Imaging symmetric and antisymmetric behavior of orbital-angular-momentum-entangled two-photon states (2024)

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  Figure 1

  Polar diagrams showing the properties of the postselected SPDC state for (a) ${\ell }_p=0$ and (b) ${\ell }_p=1$. The radius of the polar diagram (indicated in the horizontal axis) represents the SPDC topological charge projection value $\ell$. Along a circumference for a specific $\ell$, the symmetric and antisymmetric coefficients define a point, and if this lies on the horizontal (vertical) axis, the postselected state is purely symmetric (antisymmetric). Points defined by the same interprism angle $\varphi$ are joined together smoothly to form the spiral shapes shown.

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  Figure 2

  Experimental setup. OAM-entangled photon pairs are generated by the SPDC process. The signal photon is transmitted through a Dove-prism pair (${\mathrm{DP}}_1$ and ${\mathrm{DP}}_2$) to manipulate the properties of the resulting photon pair, and the photon pairs interfere in a beam splitter (BS). While one of the photons emerging from the BS is spatially resolved upon detection by an ICCD camera, upon transmission through the image-preserving optical delay line (DL), the other photon is projected to a user-selected topological charge value prior to detection by an avalanche photodiode (APD).

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  Figure 3

  HOM interference behavior for ${\ell }_p=\ell =0$ and $\varphi ={45}^{\circ }$. Panel (a) shows the spatially resolved HOM interferograms for a selection of seven different signal-idler $\tau$ values, with the case $\tau =0$ corresponding to the central plot; the green dot in this plot indicates the pixel used for the single-pixel interferogram, as described in the main text. (b) Single-pixel HOM interferogram (black dots), along with the global interferogram obtained by summing over all ICCD pixel readouts (red dots), for temporal delay $\tau$ values ranging from $-200\mathrm{fs}$ ($-60\text{μ}\mathrm{m}$) to $+200\mathrm{fs}$ ($+60\text{μ}\mathrm{m}$). The curves are fits to sinc-squared functions for comparison purposes only, as detailed in the main text. (c) Transverse intensity of the two portions of the SPDC ring (for each of the signal and idler photons) clipped by the triangular mirror, one of which is rotated by the Dove-prism pair. The image shown was taken on the plane of the BS (${\mathrm{IP}}_1$) and has dimensions of $1.5\mathrm{mm}\times 1.4\mathrm{mm}$ ($119\times 110$ pixels of the ICCD sensor).

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  Figure 4

  HOM interference behavior for ${\ell }_p=0$, $\ell =1$, and $\varphi ={45}^{\circ }$. Panel (a) shows the spatially resolved HOM interferograms for a selection of seven different signal-idler $\tau$ values, with the case $\tau =0$ corresponding to the central plot; the green dot in this plot indicates the pixel used for the single-pixel interferogram, as described in the main text. (b) Single-pixel HOM interferogram (black dots), along with the global interferogram obtained by summing over all ICCD pixel readouts (red dots), for temporal delay $\tau$ values ranging from $-200\mathrm{fs}$ ($-60\text{μ}\mathrm{m}$) to $+200\mathrm{fs}$ ($+60\text{μ}\mathrm{m}$). The curves are fits to sinc-squared functions for comparison purposes only.

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  Figure 5

  HOM interference behavior for ${\ell }_p=0$ and $\ell =0,1,2,3,4$ and $\varphi ={45}^{\circ }$. Panel (a) shows the spatially resolved HOM interferograms at a delay value corresponding to $\mathrm{\Delta }z=-56\text{μ}\mathrm{m}$ ($\tau =-186$ fs) outside of the dip/peak area for each value of $\ell$. Panel (b) shows the HOM interferograms for $\ell =0,1,2,3,4$, demonstrating an alternation between dips (for $\ell =0,2,4$) and peaks (for $\ell =1,3$).

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  Figure 6

  (a) The red dots indicate the observed visibility for the HOM interferograms from the previous figure, while the curve indicates the best fit to the relationship in Eq.(6), yielding an interprism angle of $\varphi ={47.9}^{\circ }$. (b) Spiral polar diagram plotted for ${\ell }_p=0$ and $\varphi ={47.9}^{\circ }$.

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  Figure 7

  HOM interference behavior for ${\ell }_p=0$ and an SPDC topological charge projected in a spatially selective manner to $\ell =1$ within a circumference on the SLM plane and to $\ell =0$ outside of this circumference [see inset of panel (a)]. We show the spatially resolved HOM interferograms for two values of temporal delay, corresponding to (a) $\mathrm{\Delta }z=60\text{μ}\mathrm{m}$ ($\tau =200$ fs) and (b) $\mathrm{\Delta }z=0\text{μ}\mathrm{m}$ ($\tau =0$ fs). The resulting HOM visibility is shown in panel (c), with positive (negative) values in the internal (external) region. In panel (d), we show the interferograms resulting from adding up the pixels in the two regions denoted by green boxes in panel (c), showing a peak for the internal region and a dip for the external region.

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