Particle current statistics in driven mesoscale conductors

We propose a highly scalable method to compute the statistics of charge transfer in driven conductors. The framework can be applied in situations of nonzero temperature, strong coupling to terminals, and in the presence of nonperiodic light-matter interactions, away from equilibrium. The approach combines the so-called mesoscopic leads formalism with full counting statistics. It results in a generalized quantum master equation that dictates the dynamics of current fluctuations and higher order moments of the probability distribution function of charge exchange. For generic time-dependent quadratic Hamiltonians, we provide closed-form expressions for computing noise in the nonperturbative regime of the parameters of the system, reservoir, or system-reservoir interactions. Having access to the full dynamics of the current and its noise, the method allows us to compute the variance of charge transfer over time in nonequilibrium configurations. The dynamics reveal that in driven systems, the average noise should be defined operationally with care over which period of time is covered.

Figure 1

Mesoscopic lead description of an open quantum system. (a) A depiction of an infinite bath at temperature $T$ and chemical potential $\mu$ with spectral density function $\mathcal{J}\left(\omega \right)$ coupled locally to the $p\mathrm{th}$ fermionic site of a system. (b) The bath is discretized by a finite collection of $N$ fermionic modes with self-energies ${\varepsilon }_k$, which are coupled locally to the $p\mathrm{th}$ site of the system with strength ${\kappa }_{k,p}$. Each of the modes is subject to dissipation intended to drive the mode to thermal and chemical equilibrium state. In (a), FCS is performed with a counting field $\chi$ embedded in the reservoir. In contrast, in (b) the counting fields turn up in the internal system-modes couplings.

Figure 2

(Top) average currents $J_{\mathtt{L}}\left(t\right)$ and $-J_{\mathtt{R}}\left(t\right)$ [Eq. (5)] during multiple periods $\tau =2\pi /\omega$ of the external drive, up until the LC is reached. (Bottom) noise $D_{\mathtt{L}/\mathtt{R}}(t,t_1)$, starting at the LC $t_1=24\pi /\omega$. Integrating over the first period yields $\overline{S_{\nu }^0}$ in Eq. (12). Waiting for multiple periods and then integrating yields instead $\overline{S_{\nu }^{\infty }}$ in Eq. (13). Parameters are described in the main text.

Figure 3

$\overline{J}$, $\overline{S^0}$, and $\overline{S^{\infty }}$ [Eqs. (11, 12, 13)] as a function of the driving field amplitude $eaA/\mathrm{\Delta }$, in the limit cycle, with fixed frequency $\omega =5\mathrm{\Delta }$. (Inset) same, but as a function of $\omega /\mathrm{\Delta }$, with fixed $eaA=20\mathrm{\Delta }$. Other parameters are as in Fig. 2.