Spatial two-photon coherence of the entangled field produced by down-conversion using a partially spatially coherent pump beam

We study the spatial coherence properties of the entangled two-photon field produced by parametric down-conversion (PDC) when the pump field is, spatially, a partially coherent beam. By explicitly treating the case of a pump beam of the Gaussian Schell-model type, we show that in PDC the spatial coherence properties of the pump field get entirely transferred to the spatial coherence properties of the down-converted two-photon field. As one important consequence of this study, we find that, for two-qubit states based on the position correlations of the two-photon field, the maximum achievable entanglement, as quantified by concurrence, is bounded by the degree of spatial coherence of the pump field. These results could be important by providing a means of controlling the entanglement of down-converted photons by tailoring the degree of coherence of the pump field.

Figure 1

(a) Schematic laboratory setup that could be used to study the spatial coherence properties of the two-photon field produced by PDC using a partially coherent pump beam. (b) $1$ and $2$ represent two alternative pathways by which the down-converted signal and idler photons can pass through the holes and get detected in coincidence at detectors $D_s$ and $D_i$. In alternative $1$, the signal and idler photons go through the pair of holes located at ${\mathit{r}}_{s1}\equiv ({\mathit{\rho }}_{s1},z)$ and ${\mathit{r}}_{i1}\equiv ({\mathit{\rho }}_{i1},z)$, and in alternative $2$, they go through those located at ${\mathit{r}}_{s2}\equiv ({\mathit{\rho }}_{s2},z)$ and ${\mathit{r}}_{i2}\equiv ({\mathit{\rho }}_{i2},z)$. ${\mathit{\rho }}_{1\hspace{0.5em}(2)}$ and ${\mathit{\rho }}_{1\hspace{0.5em}(2)}^{\text{'}}$ are the two displacement parameters in alternatives $1\hspace{0.5em}(2)$ defined in Eq. (1) in terms of which we describe the coherence properties of the two-photon field.

Figure 2

Schematic representation of a partially spatially coherent pump beam; ${\sigma }_s(z)$ is the rms beam radius of the pump field at plane $z$ and ${\sigma }_{\mu }(z)$ is the rms spatial coherence width of the pump field at plane $z$. ${\mathit{\rho }}_{p1}$ and ${\mathit{\rho }}_{p2}$ are the transverse position vectors of two points within the pump beam.

Figure 3

Physical interpretation of the two-photon correlation width ${\sigma }_s^{(2)}(z)$ and the two-photon transverse coherence width ${\sigma }_{\mu }^{(2)}(z)$ in terms of the two-photon transverse position vectors. (a) The two-photon correlation width ${\sigma }_s^{(2)}(z)$ is equal to the pump beam radius ${\sigma }_s(z)$. As a result, when an idler photon is detected at position ${\mathit{\rho }}_{i1}$, the corresponding signal photon has an appreciable probability of being detected anywhere inside an area whose center is at $-{\mathit{\rho }}_{i1}$ and whose radius is twice the pump beam radius ${\sigma }_s(z)$. (b) The two-photon spatial coherence width ${\sigma }_{\mu }^{(2)}(z)$ is equal to the spatial coherence width of the pump field ${\sigma }_{\mu }(z)$; thus, for alternatives 1 and 2 of Fig. 1 to remain mutually coherent, $\left|\Delta \mathit{\rho }\right|=|{\mathit{\rho }}_1-{\mathit{\rho }}_2|$ has to be less than the spatial coherence width ${\sigma }_{\mu }(z)$ of the pump field.