Counting statistics for arbitrary cycles in quantum pumps

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Statistics of charge transport in an adiabatic pump are determined by the dynamics of the scattering matrix S(t). We show that, up to an integer offset, the statistics depend only on the corresponding path N(t)=S†σ₃ S in the coset space (the sphere for a single channel). For a general loop S(t) we solve for the noise-minimizing pumping strategy. The average current is given by the area enclosed by N(t) in the coset space; its minimal noise by the area of a minimal surface (soap film) spanned by N(t) in the space of all matrices. We formulate conditions for quantization of the pumped charge.