Lateral effects in fermion antibunching

Lateral effects are analyzed in the antibunching of a beam of free noninteracting fermions. The emission of particles from a source is dynamically described in a three-dimensional full quantum-field-theoretical framework. The size of the source and the detectors, as well as the temperature of the source are taken into account and the behavior of the visibility is scrutinized as a function of these parameters.

Figure 1

Coincidence between two detectors in the beam of emitted particles: The interference of the two alternatives yields antibunching.

Figure 2

(Color online) Normalized two-particle distribution function ${\overline{C}}_{\text{st}}\left(0,0,{\overline{z}}_1;0,0,160w\right)$ vs the longitudinal detector coordinate ${\overline{z}}_1$. ${\overline{C}}_{\text{st}}$ is evaluated in the stationary state on the basis of the expressions (6.22, 6.28), with the Gaussian detectors located at ${\overline{\mathit{r}}}_1=\left(0,0,{\overline{z}}_1\right)$ and ${\overline{\mathit{r}}}_2=\left(0,0,160w\right)$. We focus here on the effect of the longitudinal resolution of the detector, $d$. The parameters are $k_0=20w^{-1}$, $\delta k_z=0.5w^{-1}$, $a=0$, and from bottom to top (in the dip) $d/w=0,1,2,3,4,5$ (in units $\hslash =1$ and $m=1$). We set $\beta =5m{w}^2/{\hslash }^2$ (a low temperature) and the Fermi level $\mu ={\left(k_0+\delta k_z\right)}^2/2m\simeq 210{\hslash }^2/m{w}^2$ (just above the momentum window). Note that in this case $p\simeq 400$ in (6.17) and the condition (6.23) imposes $w_z\ll 20w$. The values based on the numerical integrations of (6.15, 6.25) without the Gaussian approximation are also shown by dots for $d=0$ and $5w$. These were checked to be independent of $w_z$ for small $w_z$.

Figure 3

(Color online) (a) Same as in Fig. 2 but with $d=0$ and from bottom to top $a/w=0,1,2,3,4,5$. We study here the effect of the lateral size of the detector, $a$. Note that $p\simeq 225$ for $a=5w$ in (6.17) and the condition (6.23) imposes $w_z\ll 11w$. The values based on the numerical integrations of (6.15, 6.25) without the Gaussian approximation are also shown by dots for $a=0$ and $5w$ and were checked to be independent of $w_z$ for small $w_z$. (b) Depth of the antibunching dip, $1-{\overline{C}}_{\text{st}}\left(0,0,160w;0,0,160w\right)$, as a function of $a$. All the parameters are the same as in (a).

Figure 4

(Color online) (a) Same as in Fig. 2 but with $a=0$, $d=0$, and from bottom to top $\beta /\left(m{w}^2/{\hslash }^2\right)=5,0.2,0.05$. We analyze here the effect of the temperature, ${\beta }^{-1}$. The values based on the numerical integrations of (6.15, 6.25) without the Gaussian approximation are also shown for $\beta /\left(m{w}^2/{\hslash }^2\right)=5$ and 0.05 by dots, and were checked to be independent of $w_z$ for small $w_z$. (b) Particle emission (from the oven into the beam) is proportional to the overlap of the form factor of the monochromator (bell-shaped curve) and the Fermi distribution function in the oven (steplike function). The effective width of the overlap is inversely proportional to the width of the (antibunching) dip in (a).