Interference between two indistinguishable electrons from independent sources

Anomalous Aharonov-Bohm Interference in the Presence of Edge Reconstruction

Interferometry is a vital tool for studying fundamental features in the quantum Hall effect. For instance, Aharonov-Bohm interference in a quantum Hall interferometer can probe the wave-particle duality of electrons and quasiparticles. Here, we report an unusual Aharonov-Bohm interference of the outermost edge mode in a quantum Hall Fabry-Pérot interferometer, whose Coulomb interactions were suppressed with a grounded drain in the interior bulk of the interferometer. In a descending bulk filling factor from ν_{b}=3 to ν_{b}≈(5/3), the magnetic field periodicity, which corresponded to a single "flux quantum," agreed accurately with the enclosed area of the interferometer. However, in the filling range, ν_{b}≈(5/3) to ν_{b}=1, the field periodicity increased markedly, a priori suggesting a drastic shrinkage of the Aharonov-Bohm area. Moreover, the modulation gate voltage periodicity decreased abruptly at this range. We attribute these unexpected observations to edge reconstruction, leading to area changing with the field and a modified modulation gate-edge capacitance. These reproducible results support future interference experiments with a quantum Hall Fabry-Pérot interferometer.

Observation of electronic modes in open cavity resonator

The resemblance between electrons and optical waves has strongly driven the advancement of mesoscopic physics, evidenced by the widespread use of terms such as fermion or electron optics. However, electron waves have yet to be understood in open cavity structures which have provided contemporary optics with rich insight towards non-Hermitian systems and complex interactions between resonance modes. Here, we report the realization of an open cavity resonator in a two-dimensional electronic system. We studied the resonant electron modes within the cavity and resolved the signatures of longitudinal and transverse quantization, showing that the modes are robust despite the cavity being highly coupled to the open background continuum. The transverse modes were investigated by applying a controlled deformation to the cavity, and their spatial distributions were further analyzed using magnetoconductance measurements and numerical simulation. These results lay the groundwork to exploring matter waves in the context of modern optical frameworks.

Arbitrary entanglement of three qubits via linear optics

We present a linear-optical scheme for generating an arbitrary state of three qubits. It requires only three independent particles in the input and post-selection of the coincidence type at the output. The success probability of the protocol is equal for any desired state. Furthermore, the optical design remains insensitive to particle statistics (bosons, fermions or anyons). This approach builds upon the no-touching paradigm, which demonstrates the utility of particle indistinguishability as a resource of entanglement for practical applications.

Two-particle time-domain interferometry in the fractional quantum Hall effect regime

Quasi-particles are elementary excitations of condensed matter quantum phases. Demonstrating that they keep quantum coherence while propagating is a fundamental issue for their manipulation for quantum information tasks. Here, we consider anyons, the fractionally charged quasi-particles of the Fractional Quantum Hall Effect occurring in two-dimensional electronic conductors in high magnetic fields. They obey anyonic statistics, intermediate between fermionic and bosonic. Surprisingly, anyons show large quantum coherence when transmitted through the localized states of electronic Fabry-Pérot interferometers, but almost no quantum interference when transmitted via the propagating states of Mach-Zehnder interferometers. Here, using a novel interferometric approach, we demonstrate that anyons do keep quantum coherence while propagating. Performing two-particle time-domain interference measurements sensitive to the two-particle Hanbury Brown Twiss phase, we find 53 and 60% visibilities for anyons with charges e/5 and e/3. Our results give a positive message for the challenge of performing controlled quantum coherent braiding of anyons.

Scaling behavior of electron decoherence in a graphene Mach-Zehnder interferometer

Over the past 20 years, many efforts have been made to understand and control decoherence in 2D electron systems. In particular, several types of electronic interferometers have been considered in GaAs heterostructures, in order to protect the interfering electrons from decoherence. Nevertheless, it is now understood that several intrinsic decoherence sources fundamentally limit more advanced quantum manipulations. Here, we show that graphene offers a unique possibility to reach a regime where the decoherence is frozen and to study unexplored regimes of electron interferometry. We probe the decoherence of electron channels in a graphene quantum Hall PN junction, forming a Mach-Zehnder interferometer1,2, and unveil a scaling behavior of decay of the interference visibility with the temperature scaled by the interferometer length. It exhibits a remarkable crossover from an exponential decay at higher temperature to an algebraic decay at lower temperature where almost no decoherence occurs, a regime previously unobserved in GaAs interferometers.

Entanglement of Nambu Spinors and Bell Inequality Test without Beam Splitters

The identification of electronic entanglement in solids remains elusive so far, which is owed to the difficulty of implementing spinor-selective beam splitters with tunable polarization direction. Here, we propose to overcome this obstacle by producing and detecting a particular type of entanglement encoded in the Nambu spinor or electron-hole components of quasiparticles excited in quantum Hall edge states. Because of the opposite charge of electrons and holes, the detection of the Nambu spinor translates into a charge-current measurement, which eliminates the need for beam splitters and assures a high detection rate. Conveniently, the spinor correlation function at fixed effective polarizations derives from a single current-noise measurement, with the polarization directions of the detector easily adjusted by coupling the edge states to a voltage gate and a superconductor, both having been realized in experiments. We show that the violation of Bell inequality occurs in a large parameter region. Our Letter opens a new route for probing quasiparticle entanglement in solid-state physics exempt from traditional beam splitters.

Computational perspective on recent advances in quantum electronics: from electron quantum optics to nanoelectronic devices and systems

Quantum electronics has significantly evolved over the last decades. Where initially the clear focus was on light-matter interactions, nowadays approaches based on the electron's wave nature have solidified themselves as additional focus areas. This development is largely driven by continuous advances in electron quantum optics, electron based quantum information processing, electronic materials, and nanoelectronic devices and systems. The pace of research in all of these areas is astonishing and is accompanied by substantial theoretical and experimental advancements. What is particularly exciting is the fact that the computational methods, together with broadly available large-scale computing resources, have matured to such a degree so as to be essential enabling technologies themselves. These methods allow to predict, analyze, and design not only individual physical processes but also entire devices and systems, which would otherwise be very challenging or sometimes even out of reach with conventional experimental capabilities. This review is thus a testament to the increasingly towering importance of computational methods for advancing the expanding field of quantum electronics. To that end, computational aspects of a representative selection of recent research in quantum electronics are highlighted where a major focus is on the electron's wave nature. By categorizing the research into concrete technological applications, researchers and engineers will be able to use this review as a source for inspiration regarding problem-specific computational methods.

Homemade-HEMT-based transimpedance amplifier for high-resolution shot-noise measurements

We report a cryogenic transimpedance amplifier (TA) suitable for cross-correlation current-noise measurements. The TA comprises homemade high-electron-mobility transistors with high transconductance and low noise characteristics, fabricated in an AlGaAs/GaAs heterostructure. The low input-referred noise and wide frequency band of the TA lead to a high resolution in current-noise measurements. The TA's low input impedance suppresses unwanted crosstalk between two distinct currents from a sample, justifying the advantage of the TA for cross-correlation measurements. We demonstrate the high resolution of a TA-based experimental setup by measuring the shot noise generated at a quantum point contact in a quantum Hall system.

Controllable quantum point junction on the surface of an antiferromagnetic topological insulator

Engineering and manipulation of unidirectional channels has been achieved in quantum Hall systems, leading to the construction of electron interferometers and proposals for low-power electronics and quantum information science applications. However, to fully control the mixing and interference of edge-state wave functions, one needs stable and tunable junctions. Encouraged by recent material candidates, here we propose to achieve this using an antiferromagnetic topological insulator that supports two distinct types of gapless unidirectional channels, one from antiferromagnetic domain walls and the other from single-height steps. Their distinct geometric nature allows them to intersect robustly to form quantum point junctions, which then enables their control by magnetic and electrostatic local probes. We show how the existence of stable and tunable junctions, the intrinsic magnetism and the potential for higher-temperature performance make antiferromagnetic topological insulators a promising platform for electron quantum optics and microelectronic applications.

Fractional charge and fractional statistics in the quantum Hall effects

Quasiparticles with fractional charge and fractional statistics are key features of the fractional quantum Hall effect. We discuss in detail the definitions of fractional charge and statistics and the ways in which these properties may be observed. In addition to theoretical foundations, we review the present status of the experiments in the area. We also discuss the notions of non-Abelian statistics and attempts to find experimental evidence for the existence of non-Abelian quasiparticles in certain quantum Hall systems.

Multi-Particle Interference in an Electronic Mach-Zehnder Interferometer

The development of dynamic single-electron sources has made it possible to observe and manipulate the quantum properties of individual charge carriers in mesoscopic circuits. Here, we investigate multi-particle effects in an electronic Mach-Zehnder interferometer driven by a series of voltage pulses. To this end, we employ a Floquet scattering formalism to evaluate the interference current and the visibility in the outputs of the interferometer. An injected multi-particle state can be described by its first-order correlation function, which we decompose into a sum of elementary correlation functions that each represent a single particle. Each particle in the pulse contributes independently to the interference current, while the visibility (given by the maximal interference current) exhibits a Fraunhofer-like diffraction pattern caused by the multi-particle interference between different particles in the pulse. For a sequence of multi-particle pulses, the visibility resembles the diffraction pattern from a grid, with the role of the grid and the spacing between the slits being played by the pulses and the time delay between them. Our findings may be observed in future experiments by injecting multi-particle pulses into a Mach-Zehnder interferometer.

Fractional statistics in anyon collisions

Two-dimensional systems can host exotic particles called anyons whose quantum statistics are neither bosonic nor fermionic. For example, the elementary excitations of the fractional quantum Hall effect at filling factor ν = 1/m (where m is an odd integer) have been predicted to obey Abelian fractional statistics, with a phase ϕ associated with the exchange of two particles equal to π/m However, despite numerous experimental attempts, clear signatures of fractional statistics have remained elusive. We experimentally demonstrate Abelian fractional statistics at filling factor ν = ⅓ by measuring the current correlations resulting from the collision between anyons at a beamsplitter. By analyzing their dependence on the anyon current impinging on the splitter and comparing with recent theoretical models, we extract ϕ = π/3, in agreement with predictions.

Trapping and Counting Ballistic Nonequilibrium Electrons

We demonstrate the trapping of electrons propagating ballistically at far-above-equilibrium energies in GaAs/AlGaAs heterostructures in high magnetic field. We find low-loss transport along a gate-modified mesa edge in contrast to an effective decay of excess energy for the loop around a neighboring, mesa-confined node, enabling high-fidelity trapping. Measuring the full counting statistics via single-charge detection yields the trapping (and escape) probabilities of electrons scattered (and excited) within the node. Energetic and arrival-time distributions of captured electron wave packets are characterized by modulating tunnel barrier transmission.

Entangling three qubits without ever touching

All identical particles are inherently correlated from the outset, regardless of how far apart their creation took place. In this paper, this fact is used for extraction of entanglement from independent particles unaffected by any interactions. Specifically, we are concerned with operational schemes for generation of all tripartite entangled states, essentially the GHZ state and the W state, which prevent the particles from touching one another over the entire evolution. The protocols discussed in the paper require only three particles in linear optical setups with equal efficiency for boson, fermion or anyon statistics. Within this framework indistinguishability of particles presents itself as a useful resource of entanglement accessible for practical applications.

Auger-spectroscopy in quantum Hall edge channels and the missing energy problem

Quantum Hall edge channels offer an efficient and controllable platform to study quantum transport in one dimension. Such channels are a prospective tool for the efficient transfer of quantum information at the nanoscale, and play a vital role in exposing intriguing physics. Electric current along the edge carries energy and heat leading to inelastic scattering, which may impede coherent transport. Several experiments attempting to probe the concomitant energy redistribution along the edge reported energy loss via unknown mechanisms of inelastic scattering. Here we employ quantum dots to inject and extract electrons at specific energies, to spectrally analyse inelastic scattering inside quantum Hall edge channels. We show that the missing energy puzzle could be untangled by incorporating non-local Auger-like processes, in which energy is redistributed between spatially separate parts of the sample. Our theoretical analysis, accounting for the experimental results, challenges common-wisdom analyses which ignore such non-local decay channels.

Quantum tomography of electrical currents

In quantum nanoelectronics, time-dependent electrical currents are built from few elementary excitations emitted with well-defined wavefunctions. However, despite the realization of sources generating quantized numbers of excitations, and despite the development of the theoretical framework of time-dependent quantum electronics, extracting electron and hole wavefunctions from electrical currents has so far remained out of reach, both at the theoretical and experimental levels. In this work, we demonstrate a quantum tomography protocol which extracts the generated electron and hole wavefunctions and their emission probabilities from any electrical current. It combines two-particle interferometry with signal processing. Using our technique, we extract the wavefunctions generated by trains of Lorentzian pulses carrying one or two electrons. By demonstrating the synthesis and complete characterization of electronic wavefunctions in conductors, this work offers perspectives for quantum information processing with electrical currents and for investigating basic quantum physics in many-body systems.

Quasiparticle states of on-demand coherent electron sources

We introduce a general approach to extract the wave function of quasiparticles from the scattering matrix of a quantum conductor, which offers a unified way to study the features of quasiparticles from on-demand coherent electron sources with different configurations. We first show that the quasiparticles are particle-hole pairs in the Fermi sea, which can be indexed with the flow density [Formula: see text]. Both the excitation probability and the particle/hole components of the quasiparticles can be solely decided from the polar decomposition of the scattering matrix. By using such approach, we then investigate the quasiparticles from the electron sources based on a quantum point contact and a quantum dot (QD). We find that the quasiparticles from different electron sources have different features, which can be seen from the corresponding [Formula: see text]-dependence of the excitation probability and the particle/hole components. We further show that these features can also be characterized by the full counting statistics of the quasiparticles, which can be approximated by a binomial distribution with cumulant generating function [Formula: see text]. For the quantum-point-contact-based electron sources, both [Formula: see text] and [Formula: see text] are monotonically increasing functions of the driving strength. In contrast, for the quantum-dot-based electron sources, both [Formula: see text] and [Formula: see text] can exhibit oscillations, which can be attributed to the interplay between the charge excitation and charge relaxation processes in the QD.

A Proposal for Evading the Measurement Uncertainty in Classical and Quantum Computing: Application to a Resonant Tunneling Diode and a Mach-Zehnder Interferometer

Measuring properties of quantum systems is governed by a stochastic (collapse or state-reduction) law that unavoidably yields an uncertainty (variance) associated with the corresponding mean values. This non-classical source of uncertainty is known to be manifested as noise in the electrical current of nanoscale electron devices, and hence it can flaw the good performance of more complex quantum gates. We propose a protocol to alleviate this quantum uncertainty that consists of (i) redesigning the device to accommodate a large number of electrons inside the active region, either by enlarging the lateral or longitudinal areas of the device and (ii) re-normalizing the total current to the number of electrons. How the above two steps can be accommodated using the present semiconductor technology has been discussed and numerically studied for a resonant tunneling diode and a Mach-Zehnder interferometer, for classical and quantum computations, respectively. It is shown that the resulting protocol formally resembles the so-called collective measurements, although, its practical implementation is substantially different.

Tunneling Spectroscopy of Quantum Hall States in Bilayer Graphene p-n Junctions

We report tunneling transport in spatially controlled networks of quantum Hall (QH) edge states in bilayer graphene. By manipulating the separation, location, and spatial span of QH edge states via gate-defined electrostatics, we observe resonant tunneling between copropagating QH states across incompressible strips. Employing spectroscopic tunneling measurements and an analytical model, we characterize the energy gap, width, density of states, and compressibility of the QH edge states with high precision and sensitivity within the same device. The capability to engineer the QH edge network also provides an opportunity to build future quantum electronic devices with electrostatic manipulation of QH edge states, supported by rich underlying physics.

Coherent control of single electrons: a review of current progress

In this report we review the present state of the art of the control of propagating quantum states at the single-electron level and its potential application to quantum information processing. We give an overview of the different approaches that have been developed over the last few years in order to gain full control over a propagating single-electron in a solid-state system. After a brief introduction of the basic concepts, we present experiments on flying qubit circuits for ensemble of electrons measured in the low frequency (DC) limit. We then present the basic ingredients necessary to realise such experiments at the single-electron level. This includes a review of the various single-electron sources that have been developed over the last years and which are compatible with integrated single-electron circuits. This is followed by a review of recent key experiments on electron quantum optics with single electrons. Finally we will present recent developments in the new physics that has emerged using ultrashort voltage pulses. We conclude our review with an outlook and future challenges in the field.

Detecting Kondo Entanglement by Electron Conductance

Quantum entanglement between an impurity spin and electrons nearby is a key property of the single-channel Kondo effects. We show that the entanglement can be detected by measuring electron conductance through a double quantum dot in an orbital Kondo regime. We derive a relation between the entanglement and the conductance, when the SU(2) spin symmetry of the regime is weakly broken. The relation reflects the universal form of many-body states near the Kondo fixed point. Using it, the spatial distribution of the entanglement-hence, the Kondo cloud-can be detected, with breaking of the symmetry spatially nonuniformly by electrical means.

Hyper- and hybrid nonlocality

The controlled generation and identification of quantum correlations, usually encoded in either qubits or continuous degrees of freedom, builds the foundation of quantum information science. Recently, more sophisticated approaches, involving a combination of two distinct degrees of freedom, have been proposed to improve on the traditional strategies. Hyperentanglement describes simultaneous entanglement in more than one distinct degree of freedom, whereas hybrid entanglement refers to entanglement shared between a discrete and a continuous degree of freedom. In this work we propose a scheme that allows us to combine the two approaches, and to extend them to the strongest form of quantum correlations. Specifically, we show how two identical, initially separated particles can be manipulated to produce Bell nonlocality among their spins, among their momenta, as well as across their spins and momenta. We discuss possible experimental realizations with atomic and photonic systems.

Mach-Zehnder interferometry using spin- and valley-polarized quantum Hall edge states in graphene

Confined to a two-dimensional plane, electrons in a strong magnetic field travel along the edge in one-dimensional quantum Hall channels that are protected against backscattering. These channels can be used as solid-state analogs of monochromatic beams of light, providing a unique platform for studying electron interference. Electron interferometry is regarded as one of the most promising routes for studying fractional and non-Abelian statistics and quantum entanglement via two-particle interference. However, creating an edge-channel interferometer in which electron-electron interactions play an important role requires a clean system and long phase coherence lengths. We realize electronic Mach-Zehnder interferometers with record visibilities of up to 98% using spin- and valley-polarized edge channels that copropagate along a pn junction in graphene. We find that interchannel scattering between same-spin edge channels along the physical graphene edge can be used to form beamsplitters, whereas the absence of interchannel scattering along gate-defined interfaces can be used to form isolated interferometer arms. Surprisingly, our interferometer is robust to dephasing effects at energies an order of magnitude larger than those observed in pioneering experiments on GaAs/AlGaAs quantum wells. Our results shed light on the nature of edge-channel equilibration and open up new possibilities for studying exotic electron statistics and quantum phenomena.

Optimal Quantum Interference Thermoelectric Heat Engine with Edge States

We show theoretically that a thermoelectric heat engine, operating exclusively due to quantum-mechanical interference, can reach optimal linear-response performance. A chiral edge state implementation of a close-to-optimal heat engine is proposed in an electronic Mach-Zehnder interferometer with a mesoscopic capacitor coupled to one arm. We demonstrate that the maximum power and corresponding efficiency can reach 90% and 83%, respectively, of the theoretical maximum. The proposed heat engine can be realized with existing experimental techniques and has a performance robust against moderate dephasing.

Negative and positive cross-correlations of current noises in quantum Hall edge channels at bulk filling factor [Formula: see text]

Cross-correlation noise in electrical currents generated from a series connection of two quantum point contacts (QPCs), the injector and the detector, is described for investigating energy relaxation in quantum Hall edge channels at bulk filling factor [Formula: see text]. We address the importance of tuning the energy bias across the detector for this purpose. For a long channel with a macroscopic floating ohmic contact that thermalizes the electrons, the cross-correlation turns from negative values to the maximally positive value (identical noise in the two currents) by tuning the effective energy bias to zero. This can be understood by considering competition between the low-frequency charge fluctuation generated at the injector, which contributes positive correlation, and the partition noise at the detector, which gives negative correlation. Strikingly, even for a short channel without intentional thermalization, significantly large positive correlation is observed in contrast to negative values expected for coherent transport between the two QPCs.

Minimal Entanglement Witness from Electrical Current Correlations

Despite great efforts, an unambiguous demonstration of entanglement of mobile electrons in solid state conductors is still lacking. Investigating theoretically a generic entangler-detector setup, we here show that a witness of entanglement between two flying electron qubits can be constructed from only two current cross correlation measurements, for any nonzero detector efficiencies and noncollinear polarization vectors. We find that all entangled pure states, but not all mixed ones, can be detected with only two measurements, except the maximally entangled states, which require three. Moreover, detector settings for optimal entanglement witnessing are presented.

A 50/50 electronic beam splitter in graphene nanoribbons as a building block for electron optics

Based on the investigation of the multi-terminal conductance of a system composed of two graphene nanoribbons, in which one is on top of the other and rotated by [Formula: see text], we propose a setup for a 50/50 electronic beam splitter that neither requires large magnetic fields nor ultra low temperatures. Our findings are based on an atomistic tight-binding description of the system and on the Green function method to compute the Landauer conductance. We demonstrate that this system acts as a perfect 50/50 electronic beam splitter, in which its operation can be switched on and off by varying the doping (Fermi energy). We show that this device is robust against thermal fluctuations and long range disorder, as zigzag valley chiral states of the nanoribbons are protected against backscattering. We suggest that the proposed device can be applied as the fundamental element of the Hong-Ou-Mandel interferometer, as well as a building block of many devices in electron optics.

Dynamical Scheme for Interferometric Measurements of Full-Counting Statistics

We propose a dynamical scheme for measuring the full-counting statistics in a mesoscopic conductor using an electronic Mach-Zehnder interferometer. The conductor couples capacitively to one arm of the interferometer and causes a phase shift which is proportional to the number of transferred charges. Importantly, the full-counting statistics can be obtained from average current measurements at the outputs of the interferometer. The counting field can be controlled by varying the time delay between two separate voltage signals applied to the conductor and the interferometer, respectively. As a specific application, we consider measuring the entanglement entropy generated by partitioning electrons on a quantum point contact. Our scheme is robust against moderate environmental dephasing and may be realized thanks to recent advances in gigahertz quantum electronics.

Simultaneous manipulation and observation of multiple ro-vibrational eigenstates in solid para-hydrogen

We have experimentally performed the coherent control of delocalized ro-vibrational wave packets (RVWs) of solid para-hydrogen (p-H₂) by the wave packet interferometry (WPI) combined with coherent anti-Stokes Raman scattering (CARS). RVWs of solid p-H₂ are delocalized in the crystal, and the wave function with wave vector k ∼ 0 is selectively excited via the stimulated Raman process. We have excited the RVW twice by a pair of femtosecond laser pulses with delay controlled by a stabilized Michelson interferometer. Using a broad-band laser pulse, multiple ro-vibrational states can be excited simultaneously. We have observed the time-dependent Ramsey fringe spectra as a function of the inter-pulse delay by a spectrally resolved CARS technique using a narrow-band probe pulse, resolving the different intermediate states. Due to the different fringe oscillation periods among those intermediate states, we can manipulate their amplitude ratio by tuning the inter-pulse delay on the sub-femtosecond time scale. The state-selective manipulation and detection of the CARS signal combined with the WPI is a general and efficient protocol for the control of the interference of multiple quantum states in various quantum systems.

Fractionally Charged Zero-Energy Single-Particle Excitations in a Driven Fermi Sea

A voltage pulse of a Lorentzian shape carrying half of the flux quantum excites out of a zero-temperature Fermi sea an electron in a mixed state, which looks like a quasiparticle with an effectively fractional charge e/2. A prominent feature of such an excitation is a narrow peak in the energy distribution function lying exactly at the Fermi energy μ. Another spectacular feature is that the distribution function has symmetric tails around μ, which results in a zero-energy excitation. This sounds improbable since at zero temperature all available states below μ are fully occupied. The resolution lies in the fact that such a voltage pulse also excites electron-hole pairs, which free some space below μ and thus allow a zero-energy quasiparticle to exist. I discuss also how to address separately electron-hole pairs and a fractionally charged zero-energy excitation in an experiment.

Current Correlations from a Mesoscopic Anyon Collider

Fermions and bosons are fundamental realizations of exchange statistics, which governs the probability for two particles being close to each other spatially. Anyons in the fractional quantum Hall effect are an example for exchange statistics intermediate between bosons and fermions. We analyze a mesoscopic setup in which two dilute beams of anyons collide with each other, and relate the correlations of current fluctuations to the probability of particles excluding each other spatially. While current correlations for fermions vanish, negative correlations for anyons are a clear signature of a reduced spatial exclusion as compared to fermions.

What can we learn from noise? - Mesoscopic nonequilibrium statistical physics

Mesoscopic systems - small electric circuits working in quantum regime - offer us a unique experimental stage to explorer quantum transport in a tunable and precise way. The purpose of this Review is to show how they can contribute to statistical physics. We introduce the significance of fluctuation, or equivalently noise, as noise measurement enables us to address the fundamental aspects of a physical system. The significance of the fluctuation theorem (FT) in statistical physics is noted. We explain what information can be deduced from the current noise measurement in mesoscopic systems. As an important application of the noise measurement to statistical physics, we describe our experimental work on the current and current noise in an electron interferometer, which is the first experimental test of FT in quantum regime. Our attempt will shed new light in the research field of mesoscopic quantum statistical physics.

Entanglement discrimination in multi-rail electron-hole currents

We propose a quantum-Hall interferometer that integrates an electron-hole entangler with an analyzer working as an entanglement witness by implementing a multi-rail encoding. The witness has the ability to discriminate (and quantify) spatial-mode and occupancy entanglement. This represents a feasible alternative to limited approaches based on the violation of Bell-like inequalities.

Time-dependent simulation and analytical modelling of electronic Mach-Zehnder interferometry with edge-states wave packets

We compute the exact single-particle time-resolved dynamics of electronic Mach-Zehnder interferometers based on Landau edge-states transport, and assess the effect of the spatial localization of carriers on the interference pattern. The exact carrier dynamics is obtained by solving numerically the time-dependent Schrödinger equation with a suitable 2D potential profile reproducing the interferometer design. An external magnetic field, driving the system to the quantum Hall regime with filling factor one, is included. The injected carriers are represented by a superposition of edge states, and their interference pattern-controlled via magnetic field and/or area variation-reproduces the one of (Ji et al 2003 Nature 422 415). By tuning the system towards different regimes, we find two additional features in the transmission spectra, both related to carrier localization, namely a damping of the Aharonov-Bohm oscillations with increasing difference in the arms length, and an increased mean transmission that we trace to the energy-dependent transmittance of quantum point contacts. Finally, we present an analytical model, also accounting for the finite spatial dispersion of the carriers, able to reproduce the above effects.

Coupling Two Distant Double Quantum Dots with a Microwave Resonator

We fabricated a hybrid device with two distant graphene double quantum dots (DQDs) and a microwave resonator. A nonlinear response is observed in the resonator reflection amplitude when the two DQDs are jointly tuned to the vicinity of the degeneracy points. This observation can be well fitted by the Tavis-Cummings (T-C) model which describes two two-level systems coupling with one photonic field. Furthermore, the correlation between the DC currents in the two DQDs is studied. A nonzero cross-current correlation is observed which has been theoretically predicted to be an important sign of nonlocal coupling between two distant systems. Our results explore T-C physics in electronic transport and also contribute to the study of nonlocal transport and future implementations of remote electronic entanglement.

Edge mixing dynamics in graphene p-n junctions in the quantum Hall regime

Massless Dirac electron systems such as graphene exhibit a distinct half-integer quantum Hall effect, and in the bipolar transport regime co-propagating edge states along the p–n junction are realized. Additionally, these edge states are uniformly mixed at the junction, which makes it a unique structure to partition electrons in these edge states. Although many experimental works have addressed this issue, the microscopic dynamics of electron partition in this peculiar structure remains unclear. Here we performed shot-noise measurements on the junction in the quantum Hall regime as well as at zero magnetic field. We found that, in sharp contrast with the zero-field case, the shot noise in the quantum Hall regime is finite in the bipolar regime, but is strongly suppressed in the unipolar regime. Our observation is consistent with the theoretical prediction and gives microscopic evidence that the edge states are uniquely mixed along the p–n junction.

A graphene p–n junction can be created by connecting electrical gates that generate electron-doped and hole-doped areas in a flake. Here, the authors use shot-noise measurements to provide microscopic evidence that edge states are uniquely mixed along the junction in the quantum Hall regime.

Shot noise generated by graphene p-n junctions in the quantum Hall effect regime

Graphene offers a unique system to investigate transport of Dirac Fermions at p–n junctions. In a magnetic field, combination of quantum Hall physics and the characteristic transport across p–n junctions leads to a fractionally quantized conductance associated with the mixing of electron-like and hole-like modes and their subsequent partitioning. The mixing and partitioning suggest that a p–n junction could be used as an electronic beam splitter. Here we report the shot noise study of the mode-mixing process and demonstrate the crucial role of the p–n junction length. For short p–n junctions, the amplitude of the noise is consistent with an electronic beam-splitter behaviour, whereas, for longer p–n junctions, it is reduced by the energy relaxation. Remarkably, the relaxation length is much larger than typical size of mesoscopic devices, encouraging using graphene for electron quantum optics and quantum information processing.

Dirac fermions at a p–n junction can exhibit a wide variety of unusual properties. Here, the authors investigate the dynamics of such fermions in a graphene junction using shot noise measurements and demonstrate the crucial role of junction length.

Quantum Phase Coherence in Mesoscopic Transport Devices with Two-Particle Interaction

In this paper we demonstrate a new type of quantum phase coherence (QPC), which is generated by the two-body interaction. This conclusion is based on quantum master equation analysis for the full counting statistics of electron transport through two parallel quantum-dots with antiparallel magnetic fluxes in order to eliminate the Aharonov-Bohm interference of either single-particle or non-interacting two-particle wave functions. The interacting two-particle QPC is realized by the flux-dependent oscillation of the zero-frequency cumulants including the shot noise and skewness with a characteristic period. The accurately quantized peaks of cumulant spectrum may have technical applications to probe the two-body Coulomb interaction.

Subdecoherence time generation and detection of orbital entanglement in quantum dots

Recent experiments have demonstrated subdecoherence time control of individual single-electron orbital qubits. Here we propose a quantum-dot-based scheme for generation and detection of pairs of orbitally entangled electrons on a time scale much shorter than the decoherence time. The electrons are entangled, via two-particle interference, and transferred to the detectors during a single cotunneling event, making the scheme insensitive to charge noise. For sufficiently long detector dot lifetimes, cross-correlation detection of the dot charges can be performed with real-time counting techniques, providing for an unambiguous short-time Bell inequality test of orbital entanglement.

Hong-Ou-Mandel experiment for temporal investigation of single-electron fractionalization

Coulomb interaction has a striking effect on electronic propagation in one-dimensional conductors. The interaction of an elementary excitation with neighbouring conductors favours the emergence of collective modes, which eventually leads to the destruction of the Landau quasiparticle. In this process, an injected electron tends to fractionalize into separated pulses carrying a fraction of the electron charge. Here we use two-particle interferences in the electronic analogue of the Hong-Ou-Mandel experiment in a quantum Hall conductor at filling factor 2 to probe the fate of a single electron emitted in the outer edge channel and interacting with the inner one. By studying both channels, we analyse the propagation of the single electron and the generation of interaction-induced collective excitations in the inner channel. These complementary pieces of information reveal the fractionalization process in the time domain and establish its relevance for the destruction of the quasiparticle, which degrades into the collective modes.

A charge injected into the edge of a correlated one-dimensional system can split into separate charge packages. Freulon et al. now study this electron fractionalization on the picosecond timescale using Hong-Ou-Mandel interferometry.

Partitioning of on-demand electron pairs

The on-demand generation and separation of entangled photon pairs are key components of quantum information processing in quantum optics. In an electronic analogue, the decomposition of electron pairs represents an essential building block for using the quantum state of ballistic electrons in electron quantum optics. The scattering of electrons has been used to probe the particle statistics of stochastic sources in Hanbury Brown and Twiss experiments and the recent advent of on-demand sources further offers the possibility to achieve indistinguishability between multiple sources in Hong-Ou-Mandel experiments. Cooper pairs impinging stochastically at a mesoscopic beamsplitter have been successfully partitioned, as verified by measuring the coincidence of arrival. Here, we demonstrate the splitting of electron pairs generated on demand. Coincidence correlation measurements allow the reconstruction of the full counting statistics, revealing regimes of statistically independent, distinguishable or correlated partitioning, and have been envisioned as a source of information on the quantum state of the electron pair. The high pair-splitting fidelity opens a path to future on-demand generation of spin-entangled electron pairs from a suitably prepared two-electron quantum-dot ground state.

Detection of spin entanglement via spin-charge separation in crossed Tomonaga-Luttinger liquids

We investigate tunneling between two spinful Tomonaga-Luttinger liquids (TLLs) realized, e.g., as two crossed nanowires or quantum Hall edge states. When injecting into each TLL one electron of opposite spin, the dc current measured after the crossing differs for singlet, triplet, or product states. This is a striking new non-Fermi liquid feature because the (mean) current in a noninteracting beam splitter is insensitive to spin entanglement. It can be understood in terms of collective excitations subject to spin-charge separation. This behavior may offer an easier alternative to traditional entanglement detection schemes based on current noise, which we show to be suppressed by the interactions.

Resonant edge magnetoplasmons and their decay in graphene

We investigate resonant edge magnetoplasmons (EMPs) and their decay in graphene by high-frequency electronic measurements. From EMP resonances in disk shaped graphene, we show that the dispersion relation of EMPs is nonlinear due to interactions, giving rise to the intrinsic decay of EMP wave packets. We also identify extrinsic dissipation mechanisms due to interaction with localized states in bulk graphene from the decay time of EMP wave packets. We indicate that, owing to the linear band structure and the sharp edge potential, EMP dissipation in graphene can be lower than that in GaAs systems.

Limitations to carrier mobility and phase-coherent transport in bilayer graphene

We present transport measurements on high-mobility bilayer graphene fully encapsulated in hexagonal boron nitride. We show two terminal quantum Hall effect measurements which exhibit full symmetry broken Landau levels at low magnetic fields. From weak localization measurements, we extract gate-tunable phase-coherence times τϕ as well as the inter- and intravalley scattering times τi and τ*, respectively. While τϕ is in qualitative agreement with an electron-electron interaction-mediated dephasing mechanism, electron spin-flip scattering processes are limiting τϕ at low temperatures. The analysis of τi and τ* points to local strain fluctuation as the most probable mechanism for limiting the mobility in high-quality bilayer graphene.

Annihilation of colliding Bogoliubov quasiparticles reveals their Majorana nature

The single-particle excitations of a superconductor are coherent superpositions of electrons and holes near the Fermi level, called Bogoliubov quasiparticles. They are Majorana fermions, meaning that pairs of quasiparticles can annihilate. We calculate the annihilation probability at a beam splitter for chiral quantum Hall edge states, obtaining a 1±cosϕ dependence on the phase difference ϕ of the superconductors from which the excitations originated (with the ± sign distinguishing singlet and triplet pairing). This provides for a nonlocal measurement of the superconducting phase in the absence of any supercurrent.

Single-slit electron diffraction with Aharonov-Bohm phase: Feynman's thought experiment with quantum point contacts

In a "thought experiment," now a classic in physics pedagogy, Feynman visualizes Young's double-slit interference experiment with electrons in magnetic field. He shows that the addition of an Aharonov-Bohm phase is equivalent to shifting the zero-field wave interference pattern by an angle expected from the Lorentz force calculation for classical particles. We have performed this experiment with one slit, instead of two, where ballistic electrons within two-dimensional electron gas diffract through a small orifice formed by a quantum point contact (QPC). As the QPC width is comparable to the electron wavelength, the observed intensity profile is further modulated by the transverse waveguide modes present at the injector QPC. Our experiments open the way to realizing diffraction-based ideas in mesoscopic physics.