How to observe distinct universal conductances in tunneling to quantum Hall states: having the right contacts

We show that different universal values can be obtained for the two-terminal conductance of a fractional quantum Hall state. At large voltages, or strong coupling, the conductance of a point-like tunneling junction between an electron gas reservoir and a Laughlin FQH state at filling fraction $\nu$ saturates to a universal value $G={\frac{2\nu}{\nu+1}} {\frac{e^2}{h}}$. We use this result to show that devices with different types of contacts between the reservoir and the FQH state lead to distinct universal values of saturation conductance which are rational multiples of $e^2/h$. The particular fraction $e^2/h$ is obtained for the case of electron tunneling in and out of a FQH liquid by two point contacts. We demonstrate that the problem of tunneling between an electron gas and a fractional quantum Hall state through an impurity is exactly equivalent to the problem of tunneling between a chiral Fermi liquid and a chiral Luttinger liquid. We investigate in detail the case of tunneling to a $\nu=1/3$ FQH state which we show to be equivalent to the problem of tunneling between two $g=1/2$ chiral Luttinger liquids. This system provides an experimental realization of this important exactly solvable case. We use the results of the single impurity problem to consider the case of many tunneling centers coupled independently to an electron reservoir, which is relevant to recent experiments by A.~Chang et.~al. We derive an explicit universal expression for the voltage and temperature dependent conductance which exhibits a crossover reminiscent of a Kondo effect. This universal curve fits the experimental data over the full range of probed voltages.