Photonics Research, 2019, 7 (8): 08000862, Published Online: Jul. 17, 2019
Exceptional points of any order in a single, lossy waveguide beam splitter by photon-number-resolved detection Download: 587次
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Fig. 1. (a) Schematic of a single, lossy waveguide beam splitter excited with N indistinguishable photons prepared in the state | m ) ≔ | N − m , m ⟩ = | N − m ⟩ a | m ⟩ b , where a represents the neutral (gray) waveguide and b is the lossy (red) waveguide. (b) Mapping onto the N -photon subspace spanned by ( N + 1 ) multiphoton states | m ) , represented as a tight-binding lattice model. The coupling between adjacent “modes” is given by matrix elements of J ^ x ; the linearly increasing loss is also shown. (c) Flow of eigenvalues of H ^ N for N = 4 . R ( λ r ) shows level attraction with an EP of order five at Γ = 2 κ ; I ( λ r ) shows the emergence of slow modes past the transition. (d) Intensity I ( z ) shows the fraction of trials where the system remains in the N -photon subspace, i.e., the post-selection probability. It reflects the order of the exceptional point. The beam splitter parameters are ω 0 = κ = 1 cm − 1 , and the initial state is | ψ ( 0 ) ⟩ = | 0 ) .
Fig. 2. Evolution of spin-projections J r for the N + 1 eigenmodes in the post-selected manifold with N = 4 (left column) and N = 5 (right column) photons, considering different values of the dissipation coefficient: (a), (b) Γ = Γ c / 2 , (c), (d) Γ = 0.99 Γ c , and (e), (f) Γ = 1.5 Γ c . The vector coordinates in the ( J x , J y , J z ) space are defined by the expectation values of the J ^ α operators in each eigenstate. When Γ < Γ c , the spin projections are in the x − y plane; at the EP, they coalesce along the positive y axis; and when Γ > Γ c , they are in the x − z plane.
Fig. 3. Mode occupation dynamics in the post-selected manifold with NOON state input. (a) For N = 5 and small loss, the dynamics show asymmetric oscillations. (b) At the EP, P ( | m ) , z ) reaches a steady state with most of the weight localized in the low-loss region. (c) After the transition, the steady-state is reached more slowly. (d)–(f) show qualitatively similar results for an N = 8 NOON state input. The waveguide beam splitter parameters are set to ω 0 = κ = 1 cm − 1 .
Mario A. Quiroz-Juárez, Armando Perez-Leija, Konrad Tschernig, Blas M. Rodríguez-Lara, Omar S. Magaña-Loaiza, Kurt Busch, Yogesh N. Joglekar, Roberto de J. León-Montiel. Exceptional points of any order in a single, lossy waveguide beam splitter by photon-number-resolved detection[J]. Photonics Research, 2019, 7(8): 08000862.