Nonlinear statistics of quantum transport in chaotic cavities

In the framework of the random matrix approach, we apply the theory of Selberg’s integral to problems of quantum transport in chaotic cavities. All the moments of transmission eigenvalues are calculated analytically up to the fourth order. As a result, we derive exact explicit expressions for the skewness and kurtosis of the conductance and transmitted charge as well as for the variance of the shot-noise power in chaotic cavities. The obtained results are generally valid at arbitrary numbers of propagating channels in the two attached leads. In the particular limit of large (and equal) channel numbers, the shot-noise variance attends the universal value $1∕64\beta$ that determines a universal Gaussian statistics of shot-noise fluctuations in this case.

Figure 1

The variance of the shot-noise power in chaotic cavities as a function of the channel numbers in the leads. (a) The case of symmetric cavities $(N_1=N_2=n)$ for three RMT ensembles, when $\mathrm{var}\left(p\right)$ saturates at the universal value ${\left(64\beta \right)}^{-1}$ at large $n$. (b) The case of asymmetric cavities with preserved time-reversal symmetry $(\beta =1)$ for fixed number $N_1$ of channels in the one lead and varied one $N_2$ in the other lead. The shot-noise variance shows a well-developed maximum at $N_2\approx N_1$ and then diminishes down to zero according to $\mathrm{var}\left(p\right)\approx 2N_1(N_1-1+\frac{2}{\beta })∕\beta N_2^2$ as $N_2$ grows.