# Revisiting the Optical PT-Symmetric Dimer

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## Abstract

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## 1. Introduction

## 2. Linear $\mathcal{PT}$-Symmetric Dimer

#### 2.1. Quantum Mechanics, Linear Algebra Approach

#### 2.2. Nonlocal Oscillator, Partial Differential Equation Approach

#### 2.3. Nonlinear Oscillator, Renormalized Fields Approach

## 3. Nonlinear $\mathcal{PT}$-Symmetric Dimer

## 4. Linear $\mathcal{PT}$-Symmetric Planar $\mathit{N}$-Waveguide Coupler

## 5. Non-Hermitian Ehrenfest Theorem and Generalized Stokes Vector

## 6. Quantum $\mathcal{P}$$\mathcal{T}$-Symmetric Dimer

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

$\Re (x),\Im (x)$ | Real and imaginary parts of x, in that order |

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

$U(N)$, $SU(N)$, $SO(N)$ | Unitary group, special unitary group, special orthogonal group of degree N |

$SO(2,1)$ | Pseudo orthogonal group, Lorentz group, in 2+1 dimension. |

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**Figure 1.**Schematics of the experimental realizations of the linear $\mathcal{PT}$-symmetric dimer. (

**a**) Passive laser-engraved waveguides; (

**b**) passive waveguides with metallic scatterers; (

**c**) pumped active waveguides; (

**d**) pumped active whispering-gallery mode microcavities.

**Figure 2.**Coupling matrix eigenvalue dynamics: (

**a**) $\mathcal{PT}$-symmetric regime; (

**b**) fully-degenerate regime; and (

**c**) broken symmetry regime. The black arrows show the direction of the eigenvalues as the gain to coupling ratio increases.

**Figure 3.**Absolute field amplitude propagation in a coupler with effective symmetric loss, blue waveguide, and gain, red waveguide, in the: (

**a**) $\mathcal{PT}$-symmetric regime, $\gamma =0.5$; (

**b**) fully-degenerate regime, $\gamma =1$; and (

**c**) broken symmetry regime, $\gamma =1.5$.

**Figure 4.**Renormalized field intensity propagation in the waveguides with effective loss, ${|{\tilde{\mathcal{E}}}_{1}|}^{2}$, solid blue line, and gain, ${|{\tilde{\mathcal{E}}}_{2}|}^{2}$, dashed red line, in the (

**a**) $\mathcal{PT}$-symmetric regime, $\gamma =0.5$; (

**b**) fully-degenerate regime, $\gamma =1$; and (

**c**) broken symmetry regime, $\gamma =1.5$, for an initial field impinging just at the first waveguide.

**Figure 5.**Stokes vector propagation in the waveguide coupler with effective symmetric loss and gain, in the (

**a**) $\mathcal{PT}$-symmetric regime, $\gamma =0.5$; (

**b**) fully-degenerate regime, $\gamma =1$; and (

**c**) broken symmetry regime, $\gamma =1.5$, for the initial conditions ${\mathcal{E}}_{1}(0)=1$ and ${\mathcal{E}}_{2}(0)=0$ in black and ${\mathcal{E}}_{1}(0)=1/\sqrt{3}$ and ${\mathcal{E}}_{2}(0)=\sqrt{1-{|{\mathcal{E}}_{1}(0)|}^{2}}$ in red.

**Figure 6.**Stokes vector propagation in the passive two-waveguide coupler, $\gamma =0$, with nonlinearities (

**a**) below, $\kappa =1.5$; (

**b**) at, $\kappa =2$; and (

**c**) above, $\kappa =2.5$, the critical Kerr nonlinearity to coupling strength ratio, $\kappa =2$. The figure shows: (

**a**) stationary point; (

**b**) trajectory infinitesimally near the stationary point; and (

**c**) separatrix in black; (

**a**,

**b**) Rabi and (

**c**) Josephson oscillations in red; and (

**a**)–(

**c**) Rabi oscillations in blue.

**Figure 7.**Renormalized Stokes vector propagation in the waveguide coupler with a fixed gain to coupling ratio $\gamma =0.001$ and variable effective nonlinearity to coupling ratio (

**a**) $\kappa =1.5$; (

**b**) $\kappa =2$; and (

**c**) $\kappa =2.5$, for the same initial conditions as Figure 6.

**Figure 8.**Renormalized Stokes vector propagation in the waveguides’ waveguide coupler with a fixed effective Kerr nonlinearity to coupling ratio $\kappa =1$ and variable effective gain to coupling ratio (

**a**) $\gamma =\pi -1-0.2$; (

**b**) $\gamma ={\pi}^{-1}$; and (

**c**) $\gamma ={\pi}^{-1}+0.2$, for the same initial conditions as Figure 7.

**Figure 9.**Coupling matrix eigenvalue dynamics: (

**a**) $\mathcal{PT}$-symmetric regime; (

**b**) fully-degenerate regime; and (

**c**) broken symmetry regime. The black arrows show the direction of the eigenvalues as the gain to coupling ratio increases. Theses cases show the results for a $N=6$ waveguide coupler that provides a Bargmann parameter $j=5/2$.

**Figure 10.**Renormalized field intensity propagation for a $N=6$ waveguide coupler, Bargmann parameter $j=5/2$, in the (

**a**) $\mathcal{PT}$-symmetric regime, $\gamma =0.5$; (

**b**) fully-degenerate regime, $\gamma =1$; and (

**c**) broken symmetry regime, $\gamma =1.5$, for an initial field impinging just the first waveguide.

**Figure 11.**Renormalized Stokes vector propagation in a six-waveguide coupler, $j=5/2$, in the (

**a**) $\mathcal{PT}$-symmetric regime, $\gamma =0.5$; (

**b**) fully-degenerate regime, $\gamma =1$; and (

**c**) broken symmetry regime, $\gamma =1.5$, for the initial conditions ${\mathcal{E}}_{k}(0)={\delta}_{k,1}$ in black and the eigenstate of ${J}_{x}$ with eigenvalue $-j$, ${J}_{x}|\mathcal{E}(0)\rangle =-j|\mathcal{E}(0)\rangle $ in red.

**Figure 12.**Spontaneous generation of radiation in the waveguides with effective loss, ${S}_{1}$ solid blue line, and gain, ${S}_{2}$ dashed red line, in the (

**a**) $\mathcal{PT}$-symmetric regime, $\gamma =0.5$; (

**b**) fully-degenerate regime, $\gamma =1$; and (

**c**) broken symmetry regime, $\gamma =1.5$, for quantum vacuum fields in both waveguides.

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Huerta Morales, J.D.; Guerrero, J.; López-Aguayo, S.; Rodríguez-Lara, B.M.
Revisiting the Optical PT-Symmetric Dimer. *Symmetry* **2016**, *8*, 83.
https://doi.org/10.3390/sym8090083

**AMA Style**

Huerta Morales JD, Guerrero J, López-Aguayo S, Rodríguez-Lara BM.
Revisiting the Optical PT-Symmetric Dimer. *Symmetry*. 2016; 8(9):83.
https://doi.org/10.3390/sym8090083

**Chicago/Turabian Style**

Huerta Morales, José Delfino, Julio Guerrero, Servando López-Aguayo, and Blas Manuel Rodríguez-Lara.
2016. "Revisiting the Optical PT-Symmetric Dimer" *Symmetry* 8, no. 9: 83.
https://doi.org/10.3390/sym8090083