Density of states at disorder-induced phase transitions in a multichannel Majorana wire

An $N$-channel spinless p-wave superconducting wire is known to go through a series of $N$ topological phase transitions upon increasing the disorder strength. Here, we show that at each of those transitions the density of states shows a Dyson singularity $\nu(\varepsilon) \propto \varepsilon^{-1}|\ln\varepsilon|^{-3}$, whereas $\nu(\varepsilon) \propto \varepsilon^{|\alpha|-1}$ has a power-law singularity for small energies $\varepsilon$ away from the critical points. Using the concept of "superuniversality" [Gruzberg, Read, and Vishveshwara, Phys. Rev. B 71, 245124 (2005)], we are able to relate the exponent $\alpha$ to the wire's transport properties at zero energy and, hence, to the mean free path $l$ and the superconducting coherence length $\xi$.Comment: 4+1 pages, 3 figure

Reentrant topological phase transitions in a disordered spinless superconducting wire

In a one-dimensional spinless p-wave superconductor with coherence length \xi, disorder induces a phase transition between a topologically nontrivial phase and a trivial insulating phase at the critical mean free path l=\xi/2. Here, we show that a multichannel spinless p-wave superconductor goes through an alternation of topologically trivial and nontrivial phases upon increasing the disorder strength, the number of phase transitions being equal to the channel number N. The last phase transition, from a nontrivial phase into the trivial phase, takes place at a mean free path l = \xi/(N+1), parametrically smaller than the critical mean free path in one dimension. Our result is valid in the limit that the wire width W is much smaller than the superconducting coherence length \xi

Statistical periodicity in driven quantum systems: General formalism and application to noisy Floquet topological chains

Much recent experimental effort has focused on the realization of exotic quantum states and dynamics predicted to occur in periodically driven systems. But how robust are the sought-after features, such as Floquet topological surface states, against unavoidable imperfections in the periodic driving? In this work, we address this question in a broader context and study the dynamics of quantum systems subject to noise with periodically recurring statistics. We show that the stroboscopic time evolution of such systems is described by a noise-averaged Floquet superoperator. The eigenvectors and -values of this superoperator generalize the familiar concepts of Floquet states and quasienergies and allow us to describe decoherence due to noise efficiently. Applying the general formalism to the example of a noisy Floquet topological chain, we re-derive and corroborate our recent findings on the noise-induced decay of topologically protected end states. These results follow directly from an expansion of the end state in eigenvectors of the Floquet superoperator.Comment: 13 pages, 5 figures. This is the final, published versio

Charge of a quasiparticle in a superconductor

Non-linear charge transport in 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 $V_{SD}$ 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 $2{\Delta}/eV_{SD}$ and ${\Delta}$ the superconducting order parameter. Exceptionally, just above the gap, $eV_{SD}>2{\Delta}$, 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. Employing 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 $eV_{SD}{\approx}2{\Delta}$, we found a reproducible and clear dip in the extracted charge to $q{\approx}0.6$, which, after excluding other possibilities, we attribute to the partitioned quasiparticle charge. Such dip is supported by numerical simulations of our SIS structure

The CHEOPS mission

The CHaracterising ExOPlanet Satellite (CHEOPS) was selected in 2012, as the first small mission in the ESA Science Programme and successfully launched in December 2019. CHEOPS is a partnership between ESA and Switzerland with important contributions by ten additional ESA Member States. CHEOPS is the first mission dedicated to search for transits of exoplanets using ultrahigh precision photometry on bright stars already known to host planets. As a follow-up mission, CHEOPS is mainly dedicated to improving, whenever possible, existing radii measurements or provide first accurate measurements for a subset of those planets for which the mass has already been estimated from ground-based spectroscopic surveys and to following phase curves. CHEOPS will provide prime targets for future spectroscopic atmospheric characterisation. Requirements on the photometric precision and stability have been derived for stars with magnitudes ranging from 6 to 12 in the V band. In particular, CHEOPS shall be able to detect Earth-size planets transiting G5 dwarf stars in the magnitude range between 6 and 9 by achieving a photometric precision of 20 ppm in 6 hours of integration. For K stars in the magnitude range between 9 and 12, CHEOPS shall be able to detect transiting Neptune-size planets achieving a photometric precision of 85 ppm in 3 hours of integration. This is achieved by using a single, frame-transfer, back-illuminated CCD detector at the focal plane assembly of a 33.5 cm diameter telescope. The 280 kg spacecraft has a pointing accuracy of about 1 arcsec rms and orbits on a sun-synchronous dusk-dawn orbit at 700 km altitude. The nominal mission lifetime is 3.5 years. During this period, 20% of the observing time is available to the community through a yearly call and a discretionary time programme managed by ESA.Comment: Submitted to Experimental Astronom

Über topologische Phasen in ungeordneten p-Wellen Supraleiterdrähten

Topological phases of matter have been the subject of intense experimental and theoretical research during the last years. Prominent examples are the Quantum Hall Effect, Topological Insulators or Topological Superconductors. The latter host special excitations, the Majorana states, at their boundaries, which can be thought of as the halves of an electron that can exist separately in this special case. These Majorana states have attracted great interest as they exhibit so-called non-Abelian braiding statistics, which could make them useful tools in the search for fault-tolerant quantum computation. In this context topologically superconducting wires are particularly useful as the Majorana states are located unambiguously at the wire’s end, where they form localized end states. Topologically superconducting wires are not known to exist in nature but they can be engineered from commonly available ingredients: semiconductor or ferromagnet nano- wires and conventional superconductors. The nano-wires can inherit superconductivity by the proximity effect and can then exhibit a topologically nontrivial phase. By now, several experiments have been performed on such hybrid structures, reporting measurements that are consistent with the existence of a topologically superconducting phase in the nanowire. Most theoretical investigations on these systems, so far, have been restricted to a one-dimensional effective model: The one-dimensional p-wave superconductor, which is the prototype of a topologically superconducting wire. A nanowire, however, is in general in a quasi-one dimensional regime, with a continuous longitudinal but a quantized transverse degree of freedom. In this Thesis we study the multichannel generalization of a topologically superconducting wire by means of a two- dimensional p + ip-superconductor that is restricted to a narrow-strip geometry. Such systems can be in a topological phase, characterized by the existence of a zero-energy excitation at the wires end—the Majorana bound state. We study the effect of various geometrical terminations on the low- energy spectrum of such a wire and find that subgap states tend to accumulate around zero energy. In a density-of- states measurement, these states potentially obscure the Majorana state thereby hindering the detection of the topological phase. We further investigate the effect of disorder on a multichannel wire and find that it induces a series of phase transitions with a reentrant topological phase. Due to disorder-localized states accumulating in the superconducting gap, the low-energy spectrum for a disordered wire contains a signature of the topological phase transitions as well: a singularity in the density of states, which is the well-known Dyson- singularity.Topologische Phasen der kondensierten Materie sind Gegenstand intensiver theoretischer und experimenteller Forschung. Die wohl bekanntesten Beispiele für solche Phasen sind: der Quanten-Hall Effekt, Topologische Isolatoren und Topologische Supraleiter. Letztere weisen sich durch spezielle Anregungen, die Majoranazustände, aus, welche man sich als die Hälften eines Elektrons vorstellen kann und die der Oberfläche eines solchen topologischen Supraleiters getrennt voneinander existieren können. Die Majoranazustände haben aufgrund ihren besonderen Eigenschaften ein großes wissenschaftliches Interesse geweckt. Sie besitzen eine nicht-Abelsche Flechtstatistik, welche sie zu nützlichen Bauteilen für einen möglichen fehlertoleranten Quantencomputer macht. In diesem Zusammenhang sind vor allem topologisch supraleitende Drähte wichtig, da in diesen die Positionen der Majoranazustände als die Drahtenden, den Oberflächen eines eindimensionalen Systems, eindeutig bestimmt sind. Topologisch supraleitende Drähten treten zwar nicht in der Natur auf können aber von verfügbaren Materialien, Halbleiter- oder ferromagnetische Nanodrähten und konventionellen Supraleitern, konstruiert werden. Die Nanodrähte können aufgrund des Proximity-Effekts supraleitende Eigenschaften übernehmen und eine topologisch nichttriviale Phase aufweisen. Inzwischen wurden mehrere Experimente an solchen Hybrid- strukturen durchgeführt und von Messergebnissen berichtet, welche mit den theoretischen Vorhersagen konsistent sind. Die meisten theoretischen Arbeiten an solchen Drähten sind auf ein eindimensionales effektives Model beschränkt, den p-Wellen Supraleiter. Ein Nanodraht ist aber normalerweise ein quasi- eindimensionales System, mit einem kontinuierlichen, longitudinalen und einem quantisierten, transversalen Freiheitsgrad. In dieser Doktorarbeit untersuchten wir die Verallgemeinerung eines topologisch supraleitenden Drahtes zu einem Mehrkanalsystem, indem wir einen zweidimensionalen p + ip- Supraleiter auf die Geometrie eines schmalen Streifens beschränkten. Solche Systeme können eine topologisch nichttriviale Phase aufweisen, welche durch die Existenz einer Nullenergieanregung, ein Majoranazustand, gekennzeichnet ist. Wir haben den Effekt verschiedener geometrischer Drahtendungen auf das Niedrigenergiespektrum eines solchen Drahtes untersucht und beobachtet, dass sich innerhalb der Energielücke des Supraleit- ers Zustände ansammeln, welche sich um die Nullenergie scharen. In der Zustandsdichte könnten diese den Majoranazustand verdecken und daher die Identifizierung der topologischen Phase wesentlich erschweren. Des weiteren haben wir die topologische Phase eines Mehrkanalsystems unter dem Einfluss eines Unordnungspotentials erforscht und eine Serie topologischer Phasenübergänge bei ansteigender Unordnungsstärke gefunden. Im Niedrigenergiespektrum des Drahtes werden die Phasenübergänge von der charakteristischen Dyson-Singularität begleitet