Theory of filtered type-II parametric down-conversion in the continuous-variable domain: Quantifying the impacts of filtering

Parametric down-conversion (PDC) forms one of the basic building blocks for quantum optical experiments. However, the intrinsic multimode spectral-temporal structure of pulsed PDC often poses a severe hindrance for the direct implementation of the heralding of pure single-photon states or, for example, continuous-variable entanglement distillation experiments. To get rid of multimode effects narrowband frequency filtering is frequently applied to achieve a single-mode behavior. A rigorous theoretical description to accurately describe the effects of filtering on PDC, however, is still missing. To date, the theoretical models of filtered PDC are rooted in the discrete-variable domain and only account for filtering in the low-gain regime, where only a few photon pairs are emitted at any single point in time. In this paper we extend these theoretical descriptions and put forward a simple model, which is able to accurately describe the effects of filtering on PDC in the continuous-variable domain. This developed straightforward theoretical framework enables us to accurately quantify the tradeoff between suppression of higher-order modes, reduced purity, and lowered Einstein–Podolsky–Rosen entanglement, when narrowband filters are applied to multimode type-II PDC.

Figure 1

Narrowband filtering of a broadband multimode type-II PDC process, followed by a mode optimization, enables the creation of EPR entanglement in a single optical mode.

Figure 2

Visualization of a Schmidt decomposition: Each pair of Schmidt modes of signal and idler, depicted in panel (b), forms a spectral distribution which, combined and weighted by their $r_k$ values, forms the JSA in panel (a).

Figure 3

Covariance matrix of an EPR state with 3 dB of squeezing. Plotted is the absolute value of the individual elements. The $sinh\left(r\right)$ side peaks are characteristic for EPR entanglement.

Figure 4

PDC state used to demonstrate the individual steps of our protocol: (a) JSA $f({\omega }_s,{\omega }_i)$, (b) First five EPR-squeezing values, (c) First three optical modes of the signal and idler beams.

Figure 5

Covariance matrix of the unfiltered PDC state depicted in Fig. 4. This figures shows the absolute values of the individual elements for the first three optical modes. The submatrix for each mode is identical to a finitely squeezed EPR state.

Figure 6

A rectangular filter in the signal and idler arms cuts the JSA and lets only the central frequencies pass.

Figure 7

(a) Signal modes with respect to the applied rectangular filter in Fig. 6. (b) Individual signal mode functions multiplied by the filter.

Figure 8

The EPR squeezing in the first five modes after the filtering has been applied. The EPR squeezing in all modes is significantly reduced.

Figure 9

Filtering a PDC state introduces losses, which move the state towards vacuum and further leads to correlations between the different optical modes.

Figure 10

New basis set (a) obtained via the SVD basis optimization and (b) the global basis optimization routine. (c) Comparison of the obtained EPR-squeezing values. Both approaches deliver virtually identical results.

Figure 11

Covariance matrix of the filtered PDC state from Fig. 9 using the SVD basis optimization. Performing this basis optimization reveals that filtering effectively suppresses higher-order frequency modes.

Figure 12

Evolution of five modes with maximum squeezing obtained by the genetic algorithm. The genes are randomly initialized and the modes 1–5 are found by evolution.

Figure 13

Quantitative analysis of the impact of filtering on the exemplary PDC state presented in Fig. 4 using the three different initial EPR-squeezing distributions given in panel (a). (b) Remaining EPR squeezing in the first mode as a function of the single-mode character, for various filter bandwidths. (c) Purity of the filtered state as a function of the single-mode character. The blue shaded areas are the accessible regions using our exemplary PDC state.