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

Journal name:
Nature Photonics
Volume:
9,
Pages:
326–331
Year published:
DOI:
doi:10.1038/nphoton.2015.57
Received
Accepted
Published online

Abstract

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

Figures

  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.

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Author information

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

    • C.-L. Hung

Affiliations

  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

Contributions

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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The authors declare no competing financial interests.

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