Research ArticleQuantum Mechanics

Experimental nonlocal and surreal Bohmian trajectories

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Science Advances  19 Feb 2016:
Vol. 2, no. 2, e1501466
DOI: 10.1126/sciadv.1501466


  • Fig. 1 Bohmian trajectories in a double-slit apparatus.

    (A and B) Conceptual diagram of the result of reading out the WWM in a double-slit apparatus in the near field (A) and in the midfield (B). Color indicates the slit of origin of a Bohmian trajectory, and vertical position indicates the result of the WWM (the position x2 of the second, “pointer,” particle). When the WWM is read out in the near field, the Bohmian trajectories are perfectly correlated with the result of the WWM. When the WWM measurement is read out in the midfield, the Bohmian trajectories are only correlated with the WWM outcome near the edges of the diagram. Near the line of symmetry of the apparatus, both outcomes of the WWM are equally likely, regardless of which slit the Bohmian trajectory originates from.

  • Fig. 2 Experimental setup for measuring Bohmian trajectories: The trajectories of a single photon (photon 1) are measured, postselected on a detection of another photon (photon 2) by a single photon counting module (SPCM).

    A Sagnac interferometer-based source of entangled photons prepares two photons in a maximally entangled state that are then spectrally filtered using two band-pass (BP) filters. Photon 1 is sent into a double-slit apparatus and immediately split at a polarizing beamsplitter (PBS) to prepare the double-slit wave function. The lower arm’s polarization is changed to match the upper arm using a half waveplate (HWP). Both upper and lower arms passed through a polarizer, a Pockels cell (PC), and another polarizer, to postselect upon the detection of photon 2. The transverse velocity of photon 1 is weakly measured using a 0.7-mm-thick piece of calcite with its optic axis oriented at 42° to the normal in the horizontal plane, followed by a quarter waveplate (QWP) and a beam displacer. Finally, the wave function of photon 1 is imaged in different planes using an imaging system composed of three lenses, and its position is measured using a single-photon cooled CCD. Photon 2 is sent through an HWP and a QWP, followed by a PBS to measure its polarization in different bases.

  • Fig. 3 Observation of nonlocality in Bohmian mechanics.

    (A) The reconstructed trajectories when photon 2 is found in the state Embedded Image. The trajectories are drawn over a range of z = 1.7 m to z = 5.9 m, using 67 different planes. The state of photon 1 after postselection contains no information about the state of photon 2, and thus, interference is observed. (B) A single postselected trajectory beginning at the same initial condition, x = − 1.12 mm, for four different postselected polarization states of photon 2, Embedded Image, where Φ ∈ {0, π/2, π, 3π/2}. The pairs {0, π} and {π/2, 3π/2} correspond to measuring the polarization of photon 2 in two different bases. (Inset) The weak velocity values measured at z = 1.8 m and z = 5.9 m. The velocity distributions are initially independent of phase shift applied to photon 2 but depend strongly on it in the far field. The error bars on the individual velocity measurements are consistent with the scatter observed but are not displayed because they detract from rather than enhance its clarity.

  • Fig. 4 Observation of surreal trajectories.

    (A) The set of reconstructed trajectories for photon 1 without postselection onto a particular polarization of photon 2, corresponding to the delayed WWM of ESSW. The trajectories are plotted over the range z = 1.7 m to z = 5.9 m, using 67 different planes. A single trajectory beginning at x = −0.98 mm is plotted with a thicker, colored line. (B) The polarization of photon 2, represented by its Bloch vector, as a function of the position of photon 1 as it traverses the colored trajectory plotted in (A). The polarization of photon 2 is calculated by performing quantum state tomography (34) on photon 2 and correlating those counts with the counts observed on the single-photon camera. The photons have been entangled such that if photon 1 were to be found in the lower slit, photon 2 would be vertically polarized. This is the case at the start of the single trajectory we consider. However, as photon 1 traverses the double slit, it enters a region where the wave function emanating from the upper slit (for which photon 2 is horizontally polarized) interferes with that from the lower slit, leading to nonlocal coupling between the motion of photon 1 and the polarization of photon 2. As a consequence, the polarization of photon 2 changes over time and its final state no longer faithfully records the WWM information about photon 1.

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