We propose an experimental setup for detecting a Majorana zero mode consisting of a spinless quantum dot coupled to the end of a p-wave superconducting nanowire. The Majorana bound state at the end of the wire strongly influences the conductance through the quantum dot: Driving the wire through the topological phase transition causes a sharp jump in the conductance by a factor of 1/2. In the topological phase, the zero-temperature peak value of the dot conductance (i.e., when the dot is on resonance and symmetrically coupled to the leads) is e 2 /2h. In contrast, if the wire is in its trivial phase, the conductance peak value is e 2 /h, or if a regular fermionic zero mode occurs on the end of the wire, the conductance is 0. The system can also be used to tune Flensberg's qubit system [Phys. Rev. Lett. 106, 090503 (2011)] to the required degeneracy point.Majorana fermions, an exotic type of quasiparticle with non-Abelian statistics, are attracting a great deal of attention due to both their fundamental interest and their potential application for decoherence-free quantum computation. Several ways to realize unpaired Majorana fermions in a vortex core in a p-wave superconductor 1-6 and superfluid 7,8 have been proposed. Majorana bound states (MBSs) may also be realized at the ends of a one-dimensional p-wave superconductor 9 for which the proposed system is a semiconductor nanowire with Rashba spin-orbit interaction to which both a magnetic field and proximity-induced s-wave pairing are added. 10,11 In view of these proposals, how to detect and verify the existence of MBSs becomes a key issue. Suggestions include noise measurements, 12,13 resonant Andreev reflection by a scanning tunneling miscroscope (STM), 14 and 4π periodic Majorana-Josephson currents. [9][10][11]15 With regard to quantum computation, the braiding of Majorana bound states in a network of wires by applying a "keyboard" of individually tunable gates 16 leads to nontrivial computation. Note that all the detecting methods proposed to date, 9-15 involving electron transfer into or out of MBSs, will destroy the qubit information. In addition, such braiding cannot result in universal quantum computation; it must be supplemented by a topologically unprotected π/8 phase gate. 17 Recently, Flensberg introduced a system consisting of a quantum dot coupled to two MBSs (MBS-dot-MBS) through which this π/8 phase gate can be achieved. 18 A key point is that the system must be fine tuned so that the ground state is degenerate. 18 In this Rapid Communication, we consider a spinless quantum dot coupled to a MBS at the end of a p-wave superconducting (SC) nanowire, and study the conductance G through the dot by adding two external leads (schematic in Fig. 1). We find that the conductance is independent of the properties of the MBS, the nanowire, or the superconductor. The dependence of G on the dot properties has the same functional form whether a MBS is present or not. Therefore, the conductance behavior can be conveniently summarized by its peak value, when the ...

Since the advent of atomic force microscopy [1], mechanical resonators have been used to study a wide variety of phenomena, such as the dynamics of individual electron spins [2], persistent currents in normal metal rings [3], and the Casimir force [4,5].Key to these experiments is the ability to measure weak forces. Here, we report on force sensing experiments with a sensitivity of 12 zN/ √ Hz at a temperature of 1.2 K using a resonator made of a carbon nanotube. An ultra-sensitive method based on cross-correlated electrical noise measurements, in combination with parametric downconversion, is used to detect the low-amplitude vibrations of the nanotube induced by weak forces. The force sensitivity is quantified by applying a known capacitive force. A promising strategy for measuring lower forces is to employ resonators made of a molecular system, such as a carbon nanotube [14][15][16][17][18]. Nanotube resonators are characterized by an ultra-low mass M, which can be up to seven orders of magnitude lower than that of the ultra-soft cantilevers mentioned above [7], whereas their quality factor Q can be high [19] and their spring constant k 0 low. This has a great potential for generating an outstanding force sensitivity, whose classical limit is given byHere k B T is the thermal energy and γ the mechanical resistance [7]. This limit is set To efficiently convert weak forces into sizable displacements, we design nanotube resonators endowed with spring constants as low as ∼ 10 µN/m. This is achieved by fabri-2 cating the longest possible single-wall nanotube resonators. The fabrication process starts with the growth of nanotubes by chemical vapor deposition onto a doped silicon substrate coated with silicon oxide. Using atomic force microscopy (AFM), we select nanotubes that are straight over a distance of several micrometers, so that they do not touch the underlying substrate once they are released [21]. We use electron-beam lithography to pattern a source and a drain electrode that electrically contact and mechanically clamp the nanotube. We suspend the nanotube using hydrofluoric acid and a critical point dryer. Figure 1a shows a nanotube resonator that is 4 µm long. We characterize its resonant frequencies by driving it electrostatically and using a mixing detection method [18,22]. The lowest resonant frequency is 4.2 MHz (Fig. 1c). This gives a spring constant of 7 µN/m using an effective mass of 10 −20 kg, estimated from the size of the nanotube measured by AFM (supplementary information). This spring constant is comparable to that of the softest cantilevers fabricated so far [6]. When changing the gate voltage V DC g applied to the silicon substrate, the resonant frequency splits into two branches (Fig. 1c). These two branches correspond to the two fundamental modes; they vibrate in perpendicular directions (inset to Fig. 1c).We have developed an ultrasensitive detection method based on parametric downconversion, which (i) employs a cross-correlation measurement scheme to reduce the electrical noise ...

A Luttinger liquid is an interacting one-dimensional electronic system, quite distinct from the 'conventional' Fermi liquids formed by interacting electrons in two and three dimensions. Some of the most striking properties of Luttinger liquids are revealed in the process of electron tunnelling. For example, as a function of the applied bias voltage or temperature, the tunnelling current exhibits a non-trivial power-law suppression. (There is no such suppression in a conventional Fermi liquid.) Here, using a carbon nanotube connected to resistive leads, we create a system that emulates tunnelling in a Luttinger liquid, by controlling the interaction of the tunnelling electron with its environment. We further replace a single tunnelling barrier with a double-barrier, resonant-level structure and investigate resonant tunnelling between Luttinger liquids. At low temperatures, we observe perfect transparency of the resonant level embedded in the interacting environment, and the width of the resonance tends to zero. We argue that this behaviour results from many-body physics of interacting electrons, and signals the presence of a quantum phase transition. Given that many parameters, including the interaction strength, can be precisely controlled in our samples, this is an attractive model system for studying quantum critical phenomena in general, with wide-reaching implications for understanding quantum phase transitions in more complex systems, such as cold atoms and strongly correlated bulk materials.

We develop an approach to realizing a topological phase transition and non-Abelian braiding statistics with dynamically induced Floquet Majorana fermions (FMFs). When the periodic driving potential does not break fermion parity conservation, FMFs can encode quantum information. Quasienergy analysis shows that a stable FMF zero mode and two other satellite modes exist in a wide parameter space with large quasienergy gaps, which prevents transitions to other Floquet states under adiabatic driving. We also show that in the asymptotic limit FMFs preserve non-Abelian braiding statistics and, thus, behave like their equilibrium counterparts.