entropy*. 2022; 24(8):1031.
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**Chicago/Turabian Style**

Mandarino, Antonio.
2022. "Quantum Thermal Amplifiers with Engineered Dissipation" *Entropy* 24, no. 8: 1031.
https://doi.org/10.3390/e24081031