Biphoton wave packets in parametric down-conversion: Spectral and temporal structure and degree of entanglement

We investigate spectral and temporal features and entanglement of biphoton wave packets formed in spontaneous parametric down-conversion with a pulsed laser pump. The degree of entanglement is characterized by the experimentally measurable parameter $R$ defined as the ratio of the coincidence and single-particle spectral widths. In the frequency representation, this parameter is found as a function of the pump-pulse duration $\tau$. The function $R\left(\tau \right)$ is shown to have a minimum and even in the minimum, at rather natural conditions, the degree of entanglement is found to be very high $(R_{\min }=73)$. The Schmidt number $K$ is found analytically for both short and long pump pulses and interpolated for arbitrary pulse durations. All functional dependences of $R$ and $K$ are found to be identical and numerical coefficients are found to be rather close. Two-time temporal wave function of a biphoton state is investigated in detail, and a rather significant difference between the cases of short and long pump pulses is found to occur. In the case of long pulses, the temporal parameter $R_t$ (defined as the ratio of durations of the single-particle and coincidence signals) is shown to be very close to the Schmidt number $K$.

Figure 1

Normalized coincidence and pump spectra (narrow and wide curves) as functions of ${\lambda }_1$ at ${\lambda }_2=\left(\mathrm{a}\right)$ $800\mathrm{nm}$ and (b) $870\mathrm{nm}$.

Figure 2

Normalized coincidence and pump spectra $d{w}^{\left(c\right)}\left({\nu }_1\right)∕d{\nu }_1$ at (a) ${\nu }_2=2$ and (b) ${\nu }_2=-2$, ${\nu }_1$ and ${\nu }_2$ are in units of $10\mathrm{fs}$.

Figure 3

Single-particle (wide) and pump (narrow) spectra (densities of probability of the corresponding photon distributions).

Figure 4

The parameter $R$ and $K$ vs the control parameter $\eta$ [Eq. (14)].

Figure 5

(Color online) The function ${\mid \tilde{\Psi }({\tilde{t}}_1,{\tilde{t}}_2)\mid }^2$ (in relative units) for $\tau =50\mathrm{fs}$, (a) 3D plot and (b) the top view; ${\tilde{t}}_1$ and ${\tilde{t}}_2$ are in units of $10\mathrm{fs}$.

Figure 6

The diagonal squared wave function ${\mid \tilde{\Psi }({\tilde{t}}_1,{\tilde{t}}_1)\mid }^2$ (in relative units).

Figure 7

Normalized coincidence signals at (a) ${\tilde{t}}_2=0$ and (b) ${\tilde{t}}_2=2.8525\mathrm{ps}$.

Figure 8

The normalized time-dependent single-particle probability density; ${\tilde{t}}_1$ and ${\tilde{t}}_2$ are in units of $10\mathrm{fs}$.

Figure 9

(Color online) The same as in Fig. 5 but for $\tau =2$.

Figure 10

Localization region of the function $\tilde{\Psi }({\tilde{t}}_1,{\tilde{t}}_2)$; ${\tilde{t}}_1$, ${\tilde{t}}_2$, $t_{+}$, and $t_{-}$ are in units of $10\mathrm{fs}$; (b) is the increased section 3 of (a).

Figure 11

Squared absolute value of the expression on the right-hand side of Eq. (82).