# Thermal Quantum Correlations in Two Gravitational Cat States

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Model

#### Thermal Density Operator

## 3. Thermal Quantum Correlations

#### 3.1. Thermal Entanglement

#### 3.2. Quantum Coherence

## 4. Results and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Note

1 | We assume $\hslash =1$. |

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**Figure 1.**(Color online) Schematic representation of our model. Two particles are located in an even double well potential, whose minima are separated by a distance L. The quantity d (${d}^{\prime}$) describes the distance between the particles when each of them are at the same (different) relative minima.

**Figure 2.**(Color online) Concurrence $\mathcal{C}$ as a function of temperature T in the logarithmic scale, for $w=0.1$ (green curve), $w=1.0$ (red curve), $w=2.0$ (blue curve), $w=3.0$ (black curve) and different values of $\Delta $. It is assumed that $\Delta <w$ in each case.

**Figure 3.**(Color online) Concurrence $\mathcal{C}$ as a function of T in the logarithmic scale, for $w=0.01$ (red curve), $w=0.1$ (blue curve), $w=1.0$ (green curve) and different values of $\Delta $. It is assumed that $\Delta >w$ in each case.

**Figure 4.**(Color online) Concurrence $\mathcal{C}$ (solid curve) and the quantum coherence ${\mathcal{C}}_{{l}_{1}}$ (dashed curve) as a function of T in the logarithmic scale for parameter set $\Delta =0.01$ (red curve), $\Delta =0.1$ (blue curve) and $\Delta =0.2$ (green curve) and fixed value $w=1.0$. In each case, it is assumed that $\Delta <w$.

**Figure 5.**(Color online) Quantum coherence ${\mathcal{C}}_{{l}_{1}}$ (green curve), ${g}_{1}$ (blue curve), ${g}_{2}$ (red curve) as a function of T in the logarithmic scale for parameter set $w=1.0$ and $\Delta =0.2$. The magenta vertical solid line indicates the threshold temperature from which thermal fluctuations occur.

**Figure 6.**(Color online) Concurrence $\mathcal{C}$ (solid curve) and the quantum coherence ${\mathcal{C}}_{{l}_{1}}$ (dashed curve) as a function of T in the logarithmic scale for parameter set $w=0.05$ and $\Delta =0.05$ (red curve), $w=0.1$ and $\Delta =0.1$ (blue curve), $w=0.5$ and $\Delta =0.5$ (green curve). Here, it is assumed that $\Delta =w$.

**Figure 7.**(Color online) Concurrence $\mathcal{C}$ (solid curve) and the quantum coherence ${\mathcal{C}}_{{l}_{1}}$ (dashed curve) as a function of T in the logarithmic scale with fixed values $w=3.0$, $\Delta =100$, $\Delta =300$ and $\Delta =600$. It is assumed that $\Delta >w$ in each case.

**Figure 8.**(Color online) The plot shows the concurrence $\mathcal{C}$ (red curve) and the quantum coherence ${\mathcal{C}}_{{l}_{1}}$ (blue curve) as a function of T in the logarithmic scale, assuming $\Delta /{k}_{B}=0.5101\times {10}^{-6}$. This case lies within the $\Delta <w$ class.

**Figure 9.**(Color online) The plot shows the concurrence $\mathcal{C}$ (red curve) and the quantum coherence ${\mathcal{C}}_{{l}_{1}}$ (blue curve) as a function of T, assuming $\Delta /{k}_{B}=17.0072$. This case lies within the $\Delta >w$ class. quantum coherence (

**a**); quantum coherence after the thermal entanglement (

**b**).

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**MDPI and ACS Style**

Rojas, M.; Lobo, I.P.
Thermal Quantum Correlations in Two Gravitational Cat States. *Universe* **2023**, *9*, 71.
https://doi.org/10.3390/universe9020071

**AMA Style**

Rojas M, Lobo IP.
Thermal Quantum Correlations in Two Gravitational Cat States. *Universe*. 2023; 9(2):71.
https://doi.org/10.3390/universe9020071

**Chicago/Turabian Style**

Rojas, Moises, and Iarley P. Lobo.
2023. "Thermal Quantum Correlations in Two Gravitational Cat States" *Universe* 9, no. 2: 71.
https://doi.org/10.3390/universe9020071