Observing the Average Trajectories of Single Photons in a Two-Slit Interferometer

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Science  03 Jun 2011:
Vol. 332, Issue 6034, pp. 1170-1173
DOI: 10.1126/science.1202218

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  1. Fig. 1

    Experimental setup for measuring the average photon trajectories. Single photons from an InGaAs quantum dot are split on a 50:50 beam splitter and then outcoupled from two collimated fiber couplers that act as double slits. A polarizer prepares the photons with a diagonal polarization Embedded Image. Quarter waveplates (QWP) and half waveplates (HWP) before the polarizer allow the number of photons passing through each slit to be varied. The weak measurement is performed by using a 0.7-mm-thick piece of calcite with its optic axis at 42° in the x-z plane that rotates the polarization state to Embedded Image. A QWP and a beam displacer are used to measure the polarization of the photons in the circular basis, allowing the weak momentum value kx to be extracted. A cooled CCD measures the final x position of the photons. Lenses L1, L2, and L3 allow different imaging planes to be measured. The polarization states of the photons are represented on the Poincaré sphere, where the six compass points correspond to the polarization states Embedded Image, Embedded Image, and Embedded Image.

  2. Fig. 2

    Measured intensities (photon counts) of the two circular polarization components of |ψ〉, measured on the CCD screen (red and blue curves), as well as the weak momentum values calculated from these intensities (black) for imaging planes at (A) z = 3.2 m, (B) z = 4.5 m, (C) z = 5.6 m, and (D) z = 7.7 m. The red and blue data points are the intensity data with constant background subtracted. The errors for the momentum values were calculated by simulating the effect of Poissonian noise in the photon counts. The magenta curve shows momentum values obtained from enforcing probability density conservation between adjacent z planes. Because of the coarse-grained averaging over three imaging planes, the probability-conserving momentum values are not as sensitive as the measured weak momentum values to highly localized regions in the pattern with steep momentum gradients.

  3. Fig. 3

    The reconstructed average trajectories of an ensemble of single photons in the double-slit apparatus. The trajectories are reconstructed over the range 2.75 ± 0.05 to 8.2 ± 0.1 m by using the momentum data (black points in Fig. 2) from 41 imaging planes. Here, 80 trajectories are shown. To reconstruct a set of trajectories, we determined the weak momentum values for the transverse x positions at the initial plane. On the basis of this initial position and momentum information, the x position on the subsequent imaging plane that each trajectory lands is calculated, and the measured weak momentum value kx at this point found. This process is repeated until the final imaging plane is reached and the trajectories are traced out. If a trajectory lands on a point that is not the center of a pixel, then a cubic spline interpolation between neighboring momentum values is used.

  4. Fig. 4

    The trajectories from Fig. 3 plotted on top of the measured probability density distribution. Even though the trajectories were reconstructed by using only local knowledge, they reproduce the global propagation behavior of the interference pattern.

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