Distribution of Parametric Conductance Derivatives of a Quantum Dot

Abstract

The conductance $G$ of a quantum dot with single-mode ballistic point contacts depends sensitively on external parameters $X$, such as gate voltage and magnetic field. We calculate the joint distribution of $G$ and $\mathrm{dG}/\mathrm{dX}$ by relating it to the distribution of the Wigner-Smith time-delay matrix of a chaotic system. The distribution of $\mathrm{dG}/\mathrm{dX}$ has a singularity at zero and algebraic tails. While $G$ and $\mathrm{dG}/\mathrm{dX}$ are correlated, the ratio of $\mathrm{dG}/\mathrm{dX}$ and $\sqrt{G(1\ensuremath{-}G)}$ is independent of $G$. Coulomb interactions change the distribution of $\mathrm{dG}/\mathrm{dX}$, by inducing a transition from the grand-canonical to the canonical ensemble. All these predictions can be tested in semiconductor microstructures or microwave cavities.