Theory of Plasmon-Assisted Transmission of Entangled Photons

The recent surface plasmon entanglement experiment [E. Altewischer et al., Nature (London) 418, 304 (2002)] is theoretically analyzed. The entanglement preservation upon transmission in the nonfocused case is found to provide information about the interaction of the biphoton and the metallic film. The entanglement degradation in the focused case is explained in the framework of a fully multimode model. This phenomenon is a consequence of the polarization-selective filtering behavior of the metallic nanostructured film. It is shown that the “which-way” labels that degrade entanglement are not located in the degrees of freedom of the metallic film but rather in the spatial degrees of freedom of the photon field.

Figure 1

Photon 1 trajectory. The confocal telescope (i.e., the lenses) is present only in some parts of the experiment. The incident photon is a plane wave with wave vector ${\mathbf{k}}_1=({\mathbf{q}}_1,k_{1z})=(\mathbf{0},k)$. The first lens produces a focused beam including many plane waves: ${\mathbf{k}}_2=({\mathbf{q}}_2,k_{2z})$. The second lens gives rise again to a bundle of modes: ${\mathbf{k}}_3=({\mathbf{q}}_3,k_{3z})$. The unit vectors ${\widehat{\mathbf{p}}}_3$ and ${\widehat{\mathbf{s}}}_3$ indicate the $p$ and $s$ polarization directions, respectively, whereas ${\widehat{\mathbf{p}}}_{3\mathrm{paraxial}}={\widehat{\mathbf{q}}}_3$.

Figure 2

Hole array transmittance as a function of wavelength. The film is illuminated by an orthogonally incident plane wave (nonfocused), and it is tilted (from $0°$ to $6°$) around the diagonal. The incident field linear polarization is perpendicular to the lattice diagonal in (a) and parallel to it in (b). For clarity the curves are offset 0.25%.

Figure 3

Biphoton fringe visibility as a function of the telescope semiaperture. The visibilities are shown for two orientations of the second polarizer: ${\beta }_2=0°,45°$. The chosen wavelengths are the resonances shown in Fig. 2 for no tilt: 797 nm (corresponding to a surface mode propagating along the array diagonals), and 728 nm (corresponding to a mode propagating along the $X$ or $Y$ axes).

Figure 4

Intensities (a),(b) and polarizations (c),(d) of mode $|{\mathbf{q}}_3⟩$ after the telescope as a function of $(q_{3x},q_{3y})$. The excitation is one single linearly polarized photon ($\lambda =797\mathrm{nm}$). The incident photon polarizations are $-45°$ in (a),(c) and $90°$ in (b),(d). The telescope semiaperture is $8°$. The left color scale (arbitrary units) is shared by (a) and (b), and the right color scale corresponds to (c) and (d). In (c),(d) the color scale represents the ratio of the minor to the major polarization ellipse axes (for instance, $\pm 1$ correspond to left and right circular polarization), and the small line segments indicate the orientation of the ellipse major axis. The maxima (and minima) of all abscissa and ordinate scales correspond to a semiaperture of ${\theta }_3=0.1°$.