Majorana zero modes induced by superconducting phase bias

Majorana zero modes in condensed matter systems have been the subject of much interest in recent years. Their non-Abelian exchange statistics, making them a unique state of matter, and their potential applications in topological quantum computation, earned them attention from both theorists and experimentalists. It is generally understood that in order to form Majorana zero modes in quasi-one-dimensional topological insulators, time-reversal symmetry must be broken. The straightforward mechanisms for doing so—applying magnetic fields or coupling to ferromagnets—turned out to have many unwanted side effects, such as degradation of superconductivity and the formation of sub-gap states, which is part of the reason Majorana zero modes have been eluding direct experimental detection for a long time. Here we review several proposal that rely on controlling the phase of the superconducting order parameter, either as the sole mechanism for time-reversal-symmetry breaking, or as an additional handy knob used to reduce the applied magnetic field. These proposals hold practical promise to improve Majorana detection, and they shed light on the physics underlying the formation of the topological superconducting state.

Accounting for Signal Distortion Biases for Wide-Lane and Narrow-Lane Phase Bias Estimation with Inhomogeneous Networks

Due to different designs of receiver correlators and front ends, receiver-related pseudorange biases, called signal distortion biases (SDBs), exist. Ignoring SDBs that can reach up to 0.66 cycles and 10 ns in Melbourne-Wübbena (MW) and ionosphere-free (IF) combinations can negatively affect phase bias estimation. In this contribution, we investigate the SDBs and evaluate the impacts on wide-lane (WL) and narrow-lane (NL) phase bias estimations, and further propose an approach to eliminating these SDBs to improve phase bias estimation. Based on a large data set of 302 multi-global navigation satellite system (GNSS) experiment (MGEX) stations, including 5 receiver brands, we analyze the characteristics of these SDBs The SDB characteristics of different receiver types for different GNSS systems differ from each other. Compared to the global positioning system (GPS) and BeiDou navigation satellite system (BDS), SDBs of Galileo are not significant; those of BDS-3 are significantly superior to BDS-2; Septentrio (SEPT) receivers show the most excellent consistency among all receiver types. Then, we apply the corresponding corrections to phase bias estimation for GPS, Galileo and BDS. The experimental results reveal that the calibration can greatly improve the performance of phase bias estimation. For WL phase biases estimation, the consistencies of WL phase biases among different networks for GPS, Galileo, BDS-2 and BDS-3 improve by 89%, 77%, 76% and 78%, respectively. There are scarcely any improvements of the fixing rates for Galileo due to its significantly small SDBs, while for GPS, BDS-2 and BDS-3, the WL ambiguity fixing rates can improve greatly by 13%, 27% and 14% after SDB calibrations with improvements of WL ambiguity fixing rates, the corresponding NL ambiguity fixing rates can further increase greatly, which can reach approximately 16%, 27% and 22%, respectively. Additionally, after the calibration, both WL and NL phase bias series become more stable. The standard deviations (STDs) of WL phase bias series for GPS and BDS can improve by more than 46%, while those of NL phase bias series can yield improvements of more than 13%. Ultimately, the calibration can make more WL and NL ambiguity residuals concentrated in ranges within ±0.02 cycles. All these results demonstrate that SDBs for phase bias estimation cannot be ignored and must be considered when inhomogeneous receivers are used.

Phase shifter based on an ultrathin superconducting bilayer with a through-hole for a superconducting device

Applying a radian phase shift other than 2π is a key issue for superconducting circuits, such as flux qubits. The magnetic flux is useful when generating a phase shift. However, a quantized magnetic flux accompanying a trapped vortex in a superconductor does not have a phase shifter function. The magnetic flux generated by an external field generates noise. In this study, we propose a phase bias system that does not require an external field during operation. We confirm the phase shift of a direct current superconducting interference device (SQUID) placed on an ultrathin superconducting Nb bilayer with a through-hole by cooling it to a temperature below the superconducting transition temperature with an external field. Although the cause of the phase shift in our system is unclear, we believe that it may be caused by a fractional quantum in the bilayer. When the SQUID is replaced by a qubit, the phase shift can be applied to a phase bias.

Characterizing and Correcting Phase Biases in Short-Term, Multilooked Interferograms

Interferometric Synthetic Aperture Radar (InSAR) is widely used to measure deformation of the Earth’s surface over large areas and long time periods. A common strategy to overcome coherence loss in long-term interferograms is to use multiple multilooked shorter interferograms, which can cover the same time period but maintain coherence. However, it has recently been shown that using this strategy can introduce a bias (also referred to as “fading signals”) in the interferometric phase, particularly over vegetated areas. We isolate the signature of the phase bias by constructing daisy chain sums of short-term interferograms covering identical 1-year time periods, but using interferograms of different time spans. This shows that the shorter interferograms are more affected by this phenomenon and that different ground cover types are affected differently. We, propose a method for correcting the phase bias, based on the assumption that the bias in an interferogram is linearly related to the sum of the bias in shorter interferograms spanning the same time. We tested the algorithm over a study area in western Turkey by comparing average velocities against results from a phase linking approach that has been shown to be rather insensitive to the phase bias. Our corrected velocities agree well with those from phase linking approach. Our approach can be applied to global compilations of short-term interferograms and offer the possibility of accurate long-term velocities without a requirement for coherence in long-term interferograms.

The CODE ambiguity-fixed clock and phase bias analysis products: generation, properties, and performance

The generation and use of GNSS analysis products that allow—particularly for the needs of single-receiver applications—precise point positioning with ambiguity resolution (PPP-AR) are becoming more and more popular. A general uncertainty concerns the question on how the necessary phase bias information should be provided to the PPP-AR user. Until now, each AR-enabling clock/bias representation method had its own practice to provide the necessary bias information. We have generalized the observable-specific signal bias (OSB) representation, as introduced in Villiger (J Geod 93:1487–1500, 2019) originally exclusively for pseudorange measurements, to carrier phase measurements. The existing common clock (CC) approach has been extended in a way that OSBs allowing for flexible signal and frequency handling between multiple GNSS become possible. Advantages of the proposed OSB-based PPP-AR approach are: GNSS biases can be provided in a consistent way for phase and code measurements and it is capable of multi-GNSS and suitable for standardization. This new, extended PPP-AR approach has been implemented by the Center for Orbit Determination in Europe (CODE). CODE clock products that adhere to the integer-cycle property have been submitted to the International GNSS Service (IGS) since mid of 2018 for three analysis lines: Rapid, Final, and MGEX (Multi-GNSS Extension). Ambiguity fixing is performed not only for GPS but also for Galileo. The integer-cycle property of between-satellite clock differences is of fundamental importance when comparing satellite clock estimates among various analysis lines, or at day boundaries. Both kinds of comparisons could be exploited at a very high level of consistency. Any retrieved comparison essentially indicated a standard deviation for between-satellite clocks from CODE of the order of 5 ps (1.5 mm in range). Finally, the integer-cycle property that may be recovered between the CODE Final clock and the accompanying bias product of consecutive daily sessions (using clock estimates additionally provided for the second midnight epoch) allows us to deduce GPS satellite clock and phase bias information that is consistent and continuous with respect to carrier phase observation data over two, three, or, in principle, yet more days. Phase-based clock densification from initially estimated integer-cycle-conform clock corrections at intervals of 300 s to 30 s (5 s in case of our Final clock product) is a matter of particular interest. Based on direct product comparisons and GRACE K-band ranging (KBR) data analysis, the quality of accordingly densified clock corrections could be confirmed to be on a level similar to that of “anchor” (300 s) clock corrections.

Sampling Error of Continuous Periodic Data and its Application for Geodesy

Data acquisition for geoinformatics cannot be done continuously, but by discrete sampling of the object or phenomenon. The sampling involves errors on the knowledge of the continuous signal due to the loss of information in the sampling procedure. In the present study, an analytical formulation of the sampling error is provided, which embodies the amplitude, phase, bias and periodicity of the sampling error. The analysis is then subsequently applied for case studies: for the GRACE and GRACE-FO monthly solutions, and for different realizations of the Hungarian Gravimetric Network.