Bunching and antibunching in electronic transport

In quantum optics the $g^{\left(2\right)}$ function is a standard tool to investigate photon emission statistics. We define a $g^{\left(2\right)}$ function of real time $\tau$ for electronic transport and use it to investigate the bunching and antibunching of electron currents. Importantly, we show that super-Poissonian electron statistics do not necessarily imply electron bunching, and that sub-Poissonian statistics do not imply antibunching. We discuss the information contained in $g^{\left(2\right)}\left(\tau \right)$ for several typical examples of transport through nanostructures such as few-level quantum dots.

Figure 1

Sketch of (a) antibunched, (b) Poissonian, and (c) bunched photon (or electron) emission events.

Figure 2

Sketch of four of the five transport models discussed here. (a) The dynamical channel blockade model, (b) The double quantum dot, (c) the Anderson model ($j=2$), and (d) the Anderson model with only one singly occupied level participating in transport ($j=1$). In each case, electrons tunnel with the rate ${\Gamma }_L$ (${\Gamma }_R$) into (out of) the quantum dot systems except when the outgoing rate is modified by the factor $x$.

Figure 3

Main panel: The second-order correlation function $g_{\text{DCB}}^{\left(2\right)}\left(\tau \right)-1$ for the dynamical channel blockade model shown for three values of parameter $x=1,1/3,1/5$ corresponding to a sub-Poissonian (dashed), Poissonian (dotted), and super-Poissonian (line) Fano factor. Inset: The (shifted) Fano factor $F_{\text{DCB}}\left(0\right)-1$ for the same model as a function of $x$. The triangles mark the points for which the $g_{\text{DCB}}^{\left(2\right)}\left(\tau \right)$ is shown in the main panel. Other parameters were ${\Gamma }_L={\Gamma }_R$.

Figure 4

Main panel: The $g_{\text{DQD}}^{\left(2\right)}\left(\tau \right)$ function for the double quantum dot (DQD) model with zero detuning such that the Fano factor is sub-Poissonian, $F_{\text{DQD}}\left(0\right)\approx 0.859$ (dashed line) and detuning $\varepsilon /{\Gamma }_L=1/2$ such that the Fano factor is super-Poissonian, $F_{\text{DQD}}\left(0\right)\approx 1.174$ (solid line). Inset: The DQD Fano factor as a function of the detuning $\varepsilon$. The triangles mark the points for which the $g_{\text{DQD}}^{\left(2\right)}\left(\tau \right)-1$ is shown in the main panel. Other parameters are $T_C/{\Gamma }_L=1/4$, ${\Gamma }_R/{\Gamma }_L=1/10$.

Figure 5

Main panel: The $g_{2\text{el}}^{\left(2\right)}\left(\tau \right)$ function for the Anderson model with $j\in \left\{1,2\right\}$ single-electron channels. With a single channel ($j=1$) and rate parameter $x=1/2$ the Fano factor is sub-Poissonian [$F_{2\text{el}}\left(0\right)=3/4$, see inset], whereas the $g^{\left(2\right)}$ function clearly indicates bunching (solid line) for all times $\tau$. With two channels ($j=2$) and $x=1/5$ the Fano factor is super-Poissonian [$F_{2\text{el}}\left(0\right)=9/8$] but the $g^{\left(2\right)}$ function indicates complete antibunching. This relationship between (anti)bunching and Fano factor is completely opposite to the usual interpretation. Inset: Fano factor $F\left(0\right)-1$ as a function of $x$ for $j=1$ (line) and $j=2$ (dashed). The triangles mark the $x$ for which $g_{\text{DQD}}^{\left(2\right)}\left(\tau \right)$ is plotted in the two cases where $F\left(0\right)\ne 1$. Parameters: ${\Gamma }_L={\Gamma }_R=\Gamma =1$.