Non-Markovian effects in the quantum noise of interacting nanostructures

We present a theory of finite-frequency noise in nonequilibrium conductors. It is shown that Non-Markovian correlations are essential to describe the physics of quantum noise. In particular, we show the importance of a correct treatment of the initial system-bath correlations, and how these can be calculated using the formalism of quantum master equations. Our method is particularly important in interacting systems, and when the measured frequencies are larger than the temperature and applied voltage. In this regime, quantum-noise steps are expected in the power spectrum due to vacuum fluctuations. This is illustrated in the current noise spectrum of a single resonant level model and of a double quantum dot—charge qubit—attached to electronic reservoirs. Furthermore, the method allows for the calculation of the single-time counting statistics in quantum dots, measured in recent experiments.

Figure 1

Schematics of counting. The density operator evolves from the initial separable state at time $t_0$ (represented by two distinct ellipses) until it reaches a steady state at time $t=0$, where it is no longer in a product state (single ellipse). At time $t=0$ counting begins. The shading highlights the time interval where counting is effective. Solid circles denote tunnel vertices with counting factors $\chi \ne 0$; open circles denote standard tunneling vertices ($\chi =0$). Contractions between tunneling events in counting and noncounting intervals (dashed overline) give rise to $\Gamma (\chi ,z)$, while contractions within the counting interval (solid overline) give rise to the self-energy $\Sigma (\chi ,z)$.

Figure 2

Quantum noise spectra of the SRL model (${\Gamma }_L={\Gamma }_R=1$ in all figures). (a) $S^{(2)}(\omega )$ as a function of frequency $\omega$ in the shot noise regime ($\varepsilon =20$, $V=100$, $T=4$). In this limit, the noise develops dips at $\omega =\pm |\varepsilon \pm \frac{\mathrm{eV}}{2}|$. (b) $S^{(2)}(\omega )$ as a function of frequency $\omega$ in the quantum noise regime ($\varepsilon =10$, $V\to 0$, $T=1$). In this limit, $S^{(2)}(\omega )$ develops a quantum noise step at $\omega =\varepsilon$. (c) Charge noise $S_Q^{(2)}(\omega )$ as a function of frequency $\omega$ of a single electron transistor acting as a detector ($E_C=V=10$). When $\omega >E_C$, $S_Q^{(2)}(\omega )$ contains extra quantum noise contributing to back-action.

Figure 3

Zero-frequency limit of the non-Markovian theory. (a) Particle-current noise for ${\Gamma }_L={\Gamma }_R$, $\varepsilon /{\Gamma }_R=80$, $T/{\Gamma }_R=4$. The noise suddenly increases when the level enters the voltage bias window. (b) $S_R^{(2)}(0)$ as a function of bias voltage $V$ for ${\Gamma }_L={\Gamma }_R$, $\varepsilon =0$, $T/{\Gamma }_R=4$. While the Markovian approximation is flat at all voltages, the NM and exact solution show a structure that strongly differs in both for low voltages due to cotunneling processes.

Figure 4

Single-time full counting statistics of the SRL model (Fano factors of the right particle current $F^{(k)}(t)\equiv \langle n_R^k(t)\rangle {}_c/\langle n_R\rangle$ with ${\Gamma }_L={\Gamma }_R=0.25$, $T=1$) up to the fifth cumulant. (a) Shot noise regime ($\varepsilon =20$, $V=100$). (b) Quantum fluctuations regime ($\varepsilon =20$, $V=30$). Dotted lines correspond to the Markovian approximation.

Figure 5

Quantum noise processes in a double quantum dot. In the QNR, quantum fluctuations can discharge the system through the left/right reservoir if $\hslash \omega ⩾|{\mu }_{\mathrm{L}/\mathrm{R}}-\Delta /2|$. These correspond to the steps in Fig. 6a. When $\omega =\Delta$, quantum interference between the eigenstates gives a noise suppression.

Figure 6

Finite-frequency noise of a double quantum dot (results normalized to the dc current in the large-voltage limit $I_{\mathrm{dc}}(V=\infty )=t_c^2{\Gamma }_R/[{\Gamma }_R^2/4+4\varepsilon {}^2+t_c^2(2+{\Gamma }_R/{\Gamma }_L)]$. (a) Near linear response, $S^{(2)}(\omega )$ shows quantum noise steps at $\omega =|\Delta /2\pm V/2|$ and a dip centered at $\omega =\Delta$ (indicated with arrows for the case $V=4$ in the figure). (b) The feature at $\omega =\Delta$ originates from quantum interference between the bonding and antibonding qubit states. Here we set $V=0.1$ and vary the reservoir Fermi energy $E_F$, observing a displacement of the quantum noise step, as well as a modification of the resonance form at the qubit frequency $\Delta$ (see text). (c) Shot noise limit. Quantum noise steps are only visible for $V\lesssim \Delta ,k_{B}T$, otherwise the contribution from shot noise or thermal noise are dominant. Parameters are as follows: $\varepsilon =0$, $\Delta =2T_c=6$, ${\Gamma }_L={\Gamma }_R=T/2=0.1$.

Figure 7

(a) Effect of a gate voltage. As the relative distance between the dot levels and the lead chemical potentials is varied (here illustrated decreasing the Fermi energy $E_F$), a quantum noise step, absent when the bonding state is aligned with both chemical potentials, appears at the corresponding frequency difference. The Fano shape, however, gives an antiresonance at the qubit frequency $\Delta$. Here, $T=0.2$. (b) Effect of the temperature. As $T$ is increased, the quantum noise step is lost, since thermal noise overcomes quantum noise, giving a finite $S^{(2)}$ value at zero frequency. The Fano shape is, however, preserved for high temperatures. Here, $E_F=-2.5$. In both figures $V=0.1$, $\varepsilon =0$, $\Delta =2T_c=6$, ${\Gamma }_L={\Gamma }_R=0.1$.