Appearance of Fractional Charge in the Noise of Nonchiral Luttinger Liquids

The current noise of a voltage biased interacting quantum wire adiabatically connected to metallic leads is computed in the presence of an impurity in the wire. We find that in the weak backscattering limit the Fano factor characterizing the ratio between noise and backscattered current crucially depends on the noise frequency $\omega$ relative to the ballistic frequency $v_F/gL$, where $v_F$ is the Fermi velocity, $g$ is the Luttinger liquid interaction parameter, and $L$ is the length of the wire. In contrast to chiral Luttinger liquids the noise is not only due to the Poissonian backscattering of fractionally charged quasiparticles at the impurity, but it also depends on Andreev-type reflections at the contacts, so that the frequency dependence of the noise needs to be analyzed to extract the fractional charge $e^{*}=eg$ of the bulk excitations.

Figure 1

The periodic function $F(\omega )$, which determines the Fano factor, is shown as a function of $\omega /2\pi {\omega }_L$, for the case of an impurity at the center of the wire ($x_0=0$) and three different values of the interaction strength: $g=0.25$ (solid line), $g=0.50$ (dashed line), and $g=0.75$ (dotted line). In the regime $\omega /{\omega }_L\ll 1$, the function tends to 1 independent of the value of $g$, but for $\omega \lesssim {\omega }_L$ the curve strongly depends on the interaction parameter $g$. In particular, $g$ can be obtained as the average over one period.

Figure 2

The Fano factor $F(\omega )$ is shown for the interaction strength $g=0.25$ and three different values of the (relative) impurity position ${\xi }_0=x_0/L$: ${\xi }_0=0$ (solid line), ${\xi }_0=0.10$ (dashed line), and ${\xi }_0=0.25$ (dotted line).