# Photon Propagation through Linearly Active Dimers

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## Abstract

**:**

## 1. Introduction

## 2. Quantum Model and Configurations

## 3. Spontaneous Generation of Photons

**gain-loss**dimer,

**gain-gain**dimer yields the following expressions,

**gain-passive**dimer, we can write the spontaneous generation as:

**passive-loss**and

**loss-loss**dimer,

## 4. Photon Bunching in Spontaneous Generation

**balanced gain-loss**dimer,

**gain-gain**dimer,

**gain-passive**dimer,

**passive-loss**and

**loss-loss**dimer,

## 5. Photon Propagation

**balanced gain-loss**dimer,

**gain-gain**dimer,

**gain-passive**dimer,

**passive-loss**and

**loss-loss**dimers, the intensity only depends on the initial state and decays due to the nature of the auxiliary $\beta $ parameter,

## 6. Photon Bunching and Anti-Bunching in Photon Propagation

**balanced gain-loss**dimer, we use the spontaneous generation terms, ${n}_{j}^{\left(00\right)}\left(\zeta \right)$, provided by Equation (13), and the first order two-point correlation, ${n}_{12}^{\left(00\right)}\left(\zeta \right)$, from Equation (21). In the

**gain-gain**dimer, the expressions for ${n}_{j}^{\left(00\right)}\left(\zeta \right)$ and ${n}_{12}^{\left(00\right)}\left(\zeta \right)$ are given by Equations (14) and (22), in that order. In the

**gain-passive**dimer, we only need the definitions provided by Equations (15) and (23) for the spontaneous generation and the first order two-point correlation, respectively. Finally, for

**passive-loss**and

**loss-loss**dimers, the spontaneous generation is null, ${n}_{j}^{\left(00\right)}\left(\zeta \right)={n}_{12}^{\left(00\right)}\left(\zeta \right)=0$, such that

## 7. Conclusions

## Acknowledgments

## Conflicts of Interest

## Abbreviations

MDPI | Multidisciplinary Digital Publishing Institute. |

$\mathcal{PT}$ | Parity-Time. |

$\Re \left(\alpha \right)$ and $\Im \left(\alpha \right)$ | Real and imaginary parts of a complex number $\alpha $, in that order. |

CONACYT | Consejo Nacional de Ciencia y Tecnología |

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**Figure 1.**Schematic showing the renormalized light intensity arising from the spontaneous generation of photons through two coupled waveguides in the $\mathcal{PT}$-symmetry regime with balanced gain-loss configuration.

**Figure 2.**Instantaneously renormalized spontaneous generation, ${\tilde{n}}_{j}^{\left(00\right)}\left(\zeta \right)$, along different realizations of the effective $\mathcal{PT}$-symmetric dimer. The first row, (

**a**–

**c**), shows balanced gain-loss, second row, (

**d**–

**f**), shows gain-gain, and third row, (

**g**–

**i**), shows gain-passive configurations in the $\mathcal{PT}$-symmetric regime, the first column with $\left|\gamma \right|=0.5$, the Kato point with $\left|\gamma \right|=1$, the second column, and broken symmetry regime, and the third column with $\left|\gamma \right|=1.2$. Values for the first and second waveguides are shown with a solid blue and a dashed red lines, in that order. Note the oscillatory behavior of the spontaneous generation inside and its asymptotic behavior outside the $\mathcal{PT}$-symmetric regime.

**Figure 3.**Photon bunching shown in terms of the ${q}^{\left(00\right)}\left(\zeta \right)$ parameter for different realizations of the effective $\mathcal{PT}$-symmetric dimer. The first row, (

**a**–

**c**), shows balanced gain-loss, second row, (

**d**–

**f**), shows gain-gain, and third row, (

**g**–

**i**), shows gain-passive configurations in the $\mathcal{PT}$-symmetric regime, the first column with $\left|\gamma \right|=0.5$, the Kato point with $\left|\gamma \right|=1$, the second column, and broken symmetry regime, and the third column with $\left|\gamma \right|=1.2$.

**Figure 4.**Instantaneously renormalized mean photon number, ${\tilde{n}}_{j}^{\left(10\right)}\left(\zeta \right)$, along different realizations of the effective $\mathcal{PT}$-symmetric dimer. The first row, (

**a**–

**c**), shows balanced gain-loss, the second row, (

**d**–

**f**), shows gain-gain, the third row, (

**g**–

**i**), shows gain-passive, and the fourth row, (

**j**–

**l**), shows both passive-loss and loss-loss configurations in the $\mathcal{PT}$-symmetric regime, the first column with $\left|\gamma \right|=0.5$, the Kato point with $\left|\gamma \right|=1$, the second column, and the broken symmetry regime, and the third column with $\left|\gamma \right|=1.2$. Values for the first and second waveguides are shown with a solid blue and a dashed red lines, in that order.

**Figure 5.**Photon bunching and anti-bunching shown in terms of the ${q}^{\left(2002\right)}\left(\zeta \right)$ parameter for different realizations of the effective $\mathcal{PT}$-symmetric dimer. The first row, (

**a**–

**c**), shows balanced gain-loss, the second row, (

**d**–

**f**), shows gain-gain, the third row, (

**g**–

**i**), shows gain-passive, and the fourth row, (

**j**–

**l**), shows both passive-loss and loss-loss configurations in the $\mathcal{PT}$-symmetric regime, the first column with $\left|\gamma \right|=0.5$, the Kato point with $\left|\gamma \right|=1$, the second column, and the broken symmetry regime, the third column with $\left|\gamma \right|=1.2$.

**Table 1.**A summary of the parameters involved in the different feasible experimental realizations of the $\mathcal{PT}$-symmetric dimer .

Realization | ${\mathit{n}}_{1}$ | ${\mathit{n}}_{2}$ | n | ${\mathit{n}}_{0}$ | $\mathit{\gamma}$ | $\mathit{\beta}$ |
---|---|---|---|---|---|---|

Gain-loss | ${n}_{R}-i{n}_{I}$ | ${n}_{R}+i{n}_{I}$ | $-i\frac{{n}_{I}}{g}$ | $\frac{{n}_{R}}{g}$ | $-\frac{{n}_{I}}{g}$ | 0 |

Gain-gain | ${n}_{R}-i{n}_{I,1}$ | ${n}_{R}-i{n}_{I,2}$ | $i\frac{-{n}_{I,1}+{n}_{I,2}}{2g}$ | $\frac{{n}_{R}}{g}-i\frac{{n}_{I,1}+{n}_{I,2}}{2g}$ | $\frac{-{n}_{I,1}+{n}_{I,2}}{2g}$ | $\frac{{n}_{I,1}+{n}_{I,2}}{2g}$ |

Gain-passive | ${n}_{R}-i{n}_{I}$ | ${n}_{R}$ | $-i\frac{{n}_{I}}{2g}$ | $\frac{{n}_{R}}{g}+i\frac{{n}_{I}}{2g}$ | $-\frac{{n}_{I}}{2g}$ | $\frac{{n}_{I}}{2g}$ |

Passive-loss | ${n}_{R}$ | ${n}_{R}+i{n}_{I}$ | $-i\frac{{n}_{I}}{2g}$ | $\frac{{n}_{R}}{g}+i\frac{{n}_{I}}{2g}$ | $-\frac{{n}_{I}}{2g}$ | $-\frac{{n}_{I}}{2g}$ |

Loss-loss | ${n}_{R}+i{n}_{I,1}$ | ${n}_{R}+i{n}_{I,2}$ | $i\frac{{n}_{I,1}-{n}_{I,2}}{2g}$ | $\frac{{n}_{R}}{g}+i\frac{{n}_{I,1}+{n}_{I,2}}{2g}$ | $\frac{{n}_{I,1}-{n}_{I,2}}{2g}$ | $\frac{-{n}_{I,1}-{n}_{I,2}}{2g}$ |

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Huerta Morales, J.D.; Rodríguez-Lara, B.M. Photon Propagation through Linearly Active Dimers. *Appl. Sci.* **2017**, *7*, 587.
https://doi.org/10.3390/app7060587

**AMA Style**

Huerta Morales JD, Rodríguez-Lara BM. Photon Propagation through Linearly Active Dimers. *Applied Sciences*. 2017; 7(6):587.
https://doi.org/10.3390/app7060587

**Chicago/Turabian Style**

Huerta Morales, José Delfino, and Blas Manuel Rodríguez-Lara. 2017. "Photon Propagation through Linearly Active Dimers" *Applied Sciences* 7, no. 6: 587.
https://doi.org/10.3390/app7060587