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.

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.

The Role of Spin-Flip Collisions in a Dark-Exciton Condensate.

We show that a Bose-Einstein condensate consisting of dark excitons forms in GaAs coupled quantum wells at low temperatures. We find that the condensate extends over hundreds of micrometers, well beyond the optical excitation region, and is limited only by the boundaries of the mesa. We show that the condensate density is determined by spin-flipping collisions among the excitons, which convert dark excitons into bright ones. The suppression of this process at low temperature yields a density buildup, manifested as a temperature-dependent blueshift of the exciton emission line. Measurements under an in-plane magnetic field allow us to preferentially modify the bright exciton density and determine their role in the system dynamics. We find that their interaction with the condensate leads to its depletion. We present a simple rate-equations model, which well reproduces the observed temperature, power, and magnetic-field dependence of the exciton density.

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.

Topological Thermal Hall Conductance of Even-Denominator Fractional States.

The even-denominator fractional quantum Hall (FQH) states ν=5/2 and ν=7/2 have been long predicted to host non-Abelian quasiparticles. Their present energy-carrying neutral modes are hidden from customary conductance measurements and thus motivate thermal transport measurements, which are sensitive to all energy-carrying modes. While past "two-terminal" thermal conductance (k_{2t}T) measurements already proved the non-Abelian nature of the ν=5/2 FQH state, they might have been prone to a lack of thermal equilibration among the counterpropagating edge modes. Here, we report a novel thermal Hall conductance measurement of the ν=5/2 and ν=7/2 states, being insensitive to equilibration among edge modes. We verify the state's non-Abelian nature, with both states supporting a single upstream Majorana edge mode (hence, a particle-hole Pfaffian order). While current numerical works predict a different topological order, this contribution should motivate further theoretical work.

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.

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.

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.

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.

Approaching Ideal Visibility in Singlet-Triplet Qubit Operations Using Energy-Selective Tunneling-Based Hamiltonian Estimation.

We report energy-selective tunneling readout-based Hamiltonian parameter estimation of a two-electron spin qubit in a GaAs quantum dot array. Optimization of readout fidelity enables a single-shot measurement time of 16 µs on average, with adaptive initialization and efficient qubit frequency estimation based on real-time Bayesian inference. For qubit operation in a frequency heralded mode, we observe a 40-fold increase in coherence time without resorting to dynamic nuclear polarization. We also demonstrate active frequency feedback with quantum oscillation visibility, single-shot measurement fidelity, and gate fidelity of 97.7%, 99%, and 99.6%, respectively, showcasing the improvements in the overall capabilities of GaAs-based spin qubits. By pushing the sensitivity of the energy-selective tunneling-based spin to charge conversion to the limit, the technique is useful for advanced quantum control protocols such as error mitigation schemes, where fast qubit parameter calibration with a large signal-to-noise ratio is crucial.

Single-Shot Readout of a Driven Hybrid Qubit in a GaAs Double Quantum Dot.

We report a single-shot-based projective readout of a semiconductor hybrid qubit formed by three electrons in a GaAs double quantum dot. Voltage-controlled adiabatic transitions between the qubit operations and readout conditions allow high-fidelity mapping of quantum states. We show that a large ratio both in relaxation time vs tunneling time (≈50) and singlet-triplet splitting vs thermal energy (≈20) allows energy-selective tunneling-based spin-to-charge conversion with a readout visibility of ≈92.6%. Combined with ac driving, we demonstrate high visibility coherent Rabi and Ramsey oscillations of a hybrid qubit in GaAs. Further, we discuss the generality of the method for use in other materials, including silicon.

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.

Gated Bow-Tie Diode for Microwave to Sub-Terahertz Detection.

We propose a new design microwave radiation sensor based on a selectively doped semiconductor structure of asymmetrical shape (so-called bow-tie diode). The novelty of the design comes down to the gating of the active layer of the diode above different regions of the two-dimensional electron channel. The gate influences the sensing properties of the bow-tie diode depending on the nature of voltage detected across the ungated one as well as on the location of the gate in regard to the diode contacts. When the gate is located by the wide contact, the voltage sensitivity increases ten times as compared to the case of the ungated diode, and the detected voltage holds the same polarity of the thermoelectric electromotive force of hot electrons in an asymmetrically shaped n-n+ junction. Another remarkable effect of the gate placed by the wide contact is weak dependence of the detected voltage on frequency which makes such a microwave diode to be a proper candidate for the detection of electromagnetic radiation in the microwave and sub-terahertz frequency range. When the gate is situated beside the narrow contact, the two orders of sensitivity magnitude increase are valid in the microwaves but the voltage sensitivity is strongly frequency-dependent for higher frequencies.

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.

Experimental Study of the Exciton Gas-Liquid Transition in Coupled Quantum Wells.

We study the exciton gas-liquid transition in GaAs/AlGaAs coupled quantum wells. Below a critical temperature, T_{C}=4.8 K, and above a threshold laser power density the system undergoes a phase transition into a liquid state. We determine the density-temperature phase diagram over the temperature range 0.1-4.8 K. We find that the latent heat increases linearly with temperature at T≲1.1 K, similarly to a Bose-Einstein condensate transition, and becomes constant at 1.1≲T<4.8 K. Resonant Rayleigh scattering measurements reveal that the disorder in the sample is strongly suppressed and the diffusion coefficient sharply increases with decreasing temperature at T

A Newtonian approach to extraordinarily strong negative refraction.

Metamaterials with negative refractive indices can manipulate electromagnetic waves in unusual ways, and can be used to achieve, for example, sub-diffraction-limit focusing, the bending of light in the 'wrong' direction, and reversed Doppler and Cerenkov effects. These counterintuitive and technologically useful behaviours have spurred considerable efforts to synthesize a broad array of negative-index metamaterials with engineered electric, magnetic or optical properties. Here we demonstrate another route to negative refraction by exploiting the inertia of electrons in semiconductor two-dimensional electron gases, collectively accelerated by electromagnetic waves according to Newton's second law of motion, where this acceleration effect manifests as kinetic inductance. Using kinetic inductance to attain negative refraction was theoretically proposed for three-dimensional metallic nanoparticles and seen experimentally with surface plasmons on the surface of a three-dimensional metal. The two-dimensional electron gas that we use at cryogenic temperatures has a larger kinetic inductance than three-dimensional metals, leading to extraordinarily strong negative refraction at gigahertz frequencies, with an index as large as -700. This pronounced negative refractive index and the corresponding reduction in the effective wavelength opens a path to miniaturization in the science and technology of negative refraction.

Probing the microscopic structure of the stripe phase at filling factor 5/2.

A prominent manifestation of the competition between repulsive and attractive interactions acting on different length scales is the self-organized ordering of electrons in a stripelike fashion in material systems such as high-T_{c} superconductors. Such stripe phases are also believed to occur in two-dimensional electron systems exposed to a perpendicular magnetic field, where they cause a strong anisotropy in transport. The addition of an in-plane field even enables us to expel fractional quantum Hall states, to the benefit of such anisotropic phases. An important example represents the disappearance of the 5/2 fractional state. Here, we report the use of nuclear magnetic resonance spectroscopy to probe the electron density distribution of this emergent anisotropic phase. A surprisingly strong spatial density modulation was found. The observed behavior suggests a stripe pattern with a period of 2.6±0.6 magnetic lengths and an amplitude as large as 20% relative to the total density.

Charge fractionalization in the integer quantum Hall effect.

We report an observation, via sensitive shot noise measurements, of charge fractionalization of chiral edge electrons in the integer quantum Hall effect regime. Such fractionalization results solely from interchannel Coulomb interaction, leading electrons to decompose to excitations carrying fractional charges. The experiment was performed by guiding a partitioned current carrying edge channel in proximity to another unbiased edge channel, leading to shot noise in the unbiased edge channel without net current, which exhibited an unconventional dependence on the partitioning. The determination of the fractional excitations, as well as the relative velocities of the two original (prior to the interaction) channels, relied on a recent theory pertaining to this measurement. Our result exemplifies the correlated nature of multiple chiral edge channels in the integer quantum Hall effect regime.

Nonequilibrated counterpropagating edge modes in the fractional quantum Hall regime.

It is well established that density reconstruction at the edge of a two-dimensional electron gas takes place for hole-conjugate states in the fractional quantum Hall effect (such as v=2/3, 3/5, etc.). Such reconstruction leads, after equilibration between counterpropagating edge channels, to a downstream chiral current edge mode accompanied by upstream chiral neutral modes (carrying energy without net charge). Short equilibration length prevented thus far observation of the counterpropagating current channels-the hallmark of density reconstruction. Here, we provide evidence for such nonequilibrated counterpropagating current channels, in short regions (l=4 µm and l=0.4 µm) of fractional filling v=2/3 and, unexpectedly, v=1/3, sandwiched between two regions of integer filling v=1. Rather than a two-terminal fractional conductance, the conductance exhibited a significant ascension towards unity quantum conductance (GQ=e(2)/h) at or near the fractional plateaus. We attribute this conductance rise to the presence of a nonequilibrated channel in the fractional short regions.

Giant hyperfine interaction between a dark exciton condensate and nuclei.

We study the interaction of a dark exciton Bose-Einstein condensate with the nuclei in gallium arsenide/aluminum gallium arsenide coupled quantum wells and find clear evidence for nuclear polarization buildup that accompanies the appearance of the condensate. We show that the nuclei are polarized throughout the mesa area, extending to regions that are far away from the photoexcitation area and persisting for seconds after the excitation is switched off. Photoluminescence measurements in the presence of radio frequency radiation reveal that the hyperfine interaction between the nuclear and electron spins is enhanced by two orders of magnitude. We suggest that this large enhancement manifests the collective nature of the N-exciton condensate, which amplifies the interaction by a factor of [Formula: see text].