# Parity-Assisted Generation of Nonclassical States of Light in Circuit Quantum Electrodynamics

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## Abstract

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## 1. Introduction

## 2. The Model

## 3. Parity Symmetry ${{Z}}_{\mathbf{2}}$ and Selection Rules

## 4. Two Photon Process Mediated by a Quantum Rabi System

## 5. Copies of Density Matrices

## 6. Entanglement Swapping between Distant Superconducting Qubits

## 7. Implementation in Circuit QED

#### 7.1. Rabi System Hamiltonian

#### 7.2. Multimode Cavity Hamiltonian

#### 7.3. Complete Model

#### 7.4. Driving the Superconducting Qubit

## 8. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) energy spectrum of the Hamiltonian in Equation (1) as a function of the coupling strength g. Blue dashed lines stand for states with parity $p=+1$. Orange continuous lines correspond to states with parity $p=-1$; (

**b**) diagram of the energy levels at $g=0.6\phantom{\rule{3.33333pt}{0ex}}{\omega}_{\mathrm{cav}}$. In these numerical calculations, we use ${\omega}_{q}=0.8\phantom{\rule{3.33333pt}{0ex}}{\omega}_{\mathrm{cav}}$.

**Figure 2.**Population evolution of the Hamiltonian in Equation (2) for initial state $|\mathsf{\Psi}\left(0\right)\rangle =|2,+\rangle {\u2a02}_{\ell ,n}^{N,M}|{0}_{\ell}^{n}\rangle $ with cases $N=1$ (

**a**), $N=2$ (

**b**), $N=3$ (

**c**), and $N=4$ (

**d**) two-mode cavities. Blue continuous line is the evolution of the initial state $|\mathsf{\Psi}(0)\rangle $. (

**a**) orange dotted line denotes the population of ${|\mathsf{\Psi}\rangle}_{S}=|0,+\rangle \otimes |{1}_{{\omega}_{1}}\rangle \otimes |{1}_{{\omega}_{2}}\rangle $; (

**b**) green dotted line stands for the population of ${|\mathsf{\Psi}\rangle}_{B}=|0,+\rangle \otimes |{\mathsf{\Psi}}_{{\omega}_{1}}^{+}\rangle \otimes |{\mathsf{\Psi}}_{{\omega}_{2}}^{+}\rangle $; and (

**c**) red dotted line stands for ${|\mathsf{\Psi}\rangle}_{W}=|0,+\rangle \otimes |{W}_{{\omega}_{1}}\rangle \otimes |{W}_{{\omega}_{2}}\rangle $; (

**d**) purple dotted line stand for the ${|\mathsf{\Psi}\rangle}_{W}=|0,+\rangle \otimes |{W}_{{\omega}_{1}}\rangle \otimes |{W}_{{\omega}_{2}}\rangle $, where this W contains four modes. The parameters for these calculations can be found in the main text.

**Figure 3.**Gate sequence for the entanglement swapping protocol. At first, the quantum Rabi system is initialized from $|0,+\rangle $ to $|2,+\rangle $ via a driving acting on ${\sigma}^{z}$. Afterwards, the system evolves under the gate ${U}_{\mathrm{eff}}=exp(-it{\mathcal{H}}_{\mathrm{eff}}/\hslash )$. Then, the auxiliary two-level systems are tuned to the mode ${\omega}_{1}$ (${\omega}_{2}$). Thus, the system starts to evolve under ${\mathcal{H}}_{ES}$ to entangle the qubits.

**Figure 4.**Real and imaginary part of the reduced density matrix composed of the two qubits coupled to the field mode of frequency ${\omega}_{1}$ (

**a**) and mode ${\omega}_{2}$ (

**b**). The fidelity between the simulated state and the Bell state $|\mathsf{\Phi}\rangle =(|eg\rangle +|eg)\rangle /\sqrt{2}$ is (

**a**) $\mathcal{F}=0.9960$ and (

**b**) $\mathcal{F}=0.9976$.

**Figure 5.**Schematic illustration of our superconducting circuit implementation. Here, the quantum Rabi system is composed of a $\lambda /2$ transmission line resonator (grey resonator) interacting with a superconducting flux qubit located at the middle point to achieve the USC regime. In addition, the $\lambda /2$ resonator is coupled at its edges forming a finger pattern to two-mode transmission lines (blue resonators) through capacitive coupling. The limitation to keep up to six resonators relies on the reduction of the crosstalk between the resonators. The crosstalk induces a mutual-inductance effect that leads to a resonator–resonator coupling given by the following Hamiltonian. Furthermore, at the end of the two-mode transmission line resonator superconducting flux qubit ${Q}_{\ell}$ are coupled.

**Figure 6.**(

**a**) sketch of the current distribution of the first three resonator modes for the $\lambda /2$ transmission line resonator. The vertical black line corresponds to the position at which the artificial atom is placed. (

**b**) Energy spectrum of the Hamiltonian in Equation (41) considering the first three field modes. Orange lines corresponds to energy levels with parity $p=+1$, whereas blue dashed line stands for energy levels with parity $p=-1$.

**Figure 7.**Population evolution of the Hamiltonian in Equation (2) for the case where the multi-mode resonator contains three modes. The system is prepared in the state $|\mathsf{\Psi}\left(0\right)\rangle =|2,+\rangle {\u2a02}_{\ell ,n}^{N,M}|{0}_{\ell}^{n}\rangle $. The blue continuous line is the evolution of the initial state $|\mathsf{\Psi}(0)\rangle $. The orange dotted line denotes the population of ${|\mathsf{\Psi}\rangle}_{S}=|0,+\rangle \otimes |{1}_{{\omega}_{1}}\rangle \otimes |{1}_{{\omega}_{2}}\rangle $. The parameters for these calculations can be found in the main text.

**Table 1.**Summarized fidelity values between the states ${\rho}_{{\omega}_{\ell}}$ obtained through of the master Equation (17) with the fictitious states ${\rho}_{\mathrm{probe}}$ and ${\rho}_{\mathrm{tensor}}$ for the case where the quantum Rabi system is coupled to $n=\{1,2,3\}$ two-mode cavity.

$\mathit{N}=1$ | $\mathit{N}=2$ | $\mathit{N}=3$ | |
---|---|---|---|

$\mathcal{F}({\rho}_{{\omega}_{1}},{\rho}_{{\omega}_{2}})$ | 0.9898 | 0.9818 | 0.9832 |

${\mathcal{F}}_{S}$ | 0.9892 | - | - |

${\mathcal{F}}_{B}$ | - | 0.9945 | - |

${\mathcal{F}}_{W}$ | - | - | 0.9904 |

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**MDPI and ACS Style**

Cárdenas-López, F.A.; Romero, G.; Lamata, L.; Solano, E.; Retamal, J.C.
Parity-Assisted Generation of Nonclassical States of Light in Circuit Quantum Electrodynamics. *Symmetry* **2019**, *11*, 372.
https://doi.org/10.3390/sym11030372

**AMA Style**

Cárdenas-López FA, Romero G, Lamata L, Solano E, Retamal JC.
Parity-Assisted Generation of Nonclassical States of Light in Circuit Quantum Electrodynamics. *Symmetry*. 2019; 11(3):372.
https://doi.org/10.3390/sym11030372

**Chicago/Turabian Style**

Cárdenas-López, Francisco A., Guillermo Romero, Lucas Lamata, Enrique Solano, and Juan Carlos Retamal.
2019. "Parity-Assisted Generation of Nonclassical States of Light in Circuit Quantum Electrodynamics" *Symmetry* 11, no. 3: 372.
https://doi.org/10.3390/sym11030372