# 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

- You, J.Q.; Nori, F. Atomic physics and quantum optics using superconducting circuits. Nature
**2011**, 474, 589–597. [Google Scholar] [CrossRef] [PubMed][Green Version] - Houck, A.A.; Türeci, H.E.; Koch, J. On-chip quantum simulation with superconducting circuits. Nat. Phys.
**2012**, 8, 292–299. [Google Scholar] [CrossRef] - Devoret, M.H.; Schoelkopf, R.J. Superconducting circuits for quantum information: An outlook. Science
**2013**, 339, 1169–1174. [Google Scholar] [CrossRef] - Wallraff, A.; Schuster, D.I.; Blais, A.; Frunzio, L.; Huang, R.S.; Majer, J.; Kumar, S.; Girvin, S.M.; Schoelkopf, R.J. Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics. Nature
**2004**, 431, 162–167. [Google Scholar] [CrossRef] [PubMed][Green Version] - Blais, A.; Huang, R.S.; Wallraff, A.; Girvin, S.M.; Schoelkopf, R.J. Cavity quantum electrodynamics for superconducting electrical circuits: An architecture for quantum computation. Phys. Rev. A
**2004**, 69, 062320. [Google Scholar] [CrossRef][Green Version] - Nakamura, Y. Microwave quantum photonics in superconducting circuits. In Proceedings of the IEEE Photonics Conference 2012, Burlingame, CA, USA, 23–27 September 2012; pp. 544–545. [Google Scholar]
- Gu, X.; Kockum, A.F.; Miranowicz, A.; Liu, Y.X.; Nori, F. Microwave photonics with superconducting quantum circuits. Phys. Rep.
**2017**, 718–719, 1–102. [Google Scholar] [CrossRef] - Orlando, T.P.; Mooij, J.E.; Tian, L.; van der Wal, C.H.; Levitov, L.S.; Lloyd, S.; Mazo, J.J. Superconducting persistent-current qubit. Phys. Rev. B
**1999**, 60, 15398–15413. [Google Scholar] [CrossRef][Green Version] - Koch, J.; Terri, M.Y.; Gambetta, J.; Houck, A.A.; Schuster, D.I.; Majer, J.; Blais, A.; Devoret, M.H.; Girvin, S.M.; Schoelkopf, R.J. Charge-insensitive qubit design derived from the Cooper pair box. Phys. Rev. A
**2007**, 76, 042319. [Google Scholar] [CrossRef][Green Version] - Göppl, M.; Fragner, A.; Baur, M.; Bianchetti, R.; Filipp, S.; Leek, J.M.F.P.J.; Puebla, G.; Steffen, L.; Wallraff, A. Coplanar waveguide resonators for circuit quantum electrodynamics. J. Appl. Phys.
**2008**, 104, 113904. [Google Scholar] [CrossRef][Green Version] - Abdumalikov, A.A.; Astafiev, O.; Zagoskin, A.M.; Pashkin, Y.A.; Nakamura, Y.; Tsai, J.S. Electromagnetically induced transparency on a single artificial atom. Phys. Rev. Lett.
**2010**, 104, 193601. [Google Scholar] [CrossRef] - Lang, C.; Bozyigit, D.; Eichler, C.; Steffen, L.; Fink, J.M.; Abdumalikov, A.A., Jr.; Baur, M.; Filipp, S.; Da Silva, M.P.; Blais, A.; et al. Observation of resonant photon blockade at microwave frequencies using correlation function measurements. Phys. Rev. Lett.
**2011**, 106, 243601. [Google Scholar] [CrossRef] - Goetz, J.; Deppe, F.; Fedorov, K.G.; Eder, P.; Fischer, M.; Pogorzalek, S.; Xie, E.; Marx, A.; Gross, R. Parity-Engineered Light-Matter Interaction. Phys. Rev. Lett.
**2018**, 121, 060503. [Google Scholar] [CrossRef] - Bergeal, N.; Vijay, R.; Manucharyan, V.E.; Siddiqi, I.; Schoelkopf, R.J.; Girvin, S.M.; Devoret, M.H. Analog information processing at the quantum limit with a Josephson ring modulator. Nat. Phys.
**2010**, 6, 296–302. [Google Scholar] [CrossRef][Green Version] - Boissonneault, M.; Gambetta, J.M.; Blais, A. Improved superconducting qubit readout by qubit-induced nonlinearities. Phys. Rev. Lett.
**2010**, 105, 100504. [Google Scholar] [CrossRef] [PubMed] - Bourassa, J.; Beaudoin, F.; Gambetta, J.M.; Blais, A. Josephson-junction-embedded transmission-line resonators: From Kerr medium to in-line transmon. Phys. Rev. A
**2012**, 86, 013814. [Google Scholar] [CrossRef] - Leib, M.; Deppe, F.; Marx, A.; Gross, R.; Hartmann, M.J. Networks of nonlinear superconducting transmission line resonators. New J. Phys.
**2012**, 14, 075024. [Google Scholar] [CrossRef][Green Version] - Hoi, I.-C.; Kockum, A.F.; Palomaki, T.; Stace, T.M.; Fan, B.; Tornberg, L.; Sathyamoorthy, S.R.; Johansson, G.; Delsing, P.; Wilson, C.M. Giant Cross–Kerr Effect for Propagating Microwaves Induced by an Artificial Atom. Phys. Rev. Lett.
**2013**, 111, 053601. [Google Scholar] [CrossRef] [PubMed] - Marquardt, F. Efficient on-chip source of microwave photon pairs in superconducting circuit QED. Phys. Rev. B
**2007**, 76, 205416. [Google Scholar] [CrossRef] - Koshino, K. Down-conversion of a single photon with unit efficiency. Phys. Rev. A
**2009**, 79, 013804. [Google Scholar] [CrossRef] - Liu, Y.X.; Sun, H.C.; Peng, Z.H.; Miranowicz, A.; Tsai, J.S.; Nori, F. Controllable microwave three-wave mixing via a single three-level superconducting quantum circuit. Sci. Rep.
**2014**, 4, 7289. [Google Scholar] [CrossRef][Green Version] - Sánchez-Burillo, E.; Martín-Moreno, L.; García-Ripoll, J.J.; Zueco, D. Full two-photon down-conversion of a single photon. Phys. Rev. A
**2016**, 94, 053814. [Google Scholar] [CrossRef] - Yurke, B.; Corruccini, L.R.; Kaminsky, P.G.; Rupp, L.W.; Smith, A.D.; Silver, A.H.; Simon, R.W.; Whittaker, E.A. Observation of parametric amplification and deamplification in a Josephson parametric amplifier. Phys. Rev. A
**1989**, 39, 2519–2533. [Google Scholar] [CrossRef] - Everitt, M.J.; Clark, T.D.; Stiffell, P.B.; Vourdas, A.; Ralph, J.F.; Prance, R.J.; Prance, H. Superconducting analogs of quantum optical phenomena: Macroscopic quantum superpositions and squeezing in a superconducting quantum-interference device ring. Phys. Rev. A
**2004**, 69, 043804. [Google Scholar] [CrossRef] - Zagoskin, A.M.; Il’chev, E.; McCutcheon, M.W.; Young, J.F.; Nori, F. Controlled generation of squeezed states of microwave radiation in a superconducting resonant circuit. Phys. Rev. Lett.
**2008**, 101, 253602. [Google Scholar] [CrossRef] - Moon, K.; Girvin, S.M. Theory of microwave parametric down-conversion and squeezing using circuit QED. Phys. Rev. Lett.
**2005**, 95, 140504. [Google Scholar] [CrossRef] - Didier, N.; Qassemi, F.; Blais, A. Perfect squeezing by damping modulation in circuit quantum electrodynamics. Phys. Rev. A
**2014**, 89, 013820. [Google Scholar] [CrossRef] - Strauch, F.W. All-resonant control of superconducting resonators. Phys. Rev. Lett.
**2012**, 109, 210501. [Google Scholar] [CrossRef] [PubMed] - Zhao, Y.-J.; Wang, C.; Zhu, X.; Liu, Y.-X. Engineering entangled microwave photon states through multiphoton interactions between two cavity fields and a superconducting qubit. Sci. Rep.
**2016**, 6, 23646. [Google Scholar] [CrossRef][Green Version] - Merkel, S.T.; Wilhelm, F.K. Generation and detection of NOON states in superconducting circuits. New J. Phys.
**2010**, 12, 093036. [Google Scholar] [CrossRef][Green Version] - Strauch, F.W.; Jacobs, K.; Simmonds, R.W. Arbitrary Control of Entanglement between two Superconducting Resonators. Phys. Rev. Lett.
**2010**, 105, 050501. [Google Scholar] [CrossRef] [PubMed] - Wang, H.; Mariantoni, M.; Bialczak, R.C.; Lenander, M.; Lucero, E.; Neeley, M.; O’Connell, A.D.; Sank, D.; Weides, M.; Wenner, J.; et al. Deterministic Entanglement of Photons in Two Superconducting Microwave Resonators. Phys. Rev. Lett.
**2011**, 106, 060401. [Google Scholar] [CrossRef][Green Version] - Gasparinetti, S.; Pechal, M.; Besse, J.C.; Mondal, M.; Eichler, C.; Wallraff, A. Correlations and entanglement of microwave photons emitted in a cascade decay. Phys. Rev. Lett.
**2017**, 119, 140504. [Google Scholar] [CrossRef] [PubMed] - Campagne-Ibarcq, P.; Zalys-Geller, E.; Narla, A.; Shankar, S.; Reinhold, P.; Burkhart, L.D.; Axline, C.J.; Pfaff, W.; Frunzio, L.; Schoelkopf, R.J.; et al. Deterministic remote entanglement of superconducting circuits through microwave two-photon transitions. Phys. Rev. Lett.
**2018**, 120, 200501. [Google Scholar] [CrossRef] [PubMed] - Rosenblum, S.; Gao, Y.Y.; Reinhold, P.; Wang, C.; Axline, C.J.; Frunzio, L.; Girvin, S.M.; Jiang, L.; Mirrahimi, M.; Devoret, M.H.; et al. A CNOT gate between multiphoton qubits encoded in two cavities. Nat. Commun.
**2018**, 9, 652. [Google Scholar] [CrossRef] [PubMed] - Narla, A.; Shankar, S.; Hatridge, M.; Leghtas, Z.; Sliwa, K.M.; Zalys-Geller, E.; Mundhada, S.O.; Pfaff, W.; Frunzio, L.; Shoelkopf, R.J.; et al. Robust concurrent remote entanglement between two superconducting qubits. Phys. Rev. X
**2016**, 6, 031036. [Google Scholar] [CrossRef] - Kurpiers, P.; Magnard, P.; Walter, T.; Royer, B.; Pechal, M.; Heinsoo, J.; Salathé, Y.; Akin, A.; Storz, S.; Besse, J.C.; et al. Deterministic Quantum State Transfer and Generation of Remote Entanglement using Microwave Photons. Nature
**2018**, 558, 264–267. [Google Scholar] [CrossRef] - Bourassa, J.; Gambetta, J.M.; Abdumalikov, A.A.; Astafiev, O.; Nakamura, Y.; Blais, A. Ultrastrong coupling regime of cavity QED with phase-biased flux qubits. Phys. Rev. A
**2009**, 80, 032109. [Google Scholar] [CrossRef] - Niemczyk, T.; Deppe, F.; Huebl, H.; Menzel, E.P.; Hocke, F.; Schwarz, M.J.; García-Ripoll, J.J.; Zueco, D.; Hümmer, T.; Solano, E.; et al. Circuit quantum electrodynamics in the ultrastrong-coupling regime. Nat. Phys.
**2010**, 6, 772–776. [Google Scholar] [CrossRef][Green Version] - Forn-Díaz, P.; Lisenfeld, J.; Marcos, D.; García-Ripoll, J.J.; Solano, E.; Harmans, C.J.P.M.; Mooij, J.E. Observation of the Bloch-Siegert shift in a qubit-oscillator system in the ultrastrong coupling regime. Phys. Rev. Lett.
**2010**, 105, 237001. [Google Scholar] [CrossRef] - Andersen, C.K.; Blais, A. Ultrastrong coupling dynamics with a transmon qubit. New J. Phys.
**2017**, 19, 023022. [Google Scholar] [CrossRef][Green Version] - Forn-Díaz, P.; García-Ripoll, J.J.; Peropadre, B.; Orgiazzi, J.L.; Yurtalan, M.A.; Belyansky, R.; Wilson, C.M.; Lupascu, A. Ultrastrong coupling of a single artificial atom to an electromagnetic continuum in the nonperturbative regime. Nat. Phys.
**2017**, 13, 39–43. [Google Scholar] [CrossRef] - Martínez, J.P.; Léger, S.; Gheeraert, N.; Dassonneville, R.; Planat, L.; Foroughi, F.; Krupko, Y.; Buisson, O.; Naud, C.; Hasch-Guichard, W.; et al. A tunable Josephson platform to explore many-body quantum optics in circuit-QED. arXiv, 2018; arXiv:1802.00633. [Google Scholar]
- Casanova, J.; Romero, G.; Lizuain, I.; García-Ripoll, J.J.; Solano, E. Deep Strong Coupling Regime of the Jaynes-Cummings Model. Phys. Rev. Lett.
**2010**, 105, 263603. [Google Scholar] [CrossRef] - Yoshihara, F.; Fuse, T.; Ashhab, S.; Kakuyanagi, K.; Saito, S.; Semba, K. Superconducting qubit-oscillator circuit beyond the ultrastrong-coupling regime. Nat. Phys.
**2017**, 13, 44–47. [Google Scholar] [CrossRef] - Forn-Díaz, P.; Lamata, L.; Rico, E.; Kono, J.; Solano, E. Ultrastrong coupling regimes of light–matter interaction. arXiv, 2018; arXiv:1804.09275. [Google Scholar]
- Kockum, A.F.; Miranowicz, A.; de Liberato, S.; Savasta, S.; Nori, F. Ultrastrong coupling between light and matter. Nat. Rev. Phys.
**2019**, 1, 19–40. [Google Scholar] [CrossRef] - Rabi, I.I. On the Process of Space Quantization. Phys. Rev.
**1936**, 49, 324–328. [Google Scholar] [CrossRef] - Braak, D. Integrability of the Rabi Model. Phys. Rev. Lett.
**2011**, 107, 100401. [Google Scholar] [CrossRef] [PubMed] - Nataf, P.; Ciuti, C. Protected Quantum Computation with Multiple Resonators in Ultrastrong Coupling Circuit QED. Phys. Rev. Lett.
**2011**, 107, 190402. [Google Scholar] [CrossRef] [PubMed] - Romero, G.; Ballester, D.; Wang, Y.M.; Scarani, V.; Solano, E. Ultrafast Quantum Gates in Circuit QED. Phys. Rev. Lett.
**2012**, 108, 120501. [Google Scholar] [CrossRef] [PubMed] - Kyaw, T.H.; Felicetti, S.; Romero, G.; Solano, E.; Kwek, L.-C. Scalable quantum memory in the ultrastrong coupling regime. Sci. Rep.
**2015**, 5, 8621. [Google Scholar] [CrossRef] [PubMed][Green Version] - Felicetti, S.; Douce, T.; Romero, G.; Milman, P.; Solano, E. Parity-dependent state engineering and tomography in the ultrastrong coupling regime. Sci. Rep.
**2015**, 5, 11818. [Google Scholar] [CrossRef] [PubMed] - Kyaw, T.H.; Herrera-Martí, D.A.; Solano, E.; Romero, G.; Kwek, L.-C. Creation of quantum error correcting codes in the ultrastrong coupling regime. Phys. Rev. B
**2015**, 91, 064503. [Google Scholar] [CrossRef] - Wang, Y.M.; Zhang, J.; Wu, C.; You, J.Q.; Romero, G. Holonomic quantum computation in the ultrastrong-coupling regime of circuit QED. Phys. Rev. A
**2016**, 94, 012328. [Google Scholar] [CrossRef] - Albarrán-Arriagada, F.; Lamata, L.; Solano, E.; Romero, G.; Retamal, J.C. Spin-1 models in the ultrastrong-coupling regime of circuit QED. Phys. Rev. A
**2018**, 97, 022306. [Google Scholar] [CrossRef][Green Version] - Garziano, L.; Stassi, R.; Macrì, V.; Kockum, A.F.; Savasta, S.; Nori, F. Multiphoton quantum Rabi oscillations in ultrastrong cavity QED. Phys. Rev. A
**2015**, 92, 063830. [Google Scholar] [CrossRef] - Kockum, A.F.; Miranowicz, A.; Macrì, V.; Savasta, S.; Nori, F. Deterministic quantum nonlinear optics with single atoms and virtual photons. Phys. Rev. A
**2017**, 95, 063849. [Google Scholar] [CrossRef] - Kockum, A.F.; Macrì, V.; Garziano, L.; Savasta, S.; Nori, F. Frequency conversion in ultrastrong cavity QED. Sci. Rep.
**2017**, 7, 5313. [Google Scholar] [CrossRef] - Stassi, R.; Macrì, V.; Kockum, A.F.; di Stefano, O.; Miranowicz, A.; Savasta, S.; Nori, F. Quantum nonlinear optics without photons. Phys. Rev. A
**2017**, 96, 023818. [Google Scholar] [CrossRef] - Mlynek, J.A.; Abdumalikov, A.A., Jr.; Fink, J.M.; Steffen, L.; Baur, M.; Lang, C.; van Loo, A.F.; Wallraff, A. Demonstrating W-type entanglement of Dicke states in resonant cavity quantum electrodynamics. Phys. Rev. A
**2012**, 86, 053838. [Google Scholar] [CrossRef] - Wei, X.; Chen, M.-F. Preparation of multi-qubit W states in multiple resonators coupled by a superconducting qubit via adiabatic passage. Quantum Inf. Process.
**2013**, 14, 2419–2433. [Google Scholar] [CrossRef] - Liu, X.; Liao, Q.; Xu, X.; Fang, G.; Liu, S. One-step schemes for multiqubit GHZ states and W-class states in circuit QED. Opt. Commun.
**2016**, 359, 359–363. [Google Scholar] [CrossRef] - Çakmak, B.; Campbell, S.; Vacchini, B.; Müstecaplıoǧlu, Ė.; Paternostro, M. Robust multipartite entanglement generation via a collision model. Phys. Rev. A
**2019**, 99, 012319. [Google Scholar] [CrossRef] - Wei, X.; Chen, M.-F. Generation of N-Qubit W State in N Separated Resonators via Resonant Interaction. Int. J. Theor. Phys.
**2014**, 54, 812–820. [Google Scholar] [CrossRef] - Egger, D.J.; Wilhelm, F.K. Multimode Circuit Quantum Electrodynamics with Hybrid Metamaterial Transmission Lines. Phys. Rev. Lett.
**2013**, 111, 163601. [Google Scholar] [CrossRef] [PubMed] - Underwood, D.L.; Shanks, W.E.; Koch, J.; Houck, A.A. Low-disorder microwave cavity lattices for quantum simulation with photons. Phys. Rev. A
**2012**, 86, 023837. [Google Scholar] [CrossRef] - Wu, Y.; Yang, X. Strong-coupling theory of periodically driven two-level systems. Phys. Rev. Lett.
**2007**, 98, 013601. [Google Scholar] [CrossRef] - Brune, M.; Raimond, J.M.; Haroche, S. Theory of the Rydberg-atom two-photon micromaser. Phys. Rev. A
**1987**, 35, 154. [Google Scholar] [CrossRef] - Brune, M.; Raimond, J.M.; Goy, P.; Davidovich, L.; Haroche, S. Realization of a two-photon maser oscillator. Phys. Rev. Lett.
**1987**, 59, 1899–1902. [Google Scholar] [CrossRef] [PubMed] - Dür, W.; Vidal, G.; Cirac, J.I. Three qubits can be entangled in two inequivalent ways. Phys. Rev. A
**2000**, 62, 062314. [Google Scholar] [CrossRef] - Macri, V.; Nori, F.; Kockum, A.F. Simple preparation of Bell and GHZ states using ultrastrong-coupling circuit QED. Phys. Rev. A
**2018**, 98, 062327. [Google Scholar] [CrossRef] - Beaudoin, F.; Gambetta, J.M.; Blais, A. Dissipation and ultrastrong coupling in circuit QED. Phys. Rev. A
**2011**, 84, 043832. [Google Scholar] [CrossRef] - Ridolfo, A.; Leib, M.; Savasta, S.; Hartmann, M.J. Photon Blockade in the Ultrastrong Coupling Regime. Phys. Rev. Lett.
**2012**, 109, 193602. [Google Scholar] [CrossRef] - Settineri, A.; Macri, V.; Ridolfo, A.; di Stefano, O.; Kockum, A.F.; Nori, F.; Savasta, S. Dissipation and Thermal Noise in Hybrid Quantum Systems in the Ultrastrong Coupling Regime. Phys. Rev. A
**2019**, 98, 053834. [Google Scholar] [CrossRef] - Reuther, G.M.; Zueco, D.; Deppe, F.; Hoffmann, E.; Menzel, E.P.; Weißl, T.; Mariantoni, M.; Kohler, S.; Marx, A.; Solano, E.; et al. Two-resonator circuit quantum electrodynamics: Dissipative theory. Phys. Rev. B
**2010**, 81, 144510. [Google Scholar] [CrossRef][Green Version] - Forn-Díaz, P.; Romero, G.; Harmans, C.J.P.M.; Solano, E.; Mooij, J.E. Broken selection rule in the quantum Rabi model. Sci. Rep.
**2016**, 6, 26720. [Google Scholar] [CrossRef][Green Version] - Boissonneault, M.; Gambetta, J.M.; Blais, A. Dispersive regime of circuit QED: Photon-dependent qubit dephasing and relaxation rates. Phys. Rev. A
**2009**, 79, 013819. [Google Scholar] [CrossRef] - Masluk, N.A. Reducing the Losses of the Fluxonium Artificial Atom. Ph.D. Thesis, Yale University, New Haven, CT, USA, 2013. [Google Scholar]
- Yurke, B.; Denker, J.S. Quantum network theory. Phys. Rev. A
**1984**, 29, 1419–1437. [Google Scholar] [CrossRef] - Devoret, M.H. Quantum Fluctuations in electrical circuits. In Les Houches Session LXIII; Reynaud, S., Giacobino, E., Zinn-Justin, J., Eds.; Elsevier: Amsterdam, The Netherlands, 1997; pp. 351–386. [Google Scholar]
- Ashhab, S.; Johansson, J.R.; Zagoskin, A.M.; Nori, F. Two-level systems driven by large-amplitude fields. Phys. Rev. A
**2007**, 75, 063414. [Google Scholar] [CrossRef]

**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