We investigate theoretical aspects of the detection of Majorana bound states in Josephson junctions using a semiclassical model of the junction. The influence of a 4π-periodic supercurrent contribution can be detected through its effect on the width of the Shapiro steps and the Fourier spectrum of the voltage. We explain how the inclusion of a capacitance term results in a strong quenching of the odd steps when the junction is underdamped, which may be used to effectively detect Majorana bound states. Furthermore, in presence of capacitance the first and third steps are quenched to a different degree, as observed experimentally. We examine the emission spectrum of phase-locked solutions, showing that the presence of period-doubling may complicate the measurement of the 4π-periodic contribution from the Fourier spectrum. Finally, we study the voltage response in the quasiperiodic regime and indicate how the Fourier spectra and the first-return maps in this regime reflect the change of periodicity in the supercurrent.

Simultaneous breaking of inversion- and time-reversal symmetry in Josephson junction leads to a possible violation of the I(ϕ) = −I(−ϕ) equality for the current-phase relation. This is known as anomalous Josephson effect and it produces a phase shift ϕ0 in sinusoidal current-phase relations. In ballistic Josephson junctions with non-sinusoidal current phase relation the observed phenomenology is much richer, including the supercurrent diode effect and the magnetochiral anisotropy of Josephson inductance. In this work, we present measurements of both effects on arrays of Josephson junctions defined on epitaxial Al/InAs heterostructures. We show that the orientation of the current with respect to the lattice affects the magnetochiral anisotropy, possibly as the result of a finite Dresselhaus component. In addition, we show that the two-fold symmetry of the Josephson inductance reflects in the activation energy for phase slips.

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Preparation and transfer of quantum states is a fundamental task in quantum information. We propose a protocol to prepare a state in the left and center quantum dots of a triple dot array and transfer it directly to the center and right dots. Initially the state in the left and center dots is prepared combining the exchange interaction and magnetic field gradients. Once in the desired state, ac gate voltages in the outer dots are switched on, allowing to select a given photoassisted long-range path and to transfer the prepared state directly from one edge to the other with high fidelity. We investigate the effect of charge noise on the protocol and propose a configuration in which the transfer can be performed with high fidelity. Our proposal can be experimentally implemented and is a promising avenue for transferring quantum states between two spatially separated two-level systems.arXiv:1810.06517v2 [cond-mat.mes-hall]