Remote polarization-entanglement generation by local dephasing and frequency up-conversion

We introduce a scheme for remote entanglement generation for the photon polarization. The technique is based on transferring the initial frequency correlations to specific polarization-frequency correlations by local dephasing and their subsequent removal by frequency up-conversion. On fundamental level, our theoretical results show how to create and transfer entanglement, to particles which never interact, by means of local operations. This possibility stems from the multipath interference and its control in frequency space. For applications, the developed techniques and results allow for the remote generation of entanglement with distant parties without Bell state measurements and open the perspective to probe frequency-frequency entanglement by measuring the polarization state of the photons.

Figure 1

A schematic picture of the protocol and corresponding optical setup. (a) The state after parametric down-conversion. The polarization state (black circles) is factorized and the frequency state (grey circles) is entangled. See Eqs. (1) and (2). (b) The polarization is coupled with the frequency in a quartz plate. See Eq. (11). (c) The interaction time is fixed. See Eq. (16). (d) Parametric up-conversion erases the frequency information and produces the entangled polarization state. See Eq. (17).

Figure 2

Schematic description of interference of mode paths for local up-conversion of both photons. (a) Frequency component $|{\omega }_1,{\omega }_2\rangle +|{\omega }_2,{\omega }_1\rangle$, which is associated to polarization state $|{\psi }_0\rangle$ in Eq. (16). (b) Frequency component $|{\omega }_1,{\omega }_2\rangle -|{\omega }_2,{\omega }_1\rangle$ which is associated to polarization state $|{\psi }_1\rangle$ in Eq. (16). After up-conversion of only one of the photons, initial vectors have four mode paths in the middle of the figure with corresponding signs due initial relative phases. After the up-conversion of the other photon, mode paths interfere constructively in (a) and destructively in (b).

Figure 3

Polarization entanglement as a function of path difference for the double-peak case and changing the widths of the frequency distribution. Here ${\lambda }_0=780$ nm, $k=-1$, and $\mathrm{\Delta }\mathrm{\Omega }=3\text{nm}$; the dot-dashed green line shows when the width (FWHM) of the local peaks is 0.125 nm, the solid blue line when local the widths are 0.25 nm, and the dashed red line when the local widths are 0.5 nm.

Figure 4

Polarization entanglement as a function of path difference for the double-peak case and reducing the amount of initial correlations. Here ${\lambda }_0=780$ nm, $n_H=1.51004,n_V=1.54360$, the separation between two local peaks is 3 nm, and local widths (FWHM) are 0.5 nm; the solid blue line shows when $k=-0.999$, the dashed red line when $k=-0.99$, and the dot-dashed green line when $k=-0.9$.

Figure 5

Polarization entanglement of single-peak case as a function of path difference for different widths of the initial frequency distribution with $k=-1$. Here ${\lambda }_0=780$ nm. The dot-dashed green line shows when the local width (FWHM) is 2 nm, the solid blue line when the local width is 1 nm, and the dashed red line when the local width is 0.5 nm.

Figure 6

Polarization entanglement of the single-peak case as a function of path difference for different widths of the initial frequency distribution with $k=-0.99$. Here ${\lambda }_0=780$ nm, $n_H=1.51004,n_V=1.54360$; the dot-dashed green line shows when the local width (FWHM) is 2 nm, the solid blue line when the local width is 1 nm, and the dashed red line when the local width is 0.5 nm.