Effects of frequency correlation in linear optical entangling gates operated with independent photons

Bose-Einstein coalescence of independent photons at the surface of a beam splitter is the physical process that allows linear optical quantum gates to be built. When distinct parametric down-conversion events are used as an independent photon source, distinguishability arises form the energy correlation of each photon with its twin. We derive upper bound for the entanglement which can be generated under these conditions.

Figure 1

(Color online) Schematic Hong-Ou-Mandel interferometer for independent photons. $S_{\nu }\left({\nu }_p\right)$ and $S_{\mu }\left({\mu }_p\right)$ represent the pump spectra for the upper and lower beam, respectively. These produce a photon pair on the modes $a_1^{†}$ and $a_2^{†}$, $a_3^{†}$ and $a_4^{†}$, respectively. Measurement of a photon in the $j\text{th}$ detector is described by the destruction operator $E_j$.

Figure 2

(Color online) Visibility $v_0$ as a function of filter bandwidth $\sigma$ in absence of phase dispersion. Top: both pump spectra are Gaussian profiles with the same FWHM ${\sigma }_{\nu }={\sigma }_{\mu }$. Different plots correspond to different values of ${\sigma }_{\nu }$. Bottom: the two Gaussian spectra have different FWHM’s. Plots relative to different ratios ${\sigma }_{\mu }∕{\sigma }_{\nu }$ are shown for the case ${\sigma }_{\nu }=3\mathrm{nm}$.

Figure 3

(Color online) Plot of the tangle in the polarization state of Eq. (23) as a function of the HOM visibility, normalized to its maximum value $v_{\mathit{id}}=0.8$.

Figure 4

(Color online) Comparison of the plots of the relative visibility and the tangle in the polarization state in Eq. (23) as a function of the pump bandwidth in the case of a Gaussian pump. Different curves correspond to different values of the detection bandwidth.

Figure 5

(Color online) Plot of maximum achievable visibility as a function of crystal length in type-II degenerate $400\mathrm{nm}\to 800\mathrm{nm}$ PDC, for two widely adopted materials, LBO and BBO. The pump bandwidth is $5\mathrm{nm}$, the detection bandwidth is $2\mathrm{nm}$. This choice gives near unity visibility in the ideal case. Sellmeier coefficients are taken from Ref. 32.