The Josephson diode effect - an asymmetry of critical currents at opposite polarities - has attracted much attention recently. One of its simplest realizations is an asymmetric SQUID with multiple Josephson harmonics tuned away from half-integer flux (in units of the superconducting flux quantum). Here we predict a dual version of this effect in a gate-tunable Cooper pair transistor placed in series with a highly resistive environment. The dual "Bloch diode effect"manifests itself as an asymmetry of critical voltages at opposite polarities when the Cooper pair transistor is tuned away from half-integer gate charge (in units of the Cooper pair charge). It arises from an asymmetry in the dispersion of the transistor's Bloch bands. A highly resistive environment can be realized with a Josephson junction array, suggesting that such a Bloch diode could be implemented using conventional superconducting quantum circuits.

We consider the propagation of electrons in a lattice with an anisotropic dispersion in the x-y plane (lattice constant a), such that it supports open orbits along the x axis in an out-of-plane magnetic field B. We show that a point source excites a "breathing mode,"a state that periodically spreads out and refocuses after having propagated over a distance ?=(eaB/h)-1 in the x direction. Unlike known magnetic focusing effects, governed by the classical cyclotron radius, this is an intrinsically quantum mechanical effect with a focal length.