Abstract

In this paper we investigate a ground state quantum phase transition of nonlinear light in a generalized finite size Dicke model, which includes a Kerr-type nonlinear field term along with the counter-rotating interaction. The numerical solution is presented. We show that, in the ground state, the intracavity photons exhibit a third-order QPT from the bunching to the antibunching quantum phase. This phase transition stems from the competition between the atom-induced coupling and the effective photon–photon interaction. We also demonstrate that the general properties of the phase transition do not qualitatively alter by the size of the atomic ensemble and by the detuning between atoms and light, as well.

© 2011 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Imamoğlu, H. Schmidt, G. Woods, and M. Deutsch, “Strongly interacting photons in a nonlinear cavity,” Phys. Rev. Lett. 79, 1467–1470 (1997).
    [CrossRef]
  2. A. D. Greentree, C. Tahan, J. H. Cole, and L. C. L. Hollenberg, “Quantum phase transitions of light,” Nat. Phys. 2, 856–861(2006).
    [CrossRef]
  3. M. J. Hartman, F. G. S. L. Brandão, and M. B. Plenio, “Strongly interacting polaritons in coupled arrays of cavities,” Nat. Phys. 2, 849–855 (2006).
    [CrossRef]
  4. M. J. Hartmann, F. G. S. L. Brandão, and M. B. Plenio, “Quantum many-body phenomena in coupled cavity arrays,” Laser Photon. Rev. 2, 527–556 (2008).
    [CrossRef]
  5. S. C. Lei, T. K. Ng, and R. K. Lee, “Photonic analogue of Josephson effect in a dual-species optical-lattice cavity,” Opt. Express 18, 14586–14597 (2010).
    [CrossRef] [PubMed]
  6. X. Guo and Z. Ren, “Photonic tunneling effect between two coupled single-atom laser cavities imbedded within a photonic crystal platform,” Phys. Rev. A 83, 013809 (2011).
    [CrossRef]
  7. C. Emary and T. Brandes, “Quantum chaos triggered by precursors of a quantum phase transition: the Dicke model,” Phys. Rev. Lett. 90, 044101 (2003).
    [CrossRef] [PubMed]
  8. N. Lambert, C. Emary, and T. Brandes, “Entanglement and the phase transition in single-mode superradiance,” Phys. Rev. Lett. 92, 073602 (2004).
    [CrossRef] [PubMed]
  9. S. Morrison and A. S. Parkins, “Dynamical quantum phase transitions in the dissipative Lipkin–Meshkov–Glick model with proposed realization in optical cavity QED,” Phys. Rev. Lett. 100, 040403 (2008).
    [CrossRef] [PubMed]
  10. J. Larson, “Circuit QED scheme for the realization of the Lipkin–Meshkov–Glick model,” Europhys. Lett. 90, 54001 (2010).
    [CrossRef]
  11. J. Larson and M. Horsdal, “Photonic Josephson effect, phase transitions, and chaos in optomechanical systems,” arXiv:1009.2945 (2010).
  12. J. Reslen, L. Quiroga, and N. F. Johnson, “Direct equivalence between quantum phase transition phenomena in radiation-matter and magnetic systems: scaling of entanglement,” Europhys. Lett. 69, 8–14 (2005).
    [CrossRef]
  13. G. Liberti, F. Plastina, and F. Piperno, “Scaling behavior of the adiabatic Dicke model,” Phys. Rev. A 74, 022324(2006).
    [CrossRef]
  14. J. Vidal and S. Dusuel, “Finite-size scaling exponents in the Dicke model,” Europhys. Lett. 74, 817–822 (2006).
    [CrossRef]
  15. D. Nagy, G. Konya, G. Szirmai, and P. Domokos, “Dicke-model phase transition in the quantum motion of a Bose–Einstein condensate,” Phys. Rev. Lett. 104, 130401 (2010).
    [CrossRef] [PubMed]
  16. R. H. Dicke, “Coherence in spontaneous radiation processes,” Phys. Rev. 93, 99–110 (1954).
    [CrossRef]
  17. K. Hepp and E. H. Lieb, “On the superradiant phase transition for molecules in a quantized radiation field: the Dicke maser model,” Ann. Phys. 76, 360–404 (1973).
    [CrossRef]
  18. Y. K. Wang and F. T. Hioe, “Phase transition in the Dicke model of superradiance,” Phys. Rev. A 7, 831–836 (1973).
    [CrossRef]
  19. K. Baumann, C. Guerlin, F. Brennecke, and T. Esslinger, “Dicke quantum phase transition with a superfluid gas in an optical cavity,” Nature 464, 1301–1306 (2010).
    [CrossRef] [PubMed]
  20. Q. H. Chen, T. Liu, Y. Y. Zhang, and K. L. Wang, “Quantum phase transition in coupled two-level atoms in a single-mode cavity,” Phys. Rev. A 82, 053841 (2010).
    [CrossRef]
  21. B. M. Rodríguez-Lara and R. K. Lee, “Quantum phase transition of nonlinear light in the finite size Dicke Hamiltonian,” J. Opt. Soc. Am. B 27, 2443–2450 (2010).
    [CrossRef]
  22. P. D. Drummond and D. F. Walls, “Quantum theory of optical bistability. I: nonlinear polarizability model,” J. Phys. A 13, 725–741 (1980).
    [CrossRef]
  23. S. Ferretti, L. C. Andreani, H. E. Türeci, and D. Gerace, “Photon correlations in a two-site nonlinear cavity system under coherent drive and dissipation,” Phys. Rev. A 82, 013841(2010).
    [CrossRef]
  24. B. S. Garbow, J. M. Boyle, J. J. Dongarra, and C. B. Moler, Matrix Eigensystem Routines–EISPACK Guide Extension, Lecture Notes in Computer Science Volume 51 (Springer, 1977).
  25. P. Nataf and C. Ciuti, “No-go theorem for superradiant quantum phase transitions in cavity QED and counter-example in circuit QED,” Nat. Commun. 1, 72 (2010).
    [CrossRef] [PubMed]
  26. R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. 130, 2529–2539 (1963).
    [CrossRef]
  27. M. Orszag, Quantum Optics: Including Noise Reduction, Trapped Ions, Quantum Trajectories, and Decoherence(Springer-Verlag, 2000).
  28. M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University Press, 1997).
  29. X. Guo and S. Lü, “Controllable optical bistability in photonic-crystal one-atom laser,” Phys. Rev. A 80, 043826 (2009).
    [CrossRef]

2011 (1)

X. Guo and Z. Ren, “Photonic tunneling effect between two coupled single-atom laser cavities imbedded within a photonic crystal platform,” Phys. Rev. A 83, 013809 (2011).
[CrossRef]

2010 (8)

S. C. Lei, T. K. Ng, and R. K. Lee, “Photonic analogue of Josephson effect in a dual-species optical-lattice cavity,” Opt. Express 18, 14586–14597 (2010).
[CrossRef] [PubMed]

J. Larson, “Circuit QED scheme for the realization of the Lipkin–Meshkov–Glick model,” Europhys. Lett. 90, 54001 (2010).
[CrossRef]

D. Nagy, G. Konya, G. Szirmai, and P. Domokos, “Dicke-model phase transition in the quantum motion of a Bose–Einstein condensate,” Phys. Rev. Lett. 104, 130401 (2010).
[CrossRef] [PubMed]

K. Baumann, C. Guerlin, F. Brennecke, and T. Esslinger, “Dicke quantum phase transition with a superfluid gas in an optical cavity,” Nature 464, 1301–1306 (2010).
[CrossRef] [PubMed]

Q. H. Chen, T. Liu, Y. Y. Zhang, and K. L. Wang, “Quantum phase transition in coupled two-level atoms in a single-mode cavity,” Phys. Rev. A 82, 053841 (2010).
[CrossRef]

B. M. Rodríguez-Lara and R. K. Lee, “Quantum phase transition of nonlinear light in the finite size Dicke Hamiltonian,” J. Opt. Soc. Am. B 27, 2443–2450 (2010).
[CrossRef]

S. Ferretti, L. C. Andreani, H. E. Türeci, and D. Gerace, “Photon correlations in a two-site nonlinear cavity system under coherent drive and dissipation,” Phys. Rev. A 82, 013841(2010).
[CrossRef]

P. Nataf and C. Ciuti, “No-go theorem for superradiant quantum phase transitions in cavity QED and counter-example in circuit QED,” Nat. Commun. 1, 72 (2010).
[CrossRef] [PubMed]

2009 (1)

X. Guo and S. Lü, “Controllable optical bistability in photonic-crystal one-atom laser,” Phys. Rev. A 80, 043826 (2009).
[CrossRef]

2008 (2)

S. Morrison and A. S. Parkins, “Dynamical quantum phase transitions in the dissipative Lipkin–Meshkov–Glick model with proposed realization in optical cavity QED,” Phys. Rev. Lett. 100, 040403 (2008).
[CrossRef] [PubMed]

M. J. Hartmann, F. G. S. L. Brandão, and M. B. Plenio, “Quantum many-body phenomena in coupled cavity arrays,” Laser Photon. Rev. 2, 527–556 (2008).
[CrossRef]

2006 (4)

A. D. Greentree, C. Tahan, J. H. Cole, and L. C. L. Hollenberg, “Quantum phase transitions of light,” Nat. Phys. 2, 856–861(2006).
[CrossRef]

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

G. Liberti, F. Plastina, and F. Piperno, “Scaling behavior of the adiabatic Dicke model,” Phys. Rev. A 74, 022324(2006).
[CrossRef]

J. Vidal and S. Dusuel, “Finite-size scaling exponents in the Dicke model,” Europhys. Lett. 74, 817–822 (2006).
[CrossRef]

2005 (1)

J. Reslen, L. Quiroga, and N. F. Johnson, “Direct equivalence between quantum phase transition phenomena in radiation-matter and magnetic systems: scaling of entanglement,” Europhys. Lett. 69, 8–14 (2005).
[CrossRef]

2004 (1)

N. Lambert, C. Emary, and T. Brandes, “Entanglement and the phase transition in single-mode superradiance,” Phys. Rev. Lett. 92, 073602 (2004).
[CrossRef] [PubMed]

2003 (1)

C. Emary and T. Brandes, “Quantum chaos triggered by precursors of a quantum phase transition: the Dicke model,” Phys. Rev. Lett. 90, 044101 (2003).
[CrossRef] [PubMed]

1997 (1)

A. Imamoğlu, H. Schmidt, G. Woods, and M. Deutsch, “Strongly interacting photons in a nonlinear cavity,” Phys. Rev. Lett. 79, 1467–1470 (1997).
[CrossRef]

1980 (1)

P. D. Drummond and D. F. Walls, “Quantum theory of optical bistability. I: nonlinear polarizability model,” J. Phys. A 13, 725–741 (1980).
[CrossRef]

1973 (2)

K. Hepp and E. H. Lieb, “On the superradiant phase transition for molecules in a quantized radiation field: the Dicke maser model,” Ann. Phys. 76, 360–404 (1973).
[CrossRef]

Y. K. Wang and F. T. Hioe, “Phase transition in the Dicke model of superradiance,” Phys. Rev. A 7, 831–836 (1973).
[CrossRef]

1963 (1)

R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. 130, 2529–2539 (1963).
[CrossRef]

1954 (1)

R. H. Dicke, “Coherence in spontaneous radiation processes,” Phys. Rev. 93, 99–110 (1954).
[CrossRef]

Andreani, L. C.

S. Ferretti, L. C. Andreani, H. E. Türeci, and D. Gerace, “Photon correlations in a two-site nonlinear cavity system under coherent drive and dissipation,” Phys. Rev. A 82, 013841(2010).
[CrossRef]

Baumann, K.

K. Baumann, C. Guerlin, F. Brennecke, and T. Esslinger, “Dicke quantum phase transition with a superfluid gas in an optical cavity,” Nature 464, 1301–1306 (2010).
[CrossRef] [PubMed]

Boyle, J. M.

B. S. Garbow, J. M. Boyle, J. J. Dongarra, and C. B. Moler, Matrix Eigensystem Routines–EISPACK Guide Extension, Lecture Notes in Computer Science Volume 51 (Springer, 1977).

Brandão, F. G. S. L.

M. J. Hartmann, F. G. S. L. Brandão, and M. B. Plenio, “Quantum many-body phenomena in coupled cavity arrays,” Laser Photon. Rev. 2, 527–556 (2008).
[CrossRef]

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

Brandes, T.

N. Lambert, C. Emary, and T. Brandes, “Entanglement and the phase transition in single-mode superradiance,” Phys. Rev. Lett. 92, 073602 (2004).
[CrossRef] [PubMed]

C. Emary and T. Brandes, “Quantum chaos triggered by precursors of a quantum phase transition: the Dicke model,” Phys. Rev. Lett. 90, 044101 (2003).
[CrossRef] [PubMed]

Brennecke, F.

K. Baumann, C. Guerlin, F. Brennecke, and T. Esslinger, “Dicke quantum phase transition with a superfluid gas in an optical cavity,” Nature 464, 1301–1306 (2010).
[CrossRef] [PubMed]

Chen, Q. H.

Q. H. Chen, T. Liu, Y. Y. Zhang, and K. L. Wang, “Quantum phase transition in coupled two-level atoms in a single-mode cavity,” Phys. Rev. A 82, 053841 (2010).
[CrossRef]

Ciuti, C.

P. Nataf and C. Ciuti, “No-go theorem for superradiant quantum phase transitions in cavity QED and counter-example in circuit QED,” Nat. Commun. 1, 72 (2010).
[CrossRef] [PubMed]

Cole, J. H.

A. D. Greentree, C. Tahan, J. H. Cole, and L. C. L. Hollenberg, “Quantum phase transitions of light,” Nat. Phys. 2, 856–861(2006).
[CrossRef]

Deutsch, M.

A. Imamoğlu, H. Schmidt, G. Woods, and M. Deutsch, “Strongly interacting photons in a nonlinear cavity,” Phys. Rev. Lett. 79, 1467–1470 (1997).
[CrossRef]

Dicke, R. H.

R. H. Dicke, “Coherence in spontaneous radiation processes,” Phys. Rev. 93, 99–110 (1954).
[CrossRef]

Domokos, P.

D. Nagy, G. Konya, G. Szirmai, and P. Domokos, “Dicke-model phase transition in the quantum motion of a Bose–Einstein condensate,” Phys. Rev. Lett. 104, 130401 (2010).
[CrossRef] [PubMed]

Dongarra, J. J.

B. S. Garbow, J. M. Boyle, J. J. Dongarra, and C. B. Moler, Matrix Eigensystem Routines–EISPACK Guide Extension, Lecture Notes in Computer Science Volume 51 (Springer, 1977).

Drummond, P. D.

P. D. Drummond and D. F. Walls, “Quantum theory of optical bistability. I: nonlinear polarizability model,” J. Phys. A 13, 725–741 (1980).
[CrossRef]

Dusuel, S.

J. Vidal and S. Dusuel, “Finite-size scaling exponents in the Dicke model,” Europhys. Lett. 74, 817–822 (2006).
[CrossRef]

Emary, C.

N. Lambert, C. Emary, and T. Brandes, “Entanglement and the phase transition in single-mode superradiance,” Phys. Rev. Lett. 92, 073602 (2004).
[CrossRef] [PubMed]

C. Emary and T. Brandes, “Quantum chaos triggered by precursors of a quantum phase transition: the Dicke model,” Phys. Rev. Lett. 90, 044101 (2003).
[CrossRef] [PubMed]

Esslinger, T.

K. Baumann, C. Guerlin, F. Brennecke, and T. Esslinger, “Dicke quantum phase transition with a superfluid gas in an optical cavity,” Nature 464, 1301–1306 (2010).
[CrossRef] [PubMed]

Ferretti, S.

S. Ferretti, L. C. Andreani, H. E. Türeci, and D. Gerace, “Photon correlations in a two-site nonlinear cavity system under coherent drive and dissipation,” Phys. Rev. A 82, 013841(2010).
[CrossRef]

Garbow, B. S.

B. S. Garbow, J. M. Boyle, J. J. Dongarra, and C. B. Moler, Matrix Eigensystem Routines–EISPACK Guide Extension, Lecture Notes in Computer Science Volume 51 (Springer, 1977).

Gerace, D.

S. Ferretti, L. C. Andreani, H. E. Türeci, and D. Gerace, “Photon correlations in a two-site nonlinear cavity system under coherent drive and dissipation,” Phys. Rev. A 82, 013841(2010).
[CrossRef]

Glauber, R. J.

R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. 130, 2529–2539 (1963).
[CrossRef]

Greentree, A. D.

A. D. Greentree, C. Tahan, J. H. Cole, and L. C. L. Hollenberg, “Quantum phase transitions of light,” Nat. Phys. 2, 856–861(2006).
[CrossRef]

Guerlin, C.

K. Baumann, C. Guerlin, F. Brennecke, and T. Esslinger, “Dicke quantum phase transition with a superfluid gas in an optical cavity,” Nature 464, 1301–1306 (2010).
[CrossRef] [PubMed]

Guo, X.

X. Guo and Z. Ren, “Photonic tunneling effect between two coupled single-atom laser cavities imbedded within a photonic crystal platform,” Phys. Rev. A 83, 013809 (2011).
[CrossRef]

X. Guo and S. Lü, “Controllable optical bistability in photonic-crystal one-atom laser,” Phys. Rev. A 80, 043826 (2009).
[CrossRef]

Hartman, M. J.

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

Hartmann, M. J.

M. J. Hartmann, F. G. S. L. Brandão, and M. B. Plenio, “Quantum many-body phenomena in coupled cavity arrays,” Laser Photon. Rev. 2, 527–556 (2008).
[CrossRef]

Hepp, K.

K. Hepp and E. H. Lieb, “On the superradiant phase transition for molecules in a quantized radiation field: the Dicke maser model,” Ann. Phys. 76, 360–404 (1973).
[CrossRef]

Hioe, F. T.

Y. K. Wang and F. T. Hioe, “Phase transition in the Dicke model of superradiance,” Phys. Rev. A 7, 831–836 (1973).
[CrossRef]

Hollenberg, L. C. L.

A. D. Greentree, C. Tahan, J. H. Cole, and L. C. L. Hollenberg, “Quantum phase transitions of light,” Nat. Phys. 2, 856–861(2006).
[CrossRef]

Horsdal, M.

J. Larson and M. Horsdal, “Photonic Josephson effect, phase transitions, and chaos in optomechanical systems,” arXiv:1009.2945 (2010).

Imamoglu, A.

A. Imamoğlu, H. Schmidt, G. Woods, and M. Deutsch, “Strongly interacting photons in a nonlinear cavity,” Phys. Rev. Lett. 79, 1467–1470 (1997).
[CrossRef]

Johnson, N. F.

J. Reslen, L. Quiroga, and N. F. Johnson, “Direct equivalence between quantum phase transition phenomena in radiation-matter and magnetic systems: scaling of entanglement,” Europhys. Lett. 69, 8–14 (2005).
[CrossRef]

Konya, G.

D. Nagy, G. Konya, G. Szirmai, and P. Domokos, “Dicke-model phase transition in the quantum motion of a Bose–Einstein condensate,” Phys. Rev. Lett. 104, 130401 (2010).
[CrossRef] [PubMed]

Lambert, N.

N. Lambert, C. Emary, and T. Brandes, “Entanglement and the phase transition in single-mode superradiance,” Phys. Rev. Lett. 92, 073602 (2004).
[CrossRef] [PubMed]

Larson, J.

J. Larson, “Circuit QED scheme for the realization of the Lipkin–Meshkov–Glick model,” Europhys. Lett. 90, 54001 (2010).
[CrossRef]

J. Larson and M. Horsdal, “Photonic Josephson effect, phase transitions, and chaos in optomechanical systems,” arXiv:1009.2945 (2010).

Lee, R. K.

Lei, S. C.

Liberti, G.

G. Liberti, F. Plastina, and F. Piperno, “Scaling behavior of the adiabatic Dicke model,” Phys. Rev. A 74, 022324(2006).
[CrossRef]

Lieb, E. H.

K. Hepp and E. H. Lieb, “On the superradiant phase transition for molecules in a quantized radiation field: the Dicke maser model,” Ann. Phys. 76, 360–404 (1973).
[CrossRef]

Liu, T.

Q. H. Chen, T. Liu, Y. Y. Zhang, and K. L. Wang, “Quantum phase transition in coupled two-level atoms in a single-mode cavity,” Phys. Rev. A 82, 053841 (2010).
[CrossRef]

Lü, S.

X. Guo and S. Lü, “Controllable optical bistability in photonic-crystal one-atom laser,” Phys. Rev. A 80, 043826 (2009).
[CrossRef]

Moler, C. B.

B. S. Garbow, J. M. Boyle, J. J. Dongarra, and C. B. Moler, Matrix Eigensystem Routines–EISPACK Guide Extension, Lecture Notes in Computer Science Volume 51 (Springer, 1977).

Morrison, S.

S. Morrison and A. S. Parkins, “Dynamical quantum phase transitions in the dissipative Lipkin–Meshkov–Glick model with proposed realization in optical cavity QED,” Phys. Rev. Lett. 100, 040403 (2008).
[CrossRef] [PubMed]

Nagy, D.

D. Nagy, G. Konya, G. Szirmai, and P. Domokos, “Dicke-model phase transition in the quantum motion of a Bose–Einstein condensate,” Phys. Rev. Lett. 104, 130401 (2010).
[CrossRef] [PubMed]

Nataf, P.

P. Nataf and C. Ciuti, “No-go theorem for superradiant quantum phase transitions in cavity QED and counter-example in circuit QED,” Nat. Commun. 1, 72 (2010).
[CrossRef] [PubMed]

Ng, T. K.

Orszag, M.

M. Orszag, Quantum Optics: Including Noise Reduction, Trapped Ions, Quantum Trajectories, and Decoherence(Springer-Verlag, 2000).

Parkins, A. S.

S. Morrison and A. S. Parkins, “Dynamical quantum phase transitions in the dissipative Lipkin–Meshkov–Glick model with proposed realization in optical cavity QED,” Phys. Rev. Lett. 100, 040403 (2008).
[CrossRef] [PubMed]

Piperno, F.

G. Liberti, F. Plastina, and F. Piperno, “Scaling behavior of the adiabatic Dicke model,” Phys. Rev. A 74, 022324(2006).
[CrossRef]

Plastina, F.

G. Liberti, F. Plastina, and F. Piperno, “Scaling behavior of the adiabatic Dicke model,” Phys. Rev. A 74, 022324(2006).
[CrossRef]

Plenio, M. B.

M. J. Hartmann, F. G. S. L. Brandão, and M. B. Plenio, “Quantum many-body phenomena in coupled cavity arrays,” Laser Photon. Rev. 2, 527–556 (2008).
[CrossRef]

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

Quiroga, L.

J. Reslen, L. Quiroga, and N. F. Johnson, “Direct equivalence between quantum phase transition phenomena in radiation-matter and magnetic systems: scaling of entanglement,” Europhys. Lett. 69, 8–14 (2005).
[CrossRef]

Ren, Z.

X. Guo and Z. Ren, “Photonic tunneling effect between two coupled single-atom laser cavities imbedded within a photonic crystal platform,” Phys. Rev. A 83, 013809 (2011).
[CrossRef]

Reslen, J.

J. Reslen, L. Quiroga, and N. F. Johnson, “Direct equivalence between quantum phase transition phenomena in radiation-matter and magnetic systems: scaling of entanglement,” Europhys. Lett. 69, 8–14 (2005).
[CrossRef]

Rodríguez-Lara, B. M.

Schmidt, H.

A. Imamoğlu, H. Schmidt, G. Woods, and M. Deutsch, “Strongly interacting photons in a nonlinear cavity,” Phys. Rev. Lett. 79, 1467–1470 (1997).
[CrossRef]

Scully, M. O.

M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University Press, 1997).

Szirmai, G.

D. Nagy, G. Konya, G. Szirmai, and P. Domokos, “Dicke-model phase transition in the quantum motion of a Bose–Einstein condensate,” Phys. Rev. Lett. 104, 130401 (2010).
[CrossRef] [PubMed]

Tahan, C.

A. D. Greentree, C. Tahan, J. H. Cole, and L. C. L. Hollenberg, “Quantum phase transitions of light,” Nat. Phys. 2, 856–861(2006).
[CrossRef]

Türeci, H. E.

S. Ferretti, L. C. Andreani, H. E. Türeci, and D. Gerace, “Photon correlations in a two-site nonlinear cavity system under coherent drive and dissipation,” Phys. Rev. A 82, 013841(2010).
[CrossRef]

Vidal, J.

J. Vidal and S. Dusuel, “Finite-size scaling exponents in the Dicke model,” Europhys. Lett. 74, 817–822 (2006).
[CrossRef]

Walls, D. F.

P. D. Drummond and D. F. Walls, “Quantum theory of optical bistability. I: nonlinear polarizability model,” J. Phys. A 13, 725–741 (1980).
[CrossRef]

Wang, K. L.

Q. H. Chen, T. Liu, Y. Y. Zhang, and K. L. Wang, “Quantum phase transition in coupled two-level atoms in a single-mode cavity,” Phys. Rev. A 82, 053841 (2010).
[CrossRef]

Wang, Y. K.

Y. K. Wang and F. T. Hioe, “Phase transition in the Dicke model of superradiance,” Phys. Rev. A 7, 831–836 (1973).
[CrossRef]

Woods, G.

A. Imamoğlu, H. Schmidt, G. Woods, and M. Deutsch, “Strongly interacting photons in a nonlinear cavity,” Phys. Rev. Lett. 79, 1467–1470 (1997).
[CrossRef]

Zhang, Y. Y.

Q. H. Chen, T. Liu, Y. Y. Zhang, and K. L. Wang, “Quantum phase transition in coupled two-level atoms in a single-mode cavity,” Phys. Rev. A 82, 053841 (2010).
[CrossRef]

Zubairy, M. S.

M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University Press, 1997).

Ann. Phys. (1)

K. Hepp and E. H. Lieb, “On the superradiant phase transition for molecules in a quantized radiation field: the Dicke maser model,” Ann. Phys. 76, 360–404 (1973).
[CrossRef]

Europhys. Lett. (3)

J. Reslen, L. Quiroga, and N. F. Johnson, “Direct equivalence between quantum phase transition phenomena in radiation-matter and magnetic systems: scaling of entanglement,” Europhys. Lett. 69, 8–14 (2005).
[CrossRef]

J. Vidal and S. Dusuel, “Finite-size scaling exponents in the Dicke model,” Europhys. Lett. 74, 817–822 (2006).
[CrossRef]

J. Larson, “Circuit QED scheme for the realization of the Lipkin–Meshkov–Glick model,” Europhys. Lett. 90, 54001 (2010).
[CrossRef]

J. Opt. Soc. Am. B (1)

J. Phys. A (1)

P. D. Drummond and D. F. Walls, “Quantum theory of optical bistability. I: nonlinear polarizability model,” J. Phys. A 13, 725–741 (1980).
[CrossRef]

Laser Photon. Rev. (1)

M. J. Hartmann, F. G. S. L. Brandão, and M. B. Plenio, “Quantum many-body phenomena in coupled cavity arrays,” Laser Photon. Rev. 2, 527–556 (2008).
[CrossRef]

Nat. Commun. (1)

P. Nataf and C. Ciuti, “No-go theorem for superradiant quantum phase transitions in cavity QED and counter-example in circuit QED,” Nat. Commun. 1, 72 (2010).
[CrossRef] [PubMed]

Nat. Phys. (2)

A. D. Greentree, C. Tahan, J. H. Cole, and L. C. L. Hollenberg, “Quantum phase transitions of light,” Nat. Phys. 2, 856–861(2006).
[CrossRef]

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

Nature (1)

K. Baumann, C. Guerlin, F. Brennecke, and T. Esslinger, “Dicke quantum phase transition with a superfluid gas in an optical cavity,” Nature 464, 1301–1306 (2010).
[CrossRef] [PubMed]

Opt. Express (1)

Phys. Rev. (2)

R. H. Dicke, “Coherence in spontaneous radiation processes,” Phys. Rev. 93, 99–110 (1954).
[CrossRef]

R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. 130, 2529–2539 (1963).
[CrossRef]

Phys. Rev. A (6)

S. Ferretti, L. C. Andreani, H. E. Türeci, and D. Gerace, “Photon correlations in a two-site nonlinear cavity system under coherent drive and dissipation,” Phys. Rev. A 82, 013841(2010).
[CrossRef]

Q. H. Chen, T. Liu, Y. Y. Zhang, and K. L. Wang, “Quantum phase transition in coupled two-level atoms in a single-mode cavity,” Phys. Rev. A 82, 053841 (2010).
[CrossRef]

X. Guo and S. Lü, “Controllable optical bistability in photonic-crystal one-atom laser,” Phys. Rev. A 80, 043826 (2009).
[CrossRef]

Y. K. Wang and F. T. Hioe, “Phase transition in the Dicke model of superradiance,” Phys. Rev. A 7, 831–836 (1973).
[CrossRef]

G. Liberti, F. Plastina, and F. Piperno, “Scaling behavior of the adiabatic Dicke model,” Phys. Rev. A 74, 022324(2006).
[CrossRef]

X. Guo and Z. Ren, “Photonic tunneling effect between two coupled single-atom laser cavities imbedded within a photonic crystal platform,” Phys. Rev. A 83, 013809 (2011).
[CrossRef]

Phys. Rev. Lett. (5)

C. Emary and T. Brandes, “Quantum chaos triggered by precursors of a quantum phase transition: the Dicke model,” Phys. Rev. Lett. 90, 044101 (2003).
[CrossRef] [PubMed]

N. Lambert, C. Emary, and T. Brandes, “Entanglement and the phase transition in single-mode superradiance,” Phys. Rev. Lett. 92, 073602 (2004).
[CrossRef] [PubMed]

S. Morrison and A. S. Parkins, “Dynamical quantum phase transitions in the dissipative Lipkin–Meshkov–Glick model with proposed realization in optical cavity QED,” Phys. Rev. Lett. 100, 040403 (2008).
[CrossRef] [PubMed]

D. Nagy, G. Konya, G. Szirmai, and P. Domokos, “Dicke-model phase transition in the quantum motion of a Bose–Einstein condensate,” Phys. Rev. Lett. 104, 130401 (2010).
[CrossRef] [PubMed]

A. Imamoğlu, H. Schmidt, G. Woods, and M. Deutsch, “Strongly interacting photons in a nonlinear cavity,” Phys. Rev. Lett. 79, 1467–1470 (1997).
[CrossRef]

Other (4)

J. Larson and M. Horsdal, “Photonic Josephson effect, phase transitions, and chaos in optomechanical systems,” arXiv:1009.2945 (2010).

B. S. Garbow, J. M. Boyle, J. J. Dongarra, and C. B. Moler, Matrix Eigensystem Routines–EISPACK Guide Extension, Lecture Notes in Computer Science Volume 51 (Springer, 1977).

M. Orszag, Quantum Optics: Including Noise Reduction, Trapped Ions, Quantum Trajectories, and Decoherence(Springer-Verlag, 2000).

M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University Press, 1997).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

Ground state expectation value of the z component of angular momentum, J z , as a function of the linear atom–photon coupling strength, λ, and the nonlinear coefficient, κ, in units of ω a . Here the cavity field and atomic ensemble are tuned on resonance ω p = ω a , and the size of the ensemble is fixed at (a)  N = 2 and (b)  N = 6 .

Fig. 2
Fig. 2

Ground state expectation value of the intracavity photon number, n , as a function of the linear atom–photon coupling strength, λ, and the nonlinear coefficient, κ, in units of ω a . Other parameters are the same as those in Fig. 1.

Fig. 3
Fig. 3

Ground state quantum degree of second-order coherence, g ( 2 ) ( 0 ) , varies as a function of the linear atom–photon coupling strength, λ, and the nonlinear coefficient, κ, in units of ω a . The white dashed curve and several points with letters and coordinates indicate the boundary between two phases. Here the cavity field and atomic ensemble are tuned on resonance ω p = ω a , and the size of the ensemble is fixed at (a)  N = 2 ; (b)  N = 6 .

Fig. 4
Fig. 4

Derivative of the ground state free energy F with respect to the nonlinear coefficient, d n F / d κ n , as a function of κ for fixed λ / N = 0.34 ω a , N = 6 , and in the resonance case ω p = ω a . Here n = 1 (black solid curve), n = 2 (red dashed curve), and n = 3 (blue dotted curve).

Fig. 5
Fig. 5

Ground state quantum degree of second-order coherence, g ( 2 ) ( 0 ) , as a function of the nonlinear coefficient, κ, in units of ω a . Here the linear coupling strength, λ, is fixed at which g ( 2 ) ( 0 ) reaches its maximum value. We plot both the resonance and nonresonance case, and the size of ensemble is fixed at N = 2 and N = 6 .

Fig. 6
Fig. 6

Ground state quantum degree of second-order coherence, g ( 2 ) ( 0 ) , as a function of the linear atom–photon coupling strength, λ, in units of ω a . Here the nonlinear coefficient, κ, is fixed at which g ( 2 ) ( 0 ) reaches its maximum value. Other parameters are the same as those in Fig. 5.

Equations (10)

Equations on this page are rendered with MathJax. Learn more.

H = ω p a a + ω a J z + λ N ( a J + + a J ) ,
H = ω p a a + ω a J z + λ N ( a + a ) J x + κ a a a a .
J z | N / 2 , m = m | N / 2 , m , J ± | N / 2 , m = N 2 ( N 2 + 1 ) m ( m ± 1 ) | N / 2 , m ± 1 .
a | n = n | n 1 , a | n = n + 1 | n + 1 ,
i | n | j = j δ i , j , i | a | j = j + 1 δ i , j + 1 , i | a | j = j + 1 δ i , j + 1 ,
p | J z | q = q δ p , q , p | J ± | q = N 2 ( N 2 + 1 ) q ( q ± 1 ) δ p , q ± 1 ,
H trun = ω p n I a + ω a I p J z + λ N ( a + a ) J x + κ a a a a I a ,
g ( 2 ) ( 0 ) = a a a a a a 2 ,
g ( 2 ) ( 0 ) < 1 .
g ( 2 ) ( 0 ) > 1 .

Metrics