Detecting Non-Abelian Statistics with an Electronic Mach-Zehnder Interferometer

We present a theoretical study of the influence of dephasing on shot noise in an electronic Mach-Zehnder interferometer. In contrast to phenomenological approaches, we employ a microscopic model where dephasing is induced by the fluctuations of a classical potential. This enables us to treat the influence of the environment's fluctuation spectrum on the shot noise. We compare against the results obtained from a simple classical model of incoherent transport, as well as those derived from the phenomenological dephasing terminal approach, arguing that the latter runs into a problem when applied to shot noise calculations for interferometer geometries. From our model, we find two different limiting regimes: If the fluctuations are slow as compared to the time-scales set by voltage and temperature, the usual partition noise expression T(1-T) is averaged over the fluctuating phase difference. For the case of ``fast'' fluctuations, it is replaced by a more complicated expression involving an average over transmission amplitudes. The full current noise also contains other contributions, and we provide a general formula, as well as explicit expressions and plots for specific examples.

We study transport through an electronic Mach-Zehnder interferometer recently devised at the Weizmann Institute. We show that this device can be used to probe statistics of quasiparticles in the fractional quantum Hall regime. We calculate the tunneling current through the interferometer as the function of the Aharonov-Bohm flux, temperature and voltage bias, and demonstrate that its flux-dependent component is strongly sensitive to the statistics of tunneling quasiparticles. More specifically, the flux-dependent and flux-independent contributions to the current are related by a power law, the exponent being a function of the quasiparticle statistics.

Temperature dependence of the visibility in an electronic Mach–Zehnder interferometer

Quantum interference is typically detected through the dependence of the interference signal on certain parameters (path length, Aharonov-Bohm flux, etc.), which can be varied in a controlled manner. The destruction of interference by a which-path measurement is a paradigmatic manifestation of quantum effects. Here we report on a novel measurement protocol that realizes two objectives: (i)certifying that a measured signal is the result of interference avoiding the need to vary parameters of the underlying interferometer, and (ii)certifying that the interference signal at hand is of quantum nature. In particular, it yields a null outcome in the case of classical interference. Our protocol comprises measurements of cross-correlations between the readings of which-path weakly coupled detectors positioned at the respective interferometer's arms and the current in one of the interferometer's drains. We discuss its implementation with an experimentally available platform: an electronic Mach-Zehnder interferometer (MZI) coupled electrostatically to "detectors" (quantum point contacts).

We introduce a coupled wire model for a sequence of non-Abelian quantum Hall states that generalize the Z4 parafermion Read Rezayi state. The Z4 orbifold quantum Hall states occur at filling factors \nu = 2/(2m-p) for odd integers $m$ and $p$, and have a topological order with a neutral sector characterized by the orbifold conformal field theory with central charge $c=1$ at radius $R=\sqrt{p/2}$. When $p=1$ the state is Abelian. The state with $p=3$ is the $Z_4$ Read Rezayi state, and the series of $p\ge 3$ defines a sequence of non-Abelian states that resembles the Laughlin sequence. Our model is based on clustering of electrons in groups of four, and is formulated as a two fluid model in which each wire exhibits two phases: a weak clustered phase, where charge $e$ electrons coexist with charge $4e$ bosons and a strong clustered phase where the electrons are strongly bound in groups of 4. The transition between these two phases on a wire is mapped to the critical point of the 4 state clock model, which in turn is described by the orbifold conformal field theory. For an array of wires coupled in the presence of a perpendicular magnetic field, strongly clustered wires form a charge $4e$ bosonic Laughlin state with a chiral charge mode at the edge, but no neutral mode and a gap for single electrons. Coupled wires near the critical state form quantum Hall states with a gapless neutral mode described by the orbifold theory. The coupled wire approach allows us to employ the Abelian bosonization technique to fully analyze the physics of single wire, and then to extract most topological properties of the resulting non-Abelian quantum Hall states. These include the list of quasiparticles, their fusion rules, the correspondence between bulk quasiparticles and edge topological sectors, and most of the phases associated with quasiparticles winding one another.

We consider a class of models of non-equilibrium electronic Mach-Zehnder\ninterferometers built on integer quantum Hall edges states. The models are\ncharacterized by the electron-electron interaction being restricted to the\ninner part of the interferometer and transmission coefficients of the quantum\nquantum point contacts, defining the interferometer, which may take arbitrary\nvalues from zero to one. We establish an exact solution of these models in\nterms of single-particle quantities --- determinants and resolvents of Fredholm\nintegral operators. In the general situation, the results can be obtained\nnumerically. In the case of strong charging interaction, the operators acquire\nthe block Toeplitz form. Analyzing the corresponding Riemann-Hilbert problem,\nwe reduce the result to certain singular single-channel determinants (which are\na generalization of Toeplitz determinants with Fisher-Hartwig singularities),\nand obtain an analytic result for the interference current (and, in particular,\nfor the visibility of Aharonov-Bohm oscillations). Our results, which are in\ngood agreement with experimental observations, show an intimate connection\nbetween the observed "lobe" structure in the visibility of Aharonov-Bohm\noscillations and multiple branches in the asymptotics of singular integral\ndeterminants.\n

Received 18 January 2017DOI:https://doi.org/10.1103/PhysRevB.95.039904©2017 American Physical Society

For a given parent fractional quantum Hall (FQH) state at filling fraction $\ensuremath{\nu}$, the hierarchy construction produces FQH states at nearby filling fractions ${{\ensuremath{\nu}}_{n}}$ by condensing minimally charged quasiholes or quasiparticles of the parent state into their own FQH states. The hierarchy construction has been useful for relating families of FQH states and for the experimental identification of the topological order of parent states via the properties of daughter states. We reinterpret the hierarchy construction as a two-step procedure: stacking with a second FQH state and condensing a condensable algebra of bosons. This two-step procedure can be applied to both Abelian and non-Abelian FQH states, and it does not require calculations with a wave function. We show this construction reproduces the hierarchies for the Laughlin and Pfaffian states, and can be applied further to propose hierarchies for various non-Abelian FQH states.

A non-perturbative treatment is developed for the dephasing produced by the shot noise of a one-dimensional electron channel. It is applied to two systems: a charge qubit and the electronic Mach–Zehnder interferometer (MZI), both of them interacting with an adjacent partitioned electronic channel acting as a detector. We find that the visibility (interference contrast) can display oscillations as a function of detector voltage and interaction time. This is a unique consequence of the non-Gaussian properties of the shot noise, and only occurs in the strong coupling regime, when the phase contributed by a single electron exceeds π. The resulting formula reproduces the recent surprising experimental observations reported in (I Neder et al 2006 Preprint cond-mat/0610634), and indicates a general explanation for similar visibility oscillations observed earlier in the MZI at large bias voltage. We explore in detail the full pattern of oscillations as a function of coupling strength, voltage and time, which might be observable in future experiments.

It has been shown recently that non-Abelian states and the spin-polarized and unpolarized versions of the Abelian 331 state may have identical signatures in Fabry-P\'erot interferometry in the quantum Hall effect at filling factor 5/2. We calculate the Fano factor for the shot noise in a Mach-Zehnder interferometer in the 331 states and demonstrate that it differs from the Fano factor in the proposed non-Abelian states. The Fano factor depends periodically on the magnetic flux through the interferometer. Its maximal value is $2\ifmmode\times\else\texttimes\fi{}1.4e$ for the 331 states with a symmetry between two flavors of quasiparticles. In the absence of such symmetry the Fano factor can reach $2\ifmmode\times\else\texttimes\fi{}2.3e$. On the other hand, for the Pfaffian and anti-Pfaffian states the maximal Fano factor is $2\ifmmode\times\else\texttimes\fi{}3.2e$. The period of the flux dependence of the Fano factor is one flux quantum. If only quasiparticles of one flavor can tunnel through the interferometer then the period drops to one half of the flux quantum. We also discuss transport signatures of a general Halperin state with the filling factor $2+k/(k+2)$.

Selective control of edge-channel trajectories by scanning gate microscopy

We review our recent contributions to two topics that have become of interest in the field of open, dissipative quantum systems: non‐Gaussian noise and decoherence in fermionic systems. Decoherence by non‐Gaussian noise, i.e. by an environment that cannot be approximated as a bath of harmonic oscillators, is important in nanostructures (e.g. qubits) where there might be strong coupling to a small number of fluctuators. We first revisit the pedagogical example of dephasing by classical telegraph noise. Then we address two models where the quantum nature of the noise becomes essential: “quantum telegraph noise” and dephasing by electronic shot noise. In fermionic systems, many‐body aspects and the Pauli principle have to be taken care of when describing the loss of phase coherence. This is relevant in electronic quantum transport through metallic and semiconducting structures. Specifically, we recount our recent results regarding dephasing in a chiral interacting electron liquid, as it is realized in the electronic Mach–Zehnder interferometer. This model can be solved employing the technique of bosonization as well as a physically transparent semiclassical method (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Using exact numerical diagonalization we have studied correlated many-electron ground states in a partially filled second Landau level. We consider filling fractions ? = 1/2 and 2/5, for which incompressible quantum liquids with non-Abelian anion statistics have been proposed. Our calculations include finite layer width, Landau level mixing and arbitrary deformation of the interaction pseudopotential. Computed energies, gaps, and correlation functions support the non-Abelian ground states at both ? = 1/2 (“Pfaffian”) and ? = 2/5 (“parafermion” state).

The charging of a quantum box connected to a lead by a single-mode point contact is solved for arbitrary temperatures, tunneling amplitudes, and gate voltages, using a variant of Wilson's numerical renormalization group. The charge inside the box and the capacitance of the junction are calculated on equal footing for all physical regimes, including weak tunneling, near perfect transmission, and the crossover regime in between. At the charge plateaus, perturbation theory is found to break down at fairly small tunneling amplitudes. Near perfect transmission, we confirm Matveev's scenario for the smearing of the Coulomb-blockade staircase. A surprising reentrance of the Coulomb-blockade staircase is found for large tunneling amplitudes. At the degeneracy points, we obtain two-channel Kondo behavior directly from the Coulomb-blockade Hamiltonian, without the restriction to two charge configurations or the introduction of an effective cutoff.

Two-dimensional electron systems in GaAs/AlGaAs heterostructures have provided a platform for investigating numerous phenomena in condensed matter physics. The quantum Hall effect is a particularly remarkable phenomenon due to its topological properties, including chiral edge states with quantized conductance. This report describes progress made in interference measurements of these edge states in electronic Fabry-Perot interferometers. Previous interference experiments in the quantum Hall regime have been stymied by Coulomb charging effects and poor quantum coherence. These Coulomb charging effects have been dramatically suppressed by the implementation of a novel GaAs/AlGaAs heterostructure which utilizes auxiliary screening wells in addition to the primary GaAs quantum well. Using this heterostructure, Aharonov-Bohm interference is measured in very small devices which have greatly improved coherence. Robust Aharonov-Bohm interference is reported at fractional quantum Hall states nu = 1/3 and nu = 2/3. Discrete jumps in phase at nu = 1/3 consistent with anyonic braiding statistics are observed. The report concludes with proposed future experiments, including extending these results to possible non-Abelian quantum Hall states.