Measuring statistics-induced entanglement entropy with a Hong-Ou-Mandel interferometer.

Despite its ubiquity in quantum computation and quantum information, a universally applicable definition of quantum entanglement remains elusive. The challenge is further accentuated when entanglement is associated with other key themes, e.g., quantum interference and quantum statistics. Here, we introduce two novel motifs that characterize the interplay of entanglement and quantum statistics: an 'entanglement pointer' and a 'statistics-induced entanglement entropy'. The two provide a quantitative description of the statistics-induced entanglement: (i) they are finite only in the presence of quantum entanglement underlined by quantum statistics and (ii) their explicit form depends on the quantum statistics of the particles (e.g., fermions, bosons, and anyons). We have experimentally implemented these ideas by employing an electronic Hong-Ou-Mandel interferometer fed by two highly diluted electron beams in an integer quantum Hall platform. Performing measurements of auto-correlation and cross-correlation of current fluctuations of the scattered beams (following 'collisions'), we quantify the statistics-induced entanglement by experimentally accessing the entanglement pointer and the statistics-induced entanglement entropy. Our theoretical and experimental approaches pave the way to study entanglement in various correlated platforms, e.g., those involving anyonic Abelian and non-Abelian states.

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.

Heat conductance of the quantum Hall bulk.

The quantum Hall effect is a prototypical realization of a topological state of matter. It emerges from a subtle interplay between topology, interactions and disorder1-9. The disorder enables the formation of localized states in the bulk that stabilize the quantum Hall states with respect to the magnetic field and carrier density3. Still, the details of the localized states and their contribution to transport remain beyond the reach of most experimental techniques10-31. Here we describe an extensive study of the bulk's heat conductance. Using a novel 'multiterminal' short device (on a scale of 10 µm), we separate the longitudinal thermal conductance, [Formula: see text] (owing to the bulk's contribution), from the topological transverse value [Formula: see text] by eliminating the contribution of the edge modes24. When the magnetic field is tuned away from the conductance plateau centre, the localized states in the bulk conduct heat efficiently ([Formula: see text]), whereas the bulk remains electrically insulating. Fractional states in the first excited Landau level, such as the [Formula: see text] and [Formula: see text], conduct heat throughout the plateau with a finite [Formula: see text]. We propose a theoretical model that identifies the localized states as the cause of the finite heat conductance, agreeing qualitatively with our experimental findings.

Electron Pairing of Interfering Interface-Based Edge Modes.

The remarkable Cooper-like pairing phenomenon in the Aharonov-Bohm interference of a Fabry-Perot interferometer-operating in the integer quantum Hall regime-remains baffling. Here, we report the interference of paired electrons employing "interface edge modes." These modes are born at the interface between the bulk of the Fabry-Perot interferometer and an outer gated region tuned to a lower filling factor. Such a configuration allows toggling the spin and the orbital of the Landau level of the edge modes at the interface. We find that electron pairing occurs only when the two modes (the interfering outer and the first inner) belong to the same spinless Landau level.

Partitioning of diluted anyons reveals their braiding statistics.

Correlations of partitioned particles carry essential information about their quantumness1. Partitioning full beams of charged particles leads to current fluctuations, with their autocorrelation (namely, shot noise) revealing the particles' charge2,3. This is not the case when a highly diluted beam is partitioned. Bosons or fermions will exhibit particle antibunching (owing to their sparsity and discreteness)4-6. However, when diluted anyons, such as quasiparticles in fractional quantum Hall states, are partitioned in a narrow constriction, their autocorrelation reveals an essential aspect of their quantum exchange statistics: their braiding phase7. Here we describe detailed measurements of weakly partitioned, highly diluted, one-dimension-like edge modes of the one-third filling fractional quantum Hall state. The measured autocorrelation agrees with our theory of braiding anyons in the time domain (instead of braiding in space); with a braiding phase of 2θ = 2π/3, without any fitting parameters. Our work offers a relatively straightforward and simple method to observe the braiding statistics of exotic anyonic states, such as non-abelian states8, without resorting to complex interference experiments9.

Isolated ballistic non-abelian interface channel.

The topological order of a quantum Hall state is mirrored by the gapless edge modes owing to bulk-edge correspondence. The state at the filling of ν = 5/2, predicted to host non-abelian anyons, supports a variety of edge modes (integer, fractional, neutral). To ensure thermal equilibration between the edge modes and thus accurately determine the state's nature, it is advantageous to isolate the fractional channel (1/2 and neutral modes). In this study, we gapped out the integer modes by interfacing the ν = 5/2 state with integer states ν = 2 and ν = 3 and measured the thermal conductance of the isolated-interface channel. Our measured half-quantized thermal conductance confirms the non-abelian nature of the ν = 5/2 state and its particle-hole Pfaffian topological order. Such an isolated channel may be more amenable to braiding experiments.

Distinguishing between non-abelian topological orders in a quantum Hall system.

Quantum Hall states can harbor exotic quantum phases. The nature of these states is reflected in the gapless edge modes owing to "bulk-edge" correspondence. The most studied putative non-abelian state is the spin-polarized filling factor (ν) = 5/2, which permits different topological orders that can be abelian or non-abelian. We developed a method that interfaces the studied quantum state with another state and used it to identify the topological order of ν = 5/2 state. The interface between two half-planes, one hosting the ν = 5/2 state and the other an integer ν = 3 state, supports a fractional ν = 1/2 charge mode and a neutral Majorana mode. The counterpropagating chirality of the Majorana mode, probed by measuring partition noise, is consistent with the particle-hole Pfaffian (PH-Pf) topological order and rules out the anti-Pfaffian order.

Energy Relaxation in Edge Modes in the Quantum Hall Effect.

Studies of energy flow in quantum systems complement the information provided by common conductance measurements. The quantum limit of heat flow in one-dimensional ballistic modes was predicted, and experimentally demonstrated, to have a universal value for bosons, fermions, and fractionally charged anyons. A fraction of this value is expected in non-Abelian states; harboring counterpropagating edge modes. In such exotic states, thermal-energy relaxation along the edge is expected, and can shed light on their topological nature. Here, we introduce a novel experimental setup that enables a direct observation of thermal-energy relaxation in chiral 1D edge modes in the quantum Hall effect. Edge modes, emanating from a heated reservoir, are partitioned by a quantum point contact (QPC) constriction, which is located at some distance along their path. The resulting low frequency noise, measured downstream, allows determination of the "effective temperature" of the edge mode at the location of the QPC. An expected, prominent energy relaxation was found in hole-conjugate states. However, relaxation was also observed in particlelike states, where heat is expected to be conserved. We developed a model, consisting of distance-dependent energy loss, which agrees with the observations; however, we cannot exclude energy redistribution mechanisms, which are not accompanied with energy loss.

Melting of Interference in the Fractional Quantum Hall Effect: Appearance of Neutral Modes.

We attempted to measure interference of the outer edge mode in the fractional quantum hall regime with an electronic Mach-zehnder interferometer. The visibility of the interferometer wore off as we approached ν_{B}=1 and the transmission of the quantum point contacts (QPCs) of the interferometer simultaneously developed a v=1/3 conductance plateau accompanied by shot noise. The appearance of shot noise on this plateau indicates the appearance of nontopological neutral modes resulting from edge reconstruction. We have confirmed the presence of upstream neutral modes measuring upstream noise emanating from the QPC. The lack of interference throughout the lowest Landau level was correlated with a proliferation of neutral modes.

Concomitant opening of a bulk-gap with an emerging possible Majorana zero mode.

Majorana quasiparticles are generally detected in a 1D topological superconductor by tunneling electrons into its edge, with an emergent zero-bias conductance peak (ZBCP). However, such a ZBCP can also result from other mechanisms, hence, additional verifications are required. Since the emergence of a Majorana must be accompanied by an opening of a topological gap in the bulk, two simultaneous measurements are performed: one in the bulk and another at the edge of a 1D InAs nanowire coated with epitaxial aluminum. Only under certain experimental parameters, a closing of the superconducting bulk-gap that is followed by its reopening, appears simultaneously with a ZBCP at the edge. Such events suggest the occurrence of a topologically non-trivial phase. Yet, we also find that ZBCPs are observed under different tuning parameters without simultaneous reopening of a bulk-gap. This demonstrates the importance of simultaneous probing of bulk and edge in the identification of Majorana edge-states.

Synthesizing a ν=2/3 fractional quantum Hall effect edge state from counter-propagating ν=1 and ν=1/3 states.

Topological edge-reconstruction occurs in hole-conjugate states of the fractional quantum Hall effect. The frequently studied filling factor, ν = 2/3, was originally proposed to harbor two counter-propagating modes: a downstream v = 1 and an upstream v = 1/3. However, charge equilibration between these two modes always led to an observed downstream v = 2/3 charge mode accompanied by an upstream neutral mode. Here, we present an approach to synthetize a v = 2/3 edge mode from its basic counter-propagating charged constituents, allowing a controlled equilibration between the two counter-propagating charge modes. This platform is based on a carefully designed double-quantum-well, which hosts two populated electronic sub-bands (lower and upper), with corresponding filling factors, vl and vu. By separating the 2D plane to two gated intersecting halves, each with different fillings, counter-propagating chiral modes can be formed along the intersection line. Equilibration between these modes can be controlled with the top gates' voltage and the magnetic field.

Counter-propagating charge transport in the quantum Hall effect regime.

The quantum Hall effect, observed in a two-dimensional (2D) electron gas subjected to a perpendicular magnetic field, imposes a 1D-like chiral, downstream, transport of charge carriers along the sample edges. Although this picture remains valid for electrons and Laughlin's fractional quasiparticles, it no longer holds for quasiparticles in the so-called hole-conjugate states. These states are expected, when disorder and interactions are weak, to harbor upstream charge modes. However, so far, charge currents were observed to flow exclusively downstream in the quantum Hall regime. Studying the canonical spin-polarized and spin-unpolarized v = 2/3 hole-like states in GaAs-AlGaAs heterostructures, we observed a significant upstream charge current at short propagation distances in the spin unpolarized state.

Author Correction: Observation of half-integer thermal Hall conductance.

In this Article, the publication details for references 33, 34 and 40 have been corrected online.

Theory of Disorder-Induced Half-Integer Thermal Hall Conductance.

The thermal Hall conductance in the half-filled first Landau level was recently measured to take the quantized noninteger value κ_{xy}=5/2 (in units of temperature times π^{2}k_{B}^{2}/3h), which indicates a non-Abelian phase of matter. Such exotic states have long been predicted to arise at this filling factor, but the measured value disagrees with numerical studies, which predict κ_{xy}=3/2 or 7/2. We resolve this contradiction by invoking the disorder-induced formation of mesoscopic puddles with locally κ_{xy}=3/2 or 7/2. Interactions between these puddles generate a coherent macroscopic state that exhibits a plateau with quantized κ_{xy}=5/2. The non-Abelian quasiparticles characterizing this phase are distinct from those of the microscopic puddles and, by the same mechanism, could even emerge from a system comprised of microscopic Abelian puddles.

Observation of half-integer thermal Hall conductance.

Topological states of matter are characterized by topological invariants, which are physical quantities whose values are quantized and do not depend on the details of the system (such as its shape, size and impurities). Of these quantities, the easiest to probe is the electrical Hall conductance, and fractional values (in units of e2/h, where e is the electronic charge and h is the Planck constant) of this quantity attest to topologically ordered states, which carry quasiparticles with fractional charge and anyonic statistics. Another topological invariant is the thermal Hall conductance, which is harder to measure. For the quantized thermal Hall conductance, a fractional value in units of κ0 (κ0 = π2kB2/(3h), where kB is the Boltzmann constant) proves that the state of matter is non-Abelian. Such non-Abelian states lead to ground-state degeneracy and perform topological unitary transformations when braided, which can be useful for topological quantum computation. Here we report measurements of the thermal Hall conductance of several quantum Hall states in the first excited Landau level and find that the thermal Hall conductance of the 5/2 state is compatible with a half-integer value of 2.5κ0, demonstrating its non-Abelian nature.

Nonlocal supercurrent of quartets in a three-terminal Josephson junction.

A novel nonlocal supercurrent, carried by quartets, each consisting of four electrons, is expected to appear in a voltage-biased three-terminal Josephson junction. This supercurrent results from a nonlocal Andreev bound state (ABS), formed among three superconducting terminals. While in a two-terminal Josephson junction the usual ABS, and thus the dc Josephson current, exists only in equilibrium, the ABS, which gives rise to the quartet supercurrent, persists in the nonlinear regime. In this work, we report such resonance in a highly coherent three-terminal Josephson junction made in an InAs nanowire in proximity to an aluminum superconductor. In addition to nonlocal conductance measurements, cross-correlation measurements of current fluctuations provided a distinctive signature of the quartet supercurrent. Multiple device geometries had been tested, allowing us to rule out competing mechanisms and to establish the underlying microscopic origin of this coherent nondissipative current.

Robust integer and fractional helical modes in the quantum Hall effect.

Electronic systems harboring one-dimensional helical modes, where spin and momentum are locked, have lately become an important field of its own. When coupled to a conventional superconductor, such systems are expected to manifest topological superconductivity; a unique phase hosting exotic Majorana zero modes. Even more interesting are fractional helical modes, yet to be observed, which open the route for realizing generalized parafermions. Possessing non-abelian exchange statistics, these quasiparticles may serve as building blocks in topological quantum computing. Here, we present a new approach to form protected one-dimensional helical edge modes in the quantum Hall regime. The novel platform is based on a carefully designed double-quantum-well structure in a GaAs based system hosting two electronic sub-bands; each tuned to the quantum Hall effect regime. By electrostatic gating of different areas of the structure, counter-propagating integer, as well as fractional, edge modes with opposite spins are formed. We demonstrate that due to spin-protection, these helical modes remain ballistic for large distances. In addition to the formation of helical modes, this platform can serve as a rich playground for artificial induction of compounded fractional edge modes, and for construction of edge modes based interferometers.

Transmission of heat modes across a potential barrier.

Controlling the transmission of electrical current using a quantum point contact constriction paved a way to a large variety of experiments in mesoscopic physics. The increasing interest in heat transfer in such systems fosters questions about possible manipulations of quantum heat modes that do not carry net charge (neutral modes). Here we study the transmission of upstream neutral modes through a quantum point contact in fractional hole-conjugate quantum Hall states. Employing two different measurement techniques, we were able to render the relative spatial distribution of these chargeless modes with their charged counterparts. In these states, which were found to harbor more than one downstream charge mode, the upstream neutral modes are found to flow with the inner charge mode-as theoretically predicted. These results unveil a universal upstream heat current structure and open the path for more complex engineering of heat flows and cooling mechanisms in quantum nano-electronic devices.

Observed quantization of anyonic heat flow.

The quantum of thermal conductance of ballistic (collisionless) one-dimensional channels is a unique fundamental constant. Although the quantization of the electrical conductance of one-dimensional ballistic conductors has long been experimentally established, demonstrating the quantization of thermal conductance has been challenging as it necessitated an accurate measurement of very small temperature increase. It has been accomplished for weakly interacting systems of phonons, photons and electronic Fermi liquids; however, it should theoretically also hold in strongly interacting systems, such as those in which the fractional quantum Hall effect is observed. This effect describes the fractionalization of electrons into anyons and chargeless quasiparticles, which in some cases can be Majorana fermions. Because the bulk is incompressible in the fractional quantum Hall regime, it is not expected to contribute substantially to the thermal conductance, which is instead determined by chiral, one-dimensional edge modes. The thermal conductance thus reflects the topological properties of the fractional quantum Hall electronic system, to which measurements of the electrical conductance give no access. Here we report measurements of thermal conductance in particle-like (Laughlin-Jain series) states and the more complex (and less studied) hole-like states in a high-mobility two-dimensional electron gas in GaAs-AlGaAs heterostructures. Hole-like states, which have fractional Landau-level fillings of 1/2 to 1, support downstream charged modes as well as upstream neutral modes, and are expected to have a thermal conductance that is determined by the net chirality of all of their downstream and upstream edge modes. Our results establish the universality of the quantization of thermal conductance for fractionally charged and neutral modes. Measurements of anyonic heat flow provide access to information that is not easily accessible from measurements of conductance.

Charge of a quasiparticle in a superconductor.

Nonlinear charge transport in superconductor-insulator-superconductor (SIS) Josephson junctions has a unique signature in the shuttled charge quantum between the two superconductors. In the zero-bias limit Cooper pairs, each with twice the electron charge, carry the Josephson current. An applied bias VSD leads to multiple Andreev reflections (MAR), which in the limit of weak tunneling probability should lead to integer multiples of the electron charge ne traversing the junction, with n integer larger than 2Δ/eVSD and Δ the superconducting order parameter. Exceptionally, just above the gap eVSD ≥ 2Δ, with Andreev reflections suppressed, one would expect the current to be carried by partitioned quasiparticles, each with energy-dependent charge, being a superposition of an electron and a hole. Using shot-noise measurements in an SIS junction induced in an InAs nanowire (with noise proportional to the partitioned charge), we first observed quantization of the partitioned charge q = e*/e = n, with n = 1-4, thus reaffirming the validity of our charge interpretation. Concentrating next on the bias region eVSD ~ 2Δ, we found a reproducible and clear dip in the extracted charge to q ~ 0.6, which, after excluding other possibilities, we attribute to the partitioned quasiparticle charge. Such dip is supported by numerical simulations of our SIS structure.