Proposal to Observe the Nonlocality of Bohmian Trajectories with Entangled Photons

Bohmian mechanics reproduces all statistical predictions of quantum mechanics, which ensures that entanglement cannot be used for superluminal signaling. However, individual Bohmian particles can experience superluminal influences. We propose to illustrate this point using a double double-slit setup with path-entangled photons. The Bohmian velocity field for one of the photons can be measured using a recently demonstrated weak-measurement technique. The found velocities strongly depend on the value of a phase shift that is applied to the other photon, potentially at spacelike separation.

Figure 1

Conceptual structure of the proposed experiment and calculated Bohmian trajectories. A source produces a pair of path-entangled photons such that either both photons go through the upper slits on each side or both photons go through the lower slits; see Eq. (5). A phase shift $\varphi$ is introduced behind the lower slit on the left, creating the state of Eq. (6). The resulting Bohmian trajectories for photons $A$ and $B$ are shown for two possible starting positions, corresponding to the Bohmian particles passing through the upper or lower slits, and for different values of the phase shift. One can see that the trajectory of photon $B$ depends on $\varphi$, even though the phase shift is applied to photon $A$. This shows the nonlocal character of Bohmian mechanics.

Figure 2

Two-particle detection probability for the experiment of Fig. 1. The case $\varphi =0$ is shown on the left, $\varphi =\pi$ on the right. One sees interference fringes whose location clearly depends on the value of $\varphi$. However, the corresponding single-particle probability distributions do not depend on $\varphi$, as can be seen from the marginals shown at the bottom of the figures. This ensures that superluminal communication is impossible; see also the text.

Figure 3

Proposed experimental realization of the conceptual setup shown in Fig. 1, including a weak measurement to determine the velocity field for particle $B$. Type-II spontaneous parametric down-conversion is used to produce pairs of polarization-entangled photons, which are mode filtered using single-mode fibers. The polarization entanglement is converted to path entanglement using polarizing beam splitters (PBS) and half-wave plates (HWP). Together with the phase shifter behind one of the slits for particle $A$ this creates the state of Eq. (6). The Bohmian velocity field of particle $B$ is measured by performing a weak measurement of its momentum, followed by a joint strong measurement of both particles’ positions. The weak measurement is implemented using the birefringence in a calcite crystal to couple the photon’s momentum to its polarization; see the text. The position measurements are performed by (multimode) fiber-coupled single-photon counting modules (SPCMs). For photon $B$ a polarization beam splitter and two SPCMs are used in order to have access to the polarization information that is necessary for the weak measurement.

Figure 4

Velocity profiles that could be measured using the apparatus in Fig. 3. For a fixed position of the detector on the $A$ side ($t=4$, $x_A=4$), the (Bohmian) velocity field for photon $B$ is measured at different transverse positions $x_B$. One sees that the velocity depends strongly on the phase $\varphi$, even though the phase is applied on the $A$ side. The velocity profiles are offset from each other vertically by 0.2 for clarity, with the lowest ($\varphi =\pi$) profile being at its true position.