Abstract

We study a model of two coupled two-level atoms (qubits) interacting off-resonance (at non-zero detuning) with a single mode radiation field. This system is of special interest in the field of quantum information processing (QIP) and can be realized in electron spin states in quantum dots or Rydberg atoms in optical cavities and superconducting qubits in linear resonators. We present an exact analytical solution for the time evolution of the system starting from any initial state. Utilizing this solution, we show how the entanglement sudden death (ESD), which represents a major threat to QIP, can be efficiently controlled by tuning atom-atom coupling and non-zero detuning. We demonstrate that while one of these two system parameters may not separately affect the ESD, combining the two can be very effective, as in the case of an initial correlated Bell state. However in other cases, such as a W-like initial state, they may have a competing impacts on ESD. Moreover, their combined effect can be used to create ESD in the system, as in the case of an anti-correlated initial Bell state. A clear synchronization between the population inversion collapse-revival pattern and the entanglement dynamics is observed at all system parameter combinations. Nevertheless, only for initial states that may evolve to ESD, the population inversion revival oscillations, where exchange of energy between the atoms and the field takes place, temporally coincide with the entanglement revival peaks, whereas the population collapse periods match the ESD intervals. The variation of the radiation field intensity has a clear impact on the duration of the ESD at any combination of the other system parameters.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2019 (1)

N. Chathavalappil and S. V. M. Satyanarayana, “Schemes to avoid entanglement sudden death of decohering two qubit system,” Eur. Phys. J. D 73(2), 36 (2019).
[Crossref]

2018 (2)

M. K. Tavassoly, R. Daneshmand, and N. Rustaee, “Entanglement dynamics of linear and nonlinear interaction of two two-level atoms with a quantized Phase-damped field in the dispersive regime,” Int. J. Theor. Phys. 57(6), 1645–1658 (2018).
[Crossref]

T. L. Nguyen, J.-M. Raimond, C. Sayrin, R. Cortinas, T. Cantat-Moltrecht, F. Assemat, I. Dotsenko, S. Gleyzes, S. Haroche, G. Roux, T. Jolicoeur, and M. Brune, “Towards quantum simulation with circular Rydberg atoms,” Phys. Rev. X 8(1), 011032 (2018).
[Crossref]

2017 (3)

M. Donaire, J. M. Muñoz-Castañeda, and L. Nieto, “Dipole-dipole interaction in cavity QED: The weak-coupling, nondegenerate regime,” Phys. Rev. A 96(4), 042714 (2017).
[Crossref]

G. Wendin, “Quantum information processing with superconducting circuits: a review,” Rep. Prog. Phys. 80(10), 106001 (2017).
[Crossref]

X.-M. Bai, C.-P. Gao, J.-Q. Li, N. Liu, and J.-Q. Liang, “Entanglement dynamics for two spins in an optical cavity–field interaction induced decoherence and coherence revival,” Opt. Express 25(15), 17051–17065 (2017).
[Crossref]

2016 (3)

O. de los Santos-Sánchez, C. González-Gutiérrez, and J. Récamier, “Nonlinear Jaynes–Cummings model for two interacting two-level atoms,” J. Phys. B 49(16), 165503 (2016).
[Crossref]

T. Deng, Y. Yan, L. Chen, and Y. Zhao, “Dynamics of the two-spin spin-boson model with a common bath,” J. Chem. Phys. 144(14), 144102 (2016).
[Crossref]

G. Sadiek and S. Almalki, “Entanglement dynamics in Heisenberg spin chains coupled to a dissipative environment at finite temperature,” Phys. Rev. A 94(1), 012341 (2016).
[Crossref]

2015 (1)

P. Lodahl, S. Mahmoodian, and S. Stobbe, “Interfacing single photons and single quantum dots with photonic nanostructures,” Rev. Mod. Phys. 87(2), 347–400 (2015).
[Crossref]

2014 (1)

N. Wu, A. Nanduri, and H. Rabitz, “Rabi oscillations, decoherence, and disentanglement in a qubit–spin-bath system,” Phys. Rev. A 89(6), 062105 (2014).
[Crossref]

2013 (5)

M. M. Sahrapour and N. Makri, “Tunneling, decoherence, and entanglement of two spins interacting with a dissipative bath,” J. Chem. Phys. 138(11), 114109 (2013).
[Crossref]

G. Sadiek and S. Kais, “Persistence of entanglement in thermal states of spin systems,” J. Phys. B 46(24), 245501 (2013).
[Crossref]

B. Alkurtass, H. Wichterich, and S. Bose, “Quench-induced growth of distant entanglement from product and locally entangled states in spin chains,” Phys. Rev. A 88(6), 062325 (2013).
[Crossref]

L. Duan, H. Wang, Q.-H. Chen, and Y. Zhao, “Entanglement dynamics of two qubits coupled individually to Ohmic baths,” J. Chem. Phys. 139(4), 044115 (2013).
[Crossref]

Z.-L. Xiang, S. Ashhab, J. Q. You, and F. Nori, “Hybrid quantum circuits: Superconducting circuits interacting with other quantum systems,” Rev. Mod. Phys. 85(2), 623–653 (2013).
[Crossref]

2012 (1)

Y.-S. Kim, J.-C. Lee, O. Kwon, and Y.-H. Kim, “Protecting entanglement from decoherence using weak measurement and quantum measurement reversal,” Nat. Phys. 8(2), 117–120 (2012).
[Crossref]

2011 (2)

I. Buluta, S. Ashhab, and F. Nori, “Natural and artificial atoms for quantum computation,” Rep. Prog. Phys. 74(10), 104401 (2011).
[Crossref]

Q. Xu, G. Sadiek, and S. Kais, “Dynamics of entanglement in a two-dimensional spin system,” Phys. Rev. A 83(6), 062312 (2011).
[Crossref]

2010 (4)

G. Sadiek, B. Alkurtass, and O. Aldossary, “Entanglement in a time-dependent coupled XY spin chain in an external magnetic field,” Phys. Rev. A 82(5), 052337 (2010).
[Crossref]

J. M. Torres, E. Sadurní, and T. H. Seligman, “Two interacting atoms in a cavity: exact solutions, entanglement and decoherence,” J. Phys. A: Math. Theor. 43(19), 192002 (2010).
[Crossref]

M. Saffman, T. G. Walker, and K. Mølmer, “Quantum information with Rydberg atoms,” Rev. Mod. Phys. 82(3), 2313–2363 (2010).
[Crossref]

C. Guerlin, E. Brion, T. Esslinger, and K. Mølmer, “Cavity quantum electrodynamics with a Rydberg-blocked atomic ensemble,” Phys. Rev. A 82(5), 053832 (2010).
[Crossref]

2009 (4)

S. Chan, M. D. Reid, and Z. Ficek, “Entanglement evolution of two remote and non-identical Jaynes–Cummings atoms,” J. Phys. B 42(6), 065507 (2009).
[Crossref]

C. Li, S. Xiao-Qiang, and Z. Shou, “The influences of dipole–dipole interaction and detuning on the sudden death of entanglement between two atoms in the Tavis–Cummings model,” Chin. Phys. B 18(3), 888–893 (2009).
[Crossref]

Y. Dubi and M. Di Ventra, “Relaxation times in an open interacting two-qubit system,” Phys. Rev. A 79(1), 012328 (2009).
[Crossref]

T. Yu and J. H. Eberly, “Sudden death of entanglement,” Science 323(5914), 598–601 (2009).
[Crossref]

2008 (1)

N. Burić, “Influence of the thermal environment on entanglement dynamics in small rings of qubits,” Phys. Rev. A 77(1), 012321 (2008).
[Crossref]

2007 (8)

D. I. Tsomokos, M. J. Hartmann, S. F. Huelga, and M. B. Plenio, “Entanglement dynamics in chains of qubits with noise and disorder,” New J. Phys. 9(3), 79 (2007).
[Crossref]

G.-F. Zhang and Z.-Y. Chen, “The entanglement character between atoms in the non-degenerate two photons Tavis–Cummings model,” Opt. Commun. 275(1), 274–277 (2007).
[Crossref]

M. Yönaç, T. Yu, and J. H. Eberly, “Pairwise concurrence dynamics: a four-qubit model,” J. Phys. B 40(9), S45–S59 (2007).
[Crossref]

I. Sainz and G. Björk, “Entanglement invariant for the double Jaynes-Cummings model,” Phys. Rev. A 76(4), 042313 (2007).
[Crossref]

E. K. Irish, “Generalized rotating-wave approximation for arbitrarily large coupling,” Phys. Rev. Lett. 99(17), 173601 (2007).
[Crossref]

S. H. W. Van der Ploeg, A. Izmalkov, A. M. van den Brink, U. Hübner, M. Grajcar, E. Il’ichev, H.-G. Meyer, and A. M. Zagoskin, “Controllable coupling of superconducting flux qubits,” Phys. Rev. Lett. 98(5), 057004 (2007).
[Crossref]

A. O. Niskanen, K. Harrabi, F. Yoshihara, Y. Nakamura, S. Lloyd, and J. S. Tsai, “Quantum coherent tunable coupling of superconducting qubits,” Science 316(5825), 723–726 (2007).
[Crossref]

J. Majer, J. M. Chow, J. M. Gambetta, J. Koch, B. R. Johnson, J. A. Schreier, L. Frunzio, D. I. Schuster, A. A. Houck, A. Wallraff, A. Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, “Coupling superconducting qubits via a cavity bus,” Nature 449(7161), 443–447 (2007).
[Crossref]

2006 (11)

O. Gywat, F. Meier, D. Loss, and D. D. Awschalom, “Dynamics of coupled qubits interacting with an off-resonant cavity,” Phys. Rev. B 73(12), 125336 (2006).
[Crossref]

M. Steffen, M. Ansmann, R. C. Bialczak, N. Katz, E. Lucero, R. McDermott, M. Neeley, E. M. Weig, A. N. Cleland, and J. M. Martinis, “Measurement of the entanglement of two superconducting qubits via state tomography,” Science 313(5792), 1423–1425 (2006).
[Crossref]

T. Hime, P. A. Reichardt, B. L. T. Plourde, T. L. Robertson, C.-E. Wu, A. V. Ustinov, and J. Clarke, “Solid-state qubits with current-controlled couplin,” Science 314(5804), 1427–1429 (2006).
[Crossref]

R.-F. Liu and C.-C. Chen, “Role of the Bell singlet state in the suppression of disentanglement,” Phys. Rev. A 74(2), 024102 (2006).
[Crossref]

Z. Ficek and R. Tanaś, “Dark periods and revivals of entanglement in a two-qubit system,” Phys. Rev. A 74(2), 024304 (2006).
[Crossref]

T. Yu and J. H. Eberly, “Sudden death of entanglement: classical noise effects,” Opt. Commun. 264(2), 393–397 (2006).
[Crossref]

M. Yönaç, T. Yu, and J. H. Eberly, “Sudden death of entanglement of two Jaynes–Cummings atoms,” J. Phys. B 39(15), S621–S625 (2006).
[Crossref]

I. Sainz, A. B. Klimov, and L. Roa, “Entanglement dynamics modified by an effective atomic environment,” Phys. Rev. A 73(3), 032303 (2006).
[Crossref]

J. Wang, H. Batelaan, J. Podany, and A. F. Starace, “Entanglement evolution in the presence of decoherence,” J. Phys. B 39(21), 4343–4353 (2006).
[Crossref]

Z. Huang, G. Sadiek, and S. Kais, “Time evolution of a single spin inhomogeneously coupled to an interacting spin environment,” J. Chem. Phys. 124(14), 144513 (2006).
[Crossref]

A. Abliz, H. J. Gao, X. C. Xie, Y. S. Wu, and W. M. Liu, “Entanglement control in an anisotropic two-qubit Heisenberg X Y Z model with external magnetic fields,” Phys. Rev. A 74(5), 052105 (2006).
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2005 (1)

J. B. Majer, F. G. Paauw, A. C. J. Ter Haar, C. J. P. M. Harmans, and J. E. Mooij, “Spectroscopy on two coupled superconducting flux qubits,” Phys. Rev. Lett. 94(9), 090501 (2005).
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2004 (5)

C.-P. Yang, S.-I. Chu, and S. Han, “Quantum information transfer and entanglement with SQUID qubits in cavity QED: a dark-state scheme with tolerance for nonuniform device parameter,” Phys. Rev. Lett. 92(11), 117902 (2004).
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Z. Kis and E. Paspalakis, “Arbitrary rotation and entanglement of flux SQUID qubits,” Phys. Rev. B 69(2), 024510 (2004).
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A. Blais, R.-S. Huang, A. Wallraff, S. M. Girvin, and R. J. Schoelkopf, “Cavity quantum electrodynamics for superconducting electrical circuits: An architecture for quantum computation,” Phys. Rev. A 69(6), 062320 (2004).
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A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R.-S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431(7005), 162–167 (2004).
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T. Yu and J. H. Eberly, “Finite-time disentanglement via spontaneous emission,” Phys. Rev. Lett. 93(14), 140404 (2004).
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2003 (5)

T. Yamamoto, Y. A. Pashkin, O. Astafiev, Y. Nakamura, and J.-S. Tsai, “Demonstration of conditional gate operation using superconducting charge qubits,” Nature 425(6961), 941–944 (2003).
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A. J. Berkley, H. Xu, R. C. Ramos, M. A. Gubrud, F. W. Strauch, P. R. Johnson, J. R. Anderson, A. J. Dragt, C. J. Lobb, and F. C. Wellstood, “Entangled macroscopic quantum states in two superconducting qubits,” Science 300(5625), 1548–1550 (2003).
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C.-P. Yang, S.-I. Chu, and S. Han, “Possible realization of entanglement, logical gates, and quantum-information transfer with superconducting-quantum-interference-device qubits in cavity QED,” Phys. Rev. A 67(4), 042311 (2003).
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J. Q. You and F. Nori, “Quantum information processing with superconducting qubits in a microwave field,” Phys. Rev. B 68(6), 064509 (2003).
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T. E. Tessier, I. H. Deutsch, A. Delgado, and I. Fuentes-Guridi, “Entanglement sharing in the two-atom Tavis-Cummings model,” Phys. Rev. A 68(6), 062316 (2003).
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2002 (1)

S. Khlebnikov and G. Sadiek, “Decoherence by a nonlinear environment: Canonical versus microcanonical case,” Phys. Rev. A 66(3), 032312 (2002).
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2001 (2)

S. Osnaghi, P. Bertet, A. Auffeves, P. Maioli, M. Brune, J.-M. Raimond, and S. Haroche, “Coherent control of an atomic collision in a cavity,” Phys. Rev. Lett. 87(3), 037902 (2001).
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J.-M. Raimond, M. Brune, and S. Haroche, “Manipulating quantum entanglement with atoms and photons in a cavity,” Rev. Mod. Phys. 73(3), 565–582 (2001).
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2000 (2)

S.-B. Zheng and G.-C. Guo, “Efficient scheme for two-atom entanglement and quantum information processing in cavity QED,” Phys. Rev. Lett. 85(11), 2392–2395 (2000).
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P. G. Kwiat, A. J. Berglund, J. B. Altepeter, and A. G. White, “Experimental verification of decoherence-free subspaces,” Science 290(5491), 498–501 (2000).
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1998 (2)

D. A. Lidar, I. L. Chuang, and K. B. Whaley, “Decoherence-free subspaces for quantum computation,” Phys. Rev. Lett. 81(12), 2594–2597 (1998).
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W. K. Wootters, “Entanglement of formation of an arbitrary state of two qubits,” Phys. Rev. Lett. 80(10), 2245–2248 (1998).
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1997 (1)

E. Hagley, X. Maitre, G. Nogues, C. Wunderlich, M. Brune, J.-M. Raimond, and S. Haroche, “Generation of Einstein-Podolsky-Rosen pairs of atoms,” Phys. Rev. Lett. 79(1), 1–5 (1997).
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1996 (1)

A. M. Steane, “Error correcting codes in quantum theory,” Phys. Rev. Lett. 77(5), 793–797 (1996).
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1995 (1)

P. W. Shor, “Scheme for reducing decoherence in quantum computer memory,” Phys. Rev. A 52(4), R2493–R2496 (1995).
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A. Abliz, H. J. Gao, X. C. Xie, Y. S. Wu, and W. M. Liu, “Entanglement control in an anisotropic two-qubit Heisenberg X Y Z model with external magnetic fields,” Phys. Rev. A 74(5), 052105 (2006).
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Aldossary, O.

G. Sadiek, B. Alkurtass, and O. Aldossary, “Entanglement in a time-dependent coupled XY spin chain in an external magnetic field,” Phys. Rev. A 82(5), 052337 (2010).
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Alkurtass, B.

B. Alkurtass, H. Wichterich, and S. Bose, “Quench-induced growth of distant entanglement from product and locally entangled states in spin chains,” Phys. Rev. A 88(6), 062325 (2013).
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G. Sadiek, B. Alkurtass, and O. Aldossary, “Entanglement in a time-dependent coupled XY spin chain in an external magnetic field,” Phys. Rev. A 82(5), 052337 (2010).
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Almalki, S.

G. Sadiek and S. Almalki, “Entanglement dynamics in Heisenberg spin chains coupled to a dissipative environment at finite temperature,” Phys. Rev. A 94(1), 012341 (2016).
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Altepeter, J. B.

P. G. Kwiat, A. J. Berglund, J. B. Altepeter, and A. G. White, “Experimental verification of decoherence-free subspaces,” Science 290(5491), 498–501 (2000).
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Anderson, J. R.

A. J. Berkley, H. Xu, R. C. Ramos, M. A. Gubrud, F. W. Strauch, P. R. Johnson, J. R. Anderson, A. J. Dragt, C. J. Lobb, and F. C. Wellstood, “Entangled macroscopic quantum states in two superconducting qubits,” Science 300(5625), 1548–1550 (2003).
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Ansmann, M.

M. Steffen, M. Ansmann, R. C. Bialczak, N. Katz, E. Lucero, R. McDermott, M. Neeley, E. M. Weig, A. N. Cleland, and J. M. Martinis, “Measurement of the entanglement of two superconducting qubits via state tomography,” Science 313(5792), 1423–1425 (2006).
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Ashhab, S.

Z.-L. Xiang, S. Ashhab, J. Q. You, and F. Nori, “Hybrid quantum circuits: Superconducting circuits interacting with other quantum systems,” Rev. Mod. Phys. 85(2), 623–653 (2013).
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I. Buluta, S. Ashhab, and F. Nori, “Natural and artificial atoms for quantum computation,” Rep. Prog. Phys. 74(10), 104401 (2011).
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Assemat, F.

T. L. Nguyen, J.-M. Raimond, C. Sayrin, R. Cortinas, T. Cantat-Moltrecht, F. Assemat, I. Dotsenko, S. Gleyzes, S. Haroche, G. Roux, T. Jolicoeur, and M. Brune, “Towards quantum simulation with circular Rydberg atoms,” Phys. Rev. X 8(1), 011032 (2018).
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Astafiev, O.

T. Yamamoto, Y. A. Pashkin, O. Astafiev, Y. Nakamura, and J.-S. Tsai, “Demonstration of conditional gate operation using superconducting charge qubits,” Nature 425(6961), 941–944 (2003).
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Auffeves, A.

S. Osnaghi, P. Bertet, A. Auffeves, P. Maioli, M. Brune, J.-M. Raimond, and S. Haroche, “Coherent control of an atomic collision in a cavity,” Phys. Rev. Lett. 87(3), 037902 (2001).
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O. Gywat, F. Meier, D. Loss, and D. D. Awschalom, “Dynamics of coupled qubits interacting with an off-resonant cavity,” Phys. Rev. B 73(12), 125336 (2006).
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Bai, X.-M.

Batelaan, H.

J. Wang, H. Batelaan, J. Podany, and A. F. Starace, “Entanglement evolution in the presence of decoherence,” J. Phys. B 39(21), 4343–4353 (2006).
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P. G. Kwiat, A. J. Berglund, J. B. Altepeter, and A. G. White, “Experimental verification of decoherence-free subspaces,” Science 290(5491), 498–501 (2000).
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Berkley, A. J.

A. J. Berkley, H. Xu, R. C. Ramos, M. A. Gubrud, F. W. Strauch, P. R. Johnson, J. R. Anderson, A. J. Dragt, C. J. Lobb, and F. C. Wellstood, “Entangled macroscopic quantum states in two superconducting qubits,” Science 300(5625), 1548–1550 (2003).
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Bertet, P.

S. Osnaghi, P. Bertet, A. Auffeves, P. Maioli, M. Brune, J.-M. Raimond, and S. Haroche, “Coherent control of an atomic collision in a cavity,” Phys. Rev. Lett. 87(3), 037902 (2001).
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Bialczak, R. C.

M. Steffen, M. Ansmann, R. C. Bialczak, N. Katz, E. Lucero, R. McDermott, M. Neeley, E. M. Weig, A. N. Cleland, and J. M. Martinis, “Measurement of the entanglement of two superconducting qubits via state tomography,” Science 313(5792), 1423–1425 (2006).
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I. Sainz and G. Björk, “Entanglement invariant for the double Jaynes-Cummings model,” Phys. Rev. A 76(4), 042313 (2007).
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J. Majer, J. M. Chow, J. M. Gambetta, J. Koch, B. R. Johnson, J. A. Schreier, L. Frunzio, D. I. Schuster, A. A. Houck, A. Wallraff, A. Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, “Coupling superconducting qubits via a cavity bus,” Nature 449(7161), 443–447 (2007).
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A. Blais, R.-S. Huang, A. Wallraff, S. M. Girvin, and R. J. Schoelkopf, “Cavity quantum electrodynamics for superconducting electrical circuits: An architecture for quantum computation,” Phys. Rev. A 69(6), 062320 (2004).
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A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R.-S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431(7005), 162–167 (2004).
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Bose, S.

B. Alkurtass, H. Wichterich, and S. Bose, “Quench-induced growth of distant entanglement from product and locally entangled states in spin chains,” Phys. Rev. A 88(6), 062325 (2013).
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Brion, E.

C. Guerlin, E. Brion, T. Esslinger, and K. Mølmer, “Cavity quantum electrodynamics with a Rydberg-blocked atomic ensemble,” Phys. Rev. A 82(5), 053832 (2010).
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Brune, M.

T. L. Nguyen, J.-M. Raimond, C. Sayrin, R. Cortinas, T. Cantat-Moltrecht, F. Assemat, I. Dotsenko, S. Gleyzes, S. Haroche, G. Roux, T. Jolicoeur, and M. Brune, “Towards quantum simulation with circular Rydberg atoms,” Phys. Rev. X 8(1), 011032 (2018).
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S. Osnaghi, P. Bertet, A. Auffeves, P. Maioli, M. Brune, J.-M. Raimond, and S. Haroche, “Coherent control of an atomic collision in a cavity,” Phys. Rev. Lett. 87(3), 037902 (2001).
[Crossref]

J.-M. Raimond, M. Brune, and S. Haroche, “Manipulating quantum entanglement with atoms and photons in a cavity,” Rev. Mod. Phys. 73(3), 565–582 (2001).
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E. Hagley, X. Maitre, G. Nogues, C. Wunderlich, M. Brune, J.-M. Raimond, and S. Haroche, “Generation of Einstein-Podolsky-Rosen pairs of atoms,” Phys. Rev. Lett. 79(1), 1–5 (1997).
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Buluta, I.

I. Buluta, S. Ashhab, and F. Nori, “Natural and artificial atoms for quantum computation,” Rep. Prog. Phys. 74(10), 104401 (2011).
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Buric, N.

N. Burić, “Influence of the thermal environment on entanglement dynamics in small rings of qubits,” Phys. Rev. A 77(1), 012321 (2008).
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Cantat-Moltrecht, T.

T. L. Nguyen, J.-M. Raimond, C. Sayrin, R. Cortinas, T. Cantat-Moltrecht, F. Assemat, I. Dotsenko, S. Gleyzes, S. Haroche, G. Roux, T. Jolicoeur, and M. Brune, “Towards quantum simulation with circular Rydberg atoms,” Phys. Rev. X 8(1), 011032 (2018).
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Chan, S.

S. Chan, M. D. Reid, and Z. Ficek, “Entanglement evolution of two remote and non-identical Jaynes–Cummings atoms,” J. Phys. B 42(6), 065507 (2009).
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N. Chathavalappil and S. V. M. Satyanarayana, “Schemes to avoid entanglement sudden death of decohering two qubit system,” Eur. Phys. J. D 73(2), 36 (2019).
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Chen, C.-C.

R.-F. Liu and C.-C. Chen, “Role of the Bell singlet state in the suppression of disentanglement,” Phys. Rev. A 74(2), 024102 (2006).
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Chen, L.

T. Deng, Y. Yan, L. Chen, and Y. Zhao, “Dynamics of the two-spin spin-boson model with a common bath,” J. Chem. Phys. 144(14), 144102 (2016).
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Chen, Q.-H.

L. Duan, H. Wang, Q.-H. Chen, and Y. Zhao, “Entanglement dynamics of two qubits coupled individually to Ohmic baths,” J. Chem. Phys. 139(4), 044115 (2013).
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Chen, Z.-Y.

G.-F. Zhang and Z.-Y. Chen, “The entanglement character between atoms in the non-degenerate two photons Tavis–Cummings model,” Opt. Commun. 275(1), 274–277 (2007).
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Chow, J. M.

J. Majer, J. M. Chow, J. M. Gambetta, J. Koch, B. R. Johnson, J. A. Schreier, L. Frunzio, D. I. Schuster, A. A. Houck, A. Wallraff, A. Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, “Coupling superconducting qubits via a cavity bus,” Nature 449(7161), 443–447 (2007).
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Chu, S.-I.

C.-P. Yang, S.-I. Chu, and S. Han, “Quantum information transfer and entanglement with SQUID qubits in cavity QED: a dark-state scheme with tolerance for nonuniform device parameter,” Phys. Rev. Lett. 92(11), 117902 (2004).
[Crossref]

C.-P. Yang, S.-I. Chu, and S. Han, “Possible realization of entanglement, logical gates, and quantum-information transfer with superconducting-quantum-interference-device qubits in cavity QED,” Phys. Rev. A 67(4), 042311 (2003).
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Chuang, I. L.

D. A. Lidar, I. L. Chuang, and K. B. Whaley, “Decoherence-free subspaces for quantum computation,” Phys. Rev. Lett. 81(12), 2594–2597 (1998).
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M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 2010).

Clarke, J.

T. Hime, P. A. Reichardt, B. L. T. Plourde, T. L. Robertson, C.-E. Wu, A. V. Ustinov, and J. Clarke, “Solid-state qubits with current-controlled couplin,” Science 314(5804), 1427–1429 (2006).
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Cleland, A. N.

M. Steffen, M. Ansmann, R. C. Bialczak, N. Katz, E. Lucero, R. McDermott, M. Neeley, E. M. Weig, A. N. Cleland, and J. M. Martinis, “Measurement of the entanglement of two superconducting qubits via state tomography,” Science 313(5792), 1423–1425 (2006).
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Cortinas, R.

T. L. Nguyen, J.-M. Raimond, C. Sayrin, R. Cortinas, T. Cantat-Moltrecht, F. Assemat, I. Dotsenko, S. Gleyzes, S. Haroche, G. Roux, T. Jolicoeur, and M. Brune, “Towards quantum simulation with circular Rydberg atoms,” Phys. Rev. X 8(1), 011032 (2018).
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Daneshmand, R.

M. K. Tavassoly, R. Daneshmand, and N. Rustaee, “Entanglement dynamics of linear and nonlinear interaction of two two-level atoms with a quantized Phase-damped field in the dispersive regime,” Int. J. Theor. Phys. 57(6), 1645–1658 (2018).
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O. de los Santos-Sánchez, C. González-Gutiérrez, and J. Récamier, “Nonlinear Jaynes–Cummings model for two interacting two-level atoms,” J. Phys. B 49(16), 165503 (2016).
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Delgado, A.

T. E. Tessier, I. H. Deutsch, A. Delgado, and I. Fuentes-Guridi, “Entanglement sharing in the two-atom Tavis-Cummings model,” Phys. Rev. A 68(6), 062316 (2003).
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Deng, T.

T. Deng, Y. Yan, L. Chen, and Y. Zhao, “Dynamics of the two-spin spin-boson model with a common bath,” J. Chem. Phys. 144(14), 144102 (2016).
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Deutsch, I. H.

T. E. Tessier, I. H. Deutsch, A. Delgado, and I. Fuentes-Guridi, “Entanglement sharing in the two-atom Tavis-Cummings model,” Phys. Rev. A 68(6), 062316 (2003).
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Devoret, M. H.

J. Majer, J. M. Chow, J. M. Gambetta, J. Koch, B. R. Johnson, J. A. Schreier, L. Frunzio, D. I. Schuster, A. A. Houck, A. Wallraff, A. Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, “Coupling superconducting qubits via a cavity bus,” Nature 449(7161), 443–447 (2007).
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Y. Dubi and M. Di Ventra, “Relaxation times in an open interacting two-qubit system,” Phys. Rev. A 79(1), 012328 (2009).
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M. Donaire, J. M. Muñoz-Castañeda, and L. Nieto, “Dipole-dipole interaction in cavity QED: The weak-coupling, nondegenerate regime,” Phys. Rev. A 96(4), 042714 (2017).
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T. L. Nguyen, J.-M. Raimond, C. Sayrin, R. Cortinas, T. Cantat-Moltrecht, F. Assemat, I. Dotsenko, S. Gleyzes, S. Haroche, G. Roux, T. Jolicoeur, and M. Brune, “Towards quantum simulation with circular Rydberg atoms,” Phys. Rev. X 8(1), 011032 (2018).
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Dragt, A. J.

A. J. Berkley, H. Xu, R. C. Ramos, M. A. Gubrud, F. W. Strauch, P. R. Johnson, J. R. Anderson, A. J. Dragt, C. J. Lobb, and F. C. Wellstood, “Entangled macroscopic quantum states in two superconducting qubits,” Science 300(5625), 1548–1550 (2003).
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Duan, L.

L. Duan, H. Wang, Q.-H. Chen, and Y. Zhao, “Entanglement dynamics of two qubits coupled individually to Ohmic baths,” J. Chem. Phys. 139(4), 044115 (2013).
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Y. Dubi and M. Di Ventra, “Relaxation times in an open interacting two-qubit system,” Phys. Rev. A 79(1), 012328 (2009).
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T. Yu and J. H. Eberly, “Sudden death of entanglement,” Science 323(5914), 598–601 (2009).
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M. Yönaç, T. Yu, and J. H. Eberly, “Sudden death of entanglement of two Jaynes–Cummings atoms,” J. Phys. B 39(15), S621–S625 (2006).
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T. Yu and J. H. Eberly, “Sudden death of entanglement: classical noise effects,” Opt. Commun. 264(2), 393–397 (2006).
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T. Yu and J. H. Eberly, “Finite-time disentanglement via spontaneous emission,” Phys. Rev. Lett. 93(14), 140404 (2004).
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C. Guerlin, E. Brion, T. Esslinger, and K. Mølmer, “Cavity quantum electrodynamics with a Rydberg-blocked atomic ensemble,” Phys. Rev. A 82(5), 053832 (2010).
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Ficek, Z.

S. Chan, M. D. Reid, and Z. Ficek, “Entanglement evolution of two remote and non-identical Jaynes–Cummings atoms,” J. Phys. B 42(6), 065507 (2009).
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Z. Ficek and R. Tanaś, “Dark periods and revivals of entanglement in a two-qubit system,” Phys. Rev. A 74(2), 024304 (2006).
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Frunzio, L.

J. Majer, J. M. Chow, J. M. Gambetta, J. Koch, B. R. Johnson, J. A. Schreier, L. Frunzio, D. I. Schuster, A. A. Houck, A. Wallraff, A. Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, “Coupling superconducting qubits via a cavity bus,” Nature 449(7161), 443–447 (2007).
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A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R.-S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431(7005), 162–167 (2004).
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Fuentes-Guridi, I.

T. E. Tessier, I. H. Deutsch, A. Delgado, and I. Fuentes-Guridi, “Entanglement sharing in the two-atom Tavis-Cummings model,” Phys. Rev. A 68(6), 062316 (2003).
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Gambetta, J. M.

J. Majer, J. M. Chow, J. M. Gambetta, J. Koch, B. R. Johnson, J. A. Schreier, L. Frunzio, D. I. Schuster, A. A. Houck, A. Wallraff, A. Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, “Coupling superconducting qubits via a cavity bus,” Nature 449(7161), 443–447 (2007).
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Gao, C.-P.

Gao, H. J.

A. Abliz, H. J. Gao, X. C. Xie, Y. S. Wu, and W. M. Liu, “Entanglement control in an anisotropic two-qubit Heisenberg X Y Z model with external magnetic fields,” Phys. Rev. A 74(5), 052105 (2006).
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Girvin, S. M.

J. Majer, J. M. Chow, J. M. Gambetta, J. Koch, B. R. Johnson, J. A. Schreier, L. Frunzio, D. I. Schuster, A. A. Houck, A. Wallraff, A. Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, “Coupling superconducting qubits via a cavity bus,” Nature 449(7161), 443–447 (2007).
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A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R.-S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431(7005), 162–167 (2004).
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A. Blais, R.-S. Huang, A. Wallraff, S. M. Girvin, and R. J. Schoelkopf, “Cavity quantum electrodynamics for superconducting electrical circuits: An architecture for quantum computation,” Phys. Rev. A 69(6), 062320 (2004).
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Gleyzes, S.

T. L. Nguyen, J.-M. Raimond, C. Sayrin, R. Cortinas, T. Cantat-Moltrecht, F. Assemat, I. Dotsenko, S. Gleyzes, S. Haroche, G. Roux, T. Jolicoeur, and M. Brune, “Towards quantum simulation with circular Rydberg atoms,” Phys. Rev. X 8(1), 011032 (2018).
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González-Gutiérrez, C.

O. de los Santos-Sánchez, C. González-Gutiérrez, and J. Récamier, “Nonlinear Jaynes–Cummings model for two interacting two-level atoms,” J. Phys. B 49(16), 165503 (2016).
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A. J. Berkley, H. Xu, R. C. Ramos, M. A. Gubrud, F. W. Strauch, P. R. Johnson, J. R. Anderson, A. J. Dragt, C. J. Lobb, and F. C. Wellstood, “Entangled macroscopic quantum states in two superconducting qubits,” Science 300(5625), 1548–1550 (2003).
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C. Guerlin, E. Brion, T. Esslinger, and K. Mølmer, “Cavity quantum electrodynamics with a Rydberg-blocked atomic ensemble,” Phys. Rev. A 82(5), 053832 (2010).
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Guo, G.-C.

S.-B. Zheng and G.-C. Guo, “Efficient scheme for two-atom entanglement and quantum information processing in cavity QED,” Phys. Rev. Lett. 85(11), 2392–2395 (2000).
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Nature (3)

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New J. Phys. (1)

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Opt. Commun. (2)

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Opt. Express (1)

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Figures (12)

Fig. 1.
Fig. 1. Schematic diagram of two two-level coupled atoms (qubits) $q_1$ and $q_2$ in (a) an optical cavity or (b) a superconducting microwave resonator.
Fig. 2.
Fig. 2. Entanglement $E_f$ and population inversion $\langle \sigma _z \rangle$ versus the scaled time $\tau =\lambda _1 t$ with the two atoms are initially in a correlated Bell state $\psi _{Bc}=(\vert e_{1}\rangle \vert e_{2}\rangle +\vert g_{1}\rangle \vert g_{2}\rangle )/\sqrt {2}$ and the field is in a coherent state: (a) $E_f$ versus $\tau$ for $\lambda _2=0$, $\Delta =0$ and various values of the mean number of photons; (b) $E_f$ and $\langle \sigma _z \rangle$ versus $\tau$ for $\lambda _2=0$, $\Delta =0$ and $\bar {n}=100$; (c) $E_f$ versus $\tau$ for $\lambda _2=0$, $\bar {n}=100$ and various values of $\Delta$, and (d) $E_f$ versus $\tau$ for $\Delta =0$, $\bar {n}=100$ and various values of $\lambda _2$.
Fig. 3.
Fig. 3. Entanglement $E_f$ and population inversion $\langle \sigma _z \rangle$ versus the scaled time $\tau =\lambda _1 t$ with the two atoms are initially in a correlated Bell state $\psi _{Bc}=(\vert e_{1}\rangle \vert e_{2}\rangle +\vert g_{1}\rangle \vert g_{2}\rangle )/\sqrt {2}$ and the field is in a coherent state: (a) $E_f$ versus $\tau$ for $\lambda _2=5$, $\bar {n}=100$ and various values of $\Delta$, and (b) $\langle \sigma _z \rangle$ versus $\tau$ for $\bar {n}=100$ and various values of $\lambda _2$ and $\Delta$.
Fig. 4.
Fig. 4. Entanglement $E_f$ and population inversion $\langle \sigma _z \rangle$ versus the scaled time $\tau =\lambda _1 t$ with the two atoms are initially in an anti-correlated Bell state $\psi _{Ba}=(\vert g_{1}\rangle \vert e_{2}\rangle +\vert e_{1}\rangle \vert g_{2}\rangle )/\sqrt {2}$ and the field is in a coherent state: (a) $E_f$ versus $\tau$ for $\lambda _2=0$, $\Delta =0$ and various values of the mean number of photons; (b) $E_f$ and $\langle \sigma _z \rangle$ versus $\tau$ for $\lambda _2=0$, $\Delta =0$ and $\bar {n}=100$; (c) $E_f$ versus $\tau$ for $\lambda _2=0$, $\bar {n}=100$ and various values of $\Delta$, and (d) $E_f$ versus $\tau$ for $\Delta =0$, $\bar {n}=100$ and various values of $\lambda _2$.
Fig. 5.
Fig. 5. Entanglement $E_f$ and population inversion $\langle \sigma _z \rangle$ versus the scaled time $\tau =\lambda _1 t$ with the two atoms are initially in an anti-correlated Bell state $\psi _{Ba}=(\vert g_{1}\rangle \vert e_{2}\rangle +\vert e_{1}\rangle \vert g_{2}\rangle )/\sqrt {2}$ and the field is in a coherent state: (a) $E_f$ versus $\tau$ for $\bar {n}=100$ and various values of $\lambda _2$ and $\Delta$, and (b) $\langle \sigma _z \rangle$ versus $\tau$ for $\bar {n}=100$ and various values of $\lambda _2$ and $\Delta$.
Fig. 6.
Fig. 6. Entanglement $E_f$ versus the scaled time $\tau =\lambda _1 t$ with the two atoms are initially in a W-like state $\psi _{W}=(\vert g_{1}\rangle \vert g_{2}\rangle + \vert g_{1}\rangle \vert e_{2}\rangle + \vert e_{1}\rangle \vert g_{2}\rangle )/\sqrt {3}$ and the field is in a coherent state for: (a) $\lambda _2=0$, $\Delta =0$ and various values of the mean number of photons; (b) $\Delta =0$, $\bar {n}=100$ and various values of $\lambda _2$; (c) $\lambda _2=0$, $\bar {n}=100$ and various values of $\Delta$, and (d) $\bar {n}=100$ and various values of $\lambda _2$ and $\Delta$.
Fig. 7.
Fig. 7. Population inversion versus the scaled time $\tau =\lambda _1 t$ with the two atoms are initially in a W-like state $\psi _{W}=(\vert g_{1}\rangle \vert g_{2}\rangle + \vert g_{1}\rangle \vert e_{2}\rangle + \vert e_{1}\rangle \vert g_{2}\rangle )/\sqrt {3}$ and the field is in a coherent state for: (a) $\lambda _2=0$, $\Delta =0$ and various values of the mean number of photons, and (b) $\bar {n}=100$ and various values of $\lambda _2$ and $\Delta$.
Fig. 8.
Fig. 8. Entanglement in (a) and Population inversion in (b) versus the scaled time $\tau =\lambda _1 t$ and the detuning parameter $\Delta$ with the two atoms are initially in a W-like state $\psi _{W}=(\vert g_{1}\rangle \vert g_{2}\rangle + \vert g_{1}\rangle \vert e_{2}\rangle + \vert e_{1}\rangle \vert g_{2}\rangle )/\sqrt {3}$ and the field is in a coherent state for $\bar {n}=100$ and $\lambda _2=2$.
Fig. 9.
Fig. 9. Entanglement $E_f$ and population inversion $\langle \sigma _z \rangle$ versus the scaled time $\tau =\lambda _1 t$ with the two atoms are initially in a disentangled initial state $\psi _{e}=\vert e_{1}\rangle \vert e_{2}\rangle$ and the field is in a coherent state: (a) $E_f$ versus $\tau$ for $\lambda _2=0$, $\Delta =0$ and various values of the mean number of photons; (b) $\langle \sigma _z \rangle$ versus $\tau$ for $\lambda _2=0$, $\Delta =0$ and various values of the mean number of photons; (c) $E_f$ versus $\tau$ for $\bar {n}=100$ and various values of $\lambda _2$ and $\Delta$, and (d) $\langle \sigma _z \rangle$ versus $\tau$ for $\bar {n}=100$ and various values of $\lambda _2$ and $\Delta$.
Fig. 10.
Fig. 10. Entanglement in (a) and Population inversion in (b) versus the scaled time $\tau =\lambda _1 t$ and the coupling parameter $\lambda _2$ with the two atoms are initially in a disentangled state $\psi _{e}=\vert e_{1}\rangle \vert e_{2}\rangle$ and the field is in a coherent state for $\bar {n}=100$ and $\Delta =5$.
Fig. 11.
Fig. 11. Entanglement $E_f$ and population inversion $\langle \sigma _z \rangle$ versus the scaled time $\tau =\lambda _1 t$ with the two atoms are initially in a disentangled initial state $\psi _{L}=(\vert g_{1}\rangle \vert g_{2}\rangle + \vert g_{1}\rangle \vert e_{2}\rangle + \vert e_{1}\rangle \vert g_{2}\rangle + \vert e_{1}\rangle \vert e_{2}\rangle )/\sqrt {4}$ and the field is in a coherent state: (a) $E_f$ versus $\tau$ for various values of $\lambda _2$, $\Delta$ and the mean number of photons; (b) $E_f$ versus $\tau$ for $\bar {n}=100$ and various values of $\lambda _2$ and $\Delta$; (c) $\langle \sigma _z \rangle$ versus $\tau$ for $\lambda _2=0$, $\Delta =0$ and various values of mean number of photons, and (d) $\langle \sigma _z \rangle$ versus $\tau$ for $\bar {n}=100$ and various values of $\lambda _2$ and $\Delta$.
Fig. 12.
Fig. 12. (a) Entanglement versus the scaled time $\tau =\lambda _1 t$ and the coupling parameter $\lambda _2$ for $\Delta =2$, and (b) Population inversion versus the scaled time $\tau =\lambda _1 t$ and the detuning parameter $\Delta =2$ for $\lambda _2=5$, with the two atoms are initially in a disentangled initial state $\psi _{L}=(\vert g_{1}\rangle \vert g_{2}\rangle + \vert g_{1}\rangle \vert e_{2}\rangle + \vert e_{1}\rangle \vert g_{2}\rangle + \vert e_{1}\rangle \vert e_{2}\rangle )/\sqrt {4}$ and the field is in a coherent state at $\bar {n}=100$.

Equations (36)

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H = Ω a ^ a ^ + 1 2 i ω i σ ^ z ( i ) + i λ 1 i σ ^ x ( i ) ( a ^ + a ^ ) + i j , ν λ 2 , ν i j σ ^ ν ( i ) σ ^ ν ( j )
H ^ = Ω a ^ a ^ + ω 2 i = 1 , 2 σ ^ z ( i ) + λ 1 i = 1 , 2 ( a ^ σ ^ + ( i ) + a ^ σ ^ ( i ) ) + λ 2 ( σ ^ ( 1 ) σ ^ + ( 2 ) + σ ^ + ( 1 ) σ ^ ( 2 ) ) .
d O ^ d t = -i [ Q ^ , H ^ ] + O ^ t ,
d a ^ d t = i Ω a ^ i λ 1 ( σ ^ ( 1 ) + σ ^ ( 2 ) ) , d σ ^ ( 1 ) d t = i ω σ ^ ( 1 ) + i λ 1 a ^ σ ^ z ( 1 ) + i λ 2 σ ^ z ( 1 ) σ ^ ( 2 ) , d σ ^ ( 2 ) d t = i ω σ ^ ( 2 ) + i λ 1 a ^ σ ^ z ( 2 ) + i λ 2 σ ^ ( 1 ) σ ^ z ( 2 ) , d σ ^ z ( 1 ) d t = 2 i λ 1 ( a ^ σ ^ ( 1 ) a ^ σ ^ + ( 1 ) ) + 2 i λ 2 ( σ ^ ( 1 ) σ ^ + ( 2 ) σ ^ + ( 1 ) σ ^ ( 2 ) ) , d σ ^ z ( 2 ) d t = 2 i λ 1 ( a ^ σ ^ ( 2 ) a ^ σ ^ + ( 2 ) ) + 2 i λ 2 ( σ ^ + ( 1 ) σ ^ ( 2 ) σ ^ ( 1 ) σ ^ + ( 2 ) ) ,
| ψ ( 0 ) = [ a | e 1 , e 2 + b | e 1 , g 2 + c | g 1 , e 2 + d | g 1 , g 2 ] | α ,
| a | 2 + | b | 2 + | c | 2 + | d | 2 = 1 ,
| α = n Q n | n ; Q n = α n n ! exp ( | α | 2 2 ) ,
| ψ ( t ) = n [ A n ( t ) | e 1 , e 2 , n + B n + 1 ( t ) | e 1 , g 2 , n + 1 + C n + 1 ( t ) | g 1 , e 2 , n + 1 + D n + 2 ( t ) | g 1 , g 2 , n + 2 ] ,
ρ ^ red ( t ) = Tr field ρ ^ ( t ) = l l | ψ ( t ) ψ ( t ) | l .
H ^ = H ^ + H ^ i n t ,
H ^ = Ω N ^ + Δ 2 i = 1 , 2 σ ^ z ( i ) ,
H ^ i n t = λ 1 i = 1 , 2 ( a ^ σ ^ + ( i ) + a ^ σ ^ ( i ) ) + λ 2 ( σ ^ ( 1 ) σ ^ + ( 2 ) + σ ^ + ( 1 ) σ ^ ( 2 ) ) ,
N ^ = a ^ a ^ + 1 2 i = 1 , 2 σ ^ z ( i ) ,
V ^ I ( t ) = λ 1 i = 1 , 2 ( a ^ e i Δ t σ ^ + ( i ) + a ^ e i Δ t σ ^ ( i ) ) + λ 2 ( σ ^ ( 1 ) σ ^ + ( 2 ) + σ ^ + ( 1 ) σ ^ ( 2 ) ) .
i t | ψ ( t ) = V ^ I ( t ) | ψ ( t ) ,
i A ˙ n ( t ) = α e i Δ t ( B n + 1 ( t ) + C n + 1 ( t ) ) , i B ˙ n + 1 ( t ) = α e i Δ t A n ( t ) + β e i Δ t D n + 2 ( t ) + λ 2 C n + 1 ( t ) , i C ˙ n + 1 ( t ) = α e i Δ t A n ( t ) + β e i Δ t D n + 2 ( t ) + λ 2 B n + 1 ( t ) , i D ˙ n + 2 ( t ) = β e i Δ t ( B n + 1 ( t ) + C n + 1 ( t ) ) ,
i A ˙ n ( t ) = α K ( t ) e i Δ t , i D ˙ n + 2 ( t ) = β K ( t ) e i Δ t , i K ˙ ( t ) = 2 α e i Δ t A n ( t ) + 2 β e i Δ t D n + 2 ( t ) + λ 2 K ( t ) ,
K ( t ) + i λ 2 K ¨ ( t ) + [ 2 ( α 2 + β 2 ) + Δ 2 ] K ˙ ( t ) i [ 2 Δ ( α 2 β 2 ) λ 2 Δ 2 ] K ( t ) = 0 ,
K ( t ) = j = 1 3 δ j e m j t ,
δ 1 = ( B n + 1 ( 0 ) + C n + 1 ( 0 ) ) ( δ 2 + δ 3 ) , δ 2 = 1 ( m 1 m 2 ) ( m 3 m 2 ) { 2 α A n ( 0 ) [ i ( m 1 + m 3 ) λ 2 Δ ] + 2 β D n + 2 ( 0 ) [ i ( m 1 + m 3 ) λ 2 + Δ ] + [ i ( m 1 + m 3 ) ( λ 2 i m 1 ) 2 ( α 2 + β 2 ) λ 2 2 m 1 2 ] × ( B n + 1 ( 0 ) + C n + 1 ( 0 ) ) } , δ 3 = 1 ( m 1 m 3 ) ( m 2 m 3 ) { 2 α A n ( 0 ) [ i ( m 1 + m 2 ) λ 2 Δ ] + 2 β D n + 2 ( 0 ) [ i ( m 1 + m 2 ) λ 2 + Δ ] + [ i ( m 1 + m 2 ) ( λ 2 i m 1 ) 2 ( α 2 + β 2 ) λ 2 2 m 1 2 ] × ( B n + 1 ( 0 ) + C n + 1 ( 0 ) ) } ,
m 1 = ( v 1 + v 2 ) i λ 2 3 , m 2 = v 1 + v 2 2 + i 3 2 ( v 1 v 2 ) i λ 2 3 , m 3 = v 1 + v 2 2 i 3 2 ( v 1 v 2 ) i λ 2 3 ,
v 1 = [ μ 2 + ( μ 2 4 + η 3 27 ) 1 2 ] 1 3 ; v 2 = [ μ 2 ( μ 2 4 + η 3 27 ) 1 2 ] 1 3 ,
μ = i 27 [ 2 λ 2 3 + 18 λ 2 ( α 2 + β 2 Δ 2 ) + 54 Δ ( α 2 β 2 ) ] ,
η = 1 3 [ 6 ( α 2 + β 2 ) + 3 Δ 2 + λ 2 2 ] .
A n ( t ) = A n ( 0 ) i α j = 1 3 [ δ j m j + i Δ ( e ( m j + i Δ ) t 1 ) ] , B n + 1 ( t ) = 1 2 [ ( B n + 1 ( 0 ) C n + 1 ( 0 ) ) e i λ 2 t + j = 1 3 δ j e m j t ] , C n + 1 ( t ) = 1 2 [ ( C n + 1 ( 0 ) B n + 1 ( 0 ) ) e i λ 2 t + j = 1 3 δ j e m j t ] , D n + 2 ( t ) = D n + 2 ( 0 ) i β j = 1 3 [ δ j m j i Δ ( e ( m j i Δ ) t 1 ) ] ,
A n ( 0 ) = Q n a , B n + 1 ( 0 ) = Q n + 1 b , C n + 1 ( 0 ) = Q n + 1 c , D n + 2 ( 0 ) = Q n + 2 d .
ρ r e d = n = 0 ( | A n | 2 A n + 1 B n + 1 A n + 1 C n + 1 A n + 2 D n + 2 B n + 1 A n + 1 | B n + 1 | 2 B n + 1 C n + 1 B n + 2 D n + 2 C n + 1 A n + 1 C n + 1 B n + 1 | C n + 1 | 2 C n + 2 D n + 2 D n + 2 A n + 2 D n + 2 B n + 2 D n + 2 C n + 2 | D n + 2 | 2 ) ,
E f ( ρ red ) = E ( C ( ρ red ) ) ,
E ( C ( ρ red ) ) = h ( 1 + 1 C 2 ( ρ red ) 2 ) ,
h ( x ) = x log 2 x ( 1 x ) log 2 ( 1 x ) ,
C ( ρ red ) = max [ 0 , ε 1 ε 2 ε 3 ε 4 ] ,
R ρ red ρ ~ red ,
ρ ~ red = ( σ ^ y σ ^ y ) ρ red ( σ ^ y σ ^ y ) ,
ρ ^ 1 ( t ) = T r q 2 ρ ^ r e d ( t ) = ( ρ 11 ρ 12 ρ 21 ρ 22 ) ,
ρ 11 ( t ) = n = 0 | A n ( t ) | 2 + | B n + 1 ( t ) | 2 , ρ 22 ( t ) = n = 0 | C n + 1 ( t ) | 2 + | D n + 2 ( t ) | 2 , ρ 12 ( t ) = ρ 21 ( t ) = n = 0 A n + 1 ( t ) C n + 1 ( t ) + B n + 2 ( t ) D n + 2 ( t ) .
σ ^ z ( t ) = T r [ ρ ^ 1 ( t ) σ ^ z ] = n = 0 | A n ( t ) | 2 + | B n + 1 ( t ) | 2 | C n + 1 ( t ) | 2 | D n + 2 ( t ) | 2 .

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