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The even-denominator fractional quantum Hall state at a filling factor of ν= 1/2 is an intriguing many-body phase in two-dimensional electron systems, as it appears in the ground state rather than an excited Landau level. It is observed in wide quantum wells where the electrons have a bilayer charge distribution with finite tunnelling between the layers. Whether this state takes an Abelian two-component form or a non-Abelian one-component form has been debated since its experimental discovery. Here we report the observation of the ν= 1/2 state that is flanked by numerous Jain-sequence composite fermion states at ν= p/(2p± 1) up to ν= 8/17 and 9/17. As we raise the density, the ν= 1/2 state is strengthened, its energy gap increases up to 4 K and, at the same time, the 8/17 and 7/13 states abruptly become stronger than their neighbouring high-order fractions. The states at ν= 8/17 and 7/13 are the theoretically predicted, simplest daughter states of the one-component Pfaffian ν= 1/2 state. This means that our data suggest a topological phase transition of 8/17 and 7/13 states from the Jain-sequence states to the daughter states of the Pfaffian.

In neutral dense electron–hole systems at low temperatures, theory predicted Cooper-pair-like excitons exist at the Fermi energy and form a Bardeen–Cooper–Schrieffer-like condensate. Optical excitations create electron–hole systems with the density controlled via the excitation power. However, the intense optical excitations required to achieve high densities cause substantial heating that prevents the realization of simultaneously dense and cold electron–hole systems in conventional semiconductors. Here we show that the separation of electron and hole layers enables the realization of a simultaneously dense and cold electron–hole system. We find a strong enhancement of photoluminescence intensity at the Fermi energy of the neutral dense ultracold electron–hole system that demonstrates the emergence of an excitonic Fermi edge singularity due to the formation of Cooper-pair-like excitons at the Fermi energy. Our measurements also show a crossover from the hydrogen-like excitons to the Cooper-pair-like excitons with increasing density, consistent with the theoretical prediction of a smooth transition.

A central goal of condensed-matter physics is to understand how the diverse electronic and optical properties of crystalline materials emerge from the wavelike motion of electrons through periodically arranged atoms. However, more than 90 years after Bloch derived the functional forms of electronic waves in crystals1(now known as Bloch wavefunctions), rapid scattering processes have so far prevented their direct experimental reconstruction. In high-order sideband generation2–9, electrons and holes generated in semiconductors by a near-infrared laser are accelerated to a high kinetic energy by a strong terahertz field, and recollide to emit near-infrared sidebands before they are scattered. Here we reconstruct the Bloch wavefunctions of two types of hole in gallium arsenide at wavelengths much longer than the spacing between atoms by experimentally measuring sideband polarizations and introducing an elegant theory that ties those polarizations to quantum interference between different recollision pathways. These Bloch wavefunctions are compactly visualized on the surface of a sphere. High-order sideband generation can, in principle, be observed from any direct-gap semiconductor or insulator. We thus expect that the method introduced here can be used to reconstruct low-energy Bloch wavefunctions in many of these materials, enabling important insights into the origin and engineering of the electronic and optical properties of condensed matter.

Two-dimensional electrons confined to GaAs quantum wells are hallmark platforms for probing electron–electron interactions. Many key observations have been made in these systems as sample quality has improved over the years. Here, we present a breakthrough in sample quality via source-material purification and innovation in GaAs molecular beam epitaxy vacuum chamber design. Our samples display an ultra-high mobility of 44 × 106cm2V–1s–1at an electron density of 2.0 × 1011cm–2. These results imply only 1 residual impurity for every 1010Ga/As atoms. The impact of such low impurity concentration is manifold. Robust stripe and bubble phases are observed, and several new fractional quantum Hall states emerge. Furthermore, the activation gap (Δ) of the fractional quantum Hall state at the Landau-level filling (ν) = 5/2, which is widely believed to be non-Abelian and of potential use for topological quantum computing, reaches Δ≈ 820 mK. We expect that our results will stimulate further research on interaction-driven physics in a two-dimensional setting and substantially advance the field.

In 1929, Felix Bloch suggested that the paramagnetic Fermi sea of electrons should make a spontaneous transition to a fully magnetized state at very low densities, because the exchange energy gained by aligning the spins exceeds the enhancement in the kinetic energy1. However, experimental realizations of this effect have been hard to implement. Here, we report the observation of an abrupt, interaction-driven transition to full magnetization, highly reminiscent of Bloch ferromagnetism. Our platform utilizes the two-dimensional Fermi sea of composite fermions near half-filling of the lowest Landau level. We measure the Fermi wavevector—which directly provides the spin polarization—and observe a sudden transition from a partially spin-polarized to a fully spin-polarized ground state as we lower the density of the composite fermions. Our theoretical calculations that take Landau level mixing into account provide a semi-quantitative account of this phenomenon.

Exciton–polaritons are mixed light–matter particles offering a versatile solid state platform to study many-body physical effects. In this work, we demonstrate an electrically controlled polariton laser, in a compact, easy-to-fabricate and integrable configuration, based on a semiconductor waveguide. Interestingly, we show that polariton lasing can be achieved in a system without a global minimum in the polariton energy-momentum dispersion. The cavity modes for the laser emission are obtained by adding couples of specifically designed diffraction gratings on top of the planar waveguide, forming an in-plane Fabry–Perot cavity. It is due to the waveguide geometry that we can apply a transverse electric field to finely tune the laser energy and quality factor of the cavity modes. Remarkably, we exploit the system sensitivity to the applied electric field to achieve an electrically controlled population of coherent polaritons. The precise control that can be reached with the manipulation of the grating properties and of the electric field provides strong advantages to this device in terms of miniaturization and integrability, two main features for the future development of coherent sources for polaritonic technologies.

Topological physics relies on the structure of the eigenstates of the Hamiltonians. The geometry of the eigenstates is encoded in the quantum geometric tensor1—comprising the Berry curvature2(crucial for topological matter)3and the quantum metric4, which defines the distance between the eigenstates. Knowledge of the quantum metric is essential for understanding many phenomena, such as superfluidity in flat bands5, orbital magnetic susceptibility6,7, the exciton Lamb shift8and the non-adiabatic anomalous Hall effect6,9. However, the quantum geometry of energy bands has not been measured. Here we report the direct measurement of both the Berry curvature and the quantum metric in a two-dimensional continuous medium—a high-finesse planar microcavity10—together with the related anomalous Hall drift. The microcavity hosts strongly coupled exciton–photon modes (exciton polaritons) that are subject to photonic spin–orbit coupling11from which Dirac cones emerge12, and to exciton Zeeman splitting, breaking time-reversal symmetry. The monopolar and half-skyrmion pseudospin textures are measured using polarization-resolved photoluminescence. The associated quantum geometry of the bands is extracted, enabling prediction of the anomalous Hall drift, which we measure independently using high-resolution spatially resolved epifluorescence. Our results unveil the intrinsic chirality of photonic modes, the cornerstone of topological photonics13–15. These results also experimentally validate the semiclassical description of wavepacket motion in geometrically non-trivial bands9,16. The use of exciton polaritons (interacting photons) opens up possibilities for future studies of quantum fluid physics in topological systems.

The fractional quantum Hall effect (FQHE) in two-dimensional electron systems is an exotic, superfluid-like matter with an emergent topological order. From the consideration of the Aharonov–Bohm interaction between electrons and magnetic field, the ground state of a half-filled lowest Landau level is mathematically transformed to a Fermi sea of composite objects of electrons bound to two flux quanta, termed composite fermions (CFs). A strong support for the CF theories comes from experimental confirmation of the predicted Fermi surface at ν = 1/2 (where ν is the Landau level filling factor) from the detection of the Fermi wavevector in semi-classical geometrical resonance experiments. Recent developments in the theory of CFs have led to the prediction of a π Berry phase for the CF circling around the Fermi surface at half-filling. In this paper we provide experimental evidence for the detection of the Berry phase of CFs in the fractional quantum Hall effect. Our measurements of the Shubnikov–de Haas oscillations of CFs as a function carrier density at a fixed magnetic field provide strong support for the existence of a π Berry phase at ν = 1/2. We also discover that the conductivity of composite fermions at ν = 1/2 displays an anomalous linear density dependence, whose origin remains mysterious yet tantalizing.

The paper presents data on the bubble phase of two-dimensional electron systems in higher Landau levels, as an example of the application of broadband microwave and rf spectroscopy to the study of electron solids.

Transport measurements of high-mohility two-dimensional electron systems at low temperatures have revealed a large resistance anisotropy around half-filling of excited Landau levels. These results have been attributed to electronic stripe-phase formation with spontaneously broken orientational symmetry. Mechanisms which are known to break the orientational symmetry include poorly-understood crystal structure effects and an in-plane magnetic field, $B_{||}$. Here we report that a large $B_{||}$also causes the transport anisotropy to persist up to much higher temperatures. In this regime, we find that the anisotropic resistance scales sublinearly with $B_{||}/T$. These observations support the proposal that the transition from anisotropic to isotropic transport reflects a liquid crystal phase transition where local stripe order persists even in the isotropic regime.

We present a study of the interband and intersubband transitions in mixed type I and type II GaAs/AlAs superlattices. Depending on the energy and intensity of the exciting (laser) radiation, either excitons in the wide GaAs wells or separate 2DEG and 2DHG are formed in the wide and narrow GaAs wells, respectively. We observe both the direct interband transitions and the inter-well transitions, and identify them by comparison with the calculated conduction and valence subband dispersion relations and resulting spectra. Intersubband transitions of the photoexcited 2DEG are observed as well. Under the application of an external magnetic field, the interband transitions between electron and hole Landau levels are shown to be strongly dependent on the density of the photoexcited 2DEG.

We show that the initial dynamics of resonantly-excited excitons in quantum wells is controlled by several processes such as radiative recombination, spin relaxation of excitons, electrons and holes, and scattering between different in-plane momentum states of excitons. We present a unifïed analysis of the results, which provides quantitative information about these important rates, and a good physical understanding of the initial dynamics of nonequilibrium excitons. In particular, we show that the enhanced radiative recombination rate of excitons makes an important contribution to the early decay of the photoluminescence.

We report on the study of the low temperature resonant Raman scattering (RRS) in GaAs/Al0.32,Ga0.68As Bragg confining structures (BCS). The LO-phonon RRS is observed in the spectral range of the (eB: hhB) and (eB: lhB) Bragg confined excitons. The RRS profiles show an outgoing beam resonance that is much stronger than the incoming beam resonance. These profiles are analyzed by a model in which the exciton dimensionality affects the in-plane excitonic dispersion and the Fröhlich interaction dependence on the phonon wavevector. The exciton scattering, in its translational motion, is assumed to be due to heavy point defects (short range potential fluctuations). Detailed calculations of the RRS profiles result in excellent agreement with the observed spectra. From them we conclude that the Bragg confined (eB: hhB) exciton is approximately 2-dimensional.

We report on the low temperature linearly and circularly polarized photoluminescence excitation spectra, and on the temporal dependence of the polarized PL in undoped GaAs/AlxGa1-xAs quantum wells (QW). The polarization excitation spectra Plin(Et) and Pcir(Et), Elis the exciting (laser) energy, monitored at the upper energy part of the photoluminescence (PL) band show two bands that are attributed to the (e1:hh1)1S and (e1:hh1)2S exciton states. Plin(t) and Pcir(t) decay much faster than PL(t). These observations are explained by the recombination of delocalized excitons in the high energy side of the PL band. These excitons can move over large "islands" in the QW plane and have a defined [MATH]. They scatter between the islands and relax by emitting acoustic phonons within the islands where they recombine. The spectral shape of Plinand Pciris thus determined by the exciton - LA phonon scattering rate and the polarization relaxation time dependences on [MATH].

Photoluminescence (PL) image size and angular resolved spectra were studied in a GaAs/AlAs planar microcavity (MC), with an embedded GaAs quantum well (QW) that contains a photogenerated, variable density, two-dimensional electron gas (2DEG). The PL was measured at T = 7 K, under nonresonant, cw excitation. The in-plane polariton migration from the photoexcitation spot (whose radius is ~30 μm) is deduced from the PL image size. It is observed to be anisotropic in the MC plane, and for negative detuning energy values (δ = EC – EX) it extends over in-plane distances of ~300 μm. Introducing a 2DEG into the QW while maintaining the strong coupling between the excitons and the MC photon mode, reduces the migration range. On the other hand, the angular resolved PL spectra show that the emission from the k‖ ~ 0 polariton states is enhanced as the 2DEG density is increased. From these observations we propose that the polaritons migrate while they occupy the long-lived states above the relaxation bottleneck in the lower polariton (LP) branch. The efficient electron–polariton scattering processes transfer the polaritons through the bottleneck into the radiative trap (k‖ ~ 0) in the LP branch. These scattering processes occur at the point of photoexcitation in the MC plane, and therefore they decrease the migration range. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Zero length quantum wires (or point contacts) exhibit unexplained conductance structure close to 0.7 × 2e 2 /h in the absence of an applied magnetic field. We have studied the density- and temperature-dependent conductance of ultra-low-disorder GaAs/AlGaAs quantum wires with nominal lengths l=0 and 2µm, fabricated from structures free of the disorder associated with modulation doping. In a direct comparision we observe structure near 0.7 × 2e 2 /h for l = 0, whereas thel = 2µm wires show structure evolving with increasing electron density to 0.5 × 2e 2 /h in zero magnetic field, the value expected for an ideal spin-split sub-band. Our results suggest the dominant mechanism through which electrons interact can be strongly affected by the length of the 1D region.

We report on the photoluminescence in quantum wells that have a low-density (ne < 2 × 10^10 cm^−2) two-dimensional electron gas (2DEG). Mixed type I–type II GaAs/AlAs quantum wells are studied in which the 2DEG is photogenerated in the type I quantum wells and ne is determined by the excitation intensity. The photoluminescence spectrum at T = 2 K is dominated by the recombination of a complex consisting of an electron bound to an exciton (a trion). It has a Lorentzian line shape whose width increases with increasing ne. This dependence is attributed to trion–electron scattering processes. We present a model of the elastic trion–electron scattering and calculate its rate as a function of ne. The model is based on a pair Coulomb interaction, and we fit the trion linewidth dependence on ne by using the 2DEG screening wave vector (qs) as a free parameter. The intensity at the low-energy tail of the trion band increases with ne, and we attribute it to inelastic trion–electron scattering processes.

We report on our studies of magnetic breakdown (MB) in coupled GaAs/Al0.3Ga0.7As double quantum wells (DQWs) subject to crossed magnetic fields. MB is a failure of semiclassical theory that occurs when a magnetic field causes electrons to tunnel across a gap ink-space from one Fermi surface (FS) branch to another. We study MB in a two-branch FS created by subjecting a DQW to an in-plane magnetic field (B∥). The principal effect ofB∥is a distortion in the dispersion curve of the system, yielding a FS consisting of two components, a lens-shaped inner orbit and an hour-glass-shaped outer orbit. The perpendicular field (B⊥) causes Landau level formation and Shubnikov–de Haas (SdH) oscillations for each branch of the FS. At higher perpendicular fields MB occurs and electrons tunnel throughk-space from one FS orbit to the other. MB is observed by noting which peaks are present in the Fourier power spectrum of the magnetoresistance versus 1/B⊥at constantB∥. We observe MB in two DQW samples over a range ofB∥.