Scheme for Proving the Bosonic Commutation Relation Using Single-Photon Interference

We propose an experiment to directly prove the commutation relation between bosonic annihilation and creation operators, based on the recent experimental success in single-photon subtraction and addition. We devise a single-photon interferometer to realize coherent superpositions of two sequences of photon addition and subtraction. Depending on the interference outcome, the commutation relation is directly proven or a highly nonclassical state is produced. Experimental imperfections are assessed to show that the realization of the scheme is highly feasible.

Figure 1

Experimental setup. BS1 and BS2 are beam splitters of high transmittivity. A photon is added by a parametric down-converter between BS1 and BS2. A $50:50$ beam splitter (BS3) superposes the reflected fields from BS1 and BS2. The output field is selected, conditioned on registering a photon at only one of the two photodetectors PD1 and PD2. $a$, $b$, $c$ and $d$ label the field modes.

Figure 2

Wigner function $W(\beta )$ for the output state conditioned on photodetection only at PD1 (a) or only at PD2 (b) when the initial state is a coherent state of amplitude $\alpha =1$. $T=0.99$ and $\mu =1.005$ ($s=0.1$).

Figure 3

Wigner functions $W(\beta )$ for the output field of mode $a$ for photodetection only at PD1 (a) or only at PD2 (b). The average photon number of the initial thermal field is $\overline{n}=1$. $T=0.99$ and $\mu =1.005$.