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.
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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).
Birnbaum, K. M. et al. Photon blockade in an optical cavity with one trapped atom. Nature 436, 87–90 (2005).
Imamoğlu, A., Schmidt, H., Woods, G. & Deutsch, M. Strongly interacting photons in a nonlinear cavity. Phys. Rev. Lett. 79, 1467–1470 (1997).
Grangier, P., Walls, D. & Gheri, K. Comment on strongly interacting photons in a nonlinear cavity. Phys. Rev. Lett. 81, 2833 (1998).
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).
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).
Lang, R. J. & Yariv, A. An exact formulation of coupled-mode theory for coupled-cavity lasers. IEEE J. Quantum Electron. 24, 66–72 (1988).
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).
Altug, H., Englund, D. & Vučović, J. Ultrafast photonic crystal nanocavity laser. Nature Phys. 2, 484–488 (2006).
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).
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).
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).
Lee, P. A. & Ramakrishnan, T. V. Disordered electronic systems. Rev. Mod. Phys. 57, 287–337 (1985).
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).
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).
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).
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).
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).
van Oosten, D., van der Straten, P. & Stoof, H. T. C. Quantum phases in an optical lattice. Phys. Rev. A 63, 053601 (2001).
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).
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).
Krauth, W. & Trivedi, N. Mott and superfluid transitions in a strongly interacting lattice boson system. Europhys. Lett. 14, 627–632 (1991).
Krauth, W., Trivedi, N. & Ceperley, D. Superfluid-insulator transition in disordered boson systems. Phys. Rev. Lett. 67, 2307–2310 (1991).
Xie, Z. W. & Liu, W. M. Superfluid Mott-insulator transition of dipolar bosons in an optical lattice. Phys. Rev. A 70, 045602 (2004).
Albus, A., Illuminati, F. & Eisert, J. Mixtures of bosonic and fermionic atoms in optical lattices. Phys. Rev. A 68, 023606 (2003).
Lewenstein, M., Santos, L., Baranov, M. A. & Fehrmann, H. Atomic Bose–Fermi mixtures in an optical lattice. Phys. Rev. Lett. 92, 050401 (2004).
Illuminati, F. & Albus, A. High temperature atomic superfluidity in lattice Bose–Fermi mixtures. Phys. Rev. Lett. 93, 090406 (2004).
Cramer, M., Eisert, J. & Illuminati, F. Inhomogeneous atomic Bose–Fermi mixtures in cubic lattices. Phys. Rev. Lett. 93, 190405 (2004).
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).
Littlewood, P. B. et al. Models of coherent exciton condensation. J. Phys. Condens. Matter 16, S3597–S3620 (2004).
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).
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).
Tomljenovic-Hanic, S., Steel, M. J., de Sterke, C. M. & Salzman, J. Diamond based photonic crystal microcavities. Opt. Express 14, 3556 (2006).
Greentree, A. D. et al. Critical components for diamond-based quantum coherent devices. J. Phys. Condens. Matter 18, S825–S842 (2006).
Song, B.-S., Noda, S., Asano, T. & Akahane, Y. Ultra-high-Q photonic double-heterostructure nanocavity. Nature Mater. 4, 207–210 (2005).
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).
Tamarat, Ph. et al. Stark shift control of single optical centers in diamond. Phys. Rev. Lett. 97, 083002 (2006).
Wallraff, A. et al. Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics. Nature 431, 162–167 (2004).
Weidinger, M., Varcoe, B. T., Heerlein, R. & Walther, H. Trapping states in the micromaser. Phys. Rev. Lett. 82, 3795 (1999).
Trupke, M. et al. Microfabricated high-finesse optical cavity with open access and small volume. Appl. Phys. Lett. 87, 211106 (2005).
Schuster, D. I. et al. Resolving photon number states in a superconducting circuit. Preprint at <http://arxiv.org/abs/cond-mat/0608693> (2006).
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).
Kosterlitz, J. M. & Thouless, D. J. Ordering, metastability and phase transitions in two-dimensional systems. J. Phys. C 6, 1181–1203 (1973).
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).
Rokhsar, D. S. & Kotliar, B. G. Gutzwiller projection for bosons. Phys. Rev. B 44, 10328–10322 (1991).
Krauth, W., Caffarel, M. & Bouchaud, J.-P. Gutzwiller wave function for a model of strongly interacting bosons. Phys. Rev. B 45, 3137–3140 (1992).
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.
The authors declare no competing financial interests.
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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
Quantum Monte Carlo study of superradiant supersolid of light in the extended Jaynes-Cummings-Hubbard model
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