Intrinsic noise is known to be ubiquitous in Josephson junctions. We investigate a voltage biased superconducting tunnel junction including a very small number of pinholes - transport channels possessing a transmission coefficient close to unity. Although few of these pinholes contribute very little to the conductance, they can dominate current fluctuations in the low-voltage regime. We show that even fully transparent transport channels between superconductors contribute to shot noise due to the uncertainty in the number of Andreev cycles. We discuss shot noise enhancement by Multiple Andreev Reflection in such a junction and investigate whether pinholes might contribute as a microscopic mechanism of two-level current fluctuators. We discuss the connection of these results to the junction resonators observed in Josephson phase qubits.
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