Fractional Ac Josephson Effect Research Articles

Optimal conditions for observing fractional Josephson effect in topological Josephson junctions

Topological Josephson junctions (JJs), which contain Majorana bound states, are expected to exhibit 4$\pi$-periodic current-phase relation, thereby resulting in doubled Shapiro steps under microwave irradiation. We performed numerical calculations of dynamical properties of topological JJs using a modified resistively and capacitively shunted junction model and extensively investigated the progressive evolution of Shapiro steps as a function of the junction parameters and microwave power and frequency. Our calculation results indicate that the suppression of odd-integer Shapiro steps, i.e., evidence of the fractional ac Josephson effect, is enhanced significantly by the increase in the junction capacitance and IcRn product as well as the decrease in the microwave frequency even for the same portion of the 4$\pi$-periodic supercurrent. Our study provides the optimal conditions for observing the fractional ac Josephson effect; furthermore, our new model can be used to precisely quantify the topological supercurrent from the experimental data of topological JJs.

Josephson radiation in a superconductor-quantum dot-superconductor junction

We investigate the Josephson radiation emitted by a junction made of a quantum dot coupled to two conventional superconductors. Close to resonance, the particle-hole symmetric Andreev states that form in the junction are detached from the continuum above the superconducting gap in the leads, while a gap between them opens near the Fermi level. Under voltage bias, we formulate a stochastic model that accounts for non-adiabatic processes, which change the occupations of the Andreev states. This model allows calculating the current noise spectrum and determining the Fano factor. Analyzing the finite-frequency noise, we find that the model may exhibit either an integer or a fractional AC Josephson effect, depending on the bias voltage and the size of the gaps in the Andreev spectrum. Our results assess the limitations in using the fractional Josephson radiation as a probe of topology.

Experimental study of ac Josephson effect in gate-tunable (Bi1−xSbx)2Te3 thin-film Josephson junctions

We report on measurements of the ac Josephson effect in three-dimensional topological insulator (TI) Josephson junctions with Fermi energy tuning using a back gate. We successfully tune the Fermi energy into the bulk gap of the TI. We observe that the Josephson energy as a function of the gate voltage has a dip feature, indicating the presence of specular Andreev reflections. We study the ac Josephson effect with detection of both Shapiro steps and Josephson emission. The obtained results show no signature of Majorana modes. With the support of simulations, we conclude that the observation of the fractional ac Josephson effect may be complicated by both a large parallel capacitance and a small ratio of current carried in $4\ensuremath{\pi}$ period modes compared to conventional $2\ensuremath{\pi}$ modes.

Joule overheating poisons the fractional ac Josephson effect in topological Josephson junctions

Topological Josephson junctions designed on the surface of a 3D-topological insulator harbor Majorana bound states among a continuum of conventional Andreev bound states. The distinct feature of these Majorana bound states lies in the 4π-periodicity of their energy-phase relation that yields a fractional ac Josephson effect and a suppression of odd Shapiro steps under radio-frequency irradiation. Yet, recent experiments showed that a few, or only the first, odd Shapiro steps are missing, casting doubts on the interpretation. Here we show that Josephson junctions tailored on the large bandgap 3D-topological insulator Bi2Se3 exhibit a fractional ac Josephson effect acting on the first Shapiro step only. With a modified resistively shunted junction model, we demonstrate that the resilience of higher order odd Shapiro steps can be accounted for by thermal poisoning driven by Joule overheating. Furthermore, we uncover a residual supercurrent at the nodes between Shapiro lobes, which provides a direct and novel signature of the current carried by the Majorana bound states. Our findings showcase the crucial role of thermal effects in topological Josephson junctions and lend support to the Majorana origin of the partial suppression of odd Shapiro steps.

Detecting topological superconductivity using low-frequency doubled Shapiro steps

The fractional Josephson effect has been observed in many instances as a signature of a topological superconducting state containing zero-energy Majorana modes. We present a nontopological scenario which can produce a fractional Josephson effect generically in semiconductor-based Josephson junctions, namely, a resonant impurity bound state weakly coupled to a highly transparent channel. We show that the fractional ac Josephson effect can be generated by the Landau-Zener processes which flip the electron occupancy of the impurity bound state. The Josephson effect signature for Majorana modes become distinct from this nontopological scenario only at low frequency. We prove that a variant of the fractional ac Josephson effect, namely, the low-frequency doubled Shapiro steps, can provide a more reliable signature of the topological superconducting state.

Signatures of time-reversal-invariant topological superconductivity in the Josephson effect

For Josephson junctions based on s-wave superconductors, time-reversal symmetry is known to allow for powerful relations between the normal-state junction properties, the excitation spectrum, and the Josephson current. Here we provide analogous relations for Josephson junctions involving one-dimensional time-reversal-invariant topological superconductors supporting Majorana-Kramers pairs, considering both topological-topological and s-wave-topological junctions. Working in the regime where the junction is much shorter than the superconducting coherence length, we obtain a number of analytical and numerical results that hold for arbitrary normal-state conductance and the most general forms of spin-orbit coupling. The signatures of topological superconductivity we find include the fractional ac Josephson effect, which arises in topological-topological junctions provided that the energy relaxation is sufficiently slow. We also show, for both junction types, that robust signatures of topological superconductivity arise in the dc Josephson effect in the form of switches in the Josephson current due to zero-energy crossings of Andreev levels. The junction spin-orbit coupling enters the Josephson current only in the topological-topological case and in a manner determined by the switch locations, thereby allowing quantitative predictions for experiments with the normal-state conductance, the induced gaps, and the switch locations as inputs.

Proposal to probe quantum nonlocality of Majorana fermions in tunneling experiments

Topological Majorana fermion (MF) quasiparticles have been recently suggested to exist in semiconductor quantum wires with proximity induced superconductivity and a Zeeman field. Although the experimentally observed zero bias tunneling peak and a fractional ac-Josephson effect can be taken as necessary signatures of MFs, neither of them constitutes a sufficient "smoking gun" experiment. Since one pair of Majorana fermions share a single conventional fermionic degree of freedom, MFs are in a sense fractionalized excitations. Based on this fractionalization we propose a tunneling experiment that furnishes a nearly unique signature of end state MFs in semiconductor quantum wires. In particular, we show that a "teleportation"-like experiment is not enough to distinguish MFs from pairs of MFs, which are equivalent to conventional zero energy states, but our proposed tunneling experiment, in principle, can make this distinction.

Current-phase relation for Josephson effect through helical metal

Josephson junctions fabricated on the surface of three-dimensional topological insulators (TI) show a few unusual properties distinct from conventional Josephson junctions. In these devices, the Josephson coupling and the supercurrent are mediated by helical metal, the two-dimensional surface of the TI. A line junction of this kind is known to support Andreev bound states at zero energy for phase bias \pi, and consequently the so-called fractional ac Josephson effect. Motivated by recent experiments on TI-based Josephson junctions, here we describe a convenient algorithm to compute the bound state spectrum and the current-phase relation for junctions with finite length and width. We present analytical results for the bound state spectrum, and discuss the dependence of the current-phase relation on the length and width of the junction, the chemical potential of the helical metal, and temperature. A thorough understanding of the current-phase relation may help in designing topological superconducting qubits and manipulating Majorana fermions.

Fractional ac Josephson effect in unconventional superconductors

For certain orientations of Josephson junctions between two px-wave or two d-wave superconductors, the subgap Andreev bound states produce a 4π-periodic relation between the Josephson current I and the phase difference φ: I∝sin(φ/2). Consequently, the ac Josephson current has the fractional frequency eV/ℏ, where V is the dc voltage. In the tunneling limit, the Josephson current is proportional to the first power (and not the square) of the electron tunneling amplitude. Thus, the Josephson current between unconventional superconductors is carried by single electrons, rather than by Cooper pairs. The fractional ac Josephson effect can be observed experimentally by measuring the frequency spectrum of microwave radiation from the junction.

Theoretical prediction of the fractional ac Josephson effect in p- and d-wave superconductors

For certain orientations of Josephson junctions between two px-wave or two d-wave superconductors, the subgap Andreev states produce a 4p-periodic relation between the Josephson current I and the phase difference f: I µ sin( f/2). Consequently, the ac Josephson current has the fractional frequency eV/$\hbar $, where V is the dc voltage. In the tunneling limit, the Josephson current is proportional to the first power (not square) of the electron tunneling amplitude. Thus, the Josephson current is carried by single electrons, rather than by Cooper pairs.

Fractional ac Josephson effect in p- and d-wave superconductors

For certain orientations of Josephson junctions between two p x -wave or two d-wave superconductors, the subgap Andreev bound states produce a $4\pi$ -periodic relation between the Josephson current I and the phase difference $\phi$ : $I\propto\sin(\phi/2)$ . Consequently, the ac Josephson current has the fractional frequency $eV/\hbar$ , where V is the dc voltage. In the tunneling limit, the Josephson current is proportional to the first power (not square) of the electron tunneling amplitude. Thus, the Josephson current between unconventional superconductors is carried by single electrons, rather than by Cooper pairs. The fractional ac Josephson effect can be observed experimentally by measuring frequency spectrum of microwave radiation from the junction. We also study junctions between singlet s-wave and triplet p x -wave, as well as between chiral $p_x\pm ip_y$ -wave superconductors.