Interference Swapping in Scattering from a Nonlocal Quantum Target

We describe a new and distinctive interferometry in which a probe particle scatters off a superposition of locations of a single free target particle. Probe particles scattering off a single free “mirror” (in one dimension) or a single free “slit” (in two dimensions) can “swap” interference with the superposed target states. The condition for interference is loss of orthogonality of the target states and reduces, in simple examples, to transfer of orthogonality from target to probe states. We analyze experimental parameters and conditions necessary for interference to be observed.

Figure 1

(a),(b) One-dimensional model: (a) A probe (in black) approaches a target (in gray) superposed at locations $X=\pm d/2$ with relative phase $\alpha$. (b) The probe and target after scattering. (c),(d) Two-dimensional model: (c) A probe wave packet, with incident angle ${\theta }^{\mathrm{in}}$ and momentum $p^{\mathrm{in}}$, approaches a stationary target superposed at $X=\pm d/2$. (d) If the target scatters with momentum $P^{\mathrm{fin}}=M{p}^{\mathrm{in}}/m\cos {\theta }^{\mathrm{in}}$, the orthogonality of the target states is transferred to the probe.

Figure 2

Normalized angular distribution of scattered probe ($\Delta p^{\mathrm{in}}=0$) incident at ${\theta }^{\mathrm{in}}=45°$, for relative phases $\alpha =0$ and $\alpha =\pi$ and $M/m=0.5$. Insets: Visibility as a function of $M/m$ and (for $M/m=0.5$) as a function of $\Delta p^{\mathrm{in}}/p^{\mathrm{in}}$, at ${\theta }^{\mathrm{in}}=45°$, ${\theta }^{\mathrm{fin}}=60°$ [11]. Here we define visibility as $[P_{\alpha }({\theta }^{\mathrm{fin}})-P_{\alpha +\pi }({\theta }^{\mathrm{fin}})]/[P_{\alpha }({\theta }^{\mathrm{fin}})+P_{\alpha +\pi }({\theta }^{\mathrm{fin}})]$ maximized over $\alpha$, where $P_{\alpha }({\theta }^{\mathrm{fin}})$ is the probability density for the probe to scatter in the direction ${\theta }^{\mathrm{fin}}$ from the initial target state $|{\phi }_{\alpha }⟩$ of Eq. (1).