Entanglement certification in bulk nonlinear crystals for degenerate and non-degenerate SPDC: spectral filter effects on transverse spatial correlations

TL;DR

This study systematically examines how spectral filter bandwidths influence transverse spatial correlations and entanglement in bulk SPDC, revealing universal behaviors and optimal filtering conditions for quantum imaging applications.

Contribution

First comprehensive analysis of spectral filter effects on spatial correlations in bulk SPDC, identifying universal behaviors and optimal filter bandwidths for entanglement certification.

Findings

Degenerate conditional momentum width is pump-limited and filter-invariant.

Non-degenerate configurations show monotonic growth in momentum widths.

Optimal filter bandwidth is determined by the crystal's phase-matching bandwidth.

Abstract

Spatial correlations of photon pairs from spontaneous parametric down-conversion (SPDC) underpin quantum imaging and entanglement certification. We present the first systematic study of spectral filter bandwidth effects on transverse spatial correlations in bulk Type-I BBO for degenerate and non-degenerate configurations. In the far field, the degenerate conditional momentum width is pump-limited and filter-invariant, while non-degenerate configurations exhibit monotonic growth in both marginal and conditional momentum widths -- with the walk-off axis $\approx 100$ times more sensitive than the non-walk-off axis. In the near field, we identify a previously unreported flat-dip-rise profile: the conditional position width narrows by $\approx 10\%$ at an optimal bandwidth $\Delta_\mathrm{dip} \approx 1.35\,\Delta\lambda_\mathrm{SPDC}$ before rising due to geometric displacement. When the filter is placed on the idler arm, the dip shifts by the exact factor $(\lambda_i/\lambda_s)^2$. Both results are universal for any non-degenerate SPDC source, requiring only a finite crystal length, $d\theta/d\lambda \neq 0$, and incoherent spectral averaging. The Reid EPR uncertainty product is consistently smaller on the walk-off axis -- a structural advantage of bulk birefringent geometry absent in quasi-phase-matched sources. The optimal filter bandwidth $\Delta_F = \Delta_\mathrm{dip}$ is determined entirely by the intrinsic phase-matching bandwidth of the crystal and is directly readable from the X-entanglement spectral width of the source.