Abstract

We extend the idea of quantum phase transitions of light in the photonic Bose-Hubbard model with interactions to two atomic species by a self-consistent mean field theory. The excitation of two-level atoms interacting with a coherent photon field is analyzed with a finite temperature dependence of the order parameters. Four ground states of the system are found, including an isolated Mott-insulator phase and three different superfluid phases. Like two weakly coupled superconductors, our proposed dual-species lattice system shows a photonic analogue of Josephson effect. i.e., the crossovers between two superfluid states. The dynamics of the proposed two species model provides a promising quantum simulator for possible quantum information processes.

© 2010 Optical Society of America

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  1. S. Sachdev, Quantum Phase Transitions, (Cambridge University Press, 1999).
  2. M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, “Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms,” Nature 415, 39–44 (2002).
    [CrossRef] [PubMed]
  3. D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Cold Bosonic Atoms in Optical Lattices,” Phys. Rev. Lett. 81, 3108–3111 (1998).
    [CrossRef]
  4. M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S. Fisher, “Boson localization and the superfluid-insulator transition,” Phys. Rev. B 40, 546–570 (1989).
    [CrossRef]
  5. A. D. Greentree, C. Tahan, J. H. Cole, and L. C. L. Hollenberg, “Quantum phase transitions of light,” Nat. Phys. 2, 856–862 (2006).
    [CrossRef]
  6. M. J. Hartmann, F. G. S. L. Brandao, and M. B. Plenio, “Strongly interacting polaritons in coupled arrays of cavities,” Nat. Phys. 2, 849–855 (2006).
    [CrossRef]
  7. D. G. Angelakis, M. F. Santos, and S. Bose, “Photon-blockade-induced Mott transitions and XY spin models in coupled cavity arrays,” Phys. Rev. A 76, 031805 (2007).
    [CrossRef]
  8. M. J. Hartmann, F. G. S. L. Brandao, and M. B. Plenio, “Effective Spin Systems in Coupled Microcavities,” Phys. Rev. Lett. 99, 160501 (2007).
    [CrossRef] [PubMed]
  9. M. J. Hartmann, F. G. S. L. Brandao, and M. B. Plenio, “A polaritonic two-component Bose-Hubbard model,” N. J. Phys. 10, 033011 (2008).
    [CrossRef]
  10. D. Rossini, and R. Fazio, “Mott-Insulating and Glassy Phases of Polaritons in 1D Arrays of Coupled Cavities,” Phys. Rev. Lett. 99, 186401 (2007).
    [CrossRef] [PubMed]
  11. S.-C. Lei, and R.-K. Lee, “Quantum phase transitions of light in the Dicke-Bose-Hubbard model,” Phys. Rev. A 77, 033827 (2008).
    [CrossRef]
  12. T. Giamarchi, C. Rügg, and O. Tchernyshyov, “Bose-Einstein condensation in magnetic insulators,” Nat. Phys. 4, 198–204 (2008).
    [CrossRef]
  13. J. Quach, M. Makin, C.-H. Su, A. D. Greentree, and L. C. L. Hollenberg, “Band structure, phase transitions, and semiconductor analogs in one-dimensional solid light systems,” Phys. Rev. A 80, 063838 (2009).
    [CrossRef]
  14. P. Wurtz, T. Langen, T. Gericke, A. Koglbauer, and H. Ott, “Experimental Demonstration of Single-Site Addressability in a Two-Dimensional Optical Lattice,” Phys. Rev. Lett. 103, 080404 (2009).
    [CrossRef] [PubMed]
  15. E. Altman, W. Hofstetter, E. Demler, and M. D. Lukin, “Phase diagram of two-component bosons on an optical lattice,” N. J. Phys. 5, 11301–11319 (2003).
    [CrossRef]
  16. T. Mishra, B. K. Sahoo, and R. V. Pai, “Phase-separated charge-density-wave phase in the two-species extended Bose-Hubbard model,” Phys. Rev. A 78, 013632 (2008).
    [CrossRef]
  17. E. T. Jaynes, and F. W. Cummings, “Comparison of quantum and semiclassical radiation theories with application to the beam maser,” Proc. IEEE 51, 89–109 (1963).
    [CrossRef]
  18. B. D. Josephson, “The discovery of tunneling supercurrents,” Rev. Mod. Phys. 46, 251–254 (1974).
    [CrossRef]
  19. D. Gerace, H. E. Tureci, A. Imamoglu, V. Giovannetti, and R. Fazio, “The quantum-optical Josephson interferometer,” Nat. Phys. 5, 281–284 (2009).
    [CrossRef]

2009 (3)

J. Quach, M. Makin, C.-H. Su, A. D. Greentree, and L. C. L. Hollenberg, “Band structure, phase transitions, and semiconductor analogs in one-dimensional solid light systems,” Phys. Rev. A 80, 063838 (2009).
[CrossRef]

P. Wurtz, T. Langen, T. Gericke, A. Koglbauer, and H. Ott, “Experimental Demonstration of Single-Site Addressability in a Two-Dimensional Optical Lattice,” Phys. Rev. Lett. 103, 080404 (2009).
[CrossRef] [PubMed]

D. Gerace, H. E. Tureci, A. Imamoglu, V. Giovannetti, and R. Fazio, “The quantum-optical Josephson interferometer,” Nat. Phys. 5, 281–284 (2009).
[CrossRef]

2008 (4)

T. Mishra, B. K. Sahoo, and R. V. Pai, “Phase-separated charge-density-wave phase in the two-species extended Bose-Hubbard model,” Phys. Rev. A 78, 013632 (2008).
[CrossRef]

S.-C. Lei, and R.-K. Lee, “Quantum phase transitions of light in the Dicke-Bose-Hubbard model,” Phys. Rev. A 77, 033827 (2008).
[CrossRef]

T. Giamarchi, C. Rügg, and O. Tchernyshyov, “Bose-Einstein condensation in magnetic insulators,” Nat. Phys. 4, 198–204 (2008).
[CrossRef]

M. J. Hartmann, F. G. S. L. Brandao, and M. B. Plenio, “A polaritonic two-component Bose-Hubbard model,” N. J. Phys. 10, 033011 (2008).
[CrossRef]

2007 (3)

D. Rossini, and R. Fazio, “Mott-Insulating and Glassy Phases of Polaritons in 1D Arrays of Coupled Cavities,” Phys. Rev. Lett. 99, 186401 (2007).
[CrossRef] [PubMed]

D. G. Angelakis, M. F. Santos, and S. Bose, “Photon-blockade-induced Mott transitions and XY spin models in coupled cavity arrays,” Phys. Rev. A 76, 031805 (2007).
[CrossRef]

M. J. Hartmann, F. G. S. L. Brandao, and M. B. Plenio, “Effective Spin Systems in Coupled Microcavities,” Phys. Rev. Lett. 99, 160501 (2007).
[CrossRef] [PubMed]

2006 (2)

A. D. Greentree, C. Tahan, J. H. Cole, and L. C. L. Hollenberg, “Quantum phase transitions of light,” Nat. Phys. 2, 856–862 (2006).
[CrossRef]

M. J. Hartmann, F. G. S. L. Brandao, and M. B. Plenio, “Strongly interacting polaritons in coupled arrays of cavities,” Nat. Phys. 2, 849–855 (2006).
[CrossRef]

2003 (1)

E. Altman, W. Hofstetter, E. Demler, and M. D. Lukin, “Phase diagram of two-component bosons on an optical lattice,” N. J. Phys. 5, 11301–11319 (2003).
[CrossRef]

2002 (1)

M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, “Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms,” Nature 415, 39–44 (2002).
[CrossRef] [PubMed]

1998 (1)

D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Cold Bosonic Atoms in Optical Lattices,” Phys. Rev. Lett. 81, 3108–3111 (1998).
[CrossRef]

1989 (1)

M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S. Fisher, “Boson localization and the superfluid-insulator transition,” Phys. Rev. B 40, 546–570 (1989).
[CrossRef]

1974 (1)

B. D. Josephson, “The discovery of tunneling supercurrents,” Rev. Mod. Phys. 46, 251–254 (1974).
[CrossRef]

1963 (1)

E. T. Jaynes, and F. W. Cummings, “Comparison of quantum and semiclassical radiation theories with application to the beam maser,” Proc. IEEE 51, 89–109 (1963).
[CrossRef]

Altman, E.

E. Altman, W. Hofstetter, E. Demler, and M. D. Lukin, “Phase diagram of two-component bosons on an optical lattice,” N. J. Phys. 5, 11301–11319 (2003).
[CrossRef]

Angelakis, D. G.

D. G. Angelakis, M. F. Santos, and S. Bose, “Photon-blockade-induced Mott transitions and XY spin models in coupled cavity arrays,” Phys. Rev. A 76, 031805 (2007).
[CrossRef]

Bloch, I.

M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, “Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms,” Nature 415, 39–44 (2002).
[CrossRef] [PubMed]

Bose, S.

D. G. Angelakis, M. F. Santos, and S. Bose, “Photon-blockade-induced Mott transitions and XY spin models in coupled cavity arrays,” Phys. Rev. A 76, 031805 (2007).
[CrossRef]

Brandao, F. G. S. L.

M. J. Hartmann, F. G. S. L. Brandao, and M. B. Plenio, “A polaritonic two-component Bose-Hubbard model,” N. J. Phys. 10, 033011 (2008).
[CrossRef]

M. J. Hartmann, F. G. S. L. Brandao, and M. B. Plenio, “Effective Spin Systems in Coupled Microcavities,” Phys. Rev. Lett. 99, 160501 (2007).
[CrossRef] [PubMed]

M. J. Hartmann, F. G. S. L. Brandao, and M. B. Plenio, “Strongly interacting polaritons in coupled arrays of cavities,” Nat. Phys. 2, 849–855 (2006).
[CrossRef]

Bruder, C.

D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Cold Bosonic Atoms in Optical Lattices,” Phys. Rev. Lett. 81, 3108–3111 (1998).
[CrossRef]

Cirac, J. I.

D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Cold Bosonic Atoms in Optical Lattices,” Phys. Rev. Lett. 81, 3108–3111 (1998).
[CrossRef]

Cole, J. H.

A. D. Greentree, C. Tahan, J. H. Cole, and L. C. L. Hollenberg, “Quantum phase transitions of light,” Nat. Phys. 2, 856–862 (2006).
[CrossRef]

Cummings, F. W.

E. T. Jaynes, and F. W. Cummings, “Comparison of quantum and semiclassical radiation theories with application to the beam maser,” Proc. IEEE 51, 89–109 (1963).
[CrossRef]

Demler, E.

E. Altman, W. Hofstetter, E. Demler, and M. D. Lukin, “Phase diagram of two-component bosons on an optical lattice,” N. J. Phys. 5, 11301–11319 (2003).
[CrossRef]

Esslinger, T.

M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, “Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms,” Nature 415, 39–44 (2002).
[CrossRef] [PubMed]

Fazio, R.

D. Gerace, H. E. Tureci, A. Imamoglu, V. Giovannetti, and R. Fazio, “The quantum-optical Josephson interferometer,” Nat. Phys. 5, 281–284 (2009).
[CrossRef]

D. Rossini, and R. Fazio, “Mott-Insulating and Glassy Phases of Polaritons in 1D Arrays of Coupled Cavities,” Phys. Rev. Lett. 99, 186401 (2007).
[CrossRef] [PubMed]

Fisher, D. S.

M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S. Fisher, “Boson localization and the superfluid-insulator transition,” Phys. Rev. B 40, 546–570 (1989).
[CrossRef]

Fisher, M. P. A.

M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S. Fisher, “Boson localization and the superfluid-insulator transition,” Phys. Rev. B 40, 546–570 (1989).
[CrossRef]

Gardiner, C. W.

D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Cold Bosonic Atoms in Optical Lattices,” Phys. Rev. Lett. 81, 3108–3111 (1998).
[CrossRef]

Gerace, D.

D. Gerace, H. E. Tureci, A. Imamoglu, V. Giovannetti, and R. Fazio, “The quantum-optical Josephson interferometer,” Nat. Phys. 5, 281–284 (2009).
[CrossRef]

Gericke, T.

P. Wurtz, T. Langen, T. Gericke, A. Koglbauer, and H. Ott, “Experimental Demonstration of Single-Site Addressability in a Two-Dimensional Optical Lattice,” Phys. Rev. Lett. 103, 080404 (2009).
[CrossRef] [PubMed]

Giamarchi, T.

T. Giamarchi, C. Rügg, and O. Tchernyshyov, “Bose-Einstein condensation in magnetic insulators,” Nat. Phys. 4, 198–204 (2008).
[CrossRef]

Giovannetti, V.

D. Gerace, H. E. Tureci, A. Imamoglu, V. Giovannetti, and R. Fazio, “The quantum-optical Josephson interferometer,” Nat. Phys. 5, 281–284 (2009).
[CrossRef]

Greentree, A. D.

J. Quach, M. Makin, C.-H. Su, A. D. Greentree, and L. C. L. Hollenberg, “Band structure, phase transitions, and semiconductor analogs in one-dimensional solid light systems,” Phys. Rev. A 80, 063838 (2009).
[CrossRef]

A. D. Greentree, C. Tahan, J. H. Cole, and L. C. L. Hollenberg, “Quantum phase transitions of light,” Nat. Phys. 2, 856–862 (2006).
[CrossRef]

Greiner, M.

M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, “Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms,” Nature 415, 39–44 (2002).
[CrossRef] [PubMed]

Grinstein, G.

M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S. Fisher, “Boson localization and the superfluid-insulator transition,” Phys. Rev. B 40, 546–570 (1989).
[CrossRef]

Hänsch, T. W.

M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, “Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms,” Nature 415, 39–44 (2002).
[CrossRef] [PubMed]

Hartmann, M. J.

M. J. Hartmann, F. G. S. L. Brandao, and M. B. Plenio, “A polaritonic two-component Bose-Hubbard model,” N. J. Phys. 10, 033011 (2008).
[CrossRef]

M. J. Hartmann, F. G. S. L. Brandao, and M. B. Plenio, “Effective Spin Systems in Coupled Microcavities,” Phys. Rev. Lett. 99, 160501 (2007).
[CrossRef] [PubMed]

M. J. Hartmann, F. G. S. L. Brandao, and M. B. Plenio, “Strongly interacting polaritons in coupled arrays of cavities,” Nat. Phys. 2, 849–855 (2006).
[CrossRef]

Hofstetter, W.

E. Altman, W. Hofstetter, E. Demler, and M. D. Lukin, “Phase diagram of two-component bosons on an optical lattice,” N. J. Phys. 5, 11301–11319 (2003).
[CrossRef]

Hollenberg, L. C. L.

J. Quach, M. Makin, C.-H. Su, A. D. Greentree, and L. C. L. Hollenberg, “Band structure, phase transitions, and semiconductor analogs in one-dimensional solid light systems,” Phys. Rev. A 80, 063838 (2009).
[CrossRef]

A. D. Greentree, C. Tahan, J. H. Cole, and L. C. L. Hollenberg, “Quantum phase transitions of light,” Nat. Phys. 2, 856–862 (2006).
[CrossRef]

Imamoglu, A.

D. Gerace, H. E. Tureci, A. Imamoglu, V. Giovannetti, and R. Fazio, “The quantum-optical Josephson interferometer,” Nat. Phys. 5, 281–284 (2009).
[CrossRef]

Jaksch, D.

D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Cold Bosonic Atoms in Optical Lattices,” Phys. Rev. Lett. 81, 3108–3111 (1998).
[CrossRef]

Jaynes, E. T.

E. T. Jaynes, and F. W. Cummings, “Comparison of quantum and semiclassical radiation theories with application to the beam maser,” Proc. IEEE 51, 89–109 (1963).
[CrossRef]

Josephson, B. D.

B. D. Josephson, “The discovery of tunneling supercurrents,” Rev. Mod. Phys. 46, 251–254 (1974).
[CrossRef]

Koglbauer, A.

P. Wurtz, T. Langen, T. Gericke, A. Koglbauer, and H. Ott, “Experimental Demonstration of Single-Site Addressability in a Two-Dimensional Optical Lattice,” Phys. Rev. Lett. 103, 080404 (2009).
[CrossRef] [PubMed]

Langen, T.

P. Wurtz, T. Langen, T. Gericke, A. Koglbauer, and H. Ott, “Experimental Demonstration of Single-Site Addressability in a Two-Dimensional Optical Lattice,” Phys. Rev. Lett. 103, 080404 (2009).
[CrossRef] [PubMed]

Lee, R.-K.

S.-C. Lei, and R.-K. Lee, “Quantum phase transitions of light in the Dicke-Bose-Hubbard model,” Phys. Rev. A 77, 033827 (2008).
[CrossRef]

Lei, S.-C.

S.-C. Lei, and R.-K. Lee, “Quantum phase transitions of light in the Dicke-Bose-Hubbard model,” Phys. Rev. A 77, 033827 (2008).
[CrossRef]

Lukin, M. D.

E. Altman, W. Hofstetter, E. Demler, and M. D. Lukin, “Phase diagram of two-component bosons on an optical lattice,” N. J. Phys. 5, 11301–11319 (2003).
[CrossRef]

Makin, M.

J. Quach, M. Makin, C.-H. Su, A. D. Greentree, and L. C. L. Hollenberg, “Band structure, phase transitions, and semiconductor analogs in one-dimensional solid light systems,” Phys. Rev. A 80, 063838 (2009).
[CrossRef]

Mandel, O.

M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, “Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms,” Nature 415, 39–44 (2002).
[CrossRef] [PubMed]

Mishra, T.

T. Mishra, B. K. Sahoo, and R. V. Pai, “Phase-separated charge-density-wave phase in the two-species extended Bose-Hubbard model,” Phys. Rev. A 78, 013632 (2008).
[CrossRef]

Ott, H.

P. Wurtz, T. Langen, T. Gericke, A. Koglbauer, and H. Ott, “Experimental Demonstration of Single-Site Addressability in a Two-Dimensional Optical Lattice,” Phys. Rev. Lett. 103, 080404 (2009).
[CrossRef] [PubMed]

Pai, R. V.

T. Mishra, B. K. Sahoo, and R. V. Pai, “Phase-separated charge-density-wave phase in the two-species extended Bose-Hubbard model,” Phys. Rev. A 78, 013632 (2008).
[CrossRef]

Plenio, M. B.

M. J. Hartmann, F. G. S. L. Brandao, and M. B. Plenio, “A polaritonic two-component Bose-Hubbard model,” N. J. Phys. 10, 033011 (2008).
[CrossRef]

M. J. Hartmann, F. G. S. L. Brandao, and M. B. Plenio, “Effective Spin Systems in Coupled Microcavities,” Phys. Rev. Lett. 99, 160501 (2007).
[CrossRef] [PubMed]

M. J. Hartmann, F. G. S. L. Brandao, and M. B. Plenio, “Strongly interacting polaritons in coupled arrays of cavities,” Nat. Phys. 2, 849–855 (2006).
[CrossRef]

Quach, J.

J. Quach, M. Makin, C.-H. Su, A. D. Greentree, and L. C. L. Hollenberg, “Band structure, phase transitions, and semiconductor analogs in one-dimensional solid light systems,” Phys. Rev. A 80, 063838 (2009).
[CrossRef]

Rossini, D.

D. Rossini, and R. Fazio, “Mott-Insulating and Glassy Phases of Polaritons in 1D Arrays of Coupled Cavities,” Phys. Rev. Lett. 99, 186401 (2007).
[CrossRef] [PubMed]

Rügg, C.

T. Giamarchi, C. Rügg, and O. Tchernyshyov, “Bose-Einstein condensation in magnetic insulators,” Nat. Phys. 4, 198–204 (2008).
[CrossRef]

Sahoo, B. K.

T. Mishra, B. K. Sahoo, and R. V. Pai, “Phase-separated charge-density-wave phase in the two-species extended Bose-Hubbard model,” Phys. Rev. A 78, 013632 (2008).
[CrossRef]

Santos, M. F.

D. G. Angelakis, M. F. Santos, and S. Bose, “Photon-blockade-induced Mott transitions and XY spin models in coupled cavity arrays,” Phys. Rev. A 76, 031805 (2007).
[CrossRef]

Su, C.-H.

J. Quach, M. Makin, C.-H. Su, A. D. Greentree, and L. C. L. Hollenberg, “Band structure, phase transitions, and semiconductor analogs in one-dimensional solid light systems,” Phys. Rev. A 80, 063838 (2009).
[CrossRef]

Tahan, C.

A. D. Greentree, C. Tahan, J. H. Cole, and L. C. L. Hollenberg, “Quantum phase transitions of light,” Nat. Phys. 2, 856–862 (2006).
[CrossRef]

Tchernyshyov, O.

T. Giamarchi, C. Rügg, and O. Tchernyshyov, “Bose-Einstein condensation in magnetic insulators,” Nat. Phys. 4, 198–204 (2008).
[CrossRef]

Tureci, H. E.

D. Gerace, H. E. Tureci, A. Imamoglu, V. Giovannetti, and R. Fazio, “The quantum-optical Josephson interferometer,” Nat. Phys. 5, 281–284 (2009).
[CrossRef]

Weichman, P. B.

M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S. Fisher, “Boson localization and the superfluid-insulator transition,” Phys. Rev. B 40, 546–570 (1989).
[CrossRef]

Wurtz, P.

P. Wurtz, T. Langen, T. Gericke, A. Koglbauer, and H. Ott, “Experimental Demonstration of Single-Site Addressability in a Two-Dimensional Optical Lattice,” Phys. Rev. Lett. 103, 080404 (2009).
[CrossRef] [PubMed]

Zoller, P.

D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Cold Bosonic Atoms in Optical Lattices,” Phys. Rev. Lett. 81, 3108–3111 (1998).
[CrossRef]

N. J. Phys. (2)

M. J. Hartmann, F. G. S. L. Brandao, and M. B. Plenio, “A polaritonic two-component Bose-Hubbard model,” N. J. Phys. 10, 033011 (2008).
[CrossRef]

E. Altman, W. Hofstetter, E. Demler, and M. D. Lukin, “Phase diagram of two-component bosons on an optical lattice,” N. J. Phys. 5, 11301–11319 (2003).
[CrossRef]

Nat. Phys. (4)

T. Giamarchi, C. Rügg, and O. Tchernyshyov, “Bose-Einstein condensation in magnetic insulators,” Nat. Phys. 4, 198–204 (2008).
[CrossRef]

A. D. Greentree, C. Tahan, J. H. Cole, and L. C. L. Hollenberg, “Quantum phase transitions of light,” Nat. Phys. 2, 856–862 (2006).
[CrossRef]

M. J. Hartmann, F. G. S. L. Brandao, and M. B. Plenio, “Strongly interacting polaritons in coupled arrays of cavities,” Nat. Phys. 2, 849–855 (2006).
[CrossRef]

D. Gerace, H. E. Tureci, A. Imamoglu, V. Giovannetti, and R. Fazio, “The quantum-optical Josephson interferometer,” Nat. Phys. 5, 281–284 (2009).
[CrossRef]

Nature (1)

M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, “Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms,” Nature 415, 39–44 (2002).
[CrossRef] [PubMed]

Phys. Rev. A (4)

D. G. Angelakis, M. F. Santos, and S. Bose, “Photon-blockade-induced Mott transitions and XY spin models in coupled cavity arrays,” Phys. Rev. A 76, 031805 (2007).
[CrossRef]

J. Quach, M. Makin, C.-H. Su, A. D. Greentree, and L. C. L. Hollenberg, “Band structure, phase transitions, and semiconductor analogs in one-dimensional solid light systems,” Phys. Rev. A 80, 063838 (2009).
[CrossRef]

T. Mishra, B. K. Sahoo, and R. V. Pai, “Phase-separated charge-density-wave phase in the two-species extended Bose-Hubbard model,” Phys. Rev. A 78, 013632 (2008).
[CrossRef]

S.-C. Lei, and R.-K. Lee, “Quantum phase transitions of light in the Dicke-Bose-Hubbard model,” Phys. Rev. A 77, 033827 (2008).
[CrossRef]

Phys. Rev. B (1)

M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S. Fisher, “Boson localization and the superfluid-insulator transition,” Phys. Rev. B 40, 546–570 (1989).
[CrossRef]

Phys. Rev. Lett. (4)

D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Cold Bosonic Atoms in Optical Lattices,” Phys. Rev. Lett. 81, 3108–3111 (1998).
[CrossRef]

M. J. Hartmann, F. G. S. L. Brandao, and M. B. Plenio, “Effective Spin Systems in Coupled Microcavities,” Phys. Rev. Lett. 99, 160501 (2007).
[CrossRef] [PubMed]

P. Wurtz, T. Langen, T. Gericke, A. Koglbauer, and H. Ott, “Experimental Demonstration of Single-Site Addressability in a Two-Dimensional Optical Lattice,” Phys. Rev. Lett. 103, 080404 (2009).
[CrossRef] [PubMed]

D. Rossini, and R. Fazio, “Mott-Insulating and Glassy Phases of Polaritons in 1D Arrays of Coupled Cavities,” Phys. Rev. Lett. 99, 186401 (2007).
[CrossRef] [PubMed]

Proc. IEEE (1)

E. T. Jaynes, and F. W. Cummings, “Comparison of quantum and semiclassical radiation theories with application to the beam maser,” Proc. IEEE 51, 89–109 (1963).
[CrossRef]

Rev. Mod. Phys. (1)

B. D. Josephson, “The discovery of tunneling supercurrents,” Rev. Mod. Phys. 46, 251–254 (1974).
[CrossRef]

Other (1)

S. Sachdev, Quantum Phase Transitions, (Cambridge University Press, 1999).

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Figures (4)

Fig. 1.
Fig. 1.

Illustration schematic for the proposed system. An array of high-Q electromagnetic cavities is formed in the configuration of the square lattice, and each cavity contains a single two-level atom of the type A or B, which is spaced at intervals. In this configuration, the center cavity has four nearest inter-species neighbors (different atomic type) and four next-nearest intra-species neighbors (the same atomic type). εA and ε B are the transition energies for the atomic species A and B, respectively. The incident optical field interacting with the bipartite lattice is also shown in the red color.

Fig. 2.
Fig. 2.

Phase diagram on the parameter plane (gA and gB ) at zero temperature, i.e., for different atom-photon coupling strengths. Four different phases are indicated in the plot without (κ = 0 in the solid lines) and with (κ = 0.4 in the circle markers) inter-atomic species hopping effects, respectively. Other parameters used in the simulations are the same as κ′ = 0.2, ω = 2.7, μ = 0.2, εA = 0.27, and εB = 0.25.

Fig. 3.
Fig. 3.

Superfluid order parameters ψA and ψB versus the temperature T with different values of (a) gA = 0.1, gB = 2.0; and (b) gA = 2.0, gB = 2.0. Other parameters used are the same as those in Fig. 2. Here the hopping constants are κ = 0.4 and κ′ = 0.2.

Fig. 4.
Fig. 4.

Superfluid order parameters ψA and ψB versus the temperature T. All the parameters used are the same as those in Fig. 3, but with different hopping constants, κ = 0.35 and κ′ = 0.175.

Equations (52)

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H i two site = ω a ̂ 2 j a ̂ 2 j + ε A , 2 j σ ̂ A , 2 j z + g A , 2 j ( a ̂ 2 j σ ̂ A , 2 j + a ̂ 2 j σ ̂ A , 2 j ) ,
+ ω a ̂ 2 j + 1 a ̂ 2 j + 1 + ε B , 2 j + 1 σ ̂ B , 2 j + 1 z + g B , 2 j + 1 ( a ̂ 2 j + 1 σ ̂ B , 2 j + 1 + a ̂ 2 j + 1 σ ̂ B , 2 j + 1 ) ,
H = Σ i , j 2 N κ i , j a ̂ i a ̂ j + Σ i = 1 N [ μ ( n ̂ 2 i + n ̂ 2 i + 1 ) + H i two site ] ,
a ̂ σ ̂ A ( B ) ± a ̂ σ ̂ A ( B ) ± + a ̂ σ ̂ A ( B ) ± a ̂ σ ̂ A ( B ) ± .
ψ A a ̂ 2 j a ̂ 2 j ,
ψ B a ̂ 2 j + 1 a ̂ 2 j + 1 ,
J A ( B ) σ ̂ A ( B ) σ ̂ A ( B ) .
H A ( B ) a = ε A ( B ) σ ̂ A ( B ) z + g A ( B ) ψ A ( B ) [ σ ̂ A ( B ) + σ ̂ A ( B ) ] .
H A ( B ) a = E A ( B ) [ γ ̂ 1 , A ( B ) γ ̂ 1 , A ( B ) γ ̂ 0 , A ( B ) γ ̂ 0 , A ( B ) ] ,
E A ( B ) = { [ g A ( B ) ψ A ( B ) ] 2 + ε A ( B ) 2 } 1 2 ,
γ ̂ 1 , A ( B ) = 1 2 [ 1 + ε A ( B ) E A ( B ) ] f ̂ 1 , A ( B ) + 1 2 [ 1 ε A ( B ) E A ( B ) ] f ̂ 0 , A ( B ) ,
γ ̂ 0 , A ( B ) = 1 2 [ 1 ε A ( B ) E A ( B ) ] f ̂ 1 , A ( B ) + 1 2 [ 1 + ε A ( B ) E A ( B ) ] f ̂ 0 , A ( B ) .
σ ̂ A ( B ) = f ̂ 1 , A ( B ) f ̂ 0 , A ( B ) ,
σ ̂ A ( B ) = f ̂ 0 , A ( B ) f ̂ 1 , A ( B ) ,
σ ̂ A ( B ) z = f ̂ 1 , A ( B ) f ̂ 1 , A ( B ) f ̂ 0 , A ( B ) f ̂ 0 , A ( B ) .
J A ( B ) = f 1 , A ( B ) f 0 , A ( B ) ,
= 1 4 [ 1 ε A ( B ) 2 E A ( B ) 2 ] [ γ ̂ 1 , A ( B ) γ ̂ 1 , A ( B ) γ ̂ 0 , A ( B ) γ ̂ 0 , A ( B ) ] ,
= ψ A ( B ) g A ( B ) 2 E A ( B ) .
γ ̂ 1 , A ( B ) γ ̂ 1 , A ( B ) = 1 e β E A ( B ) + 1 ,
γ ̂ 0 , A ( B ) γ ̂ 0 , A ( B ) = 1 e β E A ( B ) + 1 ,
J A ( B ) = ψ A ( B ) g A ( B ) 2 E A ( B ) Tanh [ E A ( B ) 2 T ] .
H 2 j p = Σ i , j N κ i , j a ̂ i a j + Σ j = 1 N [ ( ω μ ) n ̂ 2 j + g A , 2 j J A ( a ̂ 2 j + a ̂ 2 j ) ] ,
H 2 j + 1 p = Σ i , j N κ i , j a ̂ i a ̂ j + Σ j = 1 N [ ( ω μ ) n ̂ 2 j + 1 + g B , 2 j + 1 J B ( a ̂ 2 j + 1 + a ̂ 2 j + 1 ) ] .
a ̂ A , k = Σ j N e ik · r 2 j a ̂ A , 2 j ,
a ̂ B , k = Σ j N e ik · r 2 j + 1 a ̂ B , 2 j + 1 .
H p = H 2 j p + H 2 j + 1 p ,
= [ g A J A ( a ̂ A , k = 0 + a ̂ A , k = 0 ) + g B J B ( a ̂ B , k = 0 + a ̂ B , k = 0 ) ] ,
+ Σ k Ω 0 ( k ) [ a ̂ A , k a ̂ A , k + a ̂ B , k a ̂ B , k ] + Σ k Ω 1 ( k ) [ a ̂ A , k a ̂ B , k + a ̂ B , k a ̂ A , k ] ,
Ω 0 ( k ) = 4 κ Cos ( k x ) Cos ( k y ) + ω μ ,
Ω 1 ( k ) = 2 κ [ Cos ( k x ) + Cos ( k y ) ] ,
H p = Σ k [ Ω sym ( k ) a ̂ sym , k a ̂ sym , k + Ω asym ( k ) a ̂ asym , k a ̂ asym , k ]
+ g sym J sym ( a ̂ sym , k = 0 + a ̂ sym , k = 0 ) + g asym J asym ( a ̂ asym , k = 0 + a ̂ asym , k = 0 ) ,
a ̂ sym , k ( a ̂ A , k + a ̂ B , k ) 2 ,
a ̂ asym , k ( a ̂ A , k a ̂ B , k ) 2 ,
Ω sym ( k ) = Ω 0 ( k ) + Ω 1 ( k ) ,
Ω asym ( k ) = Ω 0 ( k ) Ω 1 ( k ) .
a ̂ sym , k = 0 = ( g A J A + g B J B ) 2 Ω sym ( k = 0 ) ,
a ̂ asym , k = 0 = ( g A J A g B J B ) 2 Ω asym ( k = 0 ) .
ψ A = 1 2 [ a ̂ sym , k = 0 + a ̂ asym , k = 0 ] ,
ψ B = 1 2 [ a ̂ sym , k = 0 a ̂ asym , k = 0 ] .
F f = E A E B 2 β [ ln ( 1 + e β E A ) + ln ( 1 + e β E B ) ] ,
F p = 1 V Σ k 1 β ln [ ( 1 e β Ω A ( k ) ) ( 1 e β Ω B ( k ) ) ] ( g A J A ) 2 Ω A ( g B J B ) 2 Ω B ,
E m = 2 [ ( g A J A ) 2 Ω A + ( g B J B ) 2 Ω B ] ,
E g = 1 2 [ ( g A J A ) 2 Ω A + ( g B J B ) 2 Ω B E A E B ] .
( ψ A ψ B ) = ( Ω + 1 Ω 1 Ω 1 Ω + 1 ) × ( g A 2 4 E A Tanh ( E A 2 T ) ψ A g B 2 4 E B Tanh ( E B 2 T ) ψ B )
ψ = g 2 4 E Ω sym Tanh ( E 2 T ) ψ .
ψ = ( g 4 Ω sym ) 2 ( ε g ) 2 ,
g 2 4 Ω sym > ε .
Ω sym ( k = 0 ) = 4 κ 4 κ + ω μ .
4 ε Ω sym g 2 = Tanh ( ε 2 T c ) .
ψ A ( B ) ( 0 ) = { 0 ; for g A ( B ) 2 4 Ω + < ε A ( B ) [ g A ( B ) 4 Ω + ] 2 [ ε A ( B ) g A ( B ) ] 2 ; for g A ( B ) 2 4 Ω + > ε A ( B ) ,
δ ψ A ( B ) = { g B ( A ) 2 4 [ 1 g A ( B ) 2 4 ε A ( B ) Ω + ] Ω E B ( A ) ( 0 ) ψ B ( A ) ( 0 ) ; for ψ A ( B ) ( 0 ) = 0 Ω + Ω g B ( A ) 2 E A ( B ) ( 0 ) 2 g A ( B ) ) 4 E B ( A ) ( 0 ) 2 ψ B ( A ) ( 0 ) ψ A ( B ) ( 0 ) 3 ; for ψ A ( B ) ( 0 ) 0 ,

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