Research ArticleQUANTUM OPTICS

Quantum walks and wavepacket dynamics on a lattice with twisted photons

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Science Advances  13 Mar 2015:
Vol. 1, no. 2, e1500087
DOI: 10.1126/sciadv.1500087
  • Fig. 1 Conceptual scheme of the single-beam photonic QW in the space of OAM.

    In each traversed optical stage (QW unit), the photon can move to an OAM value m that can increase or decrease by one unit (or stay still, in the hybrid configuration). The OAM decomposition of the photonic wave function at each stage thus includes many different components, as shown in the callouts in which modes having different OAM values are represented by the corresponding helical (or “twisted”) wavefronts.

  • Fig. 2 Experimental apparatus for single-photon QW experiments.

    Frequency-doubled laser pulses at 400 nm and with 140-mW average power, obtained from the fundamental pulses (100 fs) generated by a titanium:sapphire source (Ti:Sa) at a repetition rate of 82 MHz, pump a 3-mm-thick nonlinear β-barium borate crystal (BBO1) cut for type II SPDC (see main text for a definition of all acronyms). Photon pairs at 800 nm generated through this process, cleaned from residual radiation at 400 nm using a long pass filter, pass through an HWP and the BBO2 crystal (cut as BBO1, but 1.5 mm thick) to compensate both spatial and temporal walk-off introduced by BBO1. Next, the two photons are split by a PBS; one is sent directly to the avalanche single-photon detector (APD) D1, whereas the other is coupled into an SMF. At the exit of the fiber, the photon goes through N identical subsequent QW steps (N = 5 in the figure), is then analyzed in both polarization and OAM, and is finally detected with APD D2, in coincidence with D1. Before entering the first QW step, an SLM 1 and an HWP-QWP set are used to prepare the photon initial state in the OAM and SAM spaces, respectively. At the exit of the last step, the polarization projection on the state |φfc is performed with a second HWP-QWP set followed by a linear polarizer (LP). The OAM state is then analyzed by diffraction on SLM 2, followed by coupling into a SMF. The projection state |ψfw corresponding to each OAM eigenvalue m was thus fixed by the hologram pattern displayed on SLM 2. Before detection, interferential filters (IF) centered at 800 nm and with a bandwidth of 3.6 nm were used for spectral cleaning. As shown in the legend, a single QW step consists of a QWP (optical axis at 45° from the horizontal), a QP with q = 1/2 (axis at 0°), and an HWP (axis at 0°); the HWP was not included in the wavepacket and two-photon experiments.

  • Fig. 3 Four-step QW for a single photon with localized input.

    (A to D) Experimental results, including both intermediate and final probabilities for different OAM states in the evolution (summed over different polarizations). The intermediate probabilities at step n are obtained by switching off all QPs that follow that step, that is, setting δ = 0. (A) and (B) refer to the Graphicstandard case with two different input states for the coin subsystem, (α, β) = (0, 1) and 1/Embedded Image (1, i), respectively. (C) and (D) refer to the hybrid case with δ = 1.57, with the same initial coin-states. (E to H) Corresponding theoretical predictions. Poissonian statistical uncertainties at ±1 standard deviation are shown as transparent volumes in (A) to (E). The similarities between experimental and predicted final OAM distributions are 94.7 ± 0.4%, 93.4 ± 0.5%, 99.7 ± 0.1%, and 99.2 ± 0.2%, respectively. Panels on the same column refer to the same configuration and initial states. The color scale reflects the number of steps.

  • Fig. 4 Wavepacket propagation in a five-step QW.

    (A and B) Experimental results, showing the step-by-step evolution of the OAM distribution of a single photon prepared in a Gaussian wavepacket with σ = 2, in the SAM band s = 1 (summed over different polarizations). (A) and (B) correspond to the two cases k0 = π (maximal group velocity) and k0 = π/2 (vanishing group velocity), respectively. The latter configuration shows some spreading of the Gaussian envelope, governed by the group velocity dispersion. Poissonian statistical uncertainties at ±1 SD are shown as transparent volumes. (C and D) Theoretical predictions corresponding to the same cases. At the fifth step, the similarities between experimental and theoretical OAM distributions are 98.2 ± 0.4% and 99.0 ± 0.2%, respectively. The color scale reflects the number of steps.

  • Fig. 5 QW wavepacket dispersion properties in the Brillouin zone.

    (A) Poincaré sphere representation of the polarization (or SAM) eigenstates |φ1(k)〉 prepared in our experiments, for different values of the quasi-momentum k in the irreducible Brillouin zone (0, π) taken in steps of π/8 (blue dots). These states lie on a maximal circle (blue line) of the sphere. (B) Mean OAM after a five-step QW for a single photon prepared in a Gaussian wavepacket with σ = 2 and s = 1, with different values of average quasi-momentum k0 in the range (0, π). Blue and purple points are associated to experimental data and theoretical predictions, respectively; Poissonian statistical uncertainties are too small to be shown in the graph. (C to G) Final OAM distribution associated to some of these cases (summed over different polarizations). Panels refer to k0 = 0, π/4, π/2, 3π/4, π, respectively. (H) OAM distribution after a five-step QW for a wavepacket whose coin is prepared Graphicin the superposition state (|φ1(0)〉 + |φ2(0)〉)/Embedded Image. As predicted by the theory, it splits into two components propagating in opposite directions, thus generating a maximally entangled SAM-OAM state. In (C) to (H), Poissonian statistical uncertainties at ±1 SD are shown by error bars. The similarities between experimental and theoretical OAM distributions are 98.9 ± 0.2%, 96.2 ± 0.4%, 98.4 ± 0.3%, 93.2 ± 0.6%, 99.1 ± 0.2%, and 97.3 ± 0.4%, respectively.

  • Fig. 6 Three-step QW for two identical photons.

    In this case, only final OAM probabilities are shown (summed over different polarizations). (A to C) Case of standard walk. (A) Experimental results. Vertical bars represent estimated joint probabilities for the OAM of the two photons. Because the two measured photons detected after the BS splitting are physically equivalent, their counts are averaged together, so that (m1,m2) and (m2,m1) pairs actually refer to the same piece of data. Even values of m1 and m2 are not included because they correspond to sites that cannot be occupied after an odd number of steps. (B) Theoretical predictions for the case of indistinguishable photons. (C) Theoretical predictions for the case of distinguishable photons, shown to highlight the effect of two-photon interference (Hong-Ou-Mandel effect) in the final probabilities. It can be seen that the experimental results agree better with the theory for indistinguishable photons. (D to F) Case of hybrid walk (with δ = 1.46). (D), (E), and (F) refer, respectively, to experimental data, indistinguishable photon theory, and distinguishable photon theory, as in the previous case. The QW step in these two-photon experiments is implemented with a QP and a QWP. Again, our experiment is in good agreement with the theory based on indistinguishable photons, proving that two-photon interferences are successfully implemented in our experiment. The similarities between experimental and predicted quantum distributions (IPT model) are 98.2 ± 0.4% and 95.8 ± 0.3% for the standard and the hybrid walk, respectively. The similarities with the DPT model are, instead, 96.4 and 91.8%, respectively. The color scale (common to all panels referring to the same case) reflects the vertical scale, to help compare the patterns.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/1/2/e1500087/DC1

    Fig. S1. Band structure of the QW system.

    Fig. S2. Holograms for the preparation of the OAM initial state before the QW process.

    Fig. S3. Supplementary data for the four-step quantum walk for a single photon, with various input polarization states.

    Fig. S4. Two-photon QW apparatus.

    Fig. S5. Experimental verification of the indistinguishability of the two-photon source through polarization Hong-Ou-Mandel (HOM) interference.

    Fig. S6. Experimental violation of correlation inequalities for two photons that have completed the standard QW (δ = π).

    Fig. S7. Experimental violation of correlation inequalities for two photons that have completed the hybrid QW (δ = π/2).

    Table S1. Power coefficients of the various p-index terms appearing in the expansion of the beam emerging from a QP (with q = 1/2) in the LG mode basis, assuming that the input is an L-polarized LG mode with p = 0 and the given OAM m value.

  • Supplementary Materials

    This PDF file includes:

    • Fig. S1. Band structure of the QW system.
    • Fig. S2. Holograms for the preparation of the OAM initial state before the QW process.
    • Fig. S3. Supplementary data for the four-step quantum walk for a single photon, with various input polarization states.
    • Fig. S4. Two-photon QW apparatus.
    • Fig. S5. Experimental verification of the indistinguishability of the two-photon source through polarization Hong-Ou-Mandel (HOM) interference.
    • Fig. S6. Experimental violation of correlation inequalities for two photons that have completed the standard QW (δ = π).
    • Fig. S7. Experimental violation of correlation inequalities for two photons that have completed the hybrid QW (δ = π/2).
    • Table S1. Power coefficients of the various p-index terms appearing in the expansion of the beam emerging from a QP (with q = 1/2) in the LG mode basis, assuming that the input is an L-polarized LG mode with p = 0 and the given OAM m value.

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