ac conductance and nonsymmetrized noise at finite frequency in quantum wires and carbon nanotubes

We calculate the ac conductance and the finite-frequency nonsymmetrized noise in interacting quantum wires and single-wall carbon nanotubes in the presence of an impurity. We observe a strong asymmetry in the frequency spectrum of the nonsymmetrized excess noise, even in the presence of the metallic leads. We find that this asymmetry is proportional to the differential excess ac conductance of the system, defined as the difference between the ac differential conductances at finite and zero voltage, and thus disappears for a linear system. In the quantum regime, for temperatures much smaller than the frequency and the applied voltage, we find that the emission noise is exactly equal to the impurity partition noise. For the case of a weak impurity we expand our results for the ac conductance and the noise perturbatively. In particular, if the impurity is located in the middle of the wire or at one of the contacts, our calculations show that the noise exhibits oscillations with respect to frequency, whose period is directly related to the value of the interaction parameter $g$.

Figure 1

A quantum wire with an impurity located at position $x_i$, adiabatically coupled to metallic leads and to a metallic gate at chemical potential ${\mu }_3=e{V}_3$. The leads are held at different chemical potentials ${\mu }_1=e{V}_1$ and ${\mu }_2=e{V}_2$.

Figure 2

Real parts of the excess ac conductance $\Delta G_{11}\left(\omega \right)$ in units of $\Delta G_{11}\left(\omega =0\right)=-\left(\partial I_B/\partial V-\partial I_B/\partial V{\mid }_{V=0}\right)$ for $g=0.25$ (full line), $g=0.5$ (dashed line) and $g=0.75$ (dash-dotted line). The other parameters are $x_i=0$, $T=0$, $\lambda /eV=0.01$, and $g\hslash {\omega }_L/eV=0.05$. The excess ac conductance for $g=1$ is zero (not shown), consistent with the fact that the excess noise is symmetric in this case.

Figure 3

The zero-frequency excess noise $\Delta S_{11}$ as a function of applied voltage for $g=0.25$ at $k_{B}T/\hslash {\omega }_L=0$ (main plot), and $k_{B}T/\hslash {\omega }_L=5$ (inset). The other parameters are $x_i=0$ and $\lambda /\hslash {\omega }_L=0.01$. $A$, $B$, and $C$ denote three regimes of interest. The noise is renormalized by ${\lambda }^2/{\left({\omega }_L/{\omega }_c\right)}^{2g}$, where ${\omega }_c$ is a high-energy cutoff.

Figure 4

Nonsymmetrized excess noise $\Delta S_{11}$ divided by $e{I}_B$ for different values of the interactions parameter $g$ and for $x_i=0$, $T=0$, $\lambda /eV=0.01$, and $g\hslash {\omega }_L/eV=1$.

Figure 5

(Upper graph) Absorption part and (lower graph) emission part of the nonsymmetrized excess noise $\Delta S_{11}$ divided by $e{I}_B$ for $g=0.25$, $x_i=0$, $T=0$, $\lambda /eV=0.01$, and $g\hslash {\omega }_L/eV=0.01$. The dotted line describes the emission noise of an infinite system with $g=0.25$.

Figure 6

Emission excess noise $\Delta S_{11}\left(\omega \right)$ divided by $e{I}_B$ plotted over the first half period (see arrows) of oscillations (except for $g_5=1$) for different values of the interaction parameters $g$ and for $x_i=0$, $T=0$, $\lambda /eV=0.01$, and $g\hslash {\omega }_L/eV=0.001$. The inset shows the average of the emission noise over one period ($\ast$ points) and of the symmetrized excess noise (◇ points).

Figure 7

Nonsymmetrized excess noise $\Delta S_{11}$ for a nanotube divided by $e{I}_B$ for the interactions parameter $g=0.25$ and for $x_i=0$, $T=0$, $\lambda /eV=0.01$, and $g\hslash {\omega }_L/eV=0.01$. The dotted line is the emission noise of an infinite wire with $g=g^{\ast }\approx 0.8$.