Direct temporal mode measurement of photon pairs by stimulated emission

It is known that photon pairs generated from pulse-pumped spontaneous parametric processes can be described by independent temporal modes and form a multitemporal mode entangled state. However, the exact form of the temporal modes is not known even though the joint spectral intensity of photon pairs can be measured by the method of stimulated emission tomography. In this paper, we describe a feedback-iteration method which, combined with the stimulated emission method, can give rise to the exact forms of the independent temporal modes for the temporally entangled photon pairs.

Figure 1

Entangled two-photon states consisting of various temporal modes $A_k,B_k(k=1,2,3,...)$ in the signal and idler fields generated from a pulse-pumped parametric process.

Figure 2

Schematic diagram for measuring the mode functions at low gain. WS: wave shaper. The directly related light fields have the same line style (dashed or solid) and color (red or blue) and the direction of the arrow shows the flow of the iteration. The converged functions are the outputs that give the measured mode functions ${\psi }_k\left(\omega \right)\text{and}{\phi }_k\left(\omega \right)$.

Figure 3

Simulated convergent output spectral functions with their magnitudes and phases for the first three modes $k=1$, 2, 3 for the JSF given in Eq. (21). (a) Signal field ${\psi }_k\left({\mathrm{\Omega }}_s\right)$. (b) Idler field ${\phi }_k\left({\mathrm{\Omega }}_i\right)$. The green dash-dotted curves are the input spectral functions while the blue dotted and red solid curves are intermediate outputs after the iteration steps indicated in the legends. The black dashed curves are the final outputs.

Figure 4

Mode number distribution obtained by simulation for the JSF given in Eq. (21).

Figure 5

Simulated convergent output spectral functions with their magnitudes and phases for the first three modes $k=1$, 2, 3 for the JSF given in Eq. (21) but with a chirped pump phase of $e^{i{({\mathrm{\Omega }}_s+{\mathrm{\Omega }}_i)}^2/{\sigma }_p^2}$. (a) Signal field ${\psi }_k\left({\mathrm{\Omega }}_s\right)$. (b) Idler field ${\phi }_k\left({\mathrm{\Omega }}_i\right)$. The green dash-dotted curves are the input spectral functions while the blue dotted and red solid curves are intermediate outputs after the iteration steps indicated in the legends. The black dashed curves are the final outputs.

Figure 6

Mode number distribution obtained by simulation for the JSF given in Eq. (21) but with a modified phase of $e^{i{({\mathrm{\Omega }}_s+{\mathrm{\Omega }}_i)}^2/{\sigma }_p^2}$ added due to chirping of the pump field.

Figure 7

Normalized ratio $R_k^{\left(M\right)}/G{r}_1(k=1,2,3)$ as a function of iteration step $M$. The straight lines are the limiting values of $r_k/r_1$.