Research ArticleQuantum Mechanics

Engineering two-photon high-dimensional states through quantum interference

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Science Advances  26 Feb 2016:
Vol. 2, no. 2, e1501165
DOI: 10.1126/sciadv.1501165
  • Fig. 1 Diagram of the experimental setup used to demonstrate high-dimensional HOM interference.

    The HOM filter, indicated by the beige section, consists of two mirrors and a 50:50 beamsplitter. Any antisymmetric state Embedded Image will result in one photon in each of the modes A′ and B′ and the detection of coincidences; any symmetric state Embedded Image, Embedded Image, or Embedded Image will result in a superposition of two photons in either A′ or B′ and the absence of a coincidence signal. SLM, spatial light modulator; SPAD, single-photon avalanche detector; SMF, single-mode optical fiber.

  • Fig. 2 Photon coincidence counts measured as a function of the path length difference.

    The red points denote input states Embedded Image, and the green points denote input states Embedded Image. We observe a HOM interference dip/peak when the path length difference is equal to zero. The data points shown are the average of 20 readings. A Gaussian function multiplied by a sinc function was fitted to the data and is shown as solid curves.

  • Fig. 3 Coincidence counts as a function of input state.

    (A) Graph showing the variation of coincidence counts for the input states |Ψ1〉, indicated by the green points, and |Φ1〉, indicated by the red points. The theoretical variation in coincidence counts of any |Ψ1〉 input state is equal to 2 sin2(2φ). For any |Φ1〉 input state, the coincidence counts should remain zero as a function of ≻. The normalized count rate is determined by dividing the coincidence rate in the dip/peak by that outside of the dip/peak. The error bars represent 1 SD from averaging 20 readings. (B) Graph showing the variation of coincidence counts for the input states |Ψ2〉, indicated by the blue points, and |Φ2〉, indicated by the orange points. The theoretical variation in coincidence counts of any |Ψ2〉 input state is equal to 2 sin2(4φ). (C) Normalized coincidence counts for the OAM ℓ = 1 and 2 Bell states. Only the Embedded Image and Embedded Image input states result in coincidences being recorded.

  • Fig. 4 The high-dimensional density matrices corresponding to the state before and after the filter.

    Only the real parts of the density matrices are shown. (A) The state before the filter, which in principle is given by Embedded Image; there is a strong contribution from the ℓ = 1, 2, and 3 subspaces in this six-dimensional state (36 × 36 matrix). (B) The state after the filter, which in principle is given by Embedded Image; the contribution from the ℓ = 2 subspace is 3.8 ± 0.2% of its original value.

  • Fig. 5 A comparison of the density matrices of the individual two-dimensional subspaces extracted from Fig. 4.

    Only real parts are shown in this 4 × 4 matrix of the two-dimensional subspace. (A to F) The first two columns (A, C, and E) show the subspaces before the filter; the final two columns (B, D, and F) show the subspaces after the filter. Experimental results are shown in the left column of each pair, and theoretical predictions are shown in the right column of each pair. The ℓ = 1 subspace is shown in green, ℓ = 2 in blue, and ℓ = 3 in red. Darker bars indicate negative values, whereas gray bars indicate absolute values less than 0.05.

  • Table 1 The trace of each subspace ρ before and after the filter.

    This provides the probability to project onto a particular subspace spanned by the states with OAM values ±ℓ.

    SubspaceTr[ρ] before the filterTr[ρ] after the filter
    ρ10.370 ± 0.0020.556 ± 0.001
    ρ20.319 ± 0.0020.018 ± 0.001
    ρ30.278 ± 0.0020.409 ± 0.001

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