We consider a two-dimensional electron gas with strong spin-orbit coupling contacted by two superconducting leads, forming a Josephson junction. We show that in the presence of an in-plane Zeeman field, the quasi-one-dimensional region between the two superconductors can support a topological superconducting phase hosting Majorana bound states at its ends. We study the phase diagram of the system as a function of the Zeeman field and the phase difference between the two superconductors (treated as an externally controlled parameter). Remarkably, at a phase difference of π, the topological phase is obtained for almost any value of the Zeeman field and chemical potential. In a setup where the phase is not controlled externally, we find that the system undergoes a first-order topological phase transition when the Zeeman field is varied. At the transition, the phase difference in the ground state changes abruptly from a value close to zero, at which the system is trivial, to a value close to π, at which the system is topological. The critical current through the junction exhibits a sharp minimum at the critical Zeeman field and is therefore a natural diagnostic of the transition. We point out that in the presence of a symmetry under a mirror reflection followed by time reversal, the system belongs to a higher symmetry class, and the phase diagram as a function of the phase difference and the Zeeman field becomes richer.

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Current descriptions of the ab initio density matrix renormalization group (DMRG) algorithm use two superficially different languages: an older language of the renormalization group and renormalized operators, and a more recent language of matrix product states and matrix product operators. The same algorithm can appear dramatically different when written in the two different vocabularies. In this work, we carefully describe the translation between the two languages in several contexts. First, we describe how to efficiently implement the ab initio DMRG sweep using a matrix product operator based code, and the equivalence to the original renormalized operator implementation. Next we describe how to implement the general matrix product operator/matrix product state algebra within a pure renormalized operator-based DMRG code. Finally, we discuss two improvements of the ab initio DMRG sweep algorithm motivated by matrix product operator language: Hamiltonian compression, and a sum over operators representation that allows for perfect computational parallelism. The connections and correspondences described here serve to link the future developments with the past and are important in the efficient implementation of continuing advances in ab initio DMRG and related algorithms.

We propose a setup to realize time-reversal invariant topological superconductors in quantum wires, proximity coupled to conventional superconductors. We consider a model of quantum wire with strong spin-orbit coupling and proximity coupling to two s-wave superconductors. When the relative phase between the two superconductors is φ = π a Kramers' pair of Majorana zero modes appears at each edge of the wire. We study the robustness of the phase in presence of both timereversal invariant and time-reversal breaking perturbations. In addition, we show that the system forms a natural realization of a fermion parity pump, switching the local fermion parity of both edges when the relative phase between the superconductors is changed adiabatically by 2π.Introduction.−Over the last few decades, it has been realized that there is a deep and unexpected relation between the properties of matter and topology. At zero temperature, there exist phases of matter that are distinguished by an underlying topological structure encoded in their ground-state wave functions. These phases are often characterized by a finite energy gap in their bulk, and protected gapless edge states with unusual properties. The bulk can either be insulating, as in the case of the recently discovered topological insulators (TI) [1, 2], or superconducting [3][4][5]. Phases of the latter type, known as "topological superconductors" (TSC), support anomalous zero-energy Andreev edge states which are robust as long as the bulk quasi-particle gap remains open. These edge states have attracted much attention due to their possible future applications for topologically protected quantum information processing [6]. Recently, it has been predicted that the one-dimensional variant of a TSC can be realized by proximity-coupling a semiconducting quantum wire to a superconductor (SC) [7][8][9][10][11]. The resulting TSC phase has particle-hole symmetric modes at zero energy, localized at the edges of the wire, known as Majorana zero modes [12]. Signatures of such zero modes have been observed in recent experiments [13][14][15].In the presence of time-reversal invariance (TRI), different types of TSC can arise [16, 17] [26] are possible candidates for these phases. In addition, it has been proposed that proximity-coupling an unconventional superconductor to a quantum wire can stabilize a one dimensional TSC phase which supports a Kramers pair of Majorana zero modes at its edge [27][28][29], protected by TRI. This edge modes are characterized by an anomalous relation between the fermion parity of the edge and time-reversal symmetry [16].In this Letter, we propose a different setup to realize TRI TSC in quantum wires. The setup is shown schematically in Fig. 1a. A quantum wire with strong spin-orbit coupling is proximity coupled to two s-wave superconductors from either side [30]. We assume that the relative phase φ between the two superconductors can be controlled externally, e.g. by connecting the two superconducting leads and threading a flux Φ through the resulting...

Quantum metrology uses tools from quantum information science to improve measurement signal-to-noise ratios. The challenge is to increase sensitivity while reducing susceptibility to noise, tasks that are often in conflict. Lock-in measurement is a detection scheme designed to overcome this difficulty by spectrally separating signal from noise. Here we report on the implementation of a quantum analogue to the classical lock-in amplifier. All the lock-in operations--modulation, detection and mixing--are performed through the application of non-commuting quantum operators to the electronic spin state of a single, trapped Sr(+) ion. We significantly increase its sensitivity to external fields while extending phase coherence by three orders of magnitude, to more than one second. Using this technique, we measure frequency shifts with a sensitivity of 0.42 Hz Hz(-1/2) (corresponding to a magnetic field measurement sensitivity of 15 pT Hz(-1/2)), obtaining an uncertainty of less than 10 mHz (350 fT) after 3,720 seconds of averaging. These sensitivities are limited by quantum projection noise and improve on other single-spin probe technologies by two orders of magnitude. Our reported sensitivity is sufficient for the measurement of parity non-conservation, as well as the detection of the magnetic field of a single electronic spin one micrometre from an ion detector with nanometre resolution. As a first application, we perform light shift spectroscopy of a narrow optical quadrupole transition. Finally, we emphasize that the quantum lock-in technique is generic and can potentially enhance the sensitivity of any quantum sensor.

We consider a one-dimensional, time-reversal-invariant system with attractive interactions and spin-orbit coupling. Such a system is gapless due to the strong quantum fluctuations of the superconducting order parameter. However, we show that a sharply defined topological phase with protected, exponentially localized edge states exists. If one of the spin components is conserved, the protection of the edge modes can be understood as a consequence of the presence of a spin gap. In the more general case, the localization of the edge states arises from a gap to single particle excitations in the bulk. We consider specific microscopic models and demonstrate both analytically and numerically (using density matrix renormalization group calculations) that they can support the topologically non-trivial phase.

We consider a model for a one-dimensional quantum wire with Rashba spin-orbit coupling and repulsive interactions, proximity coupled to a conventional s-wave superconductor. Using a combination of Hartree-Fock and density matrix renormalization group calculations, we show that for sufficiently strong interactions in the wire, a time-reversal invariant topological superconducting phase can be stabilized in the absence of an external magnetic field. This phase supports two zeroenergy Majorana bound states at each end, which are protected by time-reversal symmetry. The mechanism for the formation of this phase is a reversal of the sign of the effective pair potential in the wire, due to the repulsive interactions. We calculate the differential conductance into the wire and its dependence on an applied magnetic field using the scattering-matrix formalism. The behavior of the zero-bias anomaly as a function of the field direction can serve as a distinct experimental signature of the topological phase.

Cooperative effects in neural networks appear because a neuron fires only if a minimal number m of its inputs are excited. The multiple inputs requirement leads to a percolation model termed quorum percolation. The connectivity undergoes a phase transition as m grows, from a network-spanning cluster at low m to a set of disconnected clusters above a critical m. Both numerical simulations and the model reproduce the experimental results well. This allows a robust quantification of biologically relevant quantities such as the average connectivityk and the distribution of connections p k . * * * We are grateful to J.-P. Eckmann for fruitful discussions and insight.