Charge response of the Majorana toric code

At zero temperature, a two dimensional lattice of Majorana zero modes on mesoscopic superconducting islands has a topologically ordered toric code phase. Recently, a Landau field theory has been proposed for the system that captures its different phases and the associated phase-transitions. It was shown that with the increase of Josephson tunneling between the islands, a continuous symmetry-breaking 3D-XY transition gets transformed into a discrete symmetry-breaking 3D-Ising transition through a couple of tricritical points and first order transitions. Using the proposed field theory, we analyze the charge-response of the system at the different continuous phase-transitions. We calculate the universal conductivity at the 3D-XY transitions and the change in the superconducting density at the Ising transition using 1/N expansion. Furthermore, by computing a one-loop correction to the field theory, we show that an additional tricritical point is likely to be present in the phase-diagram. Finally, we provide a mean-field calculation that supports the earlier proposed field theory.Comment: Published versio

Topological Quantum Computing

This set of lecture notes forms the basis of a series of lectures delivered at the 48th IFF Spring School 2017 on Topological Matter: Topological Insulators, Skyrmions and Majoranas at Forschungszentrum Juelich, Germany. The first part of the lecture notes covers the basics of abelian and non-abelian anyons and their realization in the Kitaev's honeycomb model. The second part discusses how to perform universal quantum computation using Majorana fermions.Comment: In Topological Matter: Topological Insulators, Skyrmions and Majoranas, Lecture notes of the 48th IFF Spring School 2017, eds. S. Bluegel, Y. Mokrusov, T. Schaepers, and Y. Ando (Forschungszentrum Juelich, Key Technologies, Vol. 139, 2017), Sec. D

Resonant production of the sterile neutrino dark matter and fine-tunings in the [nu]MSM

The generation of lepton asymmetry below the electroweak scale has a considerable impact on production of dark matter sterile neutrinos. Oscillations or decays of the heavier sterile neutrinos in the neutrino minimal standard model can give rise to the requisite lepton asymmetry, provided the masses of the heavier neutrinos are sufficiently degenerate. We study the renormalization group evolution of the mass difference of these singlet fermions to understand the degree of necessary fine-tuning. We construct an example of the model that can lead to a technically natural realization of this low-energy degeneracy.Comment: 8 pages, 5 figure

Quantum phase transitions of the Majorana toric code in the presence of finite Cooper-pair tunneling

The toric code based on Majorana fermions on mesoscopic superconducting islands is a promising candidate for quantum information processing. In the limit of vanishing Cooper-pair tunneling, it has been argued that the phase transition separating the topologically ordered phase of the toric code from the trivial one is in the universality class of (2+1)D-XY. On the other hand, in the limit of infinitely large Cooper-pair tunneling, the phase transition is in the universality class of (2+1)D-Ising. In this work, we treat the case of finite Cooper-pair tunneling and address the question of how the continuous XY symmetry breaking phase transition turns into a discrete $\mathbb{Z}_2$ symmetry breaking one when the Cooper-pair tunneling rate is increased. We show that this happens through a couple of tricritical points and first order phase transitions. Using a Jordan-Wigner transformation, we map the problem to that of spins coupled to quantum rotors and subsequently, propose a Landau field theory for this model that matches the known results in the respective limits. We calculate the effective field theories and provide the relevant critical exponents for the different phase transitions. Our results are relevant for predicting the stability of the topological phase in realistic experimental implementations.Comment: 5 pages, 2 figure

Asymmetric frequency conversion in nonlinear systems driven by a biharmonic pump

A novel mechanism of asymmetric frequency conversion is investigated in nonlinear dispersive devices driven parametrically with a biharmonic pump. When the relative phase between the first and second harmonics combined in a two-tone pump is appropriately tuned, nonreciprocal frequency conversion, either upward or downward, can occur. Full directionality and efficiency of the conversion process is possible, provided that the distribution of pump power over the harmonics is set correctly. While this asymmetric conversion effect is generic, we describe its practical realization in a model system consisting of a current-biased, resistively-shunted Josephson junction (RSJ). Here, the multiharmonic Josephson oscillations, generated internally from the static current bias, provide the pump drive.Comment: 5+ pages, 4 pages supplement. Expanded and modified discussion, additional references and a new appendix in supplemental material detailing the calculation of Josephson harmonics in the RS

Entanglement entropy in critical quantum spin chains with boundaries and defects

Entanglement entropy (EE) in critical quantum spin chains described by 1+1D conformal field theories contains signatures of the universal characteristics of the field theory. Boundaries and defects in the spin chain give rise to universal contributions in the EE. In this work, we analyze these universal contributions for the critical Ising and XXZ spin chains for different conformal boundary conditions and defects. For the spin chains with boundaries, we use the boundary states for the corresponding continuum theories to compute the subleading contribution to the EE analytically and provide supporting numerical computation for the spin chains. Subsequently, we analyze the behavior of EE in the presence of conformal defects for the two spin chains and describe the change in both the leading logarithmic and subleading terms in the EE.Comment: 14 pages, 6 figures, Chapter in Entanglement in Spin Chains - Theory and Quantum Technology Applications, Springe

Quantum Electronic Circuit Simulation of Generalized sine-Gordon Models

Investigation of strongly interacting, nonlinear quantum field theories (QFT-s) remains one of the outstanding challenges of modern physics. Here, we describe analog quantum simulators for nonlinear QFT-s using mesoscopic superconducting circuit lattices. Using the Josephson effect as the source of nonlinear interaction, we investigate generalizations of the quantum sine-Gordon model. In particular, we consider a two-field generalization, the double sine-Gordon model. In contrast to the sine-Gordon model, this model can be purely quantum integrable, when it does not admit a semi-classical description - a property that is generic to many multi-field QFT-s. The primary goal of this work is to investigate different thermodynamic properties of the double sine-Gordon model and propose experiments that can capture its subtle quantum integrability. First, we analytically compute the mass-spectrum and the ground state energy in the presence of an external `magnetic' field using Bethe ansatz and conformal perturbation theory. Second, we calculate the thermodynamic Bethe ansatz equations for the model and analyze its finite temperature properties. Third, we propose experiments to verify the theoretical predictions.Comment: 17 pages, 5 figures, journal version, added appendix on TBA derivatio

Quantum-limited Parametric Amplification with Josephson Circuits in the Regime of Pump Depletion

Linear parametric amplification is a key operation in information processing. Our interest here is quantum-limited parametric amplification, $i.e.$, amplification of quantum signals while adding the minimum amount of noise allowed by quantum mechanics, which is essential for any viable implementation of quantum information processing. We describe parametric amplifiers based on the dispersive nonlinearity of Josephson junctions driven with appropriate tones playing the role of pumps. We discuss two defining characteristics in the architecture of these amplifiers: the number of modes occupied by the signal, idler and pump waves and the number of independent ports through which these waves enter into the circuit. The scattering properties of these amplifiers is also reviewed. The main focus of this work are the computations of the dynamic range and phase-space distributions of the fluctuations of the modes of the amplifiers.Comment: Review of parametric amplifiers with Josephson circuits (see arXiv:1605.00539 for a shorter earlier version). Added new sections with calculations of dynamic range, Fokker-Planck equation analysis of amplifiers and appendices on quantized transmission lines and input-output theory; Published versio