# Heat Modulation on Target Thermal Bath via Coherent Auxiliary Bath

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## Abstract

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## 1. Introduction

## 2. Model

## 3. Modulation of Heat Current via Auxiliary Bath

#### 3.1. Initial States of System and Baths

#### 3.2. Thermal Modulation with Thermal Auxiliary Bath

#### 3.3. Thermal Modulation with CAB

#### 3.3.1. Effects of Relative Phase on THC

#### 3.3.2. Effects of Coherence Magnitude on THC

#### 3.3.3. Maximum and Minimum of THC and Modulation Width

#### 3.3.4. Effect of Temperature on THC

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Schematic of heat modulation on the target thermal bath (TTB). The internal interaction of tripartite system (${S}_{a,b,c}$ ) is first implemented in $M1$ process; then the subsystems ${S}_{a}$ and ${S}_{c}$ are coupled to the $n$th ancilla (prepared in ${\rho}_{L}$) in coherent auxiliary bath (CAB) and the $n$th thermal atom (prepared in ${\rho}_{R}$) in TTB, respectively, in $M2$ process. After that, the two steps of $M1$ and $M2$ are implemented repeatedly, and ancilla (thermal atom) in CAB (TTB) interacting with ${S}_{a}$ (${S}_{c}$) are refreshed by the next one in each round. Thus, the steady heat current between the system and TTB, after many rounds, is established.

**Figure 2.**The heat current ${J}^{\mathit{SS}}$ as a function of the coupling strength ${\lambda}_{L}$, ${\lambda}_{L}\in \left[0,\pi \right]$, in terms of Equation (9) (red dotted curve) and the corresponding curve of fitting function given in Equation (15) with $A=0.567$, ${\phi}_{0}=0.055$ and $\Delta =3.445$ (blue dotted curve). The other parameters are chosen as: ${\lambda}_{R}=0.4\pi $, ${\xi}_{m}=0.15\pi $, $\tau =0.01$, $\omega =10$ and $\beta =0.01$.

**Figure 3.**The THC ${J}^{\mathit{SS}}$ as a periodical function of $\theta $ and ${\lambda}_{L}$ with $\theta \in \left[0,3\pi \right]$ and ${\lambda}_{L}\in \left[0,0.15\pi \right]$. The other parameters are set as: $\alpha =1$, ${\lambda}_{R}=0.4\pi $, ${\xi}_{m}=0.15\pi $, $\tau =0.01$, $\omega =10$ and $\beta =0.01$.

**Figure 4.**The variations of THC ${J}^{\mathit{SS}}$ as $\theta $ and ${\lambda}_{L}$. (

**a**) The phase diagram of the quantum machine working as a multifunctional device in the parametric regimes of $\theta $ and ${\lambda}_{L}$: RR, SR, HPSR, HPIR and HPAR; (

**b**) the variations of ${J}^{\mathit{SS}}$ with ${\lambda}_{L}$ for some fixed relative phases $\theta $. The other parameters are the same as the ones in Figure 3. The purple (white) dotted line in (

**a**) represents working points with ${J}^{\mathit{SS}}={J}_{\mathit{ref}}$ (${J}^{\mathit{SS}}=0$) also corresponding to the purple (black) solid line in (

**b**).

**Figure 5.**The variation of ${J}^{\mathit{SS}}$ with $\alpha $ and ${\lambda}_{L}$. (

**a**) The phase diagram of the quantum machine behaving as a multifunctional device in parameter space: $0\le \alpha \le 1$ and $0\le {\lambda}_{L}\le \pi /2$; (

**b**) the THC ${J}^{\mathit{SS}}$ as function of ${\lambda}_{L}$ for some selected $\alpha $. The other parameters are chosen by ${\lambda}_{R}=0.4\pi $, ${\xi}_{m}=0.15\pi $, $\tau =0.01$, $\omega =10$ and $\beta =0.01$.

**Figure 6.**(

**a**) The maximum and the minimum of THC, ${J}_{\mathrm{max}}^{\mathit{SS}}$ and ${J}_{\mathrm{min}}^{\mathit{SS}}$, and function regions of quantum machine; (

**b**) the modulation width of THC, $D={J}_{\mathrm{max}}^{\mathit{SS}}-{J}_{\mathrm{min}}^{\mathit{SS}}$, in in the full parametric space: ${\lambda}_{L}\in \left[0,\pi /2\right]$, $\alpha \in \left[0,1\right]$ and $\theta \in \left[0,\pi \right]$. The other parameters are set as ${\lambda}_{R}=0.4\pi $, ${\xi}_{m}=0.15\pi $, $\tau =0.01$, $\omega =10$ and $\beta =0.01$.

**Figure 7.**The heat current ${J}^{\mathit{SS}}$ as a function of the temperature of TTB ${T}_{R}$ and the coupling strength ${\lambda}_{L}$ with ${\lambda}_{L}\in \left[0,0.5\pi \right]$ for the CAB with $\alpha =1$ and $\theta =0.45\pi $ in (

**a**), and the thermal auxiliary bath in (

**b**). Here, the white solid line in (a) represents working points with ${J}^{\mathit{SS}}=0$, and the black solid line in (a,b) indicate the position with equal temperatures ${T}_{L}={T}_{R}=100$. The other parameters: ${\lambda}_{R}=0.4\pi $, ${\xi}_{m}=0.15\pi $, $\tau =0.01$, $\omega =10$ and $\beta =0.01$.

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Yu, W.-L.; Li, T.; Li, H.; Zhang, Y.; Zou, J.; Wang, Y.-D.
Heat Modulation on Target Thermal Bath via Coherent Auxiliary Bath. *Entropy* **2021**, *23*, 1183.
https://doi.org/10.3390/e23091183

**AMA Style**

Yu W-L, Li T, Li H, Zhang Y, Zou J, Wang Y-D.
Heat Modulation on Target Thermal Bath via Coherent Auxiliary Bath. *Entropy*. 2021; 23(9):1183.
https://doi.org/10.3390/e23091183

**Chicago/Turabian Style**

Yu, Wen-Li, Tao Li, Hai Li, Yun Zhang, Jian Zou, and Ying-Dan Wang.
2021. "Heat Modulation on Target Thermal Bath via Coherent Auxiliary Bath" *Entropy* 23, no. 9: 1183.
https://doi.org/10.3390/e23091183