Temporal distinguishability of an $N$-photon state and its characterization by quantum interference

We present a multimode model to describe an arbitrary $N$-photon state with a wide spectral range and some arbitrary temporal distribution. In general, some of the $N$ photons are spread out in time while others may overlap and become indistinguishable. From this model, we find that the temporal (in)distinguishability of photons is related to the exchange symmetry of the multiphoton wave function. We find that simple multiphoton detection scheme gives rise to a more general photon bunching effect with the famous two-photon effect as a special case. We then send this $N$-photon state into a recently discovered multiphoton interference scheme. We calculate the visibility of the multiphoton interference scheme and find that it is related to the temporal distinguishability of the $N$ photons. Maximum visibility of one is achieved for the indistinguishable $N$-photon state whereas the visibility degrades when some of the photons are separated and become distinguishable. Thus we can identify an experimentally measurable quantity that may quantitatively define the degree of indistinguishability of an $N$-photon state. This presents a quantitative demonstration of the complementary principle of quantum interference.

Figure 1

Generation of an $N$-photon state from single-photon states by beam splitters.

Figure 2

A NOON-state projection measurement. ${\delta }_k=2k\pi ∕N$ is the phase difference between $H$ and $V$. ${\widehat{b}}_k\propto {\widehat{E}}_H-{\widehat{E}}_V{e}^{i{\delta }_k}$.

Figure 3

(a) A temporal distribution with well separated groups of $V$ photons and (b) the corresponding normalized $P_{N+1}$ as the position of the $H$ photon is scanned.