Quantum versus classical counting in non-Markovian master equations

We discuss the description of full counting statistics in quantum transport with a non-Markovian master equation. We focus on differences arising from whether charge is considered as a classical or a quantum degree of freedom. These differences manifest themselves in the inhomogeneous term of the master equation which describes initial correlations. We describe the influence on current and in particular, the finite-frequency shotnoise. We illustrate these ideas by studying transport through a quantum dot and give results that include both sequential and cotunneling processes. Importantly, the noise spectra derived from the classical description are essentially frequency independent and all quantum noise effects are absent. These effects are fully recovered when charge is considered as a quantum degree of freedom.

Figure 1

The finite-frequency total noise $S\left(\omega \right)$ as a function of $\omega$ for the infinite-$U$ Anderson model QD with level positions ${\epsilon }_{\uparrow }=-15k_{B}T$, ${\epsilon }_{\downarrow }=5k_{B}T$ and chemical potentials ${\mu }_L=-{\mu }_R=\frac{1}{2}\mathit{eV}$; $V/k_{B}T=2,15,25,50$. Results are shown for the noise calculated with both quantum and classical counting from fourth-order (cotunneling) LPT kernels (solid black and dashed blue lines, respectively). Both counting theories give the same noise at $\omega =0$ but as frequency increases, differences start to develop. This is most evident at small biases where, whereas the classical noise is essentially flat as a function of frequency, the quantum result shows marked steps at $\omega =|{\mu }_{\alpha }-{\epsilon }_{\uparrow }|$ (steps at $\omega =|{\mu }_{\alpha }-{\epsilon }_{\downarrow }|$ are also visible at $V=50k_{B}T$; see text). For $V=2k_{B}T$, the difference between between the two results is approximately two orders-of-magnitude at high frequencies. At higher bias the system enters the shotnoise regime and the relative importance of the quantum contributions lessens. Other parameters were ${\Gamma }_L={\Gamma }_R=\frac{1}{4}{k}_{B}T$ and bandwidth $X_C=300k_{B}T$.

Figure 2

As Fig. 1 but with finite-frequency noise $S\left(\omega \right)$ plotted as a function of applied bias for $\omega /k_{B}T=0,4,10,23$. Quantum- and classical-counting results with fourth-order LPT (cotunneling) kernels (black solid and blue dashed, respectively) are show alongside results from quantum counting with second-order LPT (sequential) kernels (green dotted lines). The quantum results are dominated by sequential processes with shotnoise step at $V=30k_{B}T$ and quantum noise step up at $V=30k_{B}T-2\omega$ ($\omega <15k_{B}T$) and step down at $V=-30k_{B}T+2\omega$ for high frequencies ($\omega >15k_{B}T$) instead. With the parameters cotunnellng predominantly serves to increase noise, both at zero and finite frequency, at low frequencies. In contrast to the quantum case, the classical results change very little with increasing frequency.

Figure 3

Transport through the infinite-$U$ Anderson model with both dots levels well below transport window: ${\epsilon }_{\uparrow }=-20k_{B}T$, ${\epsilon }_{\downarrow }=-30k_{B}T$; other parameters as in Fig. 1. The inset shows the differential conductance $G=d\langle I\rangle /d\left(V\right)$ as a function of bias. A clear jump is observed around the inelastic cotunneling energy $E_{\mathrm{inelastic}}=|{\epsilon }_{\downarrow }-{\epsilon }_{\uparrow }|=10k_{B}T$. The main panel compares the derivative of the shotnoise, $dS\left(\omega \right)/d\omega$, as a function of $\omega$ for both sequential (LPT2) and cotunneling (LPT4) kernels near linear response ($V=1k_{B}T$). Both predict a large peak when the measurement frequency is approximately equal to ${\mu }_{L/R}-{\epsilon }_{\downarrow }\sim 30k_{B}T$. In the LPT4 results this peak is broadened by elastic cotunneling processes, but more importantly, there is a significant enhancement in the low-bias tail which can be attributed to resonance with inelastic cotunneling process $\omega \sim E_{\mathrm{inelastic}}$.