Abstract

We examine transfer of particle entanglement and spin squeezing between atomic and photonic subsystems in optical cavities coupled by two-photon exchange. Each cavity contains a single atom, interacting with cavity photons with a two-photon cascade transition. Particle entanglement is characterized by evaluating optimal spin squeezing inequalities for the cases of initially separable and entangled two-photon states. It is found that particle entanglement is first generated among the photons in separate cavities and then transferred to the atoms. The underlying mechanism is recognized as an intercavity two-axis twisting spin squeezing interaction, induced by two-photon exchange, and its optimal combination with the intracavity atom–photon coupling. Relative effect of nonlocal two-photon exchange and local atom–photon interactions of cavity photons on the spin squeezing and entanglement transfer is pointed out.

© 2012 Optical Society of America

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    [CrossRef]
  2. S. Longhi, “Optical realization of two-boson tunneling dynamics,” Phys. Rev. A 83, 43835 (2011).
    [CrossRef]
  3. X.-F. Zhou, Y.-S. Zhang, and G.-C. Guo, “Pair tunneling of bosonic atoms in an optical lattice,” Phys. Rev. A 80, 013605 (2009).
    [CrossRef]
  4. M. Kitagawa and M. Ueda, “Squeezed spin states,” Phys. Rev. A 47, 5138–5143 (1993).
    [CrossRef]
  5. A. Sørensen, L. M. Duan, J. I. Cirac, and P. Zoller, “Many-particle entanglement with Bose–Einstein condensates,” Nature 409, 63–66 (2001).
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  6. M. Hillery, “An introduction to the quantum theory of nonlinear optics,” Acta Phys. Slovaca 59, 1–80 (2009).
    [CrossRef]
  7. F. DellaAnno, S. De Siena, and F. Illuminati, “Multiphoton quantum optics and quantum state engineering,” Phys. Rep. 428, 53–168 (2006).
    [CrossRef]
  8. A. Biswas and G. S. Agarwal, “Transfer of an unknown quantum state, quantum networks, and memory,” Phys. Rev. A 70, 022323 (2004).
    [CrossRef]
  9. J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi, “Quantum state transfer and entanglement distribution among distant nodes in a quantum network,” Phys. Rev. Lett. 78, 3221–3224 (1997).
    [CrossRef]
  10. L. Vernac, M. Pinard, and E. Giacobino, “Quantum state transfer from light beams to atomic ensembles,” Eur. Phys. J. D 17, 125–136 (2001).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  13. A. Banerjee, “Generation of atomic-squeezed states in an optical cavity with an injected squeezed vacuum,” Phys. Rev. A 54, 5327–5333 (1996).
    [CrossRef]
  14. G. Tóth, C. Knapp, O. Gühne, and H. J. Briegel, “Optimal spin squeezing inequalities detect bound entanglement in spin models,” Phys. Rev. Lett. 99, 250405 (2007).
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    [CrossRef]
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    [CrossRef]
  17. S. Van Enk, “Single-particle entanglement,” Phys. Rev. A 72, 64306 (2005).
    [CrossRef]
  18. F. Benatti, R. Floreanini, and U. Marzolino, “Entanglement and squeezing with identical particles: ultracold atom quantum metrology,” J. Phys. B 44, 091001 (2011).
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    [CrossRef]
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    [CrossRef]
  21. J. Vidal, “Concurrence in collective models,” Phys. Rev. A 73, 062318 (2006).
    [CrossRef]
  22. G. Vitagliano, P. Hyllus, I. L. Egusquiza, and G. Tóth, “Spin squeezing inequalities for arbitrary spin,” Phys. Rev. Lett. 107, 240502 (2011).
    [CrossRef]
  23. Y. Wu and X. Yang, “Effective two-level model for a three-level atom in the Ξ configuration,” Phys. Rev. A 56, 2443–2446 (1997).
    [CrossRef]
  24. Y. Wu, “Effective Raman theory for a three-level atom in the λ configuration,” Phys. Rev. A 54, 1586 (1996).
    [CrossRef]
  25. M. Alexanian and S. K. Bose, “Unitary transformation and the dynamics of a three-level atom interacting with two quantized field modes,” Phys. Rev. A 52, 2218–2224 (1995).
    [CrossRef]
  26. M. Alexanian, S. Bose, and L. Chow, “Trapping and Fock state generation in a two-photon micromaser,” J. Mod. Opt. 45, 2519–2532 (1998).
    [CrossRef]
  27. J. Vidal, G. Palacios, and C. Aslangul, “Entanglement dynamics in the Lipkin–Meshkov–Glick model,” Phys. Rev. A 70, 062304 (2004).
    [CrossRef]
  28. P. Ribeiro, J. Vidal, and R. Mosseri, “Exact spectrum of the Lipkin–Meshkov–Glick model in the thermodynamic limit and finite-size corrections,” Phys. Rev. E 78, 021106 (2008).
    [CrossRef]
  29. P. Ribeiro, J. Vidal, and R. Mosseri, “Thermodynamical limit of the Lipkin–Meshkov–Glick model,” Phys. Rev. Lett. 99, 050402 (2007).
    [CrossRef]
  30. V. Karassiov and A. Klimov, “An algebraic approach to solving evolution problems in some nonlinear quantum models,” Phys. Lett. A 189, 43–51 (1994).
    [CrossRef]
  31. P. Higgs, “Dynamical symmetries in a spherical geometry. I,” J. Phys. A Math. Gen. 12, 309–323 (1979).
    [CrossRef]
  32. H. J. Carmichael, “Quantum fluctuations in absorptive bistability without adiabatic elimination,” Phys. Rev. A 33, 3262–3269 (1986).
    [CrossRef]
  33. M. Alexanian, “Scattering of two coherent photons inside a one-dimensional coupled-resonator waveguide,” Phys. Rev. A 81, 015805 (2010).
    [CrossRef]
  34. J. Schreier, A. Houck, J. Koch, D. Schuster, B. Johnson, J. Chow, J. Gambetta, J. Majer, L. Frunzio, M. Devoret, S. M. Girvin, and R. J. Schoelkopf, “Suppressing charge noise decoherence in superconducting charge qubits,” Phys. Rev. B 77, 180502 (2008).
    [CrossRef]
  35. C. Yang, “Fast quantum information transfer with superconducting flux qubits coupled to a cavity,” J. Phys. A Math. Theor. 45, 205304 (2012).
    [CrossRef]
  36. D. Petrosyan, G. Bensky, G. Kurizki, I. Mazets, J. Majer, and J. Schmiedmayer, “Reversible state transfer between superconducting qubits and atomic ensembles,” Phys. Rev. A 79, 040304 (2009).
    [CrossRef]
  37. P. Forn-Díaz, J. Lisenfeld, D. Marcos, J. J. García-Ripoll, E. Solano, C. J. P. M. Harmans, and J. E. Mooij, “Observation of the Bloch–Siegert shift in a qubit-oscillator system in the ultrastrong coupling regime,” Phys. Rev. Lett. 105, 237001 (2010).
    [CrossRef]
  38. J. Casanova, G. Romero, I. Lizuain, J. J. García-Ripoll, and E. Solano, “Deep strong coupling regime of the Jaynes–Cummings model,” Phys. Rev. Lett. 105, 263603 (2010).
    [CrossRef]
  39. M. Hofheinz, E. Weig, M. Ansmann, R. Bialczak, E. Lucero, M. Neeley, A. O’Connell, H. Wang, J. Martinis, and A. Cleland, “Generation of Fock states in a superconducting quantum circuit,” Nature 454, 310–314 (2008).
    [CrossRef]
  40. D. Bozyigit, C. Lang, L. Steffen, J. Fink, C. Eichler, M. Baur, R. Bianchetti, P. Leek, S. Filipp, M. da Silva, A. Blais, and A. Wallraff, “Antibunching of microwave-frequency photons observed in correlation measurements using linear detectors,” Nat. Phys. 7, 154–158 (2010).
    [CrossRef]

2012 (1)

C. Yang, “Fast quantum information transfer with superconducting flux qubits coupled to a cavity,” J. Phys. A Math. Theor. 45, 205304 (2012).
[CrossRef]

2011 (5)

M. Alexanian, “Two-photon exchange between two three-level atoms in separate cavities,” Phys. Rev. A 83, 023814 (2011).
[CrossRef]

S. Longhi, “Optical realization of two-boson tunneling dynamics,” Phys. Rev. A 83, 43835 (2011).
[CrossRef]

J. Ma, X. Wang, C. Sun, and F. Nori, “Quantum spin squeezing,” Phys. Rep. 509, 89–165(2011).
[CrossRef]

F. Benatti, R. Floreanini, and U. Marzolino, “Entanglement and squeezing with identical particles: ultracold atom quantum metrology,” J. Phys. B 44, 091001 (2011).
[CrossRef]

G. Vitagliano, P. Hyllus, I. L. Egusquiza, and G. Tóth, “Spin squeezing inequalities for arbitrary spin,” Phys. Rev. Lett. 107, 240502 (2011).
[CrossRef]

2010 (4)

M. Alexanian, “Scattering of two coherent photons inside a one-dimensional coupled-resonator waveguide,” Phys. Rev. A 81, 015805 (2010).
[CrossRef]

P. Forn-Díaz, J. Lisenfeld, D. Marcos, J. J. García-Ripoll, E. Solano, C. J. P. M. Harmans, and J. E. Mooij, “Observation of the Bloch–Siegert shift in a qubit-oscillator system in the ultrastrong coupling regime,” Phys. Rev. Lett. 105, 237001 (2010).
[CrossRef]

J. Casanova, G. Romero, I. Lizuain, J. J. García-Ripoll, and E. Solano, “Deep strong coupling regime of the Jaynes–Cummings model,” Phys. Rev. Lett. 105, 263603 (2010).
[CrossRef]

D. Bozyigit, C. Lang, L. Steffen, J. Fink, C. Eichler, M. Baur, R. Bianchetti, P. Leek, S. Filipp, M. da Silva, A. Blais, and A. Wallraff, “Antibunching of microwave-frequency photons observed in correlation measurements using linear detectors,” Nat. Phys. 7, 154–158 (2010).
[CrossRef]

2009 (5)

D. Petrosyan, G. Bensky, G. Kurizki, I. Mazets, J. Majer, and J. Schmiedmayer, “Reversible state transfer between superconducting qubits and atomic ensembles,” Phys. Rev. A 79, 040304 (2009).
[CrossRef]

B. Öztop, M. Ö. Oktel, Ö. E. Müstecaplıoğlu, and L. You, “Quantum entanglement of spin-1 bosons with coupled ground states in optical lattices,” J. Phys. B 42, 145505 (2009).
[CrossRef]

G. Tóth, C. Knapp, O. Gühne, and H. Briegel, “Spin squeezing and entanglement,” Phys. Rev. A 79, 042334 (2009).
[CrossRef]

X.-F. Zhou, Y.-S. Zhang, and G.-C. Guo, “Pair tunneling of bosonic atoms in an optical lattice,” Phys. Rev. A 80, 013605 (2009).
[CrossRef]

M. Hillery, “An introduction to the quantum theory of nonlinear optics,” Acta Phys. Slovaca 59, 1–80 (2009).
[CrossRef]

2008 (3)

J. Schreier, A. Houck, J. Koch, D. Schuster, B. Johnson, J. Chow, J. Gambetta, J. Majer, L. Frunzio, M. Devoret, S. M. Girvin, and R. J. Schoelkopf, “Suppressing charge noise decoherence in superconducting charge qubits,” Phys. Rev. B 77, 180502 (2008).
[CrossRef]

P. Ribeiro, J. Vidal, and R. Mosseri, “Exact spectrum of the Lipkin–Meshkov–Glick model in the thermodynamic limit and finite-size corrections,” Phys. Rev. E 78, 021106 (2008).
[CrossRef]

M. Hofheinz, E. Weig, M. Ansmann, R. Bialczak, E. Lucero, M. Neeley, A. O’Connell, H. Wang, J. Martinis, and A. Cleland, “Generation of Fock states in a superconducting quantum circuit,” Nature 454, 310–314 (2008).
[CrossRef]

2007 (3)

P. Ribeiro, J. Vidal, and R. Mosseri, “Thermodynamical limit of the Lipkin–Meshkov–Glick model,” Phys. Rev. Lett. 99, 050402 (2007).
[CrossRef]

M. Cunha, J. Dunningham, and V. Vedral, “Entanglement in single-particle systems,” Proc. R. Soc. A 463, 2277–2286(2007).
[CrossRef]

G. Tóth, C. Knapp, O. Gühne, and H. J. Briegel, “Optimal spin squeezing inequalities detect bound entanglement in spin models,” Phys. Rev. Lett. 99, 250405 (2007).
[CrossRef]

2006 (2)

F. DellaAnno, S. De Siena, and F. Illuminati, “Multiphoton quantum optics and quantum state engineering,” Phys. Rep. 428, 53–168 (2006).
[CrossRef]

J. Vidal, “Concurrence in collective models,” Phys. Rev. A 73, 062318 (2006).
[CrossRef]

2005 (1)

S. Van Enk, “Single-particle entanglement,” Phys. Rev. A 72, 64306 (2005).
[CrossRef]

2004 (2)

A. Biswas and G. S. Agarwal, “Transfer of an unknown quantum state, quantum networks, and memory,” Phys. Rev. A 70, 022323 (2004).
[CrossRef]

J. Vidal, G. Palacios, and C. Aslangul, “Entanglement dynamics in the Lipkin–Meshkov–Glick model,” Phys. Rev. A 70, 062304 (2004).
[CrossRef]

2001 (2)

A. Sørensen, L. M. Duan, J. I. Cirac, and P. Zoller, “Many-particle entanglement with Bose–Einstein condensates,” Nature 409, 63–66 (2001).
[CrossRef]

L. Vernac, M. Pinard, and E. Giacobino, “Quantum state transfer from light beams to atomic ensembles,” Eur. Phys. J. D 17, 125–136 (2001).
[CrossRef]

1999 (1)

J. Hald, J. L. Sørensen, C. Schori, and E. S. Polzik, “Spin squeezed atoms: a macroscopic entangled ensemble created by light,” Phys. Rev. Lett. 83, 1319–1322 (1999).
[CrossRef]

1998 (1)

M. Alexanian, S. Bose, and L. Chow, “Trapping and Fock state generation in a two-photon micromaser,” J. Mod. Opt. 45, 2519–2532 (1998).
[CrossRef]

1997 (3)

Y. Wu and X. Yang, “Effective two-level model for a three-level atom in the Ξ configuration,” Phys. Rev. A 56, 2443–2446 (1997).
[CrossRef]

A. Kuzmich, K. Mølmer, and E. S. Polzik, “Spin squeezing in an ensemble of atoms illuminated with squeezed light,” Phys. Rev. Lett. 79, 4782–4785 (1997).
[CrossRef]

J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi, “Quantum state transfer and entanglement distribution among distant nodes in a quantum network,” Phys. Rev. Lett. 78, 3221–3224 (1997).
[CrossRef]

1996 (2)

A. Banerjee, “Generation of atomic-squeezed states in an optical cavity with an injected squeezed vacuum,” Phys. Rev. A 54, 5327–5333 (1996).
[CrossRef]

Y. Wu, “Effective Raman theory for a three-level atom in the λ configuration,” Phys. Rev. A 54, 1586 (1996).
[CrossRef]

1995 (1)

M. Alexanian and S. K. Bose, “Unitary transformation and the dynamics of a three-level atom interacting with two quantized field modes,” Phys. Rev. A 52, 2218–2224 (1995).
[CrossRef]

1994 (1)

V. Karassiov and A. Klimov, “An algebraic approach to solving evolution problems in some nonlinear quantum models,” Phys. Lett. A 189, 43–51 (1994).
[CrossRef]

1993 (1)

M. Kitagawa and M. Ueda, “Squeezed spin states,” Phys. Rev. A 47, 5138–5143 (1993).
[CrossRef]

1986 (1)

H. J. Carmichael, “Quantum fluctuations in absorptive bistability without adiabatic elimination,” Phys. Rev. A 33, 3262–3269 (1986).
[CrossRef]

1979 (1)

P. Higgs, “Dynamical symmetries in a spherical geometry. I,” J. Phys. A Math. Gen. 12, 309–323 (1979).
[CrossRef]

Agarwal, G. S.

A. Biswas and G. S. Agarwal, “Transfer of an unknown quantum state, quantum networks, and memory,” Phys. Rev. A 70, 022323 (2004).
[CrossRef]

Alexanian, M.

M. Alexanian, “Two-photon exchange between two three-level atoms in separate cavities,” Phys. Rev. A 83, 023814 (2011).
[CrossRef]

M. Alexanian, “Scattering of two coherent photons inside a one-dimensional coupled-resonator waveguide,” Phys. Rev. A 81, 015805 (2010).
[CrossRef]

M. Alexanian, S. Bose, and L. Chow, “Trapping and Fock state generation in a two-photon micromaser,” J. Mod. Opt. 45, 2519–2532 (1998).
[CrossRef]

M. Alexanian and S. K. Bose, “Unitary transformation and the dynamics of a three-level atom interacting with two quantized field modes,” Phys. Rev. A 52, 2218–2224 (1995).
[CrossRef]

Ansmann, M.

M. Hofheinz, E. Weig, M. Ansmann, R. Bialczak, E. Lucero, M. Neeley, A. O’Connell, H. Wang, J. Martinis, and A. Cleland, “Generation of Fock states in a superconducting quantum circuit,” Nature 454, 310–314 (2008).
[CrossRef]

Aslangul, C.

J. Vidal, G. Palacios, and C. Aslangul, “Entanglement dynamics in the Lipkin–Meshkov–Glick model,” Phys. Rev. A 70, 062304 (2004).
[CrossRef]

Banerjee, A.

A. Banerjee, “Generation of atomic-squeezed states in an optical cavity with an injected squeezed vacuum,” Phys. Rev. A 54, 5327–5333 (1996).
[CrossRef]

Baur, M.

D. Bozyigit, C. Lang, L. Steffen, J. Fink, C. Eichler, M. Baur, R. Bianchetti, P. Leek, S. Filipp, M. da Silva, A. Blais, and A. Wallraff, “Antibunching of microwave-frequency photons observed in correlation measurements using linear detectors,” Nat. Phys. 7, 154–158 (2010).
[CrossRef]

Benatti, F.

F. Benatti, R. Floreanini, and U. Marzolino, “Entanglement and squeezing with identical particles: ultracold atom quantum metrology,” J. Phys. B 44, 091001 (2011).
[CrossRef]

Bensky, G.

D. Petrosyan, G. Bensky, G. Kurizki, I. Mazets, J. Majer, and J. Schmiedmayer, “Reversible state transfer between superconducting qubits and atomic ensembles,” Phys. Rev. A 79, 040304 (2009).
[CrossRef]

Bialczak, R.

M. Hofheinz, E. Weig, M. Ansmann, R. Bialczak, E. Lucero, M. Neeley, A. O’Connell, H. Wang, J. Martinis, and A. Cleland, “Generation of Fock states in a superconducting quantum circuit,” Nature 454, 310–314 (2008).
[CrossRef]

Bianchetti, R.

D. Bozyigit, C. Lang, L. Steffen, J. Fink, C. Eichler, M. Baur, R. Bianchetti, P. Leek, S. Filipp, M. da Silva, A. Blais, and A. Wallraff, “Antibunching of microwave-frequency photons observed in correlation measurements using linear detectors,” Nat. Phys. 7, 154–158 (2010).
[CrossRef]

Biswas, A.

A. Biswas and G. S. Agarwal, “Transfer of an unknown quantum state, quantum networks, and memory,” Phys. Rev. A 70, 022323 (2004).
[CrossRef]

Blais, A.

D. Bozyigit, C. Lang, L. Steffen, J. Fink, C. Eichler, M. Baur, R. Bianchetti, P. Leek, S. Filipp, M. da Silva, A. Blais, and A. Wallraff, “Antibunching of microwave-frequency photons observed in correlation measurements using linear detectors,” Nat. Phys. 7, 154–158 (2010).
[CrossRef]

Bose, S.

M. Alexanian, S. Bose, and L. Chow, “Trapping and Fock state generation in a two-photon micromaser,” J. Mod. Opt. 45, 2519–2532 (1998).
[CrossRef]

Bose, S. K.

M. Alexanian and S. K. Bose, “Unitary transformation and the dynamics of a three-level atom interacting with two quantized field modes,” Phys. Rev. A 52, 2218–2224 (1995).
[CrossRef]

Bozyigit, D.

D. Bozyigit, C. Lang, L. Steffen, J. Fink, C. Eichler, M. Baur, R. Bianchetti, P. Leek, S. Filipp, M. da Silva, A. Blais, and A. Wallraff, “Antibunching of microwave-frequency photons observed in correlation measurements using linear detectors,” Nat. Phys. 7, 154–158 (2010).
[CrossRef]

Briegel, H.

G. Tóth, C. Knapp, O. Gühne, and H. Briegel, “Spin squeezing and entanglement,” Phys. Rev. A 79, 042334 (2009).
[CrossRef]

Briegel, H. J.

G. Tóth, C. Knapp, O. Gühne, and H. J. Briegel, “Optimal spin squeezing inequalities detect bound entanglement in spin models,” Phys. Rev. Lett. 99, 250405 (2007).
[CrossRef]

Carmichael, H. J.

H. J. Carmichael, “Quantum fluctuations in absorptive bistability without adiabatic elimination,” Phys. Rev. A 33, 3262–3269 (1986).
[CrossRef]

Casanova, J.

J. Casanova, G. Romero, I. Lizuain, J. J. García-Ripoll, and E. Solano, “Deep strong coupling regime of the Jaynes–Cummings model,” Phys. Rev. Lett. 105, 263603 (2010).
[CrossRef]

Chow, J.

J. Schreier, A. Houck, J. Koch, D. Schuster, B. Johnson, J. Chow, J. Gambetta, J. Majer, L. Frunzio, M. Devoret, S. M. Girvin, and R. J. Schoelkopf, “Suppressing charge noise decoherence in superconducting charge qubits,” Phys. Rev. B 77, 180502 (2008).
[CrossRef]

Chow, L.

M. Alexanian, S. Bose, and L. Chow, “Trapping and Fock state generation in a two-photon micromaser,” J. Mod. Opt. 45, 2519–2532 (1998).
[CrossRef]

Cirac, J. I.

A. Sørensen, L. M. Duan, J. I. Cirac, and P. Zoller, “Many-particle entanglement with Bose–Einstein condensates,” Nature 409, 63–66 (2001).
[CrossRef]

J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi, “Quantum state transfer and entanglement distribution among distant nodes in a quantum network,” Phys. Rev. Lett. 78, 3221–3224 (1997).
[CrossRef]

Cleland, A.

M. Hofheinz, E. Weig, M. Ansmann, R. Bialczak, E. Lucero, M. Neeley, A. O’Connell, H. Wang, J. Martinis, and A. Cleland, “Generation of Fock states in a superconducting quantum circuit,” Nature 454, 310–314 (2008).
[CrossRef]

Cunha, M.

M. Cunha, J. Dunningham, and V. Vedral, “Entanglement in single-particle systems,” Proc. R. Soc. A 463, 2277–2286(2007).
[CrossRef]

da Silva, M.

D. Bozyigit, C. Lang, L. Steffen, J. Fink, C. Eichler, M. Baur, R. Bianchetti, P. Leek, S. Filipp, M. da Silva, A. Blais, and A. Wallraff, “Antibunching of microwave-frequency photons observed in correlation measurements using linear detectors,” Nat. Phys. 7, 154–158 (2010).
[CrossRef]

De Siena, S.

F. DellaAnno, S. De Siena, and F. Illuminati, “Multiphoton quantum optics and quantum state engineering,” Phys. Rep. 428, 53–168 (2006).
[CrossRef]

DellaAnno, F.

F. DellaAnno, S. De Siena, and F. Illuminati, “Multiphoton quantum optics and quantum state engineering,” Phys. Rep. 428, 53–168 (2006).
[CrossRef]

Devoret, M.

J. Schreier, A. Houck, J. Koch, D. Schuster, B. Johnson, J. Chow, J. Gambetta, J. Majer, L. Frunzio, M. Devoret, S. M. Girvin, and R. J. Schoelkopf, “Suppressing charge noise decoherence in superconducting charge qubits,” Phys. Rev. B 77, 180502 (2008).
[CrossRef]

Duan, L. M.

A. Sørensen, L. M. Duan, J. I. Cirac, and P. Zoller, “Many-particle entanglement with Bose–Einstein condensates,” Nature 409, 63–66 (2001).
[CrossRef]

Dunningham, J.

M. Cunha, J. Dunningham, and V. Vedral, “Entanglement in single-particle systems,” Proc. R. Soc. A 463, 2277–2286(2007).
[CrossRef]

Egusquiza, I. L.

G. Vitagliano, P. Hyllus, I. L. Egusquiza, and G. Tóth, “Spin squeezing inequalities for arbitrary spin,” Phys. Rev. Lett. 107, 240502 (2011).
[CrossRef]

Eichler, C.

D. Bozyigit, C. Lang, L. Steffen, J. Fink, C. Eichler, M. Baur, R. Bianchetti, P. Leek, S. Filipp, M. da Silva, A. Blais, and A. Wallraff, “Antibunching of microwave-frequency photons observed in correlation measurements using linear detectors,” Nat. Phys. 7, 154–158 (2010).
[CrossRef]

Filipp, S.

D. Bozyigit, C. Lang, L. Steffen, J. Fink, C. Eichler, M. Baur, R. Bianchetti, P. Leek, S. Filipp, M. da Silva, A. Blais, and A. Wallraff, “Antibunching of microwave-frequency photons observed in correlation measurements using linear detectors,” Nat. Phys. 7, 154–158 (2010).
[CrossRef]

Fink, J.

D. Bozyigit, C. Lang, L. Steffen, J. Fink, C. Eichler, M. Baur, R. Bianchetti, P. Leek, S. Filipp, M. da Silva, A. Blais, and A. Wallraff, “Antibunching of microwave-frequency photons observed in correlation measurements using linear detectors,” Nat. Phys. 7, 154–158 (2010).
[CrossRef]

Floreanini, R.

F. Benatti, R. Floreanini, and U. Marzolino, “Entanglement and squeezing with identical particles: ultracold atom quantum metrology,” J. Phys. B 44, 091001 (2011).
[CrossRef]

Forn-Díaz, P.

P. Forn-Díaz, J. Lisenfeld, D. Marcos, J. J. García-Ripoll, E. Solano, C. J. P. M. Harmans, and J. E. Mooij, “Observation of the Bloch–Siegert shift in a qubit-oscillator system in the ultrastrong coupling regime,” Phys. Rev. Lett. 105, 237001 (2010).
[CrossRef]

Frunzio, L.

J. Schreier, A. Houck, J. Koch, D. Schuster, B. Johnson, J. Chow, J. Gambetta, J. Majer, L. Frunzio, M. Devoret, S. M. Girvin, and R. J. Schoelkopf, “Suppressing charge noise decoherence in superconducting charge qubits,” Phys. Rev. B 77, 180502 (2008).
[CrossRef]

Gambetta, J.

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J. Casanova, G. Romero, I. Lizuain, J. J. García-Ripoll, and E. Solano, “Deep strong coupling regime of the Jaynes–Cummings model,” Phys. Rev. Lett. 105, 263603 (2010).
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L. Vernac, M. Pinard, and E. Giacobino, “Quantum state transfer from light beams to atomic ensembles,” Eur. Phys. J. D 17, 125–136 (2001).
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J. Schreier, A. Houck, J. Koch, D. Schuster, B. Johnson, J. Chow, J. Gambetta, J. Majer, L. Frunzio, M. Devoret, S. M. Girvin, and R. J. Schoelkopf, “Suppressing charge noise decoherence in superconducting charge qubits,” Phys. Rev. B 77, 180502 (2008).
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G. Tóth, C. Knapp, O. Gühne, and H. Briegel, “Spin squeezing and entanglement,” Phys. Rev. A 79, 042334 (2009).
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G. Tóth, C. Knapp, O. Gühne, and H. J. Briegel, “Optimal spin squeezing inequalities detect bound entanglement in spin models,” Phys. Rev. Lett. 99, 250405 (2007).
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X.-F. Zhou, Y.-S. Zhang, and G.-C. Guo, “Pair tunneling of bosonic atoms in an optical lattice,” Phys. Rev. A 80, 013605 (2009).
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J. Hald, J. L. Sørensen, C. Schori, and E. S. Polzik, “Spin squeezed atoms: a macroscopic entangled ensemble created by light,” Phys. Rev. Lett. 83, 1319–1322 (1999).
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P. Forn-Díaz, J. Lisenfeld, D. Marcos, J. J. García-Ripoll, E. Solano, C. J. P. M. Harmans, and J. E. Mooij, “Observation of the Bloch–Siegert shift in a qubit-oscillator system in the ultrastrong coupling regime,” Phys. Rev. Lett. 105, 237001 (2010).
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M. Hillery, “An introduction to the quantum theory of nonlinear optics,” Acta Phys. Slovaca 59, 1–80 (2009).
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M. Hofheinz, E. Weig, M. Ansmann, R. Bialczak, E. Lucero, M. Neeley, A. O’Connell, H. Wang, J. Martinis, and A. Cleland, “Generation of Fock states in a superconducting quantum circuit,” Nature 454, 310–314 (2008).
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J. Schreier, A. Houck, J. Koch, D. Schuster, B. Johnson, J. Chow, J. Gambetta, J. Majer, L. Frunzio, M. Devoret, S. M. Girvin, and R. J. Schoelkopf, “Suppressing charge noise decoherence in superconducting charge qubits,” Phys. Rev. B 77, 180502 (2008).
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G. Vitagliano, P. Hyllus, I. L. Egusquiza, and G. Tóth, “Spin squeezing inequalities for arbitrary spin,” Phys. Rev. Lett. 107, 240502 (2011).
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J. Schreier, A. Houck, J. Koch, D. Schuster, B. Johnson, J. Chow, J. Gambetta, J. Majer, L. Frunzio, M. Devoret, S. M. Girvin, and R. J. Schoelkopf, “Suppressing charge noise decoherence in superconducting charge qubits,” Phys. Rev. B 77, 180502 (2008).
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G. Tóth, C. Knapp, O. Gühne, and H. Briegel, “Spin squeezing and entanglement,” Phys. Rev. A 79, 042334 (2009).
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G. Tóth, C. Knapp, O. Gühne, and H. J. Briegel, “Optimal spin squeezing inequalities detect bound entanglement in spin models,” Phys. Rev. Lett. 99, 250405 (2007).
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J. Schreier, A. Houck, J. Koch, D. Schuster, B. Johnson, J. Chow, J. Gambetta, J. Majer, L. Frunzio, M. Devoret, S. M. Girvin, and R. J. Schoelkopf, “Suppressing charge noise decoherence in superconducting charge qubits,” Phys. Rev. B 77, 180502 (2008).
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J. Casanova, G. Romero, I. Lizuain, J. J. García-Ripoll, and E. Solano, “Deep strong coupling regime of the Jaynes–Cummings model,” Phys. Rev. Lett. 105, 263603 (2010).
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J. Ma, X. Wang, C. Sun, and F. Nori, “Quantum spin squeezing,” Phys. Rep. 509, 89–165(2011).
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J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi, “Quantum state transfer and entanglement distribution among distant nodes in a quantum network,” Phys. Rev. Lett. 78, 3221–3224 (1997).
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D. Petrosyan, G. Bensky, G. Kurizki, I. Mazets, J. Majer, and J. Schmiedmayer, “Reversible state transfer between superconducting qubits and atomic ensembles,” Phys. Rev. A 79, 040304 (2009).
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P. Forn-Díaz, J. Lisenfeld, D. Marcos, J. J. García-Ripoll, E. Solano, C. J. P. M. Harmans, and J. E. Mooij, “Observation of the Bloch–Siegert shift in a qubit-oscillator system in the ultrastrong coupling regime,” Phys. Rev. Lett. 105, 237001 (2010).
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F. Benatti, R. Floreanini, and U. Marzolino, “Entanglement and squeezing with identical particles: ultracold atom quantum metrology,” J. Phys. B 44, 091001 (2011).
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D. Petrosyan, G. Bensky, G. Kurizki, I. Mazets, J. Majer, and J. Schmiedmayer, “Reversible state transfer between superconducting qubits and atomic ensembles,” Phys. Rev. A 79, 040304 (2009).
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A. Kuzmich, K. Mølmer, and E. S. Polzik, “Spin squeezing in an ensemble of atoms illuminated with squeezed light,” Phys. Rev. Lett. 79, 4782–4785 (1997).
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P. Forn-Díaz, J. Lisenfeld, D. Marcos, J. J. García-Ripoll, E. Solano, C. J. P. M. Harmans, and J. E. Mooij, “Observation of the Bloch–Siegert shift in a qubit-oscillator system in the ultrastrong coupling regime,” Phys. Rev. Lett. 105, 237001 (2010).
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P. Ribeiro, J. Vidal, and R. Mosseri, “Exact spectrum of the Lipkin–Meshkov–Glick model in the thermodynamic limit and finite-size corrections,” Phys. Rev. E 78, 021106 (2008).
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M. Hofheinz, E. Weig, M. Ansmann, R. Bialczak, E. Lucero, M. Neeley, A. O’Connell, H. Wang, J. Martinis, and A. Cleland, “Generation of Fock states in a superconducting quantum circuit,” Nature 454, 310–314 (2008).
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J. Ma, X. Wang, C. Sun, and F. Nori, “Quantum spin squeezing,” Phys. Rep. 509, 89–165(2011).
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M. Hofheinz, E. Weig, M. Ansmann, R. Bialczak, E. Lucero, M. Neeley, A. O’Connell, H. Wang, J. Martinis, and A. Cleland, “Generation of Fock states in a superconducting quantum circuit,” Nature 454, 310–314 (2008).
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B. Öztop, M. Ö. Oktel, Ö. E. Müstecaplıoğlu, and L. You, “Quantum entanglement of spin-1 bosons with coupled ground states in optical lattices,” J. Phys. B 42, 145505 (2009).
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B. Öztop, M. Ö. Oktel, Ö. E. Müstecaplıoğlu, and L. You, “Quantum entanglement of spin-1 bosons with coupled ground states in optical lattices,” J. Phys. B 42, 145505 (2009).
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J. Vidal, G. Palacios, and C. Aslangul, “Entanglement dynamics in the Lipkin–Meshkov–Glick model,” Phys. Rev. A 70, 062304 (2004).
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D. Petrosyan, G. Bensky, G. Kurizki, I. Mazets, J. Majer, and J. Schmiedmayer, “Reversible state transfer between superconducting qubits and atomic ensembles,” Phys. Rev. A 79, 040304 (2009).
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L. Vernac, M. Pinard, and E. Giacobino, “Quantum state transfer from light beams to atomic ensembles,” Eur. Phys. J. D 17, 125–136 (2001).
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J. Hald, J. L. Sørensen, C. Schori, and E. S. Polzik, “Spin squeezed atoms: a macroscopic entangled ensemble created by light,” Phys. Rev. Lett. 83, 1319–1322 (1999).
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P. Ribeiro, J. Vidal, and R. Mosseri, “Exact spectrum of the Lipkin–Meshkov–Glick model in the thermodynamic limit and finite-size corrections,” Phys. Rev. E 78, 021106 (2008).
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P. Ribeiro, J. Vidal, and R. Mosseri, “Thermodynamical limit of the Lipkin–Meshkov–Glick model,” Phys. Rev. Lett. 99, 050402 (2007).
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J. Casanova, G. Romero, I. Lizuain, J. J. García-Ripoll, and E. Solano, “Deep strong coupling regime of the Jaynes–Cummings model,” Phys. Rev. Lett. 105, 263603 (2010).
[CrossRef]

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D. Petrosyan, G. Bensky, G. Kurizki, I. Mazets, J. Majer, and J. Schmiedmayer, “Reversible state transfer between superconducting qubits and atomic ensembles,” Phys. Rev. A 79, 040304 (2009).
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J. Schreier, A. Houck, J. Koch, D. Schuster, B. Johnson, J. Chow, J. Gambetta, J. Majer, L. Frunzio, M. Devoret, S. M. Girvin, and R. J. Schoelkopf, “Suppressing charge noise decoherence in superconducting charge qubits,” Phys. Rev. B 77, 180502 (2008).
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J. Hald, J. L. Sørensen, C. Schori, and E. S. Polzik, “Spin squeezed atoms: a macroscopic entangled ensemble created by light,” Phys. Rev. Lett. 83, 1319–1322 (1999).
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J. Schreier, A. Houck, J. Koch, D. Schuster, B. Johnson, J. Chow, J. Gambetta, J. Majer, L. Frunzio, M. Devoret, S. M. Girvin, and R. J. Schoelkopf, “Suppressing charge noise decoherence in superconducting charge qubits,” Phys. Rev. B 77, 180502 (2008).
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J. Schreier, A. Houck, J. Koch, D. Schuster, B. Johnson, J. Chow, J. Gambetta, J. Majer, L. Frunzio, M. Devoret, S. M. Girvin, and R. J. Schoelkopf, “Suppressing charge noise decoherence in superconducting charge qubits,” Phys. Rev. B 77, 180502 (2008).
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P. Forn-Díaz, J. Lisenfeld, D. Marcos, J. J. García-Ripoll, E. Solano, C. J. P. M. Harmans, and J. E. Mooij, “Observation of the Bloch–Siegert shift in a qubit-oscillator system in the ultrastrong coupling regime,” Phys. Rev. Lett. 105, 237001 (2010).
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J. Ma, X. Wang, C. Sun, and F. Nori, “Quantum spin squeezing,” Phys. Rep. 509, 89–165(2011).
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G. Vitagliano, P. Hyllus, I. L. Egusquiza, and G. Tóth, “Spin squeezing inequalities for arbitrary spin,” Phys. Rev. Lett. 107, 240502 (2011).
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G. Tóth, C. Knapp, O. Gühne, and H. Briegel, “Spin squeezing and entanglement,” Phys. Rev. A 79, 042334 (2009).
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P. Ribeiro, J. Vidal, and R. Mosseri, “Exact spectrum of the Lipkin–Meshkov–Glick model in the thermodynamic limit and finite-size corrections,” Phys. Rev. E 78, 021106 (2008).
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J. Vidal, G. Palacios, and C. Aslangul, “Entanglement dynamics in the Lipkin–Meshkov–Glick model,” Phys. Rev. A 70, 062304 (2004).
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G. Vitagliano, P. Hyllus, I. L. Egusquiza, and G. Tóth, “Spin squeezing inequalities for arbitrary spin,” Phys. Rev. Lett. 107, 240502 (2011).
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D. Bozyigit, C. Lang, L. Steffen, J. Fink, C. Eichler, M. Baur, R. Bianchetti, P. Leek, S. Filipp, M. da Silva, A. Blais, and A. Wallraff, “Antibunching of microwave-frequency photons observed in correlation measurements using linear detectors,” Nat. Phys. 7, 154–158 (2010).
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M. Hofheinz, E. Weig, M. Ansmann, R. Bialczak, E. Lucero, M. Neeley, A. O’Connell, H. Wang, J. Martinis, and A. Cleland, “Generation of Fock states in a superconducting quantum circuit,” Nature 454, 310–314 (2008).
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J. Ma, X. Wang, C. Sun, and F. Nori, “Quantum spin squeezing,” Phys. Rep. 509, 89–165(2011).
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M. Hofheinz, E. Weig, M. Ansmann, R. Bialczak, E. Lucero, M. Neeley, A. O’Connell, H. Wang, J. Martinis, and A. Cleland, “Generation of Fock states in a superconducting quantum circuit,” Nature 454, 310–314 (2008).
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D. Bozyigit, C. Lang, L. Steffen, J. Fink, C. Eichler, M. Baur, R. Bianchetti, P. Leek, S. Filipp, M. da Silva, A. Blais, and A. Wallraff, “Antibunching of microwave-frequency photons observed in correlation measurements using linear detectors,” Nat. Phys. 7, 154–158 (2010).
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Figures (4)

Fig. 1.
Fig. 1.

Variation of spin squeezing in (left) atomic and (right) photonic subsystems with respect to time and hopping constant. ζ is scaled by λ, and ζ1 corresponds to the local-interaction dominant regimes, while ζ>1 corresponds to the hopping dominant regimes. Spin squeezing is characterized positive values in both figures. At t=0, Ineqa=1 and Ineqp=1. All the parameters plotted are dimensionless as explained in the text.

Fig. 2.
Fig. 2.

Variations of the spin squeezing in (left) atomic and (right) photonic subsystems with respect to time and hopping constant. Both inequalities are violated; therefore, we have both spin squeezed and entangled systems. Spin squeezing is characterized by negative values in both figures. At t=0, both inequalities are equal to zero. All the parameters plotted are dimensionless as explained in the text.

Fig. 3.
Fig. 3.

Variation of the variances of the quadratures for (left) initially entangled state and (right) initially nonentangled state with respect to time and hopping constant. Quadratures are squeezed for values 0.25. At t=0, (ΔX1)2=(ΔX2)2=0.75 for initially entangled state and (ΔX1)2=(ΔX2)2=0.6875 for initially nonentangled state. All the parameters plotted are dimensionless as explained in the text.

Fig. 4.
Fig. 4.

Two identical cavities, with cavity frequencies ω, coupled via two-photon exchange, which is governed by a nonlinear Kerr medium. Each cavity contains a three-level atom in cascade formation.

Equations (50)

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ξ=ΔJJ/2,
ξe2=N(ΔJn1)2Jn22+Jn321,
Jx2+Jy2+Jz2N(N+2)4,
(ΔJx)2+(ΔJy)2+(ΔJz)2N2,
Jk2+Jl2N2(N1)(ΔJm)2,
(N1)[(ΔJk)2+(ΔJl)2]Jm2+N(N2)4,
H=H(1)H0(1)+H(2)H0(2)+ζ(a12a22+a22a12),
H(i)=ωaiai+ω(σee(i)σgg(i))+μσee(i)+ησgg(i)+λ(σeg(i)ai2+σge(i)ai2),
H0(i)=ω(aiai+σee(i))σgg(i)+(Eg+Ee)/2,
Sx=12(σeg+σge),Sy=i2(σegσge),Sz=12(σeeσgg).
Sx=Sx(1)1(2)+1(1)Sx(2),Sy=Sy(1)1(2)+1(1)Sy(2),Sz=Sz(1)1(2)+1(1)Sz(2),
Lx12(a1a2+a2a1),Lyi2(a1a2a2a1),Lz12(a1a1a2a2).
Hp=ζ(L+2+L2)=2ζ(Lx2Ly2).
|ψ(t)=A(t)|g,21|g,02+B(t)|g,01|g,22+C(t)|g,01|e,02+D(t)|e,01|g,02.
|ψ(t)=(A1eiω1t+A2eiω2t)|ϕ1+(A3eiω3t+A4eiω4t)|ϕ2+(α1A1eiω1t+α2A2eiω2t)|ϕ3(α3A3eiω3t+α4A4eiω4t)|ϕ4,
|ϕ1=12(|g,21|g,02+|g,01|g,22),
|ϕ2=12(|g,21|g,02|g,01|g,22),
|ϕ3=12(|e,01|g,02+|g,01|e,02),
|ϕ4=12(|e,01|g,02|g,01|e,02).
|ψ(0)=12(|g,21|g,02+|g,01|g,22).
|ψ(t)=A1(eiω1t+α12eiω2t)|ϕ1+α1A1(eiω1teiω2t)|ϕ3,
ρap12(t)=|A|2|ϕ1ϕ1|+AB*|ϕ1ϕ3|+BA*|ϕ3ϕ1|+|B|2|ϕ3ϕ3|,
|A|2=A12[1+2α12cos(δ12t)+α14],
AB*=α1A12[1α12+α12eiδ12teiδ12t],
|B|2=2α12A12[1cos(δ12t)],
ρa12(t)=Trpρap12(t)=|A|2|g11g||g22g|+|B|22(|e11e||g22g|+|e11g||g22e|+|g11e||e22g|+|g11g||e22e|),
ρp12(t)=Traρap12(t)=|A|22(|2,02,0|+|2,00,2|+|0,22,0|+|0,20,2|)+|B|2|0,00,0|.
Ineqa45|A|20,
Ineqp|A|20,
|ψ(0)=|g,21|g,02,
|ψ(t)=A1(eiω1t+α12eiω2t)|ϕ1+α1A1(eiω1teiω2t)|ϕ3+A3(eiω3t+α32eiω4t)|ϕ2+α3A3(eiω3teiω4t)|ϕ4.
ρap12(t)=|A|2|ϕ1ϕ1|+AB*|ϕ1ϕ3|+AC*|ϕ1ϕ2|+AD*|ϕ1ϕ4|+BA*|ϕ3ϕ1|+|B|2|ϕ3ϕ3|+BC*|ϕ3ϕ2|+BD*|ϕ3ϕ4|+CA*|ϕ2ϕ1|+CB*|ϕ2ϕ3|+|C|2|ϕ2ϕ2|+CD*|ϕ2ϕ4|+DA*|ϕ4ϕ1|+DB*|ϕ4ϕ3|+DC*|ϕ4ϕ2|+|D|2|ϕ4ϕ4|,
AC*=A1A3[ei(ω3ω1)t+α32ei(ω4ω1)t+α12ei(ω3ω2)t+α12α32ei(ω4ω2)t],
AD*=α3A1A3[ei(ω3ω1)tei(ω4ω1)t+α12(ei(ω3ω2)tei(ω4ω2)t)],
BC*=α1A1A3[ei(ω3ω1)tei(ω3ω2)t+α32(ei(ω4ω1)tei(ω4ω3)t)],
BD*=α1α3A1A3[ei(ω3ω1)t+ei(ω4ω2)tei(ω4ω1)tei(ω3ω2)t],
CD*=α3A32[1ei(ω4ω3)t+α32(ei(ω3ω4)t1)],
|C|2=A32[1+2α32cos[(ω3ω4)t]+α34],
|D|2=2α32A32[1cos[(ω3ω4)t]],
ρa12(t)=(|A|2+|C|2)|g11g||g22g|+12[(|B|2+|D|2+BD*+DB*)|e11e||g22g|+(|B|2|D|2BD*+DB*)|e11g||g22e|+(|B|2|D|2+BD*DB*)|g11e||e22g|+(|B|2+|D|2BD*DB*)|g11g||e22e|],
ρp12(t)=12[(|A|2+AC*+CA*)|2,02,0|+(|A|2AC*+CA*)|2,00,2|+(|A|2+AC*CA*)|0,22,0|+(|A|2AC*CA*)|0,20,2|]+(2|B|2+|D|2)|0,00,0|.
Ineqa3|B|2|D|22(|B|2+|D|2)20
Ineqp2|A|210,
X1=12(a+a),
X2=i2(aa).
(ΔX1)2=(ΔX2)2=14+12|A|2,
(ΔX1)2=(ΔX2)2=78|A|2+14(AC*+CA*+|D|2)+12|B|2,
λ=g12g23{[Δ2+42(g˜122+g˜232)]1/2Δ}2(g˜122+g˜232),
μ=Δg˜1222(g˜122+g˜232)+(g˜122+2g˜232)[Δ2+42(g˜122+g˜232)]1/22(g˜122+g˜232),
η=Δg˜2322(g˜122+g˜232)+(2g˜122+g˜232)[Δ2+42(g˜122+g˜232)]1/22(g˜122+g˜232),

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