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Topological phase transitions and chiral inelastic transport induced by the squeezing of light

Peano, Vittorio, Houde, Martin, Brendel, Christian, Marquardt, Florian, Clerk, Aashish A.·2016·Nature Communications
DOI 10.1038/ncomms10779PMC PMC4778048
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Abstract

There is enormous interest in engineering topological photonic systems. Despite intense activity, most works on topological photonic states (and more generally bosonic states) amount in the end to replicating a well-known fermionic single-particle Hamiltonian. Here we show how the squeezing of light can lead to the formation of qualitatively new kinds of topological states. Such states are characterized by non-trivial Chern numbers, and exhibit protected edge modes, which give rise to chiral elastic and inelastic photon transport. These topological bosonic states are not equivalent to their fermionic (topological superconductor) counterparts and, in addition, cannot be mapped by a local transformation onto topological states found in particle-conserving models. They thus represent a new type of topological system. We study this physics in detail in the case of a kagome lattice model, and discuss possible realizations using nonlinear photonic crystals or superconducting circuits.

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Figure 1 Setup figure. (a) An array of nonlinear cavities forming a kagome lattice. (b) Photons hop between nearest-neighbour sites with rate J. Each cavity is driven parametrically leading to the creation of photon pairs on the same lattice site (rate νon) and on nearest-neighbour sites (rate νoff). A spatial pattern of the driving phase is imprinted on the parametric interactions, breaking the time-reversal symmetry (but preserving the rotational symmetry).
Figure 1 Setup figure. (a) An array of nonlinear cavities forming a kagome lattice. (b) Photons hop between nearest-neighbour sites with rate J. Each cavity is driven parametrically leading to the creation of photon pairs on the same lattice site (rate νon) and on nearest-neighbour sites (rate νoff). A spatial pattern of the driving phase is imprinted on the parametric interactions, breaking the time-reversal symmetry (but preserving the rotational symmetry).CC BY · PubMed Central
Figure 2 Topological Band structure. (a,b) 3D plots of the bulk band structure. The hexagonal Brillouin zone is also shown. (a) In the absence of parametric driving, neighbouring bands touch at the rotational symmetry points K, K′ and Γ. (b) The parametric driving opens a gap between subsequent bands. For the chosen parameters, there is a global band gap between the second and third band. (d) Hole and particle bands, ±Em[kx], in a strip geometry (sketched in c). The line intensity is proportional to the weight of the corresponding resonance in the photon spectral function, Supplementary Note 1. The edge states localized on the right (left) edge, plotted in green (blue), have positive (negative) velocity. Parameters: Hopping rate J=0.02ω0 (ω0 is the onsite frequency); (b,d), the parametric couplings are νon=−0.085ω0 and νoff=0.22ω0.
Figure 2 Topological Band structure. (a,b) 3D plots of the bulk band structure. The hexagonal Brillouin zone is also shown. (a) In the absence of parametric driving, neighbouring bands touch at the rotational symmetry points K, K′ and Γ. (b) The parametric driving opens a gap between subsequent bands. For the chosen parameters, there is a global band gap between the second and third band. (d) Hole and particle bands, ±Em[kx], in a strip geometry (sketched in c). The line intensity is proportional to the weight of the corresponding resonance in the photon spectral function, Supplementary Note 1. The edge states localized on the right (left) edge, plotted in green (blue), have positive (negative) velocity. Parameters: Hopping rate J=0.02ω0 (ω0 is the onsite frequency); (b,d), the parametric couplings are νon=−0.085ω0 and νoff=0.22ω0.CC BY · PubMed Central
Figure 3 Symplectic Topological phase diagrams. (a)Topological phase diagram for the parametrically driven kagome lattice model. The y (x) axis corresponds to the strength of the onsite parametric drive νon (offsite parametric drive νoff), and different colours correspond to different triplets C=(C1,C2,C3) of Chern numbers for the three bands of the model. Note that only the grey and dark-grey phases are found in the particle-conserving version of our model with a staggered field. We have fixed the hopping rate J/ω0=0.02, and the vorticity of the pump mν=1. (b) Same phase diagram, but now plotted in terms of the effective flux and effective parametric drive experienced by α quasiparticles.
Figure 3 Symplectic Topological phase diagrams. (a)Topological phase diagram for the parametrically driven kagome lattice model. The y (x) axis corresponds to the strength of the onsite parametric drive νon (offsite parametric drive νoff), and different colours correspond to different triplets C=(C1,C2,C3) of Chern numbers for the three bands of the model. Note that only the grey and dark-grey phases are found in the particle-conserving version of our model with a staggered field. We have fixed the hopping rate J/ω0=0.02, and the vorticity of the pump mν=1. (b) Same phase diagram, but now plotted in terms of the effective flux and effective parametric drive experienced by α quasiparticles.CC BY · PubMed Central
Figure 4 Topologically protected transport in a finite system. (a) A probe beam at frequency ωp inside the bulk band gap is focused on a site (marked in yellow) at the edge of a finite sample. The probability map of the light transmitted inelastically at frequency ωL−ωp (where ωL is the frequency of the drive tone applied to the auxiliary pump modes) clearly shows that the transport is chiral. (b) The elastic and inelastic transmission probability to a pair of sites along the edges (indicated in red in a) is plotted in blue and green, respectively. A cut through the bulk bands is shown to the left. (c,d) Sketch of the relevant scattering processes and energy scales. The inelastic (elastic) transmission has a larger rate when the light is injected in the hole (particle) band gap. Parameters: Hopping rate J=0.02ω0 (ω0 is the onsite frequency), parametric couplings νon=0.4ω0 and νoff=0.02ω0, optical decay rate κ=0.001ω0. (a) ωp−ωL/2=0.95ω0.
Figure 4 Topologically protected transport in a finite system. (a) A probe beam at frequency ωp inside the bulk band gap is focused on a site (marked in yellow) at the edge of a finite sample. The probability map of the light transmitted inelastically at frequency ωL−ωp (where ωL is the frequency of the drive tone applied to the auxiliary pump modes) clearly shows that the transport is chiral. (b) The elastic and inelastic transmission probability to a pair of sites along the edges (indicated in red in a) is plotted in blue and green, respectively. A cut through the bulk bands is shown to the left. (c,d) Sketch of the relevant scattering processes and energy scales. The inelastic (elastic) transmission has a larger rate when the light is injected in the hole (particle) band gap. Parameters: Hopping rate J=0.02ω0 (ω0 is the onsite frequency), parametric couplings νon=0.4ω0 and νoff=0.02ω0, optical decay rate κ=0.001ω0. (a) ωp−ωL/2=0.95ω0.CC BY · PubMed Central

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Metadata

Authors
Peano, V., Houde, M., Brendel, C., Marquardt, F., & Clerk, A. A.
Venue
Nature Communications
Publisher
Springer Science and Business Media LLC
Published
2016-03-02
Volume
7
Issue
1
ISSN
2041-1723
DOI
10.1038/ncomms10779
PMC
PMC4778048
Cited by
107
References
4
Language
en
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