In 1909, Millikan showed that the charge of electrically isolated systems is quantized in units of the elementary electron charge e. Today, the persistence of charge quantization in small, weakly connected conductors allows for circuits in which single electrons are manipulated, with applications in, for example, metrology, detectors and thermometry. However, as the connection strength is increased, the discreteness of charge is progressively reduced by quantum fluctuations. Here we report the full quantum control and characterization of charge quantization. By using semiconductor-based tunable elemental conduction channels to connect a micrometre-scale metallic island to a circuit, we explore the complete evolution of charge quantization while scanning the entire range of connection strengths, from a very weak (tunnel) to a perfect (ballistic) contact. We observe, when approaching the ballistic limit, that charge quantization is destroyed by quantum fluctuations, and scales as the square root of the residual probability for an electron to be reflected across the quantum channel; this scaling also applies beyond the different regimes of connection strength currently accessible to theory. At increased temperatures, the thermal fluctuations result in an exponential suppression of charge quantization and in a universal square-root scaling, valid for all connection strengths, in agreement with expectations. Besides being pertinent for the improvement of single-electron circuits and their applications, and for the metal-semiconductor hybrids relevant to topological quantum computing, knowledge of the quantum laws of electricity will be essential for the quantum engineering of future nanoelectronic devices.

In this work, we address the recent experiment [S. Tewari et al., arXiv:1503.05057v1], where the suppression of phase coherence of a single-electron wave packet created at the edge of a quantum Hall (QH) system at filling factor 2 has been investigated with the help of an electronic Mach-Zehnder (MZ) interferometer. The authors of the experiment have observed an unexpected behavior of phase coherence, that saturates at high energies instead of vanishing, presumably suggesting the relaxation of a wave packet to the ground state before it arrives to the MZ interferometer. Here, we theoretically investigate this situation using the model of edge states [I. P. Levkivskyi, E. V. Sukhorukov, Phys. Rev. B 78, 045322 (2008)], which accounts for the strong Coulomb interaction between the two electron channels at the edge of a QH system. We conclude that the observed phenomenon cannot be explained within this model for the reason that under an assumption of linearity of the electron spectrum at low energies the system remains integrable in terms of the collective charge excitations, and therefore full relaxation to the ground state is not possible, despite strong interactions. As a result, the degree of the phase coherence decreases with energy of the initial state in a power-law manner. Since this does not happen in the experiment, a new physical phenomenon may take place at the edge of a QH state, which deserves further investigations. We support our findings by calculating the energy distribution and the Wigner function of the outgoing non-equilibrium state of the single-electron wave packet.

Parafermions are non-Abelian anyons which generalize Majorana fermions and hold great promise for topological quantum computation. We study the braiding of Z 2n parafermions which have been predicted to emerge as bound states in fractional quantum Hall systems at filling factor ν = 1/n (n odd). Using a combination of bosonization and refermionization, we calculate the energy splitting as a function of distance and chemical potential for a pair of parafermions separated by a gapped region. Braiding of parafermions in quantum Hall edge states can be implemented by repeated fusion and nucleation of parafermion pairs. We simulate the conventional braiding protocol of parafermions numerically, taking into account the finite separation and finite chemical potential. We show that a nonzero chemical potential poses challenges for the adiabaticity of the braiding process because it leads to accidental crossings in the spectrum. To remedy this, we propose an improved braiding protocol which avoids those degeneracies. arXiv:1907.12799v1 [cond-mat.str-el]

We study dephasing in an electronic Mach-Zehnder (MZ) interferometer based on quantum Hall edge states by a micrometer-sized Ohmic contact embedded in one of its arms. We find that at the filling factor ν=1, as well as in the case where an Ohmic contact is connected to a MZ interferometer by a quantum point contact that transmits only one electron channel, the phase coherence may not be fully suppressed. Namely, if the voltage bias Δμ and the temperature T are small compared to the charging energy of the Ohmic contact E_{C}, the free fermion picture is manifested, and the visibility saturates at its maximum value. At large biases, Δμ≫E_{C}, the visibility decays in a power-law manner.