Generation of Fourier-transform-limited heralded single photons

In this paper we study the spectral (temporal) properties of heralded single photon wave packets, triggered by the detection of an idler photon in the process of parametric down conversion. The generated single photons are studied within the framework of the chronocyclic Wigner function, from which the single photon spectral width and temporal duration can be computed. We derive specific conditions on the two-photon joint spectral amplitude which result in both pure and Fourier-transform-limited heralded single photons. Likewise, we present specific source geometries which lead to the fulfillment of these conditions and show that one of these geometries leads, for a given pump bandwidth, to the temporally shortest possible heralded single photon wave packets.

Figure 1

(Color online) Heralded single photon source, based on pulsed parametric down conversion in a ${\chi }^{\left(2\right)}$ crystal.

Figure 2

Chronocyclic Wigner function (CWF) obtained numerically for a BBO crystal of $5\mathrm{mm}$ length, phase matched for collinear, degenerate type-I PDC centered at $800\mathrm{nm}$, pumped by an ultrashort pulse train with a $5\mathrm{nm}$ bandwidth. The resulting single photon time duration $\mathrm{\Delta }t=288.01\mathrm{fs}$ and single photon bandwidth $\mathrm{\Delta }\omega =61.4\mathrm{THz}$, with $\mathrm{\Delta }t\mathrm{\Delta }\omega \approx 17.7$, suggest a large departure from the transform limit.

Figure 3

(Color online) Panels (a)–(c) show examples of numerically calculated chronocyclic Wigner functions (CWF) without resorting to the Gaussian approximation and taking into account all dispersive orders, centered at a time $t=0$, for a single heralded single photon source based on (a) a KDP crystal, fulfilling asymmetric group velocity matching, (b) a BBO crystal, fulfilling symmetric group velocity matching, and (c) a BBO crystal not fulfilling any special group velocity condition. Panels (d)–(f) show, correspondingly to (a)–(c), the contour associated to one-half of the maximum value for the CWF calculated numerically (solid line) and for the CWF calculated analytically [dashed line; see Eq. (27)].