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Measuring topological invariants in small photonic lattices

Bardyn, C-E, Huber, S D, Zilberberg, O·2014·New Journal of Physics
DOI 10.1088/1367-2630/16/12/123013
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FIG. 1 (Color online) Quantum simulation in the unit cell. (a) Unit-cell configuration consisting of N u lattice sites in each spatial direction u (here u = 1 , 2 in 2D), with twisted boundary conditions defined by periodic boundary conditions with additional twist angles φ u at the boundaries. (b) Gaugeequivalent configuration with periodic boundary conditions and fluxes φ u threading the unit cell.
FIG. 1 (Color online) Quantum simulation in the unit cell. (a) Unit-cell configuration consisting of N u lattice sites in each spatial direction u (here u = 1 , 2 in 2D), with twisted boundary conditions defined by periodic boundary conditions with additional twist angles φ u at the boundaries. (b) Gaugeequivalent configuration with periodic boundary conditions and fluxes φ u threading the unit cell.CC-BY · From the PDF
FIG. 2 (Color online). Unit-cell photonic simulation of a 2D quantum Hall-like Hofstadter model with n p = p/q = 1 / 6 effective magnetic flux quanta per (square-lattice) plaquette. (a) Relevant (magnetic) unit cell containing q = 6 lattice sites, with uniform hopping amplitudes t and hopping phases φ ij (or gauge; see Eq. (6)) chosen here so that N 1 = 3, N 2 = 2. Twisted boundary conditions are indicated by the corresponding twist angles φ 1 and φ 2 . (b) Left: Spectrum E ( φ 1 , φ 2 ) (shown here as a function of φ 1 with φ 2 = 0) exhibiting q = 6 energy levels which generate the bands of the simulated infinite system as the twist angles are continuously varied. Right: Table presenting the Chern number ν extracted for each band using a coarsely discretized (7 × 7 grid) twist-angle space, along with the (estimated) maximum photon loss rate γ max allowing to extract it. Note that γ max is much smaller for the central band because the latter consists of 2 energy levels which must be spectrally resolvable-for all chosen values of φ 1 and φ 2 -for our scheme to work. (c) From left to right: Steady-state coherent-field amplitude, normalized amplitude, and phase found in cavity 4 (see (a)), for example, as a function of the twist angles (also shown is the 7 × 7 discretization grid that is used). All three plots were obtained for t = 1 and γ = 0 . 1. The driving was applied to cavity 1 with a coherent field amplitude f = 1 and frequency Ω tuned into resonance with the lowest energy level E ( φ 1 , φ 2 ) (see (b)).
FIG. 2 (Color online). Unit-cell photonic simulation of a 2D quantum Hall-like Hofstadter model with n p = p/q = 1 / 6 effective magnetic flux quanta per (square-lattice) plaquette. (a) Relevant (magnetic) unit cell containing q = 6 lattice sites, with uniform hopping amplitudes t and hopping phases φ ij (or gauge; see Eq. (6)) chosen here so that N 1 = 3, N 2 = 2. Twisted boundary conditions are indicated by the corresponding twist angles φ 1 and φ 2 . (b) Left: Spectrum E ( φ 1 , φ 2 ) (shown here as a function of φ 1 with φ 2 = 0) exhibiting q = 6 energy levels which generate the bands of the simulated infinite system as the twist angles are continuously varied. Right: Table presenting the Chern number ν extracted for each band using a coarsely discretized (7 × 7 grid) twist-angle space, along with the (estimated) maximum photon loss rate γ max allowing to extract it. Note that γ max is much smaller for the central band because the latter consists of 2 energy levels which must be spectrally resolvable-for all chosen values of φ 1 and φ 2 -for our scheme to work. (c) From left to right: Steady-state coherent-field amplitude, normalized amplitude, and phase found in cavity 4 (see (a)), for example, as a function of the twist angles (also shown is the 7 × 7 discretization grid that is used). All three plots were obtained for t = 1 and γ = 0 . 1. The driving was applied to cavity 1 with a coherent field amplitude f = 1 and frequency Ω tuned into resonance with the lowest energy level E ( φ 1 , φ 2 ) (see (b)).CC-BY · From the PDF
FIG. 1 (Color online). Unit-cell photonic simulation of an infinite quantum Hall-like system with n p = p/q = 1 / 3 effective magnetic flux quanta per (square-lattice) plaquette. We use a mixed position-Fourier representation in order to treat the case q = 3, which is prime. (a) Relevant unit cell of the Harper model containing q = 3 lattice sites, with uniform hopping amplitudes t and modulated on-site potential [cf. Eq. (1)]. Twisted boundary conditions are indicated by the corresponding twist angles φ 1 and φ 2 . (b) Left: Spectrum E ( φ 1 , φ 2 ) (shown here as a function of φ 1 with φ 2 = 0) exhibiting q = 3 energy levels generating the bands of the simulated infinite system as the twist angles are continuously varied. Right: Table indicating the Chern number ν extracted for each band using a coarsely discretized (4 × 4 grid) twistangle space, along with the (estimated) maximum photon loss rate γ max allowing to extract it. (c) From left to right: Steady-state coherent-field amplitude, normalized amplitude, and phase obtained in cavity 3 (see (a)), for example, as a function of the twist angles (showing the 4 × 4 discretization grid that is used). All three plots were obtained for t = 1 and γ = 0 . 4, driving cavity 1 with a coherent field of amplitude f = 1 and frequency Ω tuned into resonance with the central energy level E 2 ( φ 1 , φ 2 ) (see (b)).
FIG. 1 (Color online). Unit-cell photonic simulation of an infinite quantum Hall-like system with n p = p/q = 1 / 3 effective magnetic flux quanta per (square-lattice) plaquette. We use a mixed position-Fourier representation in order to treat the case q = 3, which is prime. (a) Relevant unit cell of the Harper model containing q = 3 lattice sites, with uniform hopping amplitudes t and modulated on-site potential [cf. Eq. (1)]. Twisted boundary conditions are indicated by the corresponding twist angles φ 1 and φ 2 . (b) Left: Spectrum E ( φ 1 , φ 2 ) (shown here as a function of φ 1 with φ 2 = 0) exhibiting q = 3 energy levels generating the bands of the simulated infinite system as the twist angles are continuously varied. Right: Table indicating the Chern number ν extracted for each band using a coarsely discretized (4 × 4 grid) twistangle space, along with the (estimated) maximum photon loss rate γ max allowing to extract it. (c) From left to right: Steady-state coherent-field amplitude, normalized amplitude, and phase obtained in cavity 3 (see (a)), for example, as a function of the twist angles (showing the 4 × 4 discretization grid that is used). All three plots were obtained for t = 1 and γ = 0 . 4, driving cavity 1 with a coherent field of amplitude f = 1 and frequency Ω tuned into resonance with the central energy level E 2 ( φ 1 , φ 2 ) (see (b)).CC-BY · From the PDF

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Citations

Cites

Research works printed but not linked: 30

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Metadata

Authors
Bardyn, C.-E., Huber, S. D., & Zilberberg, O.
Venue
New Journal of Physics
Publisher
IOP Publishing
Published
2014-12-08
Volume
16
Issue
12
Pages
123013
ISSN
1367-2630
DOI
10.1088/1367-2630/16/12/123013
Cited by
25
References
5
Licence
Funders
Swiss National Science Foundation

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