We report ultra-low temperature experiments on the obscure fractional quantum Hall effect (FQHE) at Landau level filling factor ν = 5/2 in a very high mobility specimen of µ = 1.7 × 10 7 cm 2 /Vs. We achieve an electron temperature as low as ∼ 4 mK, where we observe vanishing Rxx and, for the first time, a quantized Hall resistance, Rxy = h/(5/2e2 ) to within 2 ppm. Rxy at the neighboring odd-denominator states ν = 7/3 and 8/3 is also quantized. The temperature dependences of the Rxx-minima at these fractional fillings yield activation energy gaps ∆ 5/2 = 0.11 K, ∆ 7/3 = 0.10 K, and ∆ 8/3 = 0.055 K.PACS Numbers: 73.40Hm Electrons in two-dimensional systems at low temperatures and in the presence of an intense magnetic field condense into a sequence of incompressible quantum fluids with finite energy gaps for quasiparticle excitation, termed collectively the fractional quantum Hall effect (FQHE) [1]. These highly correlated electronic states occur at rational fractional filling ν = p/q of Landau levels. Their characteristic features in electronic transport experiments are vanishing resistance, R xx , and exact quantization of the concomitant Hall resistance, R xy , to h/(p/qe 2 ). Over the years, a multitude of FQHE states have been discovered -all q's being odd numbers. The only known exceptions are the states at half-filling of the second Landau level ν = 5/2 (=2+1/2) and ν = 7/2 (=3+1/2) [2-4]. Half-filled states in the lowest Landau level show no FQHE, whereas half-filled states in still higher Landau levels exhibit yet unresolved anisotropies [5]. Recent experiments in tilted magnetic field even seem to hint at a connection between the ν = 9/2 state and the state at ν = 5/2 [6].The origin of the ν = 5/2 and 7/2 states remains mysterious. Observation of odd-denominator FQHE states is intimately connected to the anti-symmetry requirement for the electronic wave function. An early, socalled hollow-core model [7] for the FQHE at ν = 5/2 and 7/2, which takes explicitly into account aspects of the modified single-particle wave functions of the second Landau level, arrived at a trial wave function. However, for a Coulomb Hamiltonian its applicability is problematic [8,9].With the advent of the composite fermion (CF) model [10] the existence of exclusively odd-denominator FQHE states is traced back to the formation of Landau levels of CFs emanating from even-denominator fillings, such as the sequence ν = p/(2p ± 1) from ν = 1/2. Evendenominator fillings themselves represent Fermi-liquid like states, resulting from the attachment of an even number of magnetic flux quanta to each electron. The obvious conflict between this theory and experiment at ν = 5/2 is resolved by invoking a CF-pairing mechanism [11][12][13][14]. In loose analogy to the formation of Cooper pairs in superconductivity such pairing creates a gapped, BCS-like ground state at ν = 5/2, called a "Pfaffian" state, which displays a FQHE. Indeed, an exact numerical diagonalization calculation by Morf [9] favors the Pfaffian state.The experimental si...

In a GaAs/AlGaAs quantum well of density 1 x 10(11) cm(-2) we observed a fractional quantum Hall effect (FQHE) at nu = 4/11 and 5/13, and weaker states at nu = 6/17, 4/13, 5/17, and 7/11. These sequences of fractions do not fit into the standard series of integral quantum Hall effects of composite fermions (CF) at nu = p/(2mp +/- 1). They rather can be regarded as the FQHE of CFs attesting to residual interactions between these composite particles. In tilted magnetic fields the nu = 4/11 state remains unchanged, strongly suggesting it to be spin polarized. The weak nu = 7/11 state vanishes quickly with tilt.

At a very low temperature of 9mK, electrons in the 2nd Landau level of an extremely high mobility two-dimensional electron system exhibit a very complex electronic behavior. With varying filling factor, quantum liquids of different origins compete with several insulating phases leading to an irregular pattern in the transport parameters. We observe a fully developed ν = 2 + 2/5 state separated from the even-denominator ν = 2 + 1/2 state by an insulating phase and a ν = 2 + 2/7 and ν = 2 + 1/5 state surrounded by such phases. A developing plateau at ν = 2 + 3/8 points to the existence of other even-denominator states.Low-temperature electron correlation in the lowest Landau level (LL) of a two-dimensional electron system (2DES) separates largely into two regions. At very low filling factor, ν ≤ 1/5, an insulating phase exists, which has now quite convincingly been determined to be a pinned electron solid [1,2,3]. At higher filling factor 1 > ν >∼ 1/5 the multiple sequences of fractional quantum Hall effect (FQHE) liquids [4,5, 6,7] dominate, which show the characteristic vanishing magneto resistance, R xx , and quantized Hall resistance, R xy , at many odd-denominator rational fractional fillings ν = p/q [8].Altogether about fifty such FQHE states have been observed in this region. Their multiple sequences can largely be described within the composite fermion (CF) model [9,10, 11,12], with the exact origin of some higher order states still being argued. The electrical behavior between FQHE states carries no particularly strong transport signature, being thought of as arising largely from the conduction of excited quasiparticles of the neighboring FQHE states, with CF liquids occurring at some even-denominator fractions.At high LL's a very different pattern seems to emerge. There charge density wave (CDW) or liquid crystal like states dominate, often referred to as electronic stripe and bubble phases [13,14,15]. Characteristically these states are pinned to the lattice, immobilizing the electrons of this LL, which leads to transport properties identical to those of the neighboring integer quantum Hall effect (IQHE) states. FQHE states are absent in these high LL's, except for the recent observation of two FQHE features in the third LL, at elevated temperatures [16]. Of course, high LL fillings typically occur at lower magnetic fields and hence at poorer resolution of potential FQHE features. However, very general theoretical arguments [17,18] based on an increasing extent of the wavefunction with increasing LL index, hence the increasing importance of exchange and the diminishing applicability of point-like interactions, clearly support this trend.It is in the 2nd LL where electron liquids and electron solids collide. The larger extent of the wavefunction as compared to the lowest LL and its additional zero allows for a much broader range of electron correlations to be favorable, leading to an ever changing competition between multiple electronic phases as the filling factor is varied and as the temperature is low...

We have investigated the influence of an increasing in-plane magnetic field on the states of half filling of Landau levels (n 11͞2, 9͞2, 7͞2, and 5͞2) of a two-dimensional electron system. In the electrically anisotropic phase at n 9͞2 and 11͞2 an in-plane magnetic field of ϳ1 2 T overcomes its initial pinning to the crystal lattice and reorients this phase. In the initially isotropic phases at n 5͞2 and 7͞2 an in-plane magnetic field induces a strong electrical anisotropy. In all cases, for high in-plane fields the high-resistance axis is parallel to the direction of the in-plane field.

Nonperturbative coupling of light with condensed matter in an optical cavity is expected to reveal a host of coherent many-body phenomena and states [1][2][3][4][5][6][7]. In addition, strong coherent light-matter interaction in a solid-state environment is of great interest to emerging quantum-based technologies [8,9]. However, creating a system that combines a long electronic coherence time, a large dipole moment, and a high cavity quality (Q) factor has been a challenging goal [10][11][12][13]. Here, we report collective ultrastrong light-matter coupling in an ultrahigh-mobility two-dimensional electron gas in a high-Q terahertz photonic-crystal cavity in a quantizing magnetic field, demonstrating a cooperativity of ∼360. The splitting of cyclotron resonance (CR) into the lower and upper polariton branches exhibited a √ ne-dependence on the electron density (ne), a hallmark of collective vacuum Rabi splitting. Furthermore, a small but definite blue shift was observed for the polariton frequencies due to the normally negligible A 2 term in the light-matter interaction Hamiltonian. Finally, the high-Q cavity suppressed the superradiant decay of coherent CR, which resulted in an unprecedentedly narrow intrinsic CR linewidth of 5.6 GHz at 2 K. These results open up a variety of new possibilities to combine the traditional disciplines of many-body condensed matter physics and cavity-based quantum optics.PACS numbers: 78.67. De, 76.40.+b, 78.47.jh Strong resonant light-matter coupling in a cavity setting is an essential ingredient in fundamental cavity quantum electrodynamics (QED) studies [14] as well as in cavity-QED-based quantum information processing [8,9]. In particular, a variety of solid-state cavity QED systems have recently been examined [15][16][17][18], not only for the purpose of developing scalable quantum technologies, but also for exploring novel many-body effects inherent to condensed matter. For example, collective √ N -fold enhancement of light-matter coupling in an N -body system [19], combined with colossal dipole moments available in solids, compared to traditional atomic systems, is promising for entering uncharted regimes of ultrastrong light-matter coupling. Nonintuitive quantum phenomena can occur in such regimes, including a "squeezed" vacuum state [1], the Dicke superradiant phase transition [2,3], the breakdown of the Purcell effect [4], and quantum vacuum radiation [5] induced by the dynamic Casimir effect [6,7].Specifically, in a cavity QED system, there are three rates that jointly characterize different light-matter coupling regimes: g, κ, and γ. The parameter g is the coupling constant, with 2g being the vacuum Rabi splitting between the two normal modes, the lower polariton (LP) and upper polariton (UP), of the coupled system. The parameter κ is the photon decay rate of the cavity; τ cav = κ −1 is the photon lifetime of the cavity, and the cavity Q = ω 0 τ cav at mode frequency ω 0 . The parameter γ is the nonresonant matter decay rate, which is usually the decoherence rate in ...

We present a spectrum of experimental data on the fractional quantum Hall effect (FQHE) states in the first excited Landau level, obtained in an ultrahigh mobility twodimensional electron system (2DES) and at very low temperatures and report the following results: For the even-denominator FQHE states, the sample dependence of the ν=5/2 state clearly shows that disorder plays an important role in determining the energy gap at ν=5/2. For the developing ν=19/8 FQHE state the temperature dependence of the R xx minimum implies an energy gap of ~5mK.The energy gaps of the odd-denominator FQHE states at ν=7/3 and 8/3 also increase with decreasing disorder, similar to the gap at 5/2 state. Unexpectedly and contrary to earlier data on lower mobility samples, in this ultra-high quality specimen, the ν=13/5 state is missing, while its particle-hole conjugate state, the ν=12/5 state, is a fully developed FQHE state. We speculate that this disappearance might indicate a spin polarization of the ν=13/5 state.Finally, the temperature dependence is studied for the two-reentrant integer quantum Hall states around ν=5/2 and is found to show a very narrow temperature range for the transition from quantized to classical value.

The quantum Hall plateau transition was studied at temperatures down to 1 mK in a random alloy disordered high mobility two-dimensional electron gas. A perfect power-law scaling with κ=0.42 was observed from 1.2K down to 12mK. This perfect scaling terminates sharply at a saturation temperature of T s~1 0mK. The saturation is identified as a finite-size effect when the quantum phase coherence length (L φ ∝T −p/2 ) reaches the sample size (W) of millimeter scale. From a size dependent study, T s ∝W −1 was observed and p=2 was obtained. The exponent of the localization length, determined directly from the measured κ and p, is ν=2.38, and the dynamic critical exponent z = 1.

We report on the observation of collective radiative decay, or superradiance, of cyclotron resonance (CR) in high-mobility two-dimensional electron gases in GaAs quantum wells using time-domain terahertz magnetospectroscopy. The decay rate of coherent CR oscillations increases linearly with the electron density in a wide range, which is a hallmark of superradiant damping. Our fully quantum mechanical theory provides a universal formula for the decay rate, which reproduces our experimental data without any adjustable parameter. These results firmly establish the many-body nature of CR decoherence in this system, despite the fact that the CR frequency is immune to electron-electron interactions due to Kohn's theorem.