Two-particle indistinguishability and identification of boson and fermion species: a Fisher information approach

Abstract

We present a study on two-particle indistinguishability and particle species identification by introducing a Fisher information (FI) approach—in which two particles pass through a two-wave mixing operation and the number of particles is counted in one of the output modes. In our study, we first show that FI can reproduce the Hong–Ou–Mandel (HOM) effect with two bosons or two fermions. In particular, it is found that even though bosons and fermions exhibit different physical behavior (i.e., “bunching” or “anti-bunching”) due to their indistinguishability, the aspects of HOM-like dip are quantitatively same. We then provide a simple method for estimating the degree of two-particle indistinguishability in a Mach–Zehnder interferometer-type setup. The presented method also enables us to identify whether the particles are bosons or fermions. Our study will provide useful primitives for various studies of boson and fermion characteristics.

Appendix A: Distinguishing an input pure state from an input mixed state by Fisher information

For simplicity, here we count the number of particles without discriminating their polarizations or spins. For bosons, we start with the single-photon of mode b in the superposed state of $\cos {\theta }\left| V\right\rangle _b+\sin {\theta }\left| H\right\rangle _b$. When the single photon is in a pure state $\alpha \left| V\right\rangle _a+\beta \left| H\right\rangle _a$, we obtain the probability distributions by counting the number of photons only in the output mode $b'$, such that:

$$\begin{aligned} P_{b'}(0|\phi )= & {} P_{b'}(2|\phi )=\frac{1-\cos {2\theta }}{4}\left( c_0+\frac{c_3}{2}\right) , \nonumber \\ P_{b'}(1|\phi )= & {} \frac{1}{2}\left( 1 + c_1 \cos {2\phi }+ c_2\cos ^2{\phi }-c_3\right) . \end{aligned}$$

(25)

With the above probability distributions, we can have the FI of the phase parameter $\phi$ as

$$\begin{aligned} F_{\text {pure}}(\phi )=2\left( c_0 + \frac{c_3}{2} \right) ^2 \sin ^2{2\phi }\left[ \frac{1}{\left( c_0+\frac{c_3}{2}\right) \left( 1-\cos {2\phi }\right) }+\frac{1}{2-2\left( c_0+\frac{c_3}{2}\right) \sin ^2{\phi }}\right] ,\nonumber \\ \end{aligned}$$

(26)

where $c_0=\frac{1}{2}(1+c_1)$, $c_1=\left| \alpha \right| ^2\cos ^2{\theta }+\left| \beta \right| ^2\sin ^2{\theta }$, $c_2=\left| \alpha \right| ^2\sin ^2{\theta }+\left| \beta \right| ^2\cos ^2{\theta }$, and $c_3=(\alpha \beta ^{*}+\beta \alpha ^{*})\sin {\theta }\cos {\theta }$. Here, note that the factor $c_3$ is related to the interference, and hence the purity, of the input state of the mode a. On the other hand, for a mixed state $\left| \alpha \right| ^2\left| V\right\rangle _a\left\langle V\right| +\left| \beta \right| ^2\left| H\right\rangle _a\left\langle H\right|$, the FI is given by

$$\begin{aligned} F_{\text {mix}}(\phi )=2c_0^2 \sin ^2{2\phi }\left[ \frac{1}{c_0 \left( 1-\cos {2\phi }\right) } +\frac{1}{2-2c_0 \sin ^2{\phi }}\right] . \end{aligned}$$

(27)

Consequently, we can figure out the difference between Eqs. (26) and  (27) in terms of the existence of the interference with $c_3$ (as long as $\phi \ne \frac{\pi }{2}$ and $\phi \ne \pi$). As an example, in Fig. 5 we depict the 3D graphs of (left) $F_{\text {pure}}(\phi )$ and (right) $F_{\text {mix}}(\phi )$ for $\phi =0.1$.

Fig. 5

The graphs of (left) $F_{\text {pure}}(\phi )$ and (right) $F_{\text {mix}}(\phi )$ evaluated for the pure and mixed bosonic input states in a-mode. Here, the phase $\phi$ is set to be 0.1

Fig. 6

The graphs of (left) $F_{\text {pure}}(\phi )$ and (right) $F_{\text {mix}}(\phi )$ evaluated for the pure and mixed fermionic input states in a-mode. Here, we also set $\phi = 0.1$

For fermions, we start with the single-electron of mode b in the superposed state of $\cos {\theta }\left| \uparrow \right\rangle _b+\sin {\theta }\left| \downarrow \right\rangle _b$. When the single-electron of mode a is in a pure state of $\alpha \left| \uparrow \right\rangle _a+\beta \left| \downarrow \right\rangle _a$, we obtain the probability distributions by counting the number of electrons only in the output mode $b'$, such that:

$$\begin{aligned} P_{b'}(0|\phi )= & {} P_{b'}(2|\phi )=\frac{1-\cos {2\theta }}{8}\left( c_2+c_3\right) , \nonumber \\ P_{b'}(1|\phi )= & {} c_0+\frac{c_2 \cos ^2{\phi }}{2} - \frac{c_3\sin ^2{\phi }}{2}. \end{aligned}$$

(28)

Then, from the above probabilities, the FI of $\phi$ is given as

$$\begin{aligned} F_{\text {pure}}(\phi )=\left( c_2+c_3\right) ^2\sin ^2{2\phi }\left[ \frac{1}{\left( c_2+c_3\right) \left( 1-\cos {2\phi }\right) }+\frac{1}{4-2\left( c_2+c_3\right) \sin ^2{\phi }}\right] .\nonumber \\ \end{aligned}$$

(29)

On the other hand, for a mixed state $\left| \alpha \right| ^2\left| \uparrow \right\rangle _a\left\langle \uparrow \right| +\left| \beta \right| ^2\left| \downarrow \right\rangle _a\left\langle \downarrow \right|$ of the single-electron, the FI is given as

$$\begin{aligned} F_{\text {mix}}(\phi )=c_2^2\sin ^2{2\phi }\left[ \frac{1}{c_2\left( 1-\cos {2\phi }\right) }+\frac{1}{4-2c_2\sin ^2{\phi }}\right] . \end{aligned}$$

(30)

As such, we can also figure out the difference of FIs in Eqs. (29) and (30) with $c_3$ (as long as $\phi \ne \frac{\pi }{2}$ and $\phi \ne \pi$). In Fig. 6, we also present the graphs of (left) $F_{\text {pure}}(\phi )$ and (right) $F_{\text {mix}}(\phi )$ for $\phi =0.1$, which exhibits the same behavior as the case of boson.³