Exploring the quantum nature of the radial degree of freedom of a photon via Hong-Ou-Mandel interference

In quantum information, quantum systems and their properties offer unprecedented opportunities. Being able to harness additional degrees of freedom adds power and flexibility to quantum algorithms and protocols. In this work, we demonstrate that the radial transverse mode of a single photon constitutes one such degree of freedom. We do so by showing that we can tune the two-photon interference, a quintessential quantum effect and the basic constituent of many quantum protocols, by manipulating its radial transverse modal profiles. Our work, in addition to allowing for greater versatility of existing protocols and significantly increasing the information channel capacity, can inspire novel quantum information tasks.

Figure 1

(a) Experimental setup for generating entangled photon pairs: The photon pair generated via spontaneous parametric down-conversion is made indistinguishable in all internal degrees of freedom by being coupled to single-mode optical fibers (SMOF), where only the Gaussian transverse mode is selected. They are then sent to the main apparatus, b, in order to demonstrate the HOM interference. (b) Experimental setup to test the Hong-Ou-Mandel interference: A quarter-wave (QWP) and half-wave (HWP) plates are used to compensate and rotate the polarization state of both signal and idler photons to horizontal (suitable to obtain a maximum efficiency at the spatial light modulators). Appropriate kinoforms are displayed on both spatial light modulators, SLM–A and SLM–B, to manipulate the radial DOF of the two photons and to render them completely indistinguishable, partially distinguishable or completely distinguishable. The insets near the SLMs show two possible types of holographic kinoforms. The two photons interfere on a 50:50 symmetric nonpolarizing beam splitter (NPBS). Then, they are collected by low-mode optical fibers (LMOF) coupled to single-photon counting modules (D${}_A$ and D${}_B$), and detected in coincidence. In order to check the HOM coalescence enhancement, an additional NPBS is inserted in arm–B (blue dashed area), and the coincidence counts are read between detector D${}_B$ and D${}_B^{\prime }$.

Figure 2

(a) Experimental data of the Hong-Ou-Mandel interference in the radial DOF (coincidence detection with $D_A$ and $D_B$): The HOM dip indicates indistinguishability between the two photons; the flat data (red) corresponds to two completely distinguishable radial states, while the other two curves (magenta and blue) correspond to a pair of partially distinguishable and indistinguishable radial states. The visibilities of the dips are $\mathcal{V}=0.014\pm 0.027$, $\mathcal{V}=0.465\pm 0.030$, and $\mathcal{V}=0.646\pm 0.026$, respectively. The inset shows that the radius of each holographic kinoform displayed on the SLM-B which determined the value of the distinguishability. (b) Experimental data of the Hong-Ou-Mandel coalescence enhancement (coincidence detection with $D_B$ and $D_{B^{\prime }}$); the enhancement increases for indistinguishable photons in the radial DOF. The enhancements due to coalescence are $\mathcal{C}=1.113\pm 0.093$, $\mathcal{C}=1.590\pm 0.060$, and $\mathcal{C}=1.907\pm ,0.047$, respectively. The error bars correspond to one standard deviation and were calculated from a Poisson distribution. Solid curves are the best theoretical Gaussian fit. Note that the normalization factor for (a) and (b) are different.

Figure 3

Experimental data of visibility of the HOM interference for two different radial modes of $p_s$ (signal) and $p_i$ (idler). The top right inset is a density plot of the three-dimensional plot. This shows that the eigenstates of the radial mode are mutually orthogonal; thus the associate Hilbert space is discrete and unbounded. Decreasing in visibility for higher $p$ eigenstates is caused by truncation introduced by optics and a relatively small active area of the SLMs.