Quantum many-body models with cold atoms coupled to photonic crystals

Journal name:
Nature Photonics
Year published:
Published online


Using cold atoms to simulate strongly interacting quantum systems is an exciting frontier of physics. However, because atoms are nominally neutral point particles, this limits the types of interaction that can be produced. We propose to use the powerful new platform of cold atoms trapped near nanophotonic systems to extend these limits, enabling a novel quantum material in which atomic spin degrees of freedom, motion and photons strongly couple over long distances. In this system, an atom trapped near a photonic crystal seeds a localized, tunable cavity mode around the atomic position. We find that this effective cavity facilitates interactions with other atoms within the cavity length, in a way that can be made robust against realistic imperfections. Finally, we show that such phenomena should be accessible using one-dimensional photonic crystal waveguides in which coupling to atoms has already been experimentally demonstrated.

At a glance


  1. From cavity-QED to atom-induced cavities in photonic crystals.
    Figure 1: From cavity-QED to atom-induced cavities in photonic crystals.

    a, Two atoms are coupled with strength gc to a single mode of a Fabry–Perot cavity, enabling an excited atom (atom 1) to transfer its excitation to atom 2 and back. The coherence of this process is reduced by the cavity decay (rate κ) and atomic spontaneous emission into free space (rate γ). b, Photonic crystals are alternating dielectric materials, shown here as oval air holes in a dielectric waveguide, with unit cell length a. A defect, such as that caused by removing or altering the hole sizes, can lead to a localized photonic mode (red). Atoms coupled to such a system may then interact via this mode in an analogous manner to that in a. c, A typical band structure of a one-dimensional photonic crystal, illustrating the guided mode frequency ωk versus the Bloch wavevector k in the first Brillouin zone. We are interested in the case where atoms coupled to the crystal have resonance frequency ωa close to the band edge frequency ωb, with Δ ≡ ωa − ωb. d, An atom near a photonic crystal can act as a defect, inducing its own cavity mode with an exponentially decaying envelope (red). A second atom can couple to this mode to produce an interaction similar to that in a and b, but where the strength now depends on the inter-atomic distance.

  2. Effective cavity mode properties.
    Figure 2: Effective cavity mode properties.

    a, Energy level diagram for the photonic crystal dressed state (blue). The dressed state energy ω is detuned by δ from the band edge into the bandgap (band shown in red). The atom is coupled to an effective cavity mode with frequency formed by superposition of modes in the band. b, The detuning δ approaches 0 when Δ/β ≪ −1 and approaches Δ when Δ/β ≫ 1. c, The photonic component of the dressed state has an exponentially decaying envelope around the atomic position. Increasing Δ decreases the length scale L of the exponential decay and the photonic part of the bound state superposition. d, The atomic excited state population of , Pe = cos2(θ) (green), increases as a function of Δ, while the population of the photon mode, Pp = sin2(θ) (red), decreases as the state switches from photonic to atomic. e, The length of the effective cavity decreases with Δ. Here L is in units of the lattice constant a, calculated for α = 10.6 and β = 4.75 × 10−7ωb, which is consistent with the ‘alligator’ photonic crystal waveguide (see main text)32.

  3. Comparison of the single-band model with numerical calculations.
    Figure 3: Comparison of the single-band model with numerical calculations.

    a, Band structure of the fundamental TE mode of the one-dimensional ‘alligator’ photonic crystal waveguide (APCW), designed for coupling to the D1 line of atomic caesium near the photonic band-edge frequency ωb/2π = 333 THz (refs 19,32). The calculated band structure has a curvature α ≈ 10.6 near the band edge at k0 = π/a. Inset: the dielectric profile of the APCW. Red circles denote the location of trapped atoms. b, Atom–atom coupling strength Uij evaluated using FDTD simulations (solid circles) and the single-band model from equation (4) (solid lines). Results are plotted for atomic detunings from the band edge Δ/2π = 400 (black), 800 (red), 1,300 (blue), 2,800 (magenta) GHz. Inset: FDTD results where the contribution from all other photonic and free-space modes in the APCW (open circles) has been estimated numerically and subtracted (see Methods).

  4. Designing interaction potentials.
    Figure 4: Designing interaction potentials.

    a, Driven (black) Λ and (black and blue) four-level system. In the Λ scheme, transition |g〉−|e〉 couples with strength g to the photonic crystal modes, while |s〉−|e〉 is pumped by a laser with detuning δL and Rabi frequency Ω. Interactions between the x-component of the effective spin can be achieved by adding level |e′〉, where the transition |s〉−|e′〉 also couples to the modes of the photonic crystal, while a second pump drives |g〉−|e′〉. b, Approximate power law interactions between atoms over a finite region can be achieved by summing the different exponential interactions associated with multiple drive fields. This is illustrated here over 50 lattice sites, where two exponentials are added to yield an η = 1/4 power law: (solid blue curve). Error f(z) − z−1/4 is given by the dashed curve. Here w1 = 0.5480, w2 = 0.5684, s1 = 0.2916 and s2 = 0.0089 could be achieved by detuning one laser from the band edge by 1.723 × 10–3ωb and the second by 1.612 × 10–6ωb for α = 0.2.


  1. Bloch, I., Dalibard, J. & Nascimbene, S. Quantum simulations with ultracold quantum gases. Nature Phys. 8, 267276 (2012).
  2. Batrouni, G. G., Scalettar, R. T., Zimanyi, G. T. & Kampf, A. P. Supersolids in the Bose–Hubbard Hamiltonian. Phys. Rev. Lett. 74, 25272530 (1995).
  3. Wigner, E. On the interaction of electrons in metals. Phys. Rev. 46, 10021011 (1934).
  4. Micheli, A., Brennen, G. K. & Zoller, P. A toolbox for lattice-spin models with polar molecules. Nature Phys. 2, 341347 (2006).
  5. Campa, A., Dauxois, T. & Ruffo, S. Statistical mechanics and dynamics of solvable models with long-range interactions. Phys. Rep. 480, 57159 (2009).
  6. Shahmoon, E., Mazets, I. & Kurizki, G. Non-additivity in laser-illuminated many-atom systems. Opt. Lett. 39, 36743677 (2014).
  7. Hauke, P. & Tagliacozzo, L. Spread of correlations in long-range interacting quantum systems. Phys. Rev. Lett. 111, 207202 (2013).
  8. Richerme, P. et al. Non-local propagation of correlations in quantum systems with long-range interactions. Nature 511, 198201 (2014).
  9. Jurcevic, P. et al. Quasiparticle engineering and entanglement propagation in a quantum many-body system. Nature 511, 202205 (2014).
  10. Lahaye, T., Menotti, C., Santos, L., Lewenstein, M. & Pfau, T. The physics of dipolar bosonic quantum gases. Rep. Prog. Phys. 72, 126401 (2009).
  11. Griesmaier, A., Werner, J., Hensler, S., Stuhler, J. & Pfau, T. Bose–Einstein condensation of chromium. Phys. Rev. Lett. 94, 160401 (2005).
  12. Lu, M., Burdick, N. Q., Youn, S. H. & Lev, B. L. Strongly dipolar Bose–Einstein condensate of dysprosium. Phys. Rev. Lett. 107, 190401 (2011).
  13. Saffman, M., Walker, T. G. & Mølmer, K. Quantum information with Rydberg atoms. Rev. Mod. Phys. 82, 23132363 (2010).
  14. Ni, K.-K. et al. A high phase-space-density gas of polar molecules. Science 322, 231235 (2008).
  15. Khitrova, G., Gibbs, H. M., Kira, M., Koch, S. W. & Scherer, A. Vacuum Rabi splitting in semiconductors. Nature Phys. 2, 8190 (2006).
  16. Vetsch, E. et al. Optical interface created by laser-cooled atoms trapped in the evanescent field surrounding an optical nanofiber. Phys. Rev. Lett. 104, 203603 (2010).
  17. Goban, A. et al. Demonstration of a state-insensitive, compensated nanofiber trap. Phys. Rev. Lett. 109, 033603 (2012).
  18. Thompson, J. D. et al. Coupling a single trapped atom to a nanoscale optical cavity. Science 340, 12021205 (2013).
  19. Goban, A. et al. Atom–light interactions in photonic crystals. Nature Commun. 5, 3808 (2014).
  20. Kimble, H. J. The quantum internet. Nature 453, 10231030 (2008).
  21. Greentree, A. D., Tahan, C., Cole, J. H. & Hollenberg, L. C. L. Quantum phase transitions of light. Nature Phys. 2, 856861 (2006).
  22. Hartmann, M. J., Brandao, F. G. S. L. & Plenio, M. B. Strongly interacting polaritons in coupled arrays of cavities. Nature Phys. 2, 849855 (2006).
  23. Angelakis, D. G., Santos, M. F. & Bose, S. Photon-blockade-induced Mott transitions and xy spin models in coupled cavity arrays. Phys. Rev. A 76, 031805 (2007).
  24. Joannopoulos, J. D., Johnson, S. G., Winn, J. N. & Meade, R. D. Photonic Crystals: Molding the Flow of Light 2nd edn (Princeton Univ. Press, 2008).
  25. Kurizki, G. Two-atom resonant radiative coupling in photonic band structures. Phys. Rev. A 42, 29152924 (1990).
  26. John, S. & Wang, J. Quantum optics of localized light in a photonic band gap. Phys. Rev. B 43, 1277212789 (1991).
  27. John, S. & Wang, J. Quantum electrodynamics near a photonic band gap: photon bound states and dressed atoms. Phys. Rev. Lett. 64, 24182421 (1990).
  28. Bay, S., Lambropoulos, P. & Mølmer, K. Atom–atom interaction in strongly modified reservoirs. Phys. Rev. A 55, 14851496 (1997).
  29. Lambropoulos, P., Nikolopoulos, G. M., Nielsen, T. R. & Bay, S. Fundamental quantum optics in structured reservoirs. Rep. Prog. Phys. 63, 455 (2000).
  30. Shahmoon, E. & Kurizki, G. Nonradiative interaction and entanglement between distant atoms. Phys. Rev. A 87, 033831 (2013).
  31. González-Tudela, A., Hung, C.-L., Chang, D. E., Cirac, J. I. & Kimble, H. J. Subwavelength vacuum lattices and atom–atom interactions in two-dimensional photonic crystals. Nature Photon. http://dx.doi.org/10.1038/nphoton.2015.54 (2015).
  32. Yu, S.-P. et al. Nanowire photonic crystal waveguides for single-atom trapping and strong light–matter interactions. Appl. Phys. Lett. 104, 111103 (2014).
  33. Agarwal, G., Gupta, S. & Puri, R. Fundamentals of Cavity Quantum Electrodynamics (World Scientific Publishing, 1995).
  34. Plenio, M. B., Huelga, S. F., Beige, A. & Knight, P. L. Cavity-loss-induced generation of entangled atoms. Phys. Rev. A 59, 24682475 (1999).
  35. Domokos, P. & Ritsch, H. Collective cooling and self-organization of atoms in a cavity. Phys. Rev. Lett. 89, 253003 (2002).
  36. Black, A. T., Chan, H. W. & Vuletić, V. Observation of collective friction forces due to spatial self-organization of atoms: from Rayleigh to Bragg scattering. Phys. Rev. Lett. 91, 203001 (2003).
  37. Baumann, K., Guerlin, C., Brennecke, F. & Esslinger, T. Dicke quantum phase transition with a superfluid gas in an optical cavity. Nature 464, 13011306 (2010).
  38. Hung, C.-L., Meenehan, S. M., Chang, D. E., Painter, O. & Kimble, H. J. Trapped atoms in one-dimensional photonic crystals. New J. Phys. 15, 083026 (2013).
  39. Markos, P. & Soukoulis, C. M. Wave Propagation: From Electrons to Photonic Crystals and Left-Handed Materials (Princeton Univ Press, 2010).
  40. Porras, D. & Cirac, J. I. Effective quantum spin systems with trapped ions. Phys. Rev. Lett. 92, 207901 (2004).
  41. Islam, R. et al. Emergence and frustration of magnetism with variable-range interactions in a quantum simulator. Science 340, 583587 (2013).
  42. Gross, M. & Haroche, S. Superradiance: an essay on the theory of collective spontaneous emission. Phys. Rep. 93, 301396 (1982).
  43. Mattioli, M., Dalmonte, M., Lechner, W. & Pupillo, G. Cluster Luttinger liquids of Rydberg-dressed atoms in optical lattices. Phys. Rev. Lett. 111, 165302 (2013).
  44. Basko, D. M., Aleiner, I. L. & Altshuler, B. L. On the problem of many-body localization. Preprint at http://arXiv.org/abs/cond-mat/0602510 (2006).
  45. Longo, P., Schmitteckert, P. & Busch, K. Few-photon transport in low-dimensional systems. Phys. Rev. A 83, 063828 (2011).
  46. Firstenberg, O. et al. Attractive photons in a quantum nonlinear medium. Nature 502, 7175 (2013).
  47. Nayak, K. P. & Hakuta, K. Photonic crystal formation on optical nanofibers using femtosecond laser ablation technique. Opt. Express 21, 24802490 (2013).
  48. Chang, D. E., Jiang, L., Gorshkov, A. V. & Kimble, H. J. Cavity QED with atomic mirrors. New J. Phys. 14, 063003 (2012).
  49. Eichenfield, M., Chan, J., Camacho, R. M., Vahala, K. J. & Painter, O. Optomechanical crystals. Nature 462, 7882 (2009).
  50. Dung, H. T., Knöll, L. & Welsch, D.-G. Resonant dipole–dipole interaction in the presence of dispersing and absorbing surroundings. Phys. Rev. A 66, 063810 (2002).

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Author information

  1. Present address: Department of Physics and Astronomy, Purdue University, West Lafayette, Indiana 47907, USA

    • C.-L. Hung


  1. ICFO-Institut de Ciencies Fotoniques, 08860 Castelldefels, Barcelona, Spain

    • J. S. Douglas,
    • H. Habibian &
    • D. E. Chang
  2. Norman Bridge Laboratory of Physics and Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA

    • C.-L. Hung &
    • H. J. Kimble
  3. Joint Quantum Institute and Joint Center for Quantum Information and Computer Science, NIST/University of Maryland, College Park, Maryland 20742, USA

    • A. V. Gorshkov


J.S.D., H.H. and C.-L.H. performed the calculations. All authors contributed ideas. J.S.D. and D.E.C. wrote the manuscript.

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