The collision of two electrons at a beam splitter provides a method for studying their coherence and indistinguishability. Its realization requires the on-demand generation and synchronization of single electrons. In this work, we demonstrate the coherent collision of single electrons, generated by voltage pulses, in a graphene Mach-Zehnder interferometer. By measuring shot noise resulting from the collisions, we unveil fundamental characteristics of colliding electrons, highlighting the complementarity between the indistinguishable and distinguishable parts of their wave functions. The former is manifested through fermionic Hong-Ou-Mandel destructive interference, whereas the latter is discerned through double-winding Aharonov-Bohm interference in the noise. The interference visibilities of around 60% enable comprehensive quantum state tomography. Our findings may place coherent operations involving flying qubits within reach in graphene.

Flying qubits encode quantum information in propagating modes instead of stationary discrete states. Although photonic flying qubits are available, the weak interaction between photons limits the efficiency of conditional quantum gates. Conversely, electronic flying qubits can use Coulomb interactions, but the weaker quantum coherence in conventional semiconductors has hindered their realization. In this work, we engineered on-demand injection of a single electronic flying qubit state and its manipulation over the Bloch sphere. The flying qubit is a Leviton propagating in quantum Hall edge channels of a high-mobility graphene monolayer. Although single-shot qubit readout and two-qubit operations are still needed for a viable manipulation of flying qubits, the coherent manipulation of an itinerant electronic state at the single-electron level presents a highly promising alternative to conventional qubits.

The collision of two electrons at a beam splitter provides a method for studying their coherence and indistinguishability. Its realization requires the on-demand generation and synchronization of single electrons. In this work, we demonstrate the coherent collision of single electrons, generated by voltage pulses, in a graphene Mach-Zehnder interferometer. By measuring shot noise resulting from the collisions, we unveil fundamental characteristics of colliding electrons, highlighting the complementarity between the indistinguishable and distinguishable parts of their wave functions. The former is manifested through fermionic Hong-Ou-Mandel destructive interference, whereas the latter is discerned through double-winding Aharonov-Bohm interference in the noise. The interference visibilities of around 60% enable comprehensive quantum state tomography. Our findings may place coherent operations involving flying qubits within reach in graphene.

Flying qubits encode quantum information in propagating modes instead of stationary discrete states. Although photonic flying qubits are available, the weak interaction between photons limits the efficiency of conditional quantum gates. Conversely, electronic flying qubits can use Coulomb interactions, but the weaker quantum coherence in conventional semiconductors has hindered their realization. In this work, we engineered on-demand injection of a single electronic flying qubit state and its manipulation over the Bloch sphere. The flying qubit is a Leviton propagating in quantum Hall edge channels of a high-mobility graphene monolayer. Although single-shot qubit readout and two-qubit operations are still needed for a viable manipulation of flying qubits, the coherent manipulation of an itinerant electronic state at the single-electron level presents a highly promising alternative to conventional qubits.

While ballistic electrons are a key tool for applications in sensing and flying qubits, sub-nanosecond propagation times and complicated interactions make control of ballistic single electrons challenging. Recent experiments have revealed Coulomb collisions of counterpropagating electrons in a beam splitter, giving time resolved control of interactions between single electrons. Here we use remote Coulomb interactions to demonstrate a scheme for sensing single ballistic electrons. We show that interactions are highly controllable via electron energy and emission timing. We use a weakly coupled "sensing" regime to characterize the nanoscale potential landscape of the beam splitter and the strength of the Coulomb interaction. We also show multielectron sensing with picosecond resolution.

Understanding environmental effects in a topological Josephson junction is vital for identifying signatures of Majorana modes. We consider a planar Josephson junction formed on the surface of a three-dimensional topological insulator, which possesses Majorana modes inside the junction and boundary modes outside. We find that tunneling between the inner and outer modes gives rise to effective coupling between the inner Majorana modes and hence induces energy splitting of their states even in the absence of the direct spatial overlap of their wave functions. The energy splitting is obtained analytically in the weak tunneling limit and is numerically investigated for an arbitrary tunneling strength. We discuss in detail the evolution of the energy splitting with an external perpendicular magnetic field and its effect on the shape of the Fraunhofer pattern.

The precise characterization and control of single-electron wave functions emitted from a single-electron source are essential for advancing electron quantum optics. Here, we introduce a method for tailoring a single-electron emission distribution using energy filtering, enabling selective control of the distribution under various energy barrier conditions of the filter. The tailored electron is studied by reconstructing its Wigner distribution in the time-energy phase space using the continuous-variable tomography method. Our results reveal that the filtering cuts the portion of the distribution below the energy-barrier height of the filter in the time-energy space. While the filtering is demonstrated in a classical regime of the emitted electrons, we expect that this study significantly contributes to the design and implementation of advanced experiments toward quantum information processing based on single electrons.

When a local impurity spin interacts with conduction electrons whose density of states (DOS) has a (pseudo)gap or diverges at the Fermi energy, a local moment (LM) phase can be favored over a Kondo phase. Theoretically studying quantum entanglement between the impurity and conduction electrons, we demonstrate that conduction electrons form an ''LM spin cloud'' in general LM phases, which corresponds to, but has fundamental difference from, the Kondo cloud screening the impurity spin in the Kondo phase. The LM cloud algebraically decays over the distance from the impurity when the DOS has a pseudogap or divergence, and exponentially when it has a hard gap. We find an ''LM cloud length'', a single length scale characterizing a universal form of the LM cloud. The findings are supported by both of analytic theories and numerical computations.