Detection of interactions via generalized factorial cumulants in systems in and out of equilibrium

We introduce time-dependent, generalized factorial cumulants $C_s^m\left(t\right)$ of the full counting statistics of electron transfer as a tool to detect interactions in nanostructures. The violation of the sign criterion ${(-1)}^{m-1}{C}_s^m\left(t\right)\ge 0$ for any time $t$, order $m$, and parameter $s$ proves the presence of interactions. For given system parameters, there is a minimal time span $t_{\text{min}}$ and a minimal order $m$ to observe the violation of the sign criterion. We demonstrate that generalized factorial cumulants are more sensitive to interactions than ordinary ones and can detect interactions even in regimes where ordinary factorial cumulants fail. We illustrate our findings with the example of a quantum dot tunnel coupled to electronic reservoirs either in or out of equilibrium.

Figure 1

(a) Equilibrium scenario: A single-level quantum dot subject to a Zeeman field is tunnel coupled to one normal lead. (b) Nonequilibrium scenario: A single-level quantum dot is tunnel coupled to two ferromagnetic leads with finite bias voltage. (c) Sketch of the states and transition rates.

Figure 2

Zeros $z_j$ for the equilibrium scenario with $\mathrm{\Delta }=k_{\text{B}}T/2, \varepsilon =-\mathrm{\Delta }$, and times $t=5/\mathrm{\Gamma },8/\mathrm{\Gamma },15/\mathrm{\Gamma }$. For times $t\le t_{\text{min}}\approx 6.83/\mathrm{\Gamma }$, the zeros remain on the real axis. For $t>t_{\text{min}}$, zeros leave the real axis and interactions can be detected by generalized factorial cumulants.

Figure 3

(Generalized) factorial cumulant $C_s^m$ normalized by $C_1^1=C^1=\langle N\rangle >0$ for the equilibrium scenario as a function of (a) time $t$ and (b) dot-level energy $\varepsilon$. The parameters are $\mathrm{\Delta }=k_{\text{B}}T/2$ and (a) $\varepsilon =-\mathrm{\Delta }$ or (b) $t=100/\mathrm{\Gamma }$. Negative values of ${(-1)}^{m-1}{C}_s^m\left(t\right)$ indicate the presence of interactions. Interactions can be detected (a) for times larger than $t_{\text{min}}\approx 6.83/\mathrm{\Gamma }$ and (b) for level positions $\varepsilon \lesssim k_{\text{B}}Tln2$.

Figure 4

(a) Parameter space and (b) minimal time span $t_{\text{min}}$ of the system in equilibrium. (a) To the left of the colored lines, the sign criterion for the respective $C_s^m\left(t\right)$ is violated at some time $t$. To the right of the dashed line, ${\mathcal{M}}_s(z,t)$ has only real zeros such that the sign criterion cannot be violated. (b) Minimal time span $t_{\text{min}}$ increases with increasing $\varepsilon$ and decreasing $\left|\mathrm{\Delta }\right|$ and diverges at the dashed line in (a) and for $\mathrm{\Delta }=0$.

Figure 5

Zeros $z_j$ for the equilibrium scenario with $\mathrm{\Delta }=k_{\text{B}}T/2, t=100/\mathrm{\Gamma }$, and $\varepsilon /k_{\text{B}}T=-1.5,0.0,1.0$. For $\varepsilon /k_{\text{B}}T\lesssim ln2$, zeros leave the real axis after some time and interactions can be detected by generalized factorial cumulants.

Figure 6

(a) Parameter space and (b) minimal time span $t_{\text{min}}$ of the system out of equilibrium. The polarizations are $p_{\text{L}}=p_{\text{R}}=p$, however, a different polarization $p_{\text{L}}$ (even $p_{\text{L}}=0$) does not change the results qualitatively. (a) Above the colored lines, the sign criterion for the respective $C_s^m\left(t\right)$ is violated at some time $t$. Below the dashed line, ${\mathcal{M}}_s(z,t)$ has only real zeros such that the sign criterion cannot be violated. (b) For $a\ge -1/3$, the minimal time $t_{\text{min}}$ increases with decreasing $p$ and decreasing $a$ and diverges at $p=0$; for $a<-1/3$, the minimal time $t_{\text{min}}$ diverges both at the dashed and the dotted-dashed line in (a).

Figure 7

Zeros $z_j$ for the nonequilibrium scenario discussed in Fig. 6 with $p_{\text{L}}=p_{\text{R}}=0.35, t=50/({\mathrm{\Gamma }}_{\text{L}}+{\mathrm{\Gamma }}_{\text{R}})$, and $a=({\mathrm{\Gamma }}_{\text{L}}-{\mathrm{\Gamma }}_{\text{R}})/({\mathrm{\Gamma }}_{\text{L}}+{\mathrm{\Gamma }}_{\text{R}})=-0.8,-0.5,0.0$. For $a=-0.8$, all zeros remain on and for $a=0.0$ all zeros leave the real axis. For $a=-0.5$, the rightmost zeros remain on the real axis, but zeros further to the left move into the complex plane. While factorial cumulants are insensitive to interactions in this case, generalized factorial cumulants can detect their presence.