Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Article
  • Published:

Quantum phase transitions of light

Nature Physics volume 2, pages 856–861 (2006)Cite this article

Abstract

The ability to conduct experiments at length scales and temperatures at which interesting and potentially useful quantum-mechanical phenomena emerge in condensed-matter or atomic systems is now commonplace. In optics, though, the weakness with which photons interact with each other makes exploring such behaviour more difficult. Here we describe an optical system that exhibits strongly correlated dynamics on a mesoscopic scale. By adding photons to a two-dimensional array of coupled optical cavities each containing a single two-level atom in the photon-blockade regime, we form dressed states, or polaritons, that are both long-lived and strongly interacting. Our results predict that at zero temperature the system will undergo a characteristic Mott insulator (excitations localized on each site) to superfluid (excitations delocalized across the lattice) quantum phase transition. Moreover, the ability to couple light to and from individual cavities of this system could be useful in the realization of tuneable quantum simulators and other quantum-mechanical devices.

This is a preview of subscription content, access via your institution

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

Figure 1: A proposed implementation of the photonic condensed-matter analogue.
Figure 2: Eigenspectrum for a single atom in a high-Q cavity, as a function of the atom–cavity detuning, centred around zero (E−n ω).
Figure 3: Boundaries between Mott lobes in the limit of low tunnelling (small κ) as a function of μ and Δ.
Figure 4: Slices showing the superfluid order parameter, ψ, as a function of the photon hopping frequency, κ, and the chemical potential, μ, for different detunings.
Figure 5: Plateaux with constant density, ρ, indicating regions with definite-number state excitations, as a function of the chemical potential, μ, and photon hopping frequency, κ, for detuning, Δ=0.

Similar content being viewed by others

References

  1. Jaynes, E. T. & Cummings, F. W. Comparison of quantum and semiclassical radiation theory with application to the beam maser. Proc. IEEE 51, 89–109 (1963).

    Article  Google Scholar 

  2. Birnbaum, K. M. et al. Photon blockade in an optical cavity with one trapped atom. Nature 436, 87–90 (2005).

    Article  ADS  Google Scholar 

  3. Imamoğlu, A., Schmidt, H., Woods, G. & Deutsch, M. Strongly interacting photons in a nonlinear cavity. Phys. Rev. Lett. 79, 1467–1470 (1997).

    Article  ADS  Google Scholar 

  4. Grangier, P., Walls, D. & Gheri, K. Comment on strongly interacting photons in a nonlinear cavity. Phys. Rev. Lett. 81, 2833 (1998).

    Article  ADS  Google Scholar 

  5. Greentree, A. D., Vaccaro, J. A., de Echaniz, S. R., Durrant, A. V. & Marangos, J. P. Prospects for photon blockade in four-level systems in the N configuration with more than one atom. J. Opt. B: Quantum Semiclass. Opt. 2, 252–259 (2000).

    Article  ADS  Google Scholar 

  6. Rebić, S., Parkins, A. S. & Tan, S. M. Photon statistics of a single-atom intracavity system involving electromagnetically induced transparency. Phys. Rev. A 65, 063804 (2002).

    Article  ADS  Google Scholar 

  7. Lang, R. J. & Yariv, A. An exact formulation of coupled-mode theory for coupled-cavity lasers. IEEE J. Quantum Electron. 24, 66–72 (1988).

    Article  ADS  Google Scholar 

  8. Ozbay, E., Bayindir, M., Bulu, I. & Cubukcu, E. Investigation of localized coupled-cavity modes in two-dimensional photonic band gap structures. IEEE J. Quantum Electron. 38, 837–843 (2002).

    Article  ADS  Google Scholar 

  9. Altug, H., Englund, D. & Vučović, J. Ultrafast photonic crystal nanocavity laser. Nature Phys. 2, 484–488 (2006).

    Article  ADS  Google Scholar 

  10. Greentree, A. D., Salzman, J., Prawer, S. & Hollenberg, L. C. L. Quantum gate for Q switching in monolithic photonic bandgap cavities containing two-level atoms. Phys. Rev. A 73, 013818 (2006).

    Article  ADS  Google Scholar 

  11. Hartmann, M. J., Brandão, F. G. S. L. & Plenio, M. B. Strongly interacting polaritons in coupled arrays of cavities. Preprint at <http://arxiv.org/abs/quant-ph/0606097> (2006).

  12. Angelakis, D. G., Santos, M. F. & Bose, S. Photon blockade induced Mott transitions and XY spin models in coupled cavity arrays. Preprint at <http://arxiv.org/abs/quant-ph/0606159> (2006).

  13. Lee, P. A. & Ramakrishnan, T. V. Disordered electronic systems. Rev. Mod. Phys. 57, 287–337 (1985).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  15. Jaksch, D., Bruder, C., Cirac, J. I., Gardiner, C. W. & Zoller, P. Cold bosonic atoms in optical lattices. Phys. Rev. Lett. 81, 3108–3111 (1998).

    Article  ADS  Google Scholar 

  16. Greiner, M., Mandel, O., Esslinger, T., Hänsch, T. W. & Bloch, I. Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms. Nature 415, 39–44 (2002).

    Article  ADS  Google Scholar 

  17. Cohen-Tannoudji, C. in Frontiers in Laser Spectroscopy (eds Balian, R., Haroche, S. & Liberman, S.) 1–104 (Elsevier Science Publishers B.V., North-Holland, Amsterdam, 1977).

    Google Scholar 

  18. Hussin, V. & Nieto, L. M. Ladder operators and coherent states for the Jaynes–Cummings model in the rotating wave approximation. J. Math. Phys. 46, 122102 (2005).

    Article  ADS  MathSciNet  Google Scholar 

  19. van Oosten, D., van der Straten, P. & Stoof, H. T. C. Quantum phases in an optical lattice. Phys. Rev. A 63, 053601 (2001).

    Article  ADS  Google Scholar 

  20. van Oosten, D., van der Straten, P. & Krishnamurthy, H. R. Mott insulators in an optical lattice with high filling factors. Phys. Rev. A 67, 033606 (2003).

    Article  ADS  Google Scholar 

  21. Sheshadri, K., Krishnamurthy, H. R., Pandit, R. & Ramakrishnan, T. V. Superfluid and insulating phases in an interacting-boson model: Mean-field theory and the RPA. Europhys. Lett. 22, 257–263 (1993).

    Article  ADS  Google Scholar 

  22. Krauth, W. & Trivedi, N. Mott and superfluid transitions in a strongly interacting lattice boson system. Europhys. Lett. 14, 627–632 (1991).

    Article  ADS  Google Scholar 

  23. Krauth, W., Trivedi, N. & Ceperley, D. Superfluid-insulator transition in disordered boson systems. Phys. Rev. Lett. 67, 2307–2310 (1991).

    Article  ADS  Google Scholar 

  24. Xie, Z. W. & Liu, W. M. Superfluid Mott-insulator transition of dipolar bosons in an optical lattice. Phys. Rev. A 70, 045602 (2004).

    Article  ADS  Google Scholar 

  25. Albus, A., Illuminati, F. & Eisert, J. Mixtures of bosonic and fermionic atoms in optical lattices. Phys. Rev. A 68, 023606 (2003).

    Article  ADS  Google Scholar 

  26. Lewenstein, M., Santos, L., Baranov, M. A. & Fehrmann, H. Atomic Bose–Fermi mixtures in an optical lattice. Phys. Rev. Lett. 92, 050401 (2004).

    Article  ADS  Google Scholar 

  27. Illuminati, F. & Albus, A. High temperature atomic superfluidity in lattice Bose–Fermi mixtures. Phys. Rev. Lett. 93, 090406 (2004).

    Article  ADS  Google Scholar 

  28. Cramer, M., Eisert, J. & Illuminati, F. Inhomogeneous atomic Bose–Fermi mixtures in cubic lattices. Phys. Rev. Lett. 93, 190405 (2004).

    Article  ADS  Google Scholar 

  29. Fehrmann, H., Baranov, M. A., Damski, B., Lewenstein, M. & Santos, L. Mean-field theory of Bose–Fermi mixtures in optical lattices. Opt. Commun. 243, 23–31 (2004).

    Article  ADS  Google Scholar 

  30. Littlewood, P. B. et al. Models of coherent exciton condensation. J. Phys. Condens. Matter 16, S3597–S3620 (2004).

    Article  Google Scholar 

  31. Olivero, P. et al. Ion beam assisted lift-off technique for three-dimensional micromachining of free standing single-crystal diamond. Adv. Mater. 17, 2427–2430 (2005).

    Article  Google Scholar 

  32. Baldwin, J. W., Zalalutdinov, M., Feygelson, T., Butler, J. E. & Houston, B. H. Fabrication of short-wavelength photonic crystals in wide-band-gap nanocrystalline diamond films. J. Vac. Sci. Technol. B 24, 50–54 (2006).

    Article  Google Scholar 

  33. Tomljenovic-Hanic, S., Steel, M. J., de Sterke, C. M. & Salzman, J. Diamond based photonic crystal microcavities. Opt. Express 14, 3556 (2006).

    Article  ADS  Google Scholar 

  34. Greentree, A. D. et al. Critical components for diamond-based quantum coherent devices. J. Phys. Condens. Matter 18, S825–S842 (2006).

    Article  Google Scholar 

  35. Song, B.-S., Noda, S., Asano, T. & Akahane, Y. Ultra-high-Q photonic double-heterostructure nanocavity. Nature Mater. 4, 207–210 (2005).

    Article  ADS  Google Scholar 

  36. Jamieson, D. N. et al. Controlled shallow single-ion implantation in silicon using an active substrate for sub-20-keV ions. Appl. Phys. Lett. 86, 202101 (2005).

    Article  ADS  Google Scholar 

  37. Tamarat, Ph. et al. Stark shift control of single optical centers in diamond. Phys. Rev. Lett. 97, 083002 (2006).

    Article  ADS  Google Scholar 

  38. Wallraff, A. et al. Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics. Nature 431, 162–167 (2004).

    Article  ADS  Google Scholar 

  39. Weidinger, M., Varcoe, B. T., Heerlein, R. & Walther, H. Trapping states in the micromaser. Phys. Rev. Lett. 82, 3795 (1999).

    Article  ADS  Google Scholar 

  40. Trupke, M. et al. Microfabricated high-finesse optical cavity with open access and small volume. Appl. Phys. Lett. 87, 211106 (2005).

    Article  ADS  Google Scholar 

  41. Schuster, D. I. et al. Resolving photon number states in a superconducting circuit. Preprint at <http://arxiv.org/abs/cond-mat/0608693> (2006).

  42. Freedman, M., Nayak, C. & Shtengel, K. Extended Hubbard model with ring exchange: A route to a non-Abelian topological phase. Phys. Rev. Lett. 94, 066401 (2005).

    Article  ADS  Google Scholar 

  43. Kosterlitz, J. M. & Thouless, D. J. Ordering, metastability and phase transitions in two-dimensional systems. J. Phys. C 6, 1181–1203 (1973).

    Article  ADS  Google Scholar 

  44. Scarola, V. W. & Das Sarma, S. Quantum phases of the extended Bose–Hubbard hamiltonian: The possibility of a supersolid state of cold atoms in optical lattices. Phys. Rev. Lett. 95, 033003 (2005).

    Article  ADS  Google Scholar 

  45. Rokhsar, D. S. & Kotliar, B. G. Gutzwiller projection for bosons. Phys. Rev. B 44, 10328–10322 (1991).

    Article  ADS  Google Scholar 

  46. Krauth, W., Caffarel, M. & Bouchaud, J.-P. Gutzwiller wave function for a model of strongly interacting bosons. Phys. Rev. B 45, 3137–3140 (1992).

    Article  ADS  Google Scholar 

Download references

Acknowledgements

The authors would like to acknowledge useful discussions with P. Eastham, M. Friesen, D. Jamieson, R. Joynt, R. Kalish, P. Littlewood, A. Martin, M. Makin, G. Milburn, J. Salzman and H. Wiseman. C.T. is funded by a USA National Science Foundation Math and Physical Sciences Distinguished International Postdoctoral Research Fellowship. This work was supported by the Australian Research Council, the Australian Government and by the US National Security Agency (NSA), Advanced Research and Development Activity (ARDA) and the Army Research Office (ARO) under Contract Nos. W911NF-04-1-0290 and W911NF-05-1-0284.

Author information

Authors and Affiliations

  1. Centre for Quantum Computer Technology, School of Physics, The University of Melbourne, Victoria, 3010, Australia

    Andrew D. Greentree, Charles Tahan, Jared H. Cole & Lloyd C. L. Hollenberg

  2. Cavendish Laboratory, University of Cambridge, JJ Thomson Ave, Cambridge, CB3 0HE, UK

    Charles Tahan

Authors
  1. Andrew D. Greentree
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Charles Tahan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jared H. Cole
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Lloyd C. L. Hollenberg
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

A.D.G. and J.H.C. conceived the original concept, which was developed by C.T. and L.C.L.H. C.T. initiated the quantum many-body analysis and A.D.G. and C.T. carried out the mean-field calculations. All authors contributed to writing the paper, and checking and interpreting the results.

Corresponding author

Correspondence to Andrew D. Greentree.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Greentree, A., Tahan, C., Cole, J. et al. Quantum phase transitions of light. Nature Phys 2, 856–861 (2006). https://doi.org/10.1038/nphys466

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/nphys466

This article is cited by

Search

Advanced search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing