Josephson Junction Chains Research Articles

Persistence of nonlinear hysteresis in fractional models of Josephson transmission lines

In this note, we depart from a model describing the transmission of electric currents in Josephson-junction chains, and provide a fractional generalization using Riesz discrete differential operators. The fractional model considered has generalized Hamiltonian- and energy-like functionals. The model and the energy functionals are fully discretized in order to investigate numerically the complex dynamics of the system when a sinusoidal perturbation at one end of the chain is imposed. As one of the most important results in this report, we establish the persistence of the nonlinear phenomena of supratransmission and infratransmission in Riesz fractional chains. Nonlinear hysteresis loops are obtained numerically for some values of the order of the fractional derivative, and numerical simulations of the propagation of monochromatic wave signals through the transmission line are presented using the nonlinear bistability of the system.

Inhomogeneous Josephson junction chains: a superconducting meta-material for superinductance optimization

We report a theoretical study of the low-frequency impedance of a Josephson junction chain whose parameters vary in space. Our goal is to find the optimal spatial profile which maximizes the total inductance of the chain without shrinking the low-frequency window where the chain behaves as an inductor. If the spatial modulation is introduced by varying the junction areas, we find that the best result is obtained for a spatially homogeneous chain, reported earlier in the literature. An improvement over the homogeneous result can be obtained by representing the junctions by SQUIDs with different loop areas, so the inductances can be varied by applying a magnetic field. Still, we find that this improvement becomes less important for longer chains.

Nonergodic metallic and insulating phases of Josephson junction chains

Strictly speaking, the laws of the conventional statistical physics, based on the equipartition postulate [Gibbs J W (1902) Elementary Principles in Statistical Mechanics, developed with especial reference to the rational foundation of thermodynamics] and ergodicity hypothesis [Boltzmann L (1964) Lectures on Gas Theory], apply only in the presence of a heat bath. Until recently this restriction was believed to be not important for real physical systems because a weak coupling to the bath was assumed to be sufficient. However, this belief was not examined seriously until recently when the progress in both quantum gases and solid-state coherent quantum devices allowed one to study the systems with dramatically reduced coupling to the bath. To describe such systems properly one should revisit the very foundations of statistical mechanics. We examine this general problem for the case of the Josephson junction chain that can be implemented in the laboratory and show that it displays a novel high-temperature nonergodic phase with finite resistance. With further increase of the temperature the system undergoes a transition to the fully localized state characterized by infinite resistance and exponentially long relaxation.

Parity effect and single-electron injection for Josephson junction chains deep in the insulating state

We have made a systematic investigation of charge transport in one-dimensional chains of Josephson junctions where the characteristic Josephson energy is much less than the single-junction Cooper-pair charging energy, ${E}_{J}\ensuremath{\ll}{E}_{CP}$. Such chains are deep in the insulating state, where superconducting phase coherence across the chain is absent, and a voltage threshold for conduction is observed at the lowest temperatures. We find that Cooper-pair tunneling in such chains is completely suppressed. Instead, charge transport is dominated by tunneling of single electrons, which is very sensitive to the presence of BCS quasiparticles on the superconducting islands of the chain. Consequently, we observe a strong parity effect, where the threshold voltage vanishes sharply at a characteristic parity temperature ${T}^{*}$, which is significantly lower than the critical temperature ${T}_{c}$. A measurable and thermally activated zero-bias conductance appears above ${T}^{*}$, with an activation energy equal to the superconducting gap, confirming the role of thermally excited quasiparticles. Conduction below ${T}^{*}$ and above the voltage threshold occurs via injection of single electrons/holes into the Cooper-pair insulator, forming a nonequilibrium steady state with a significantly enhanced effective temperature. Our results explicitly show that single-electron transport dominates deep in the insulating state of Josephson junction arrays. This conduction process has mostly been ignored in previous studies of both superconducting junction arrays and granular superconducting films below the superconductor-insulator quantum phase transition.

Kerr coefficients of plasma resonances in Josephson junction chains

We present an experimental and theoretical analysis of the self- and cross-Kerr effect of extended plasma resonances in Josephson junction chains. We calculate the Kerr coefficients by deriving and diagonalizing the Hamiltonian of a linear circuit model for the chain and then adding the Josephson non-linearity as a perturbation. The calculated Kerr-coefficients are compared with the measurement data of a chain of 200 junctions. The Kerr effect manifests itself as a frequency shift that depends linearly on the number of photons in a resonant mode. By changing the input power on a low signal level, we are able to measure this shift. The photon number is calibrated using the self-Kerr shift calculated from the sample parameters. We then compare the measured cross-Kerr shift with the theoretical prediction, using the calibrated photon number.

Mode engineering with a one-dimensional superconducting metamaterial

We propose a way to control the Josephson energy of a single Josephson junction embedded in a one-dimensional superconducting metamaterial: an inhomogeneous superconducting loop, made out of a superconducting nanowire or a chain of Josephson junctions. The Josephson energy is renormalized by the electromagnetic modes propagating along the loop. We study the behavior of the modes as well as of their frequency spectrum when the capacitance and the inductance along the loop are spatially modulated. We show that, depending on the amplitude of the modulation, the renormalized Josephson energy is either larger or smaller than the one found for a homogeneous loop. Using typical experimental parameters for Josephson junction chains and superconducting nanowires, we conclude that this mode engineering can be achieved with currently available metamaterials.

Disordered Josephson junction chains: Anderson localization of normal modes and impedance fluctuations

We study the properties of the normal modes of a chain of Josephson junctions in the simultaneous presence of disorder and absorption. We consider the superconducting regime of small phase fluctuations and focus on the case where the effects of disorder and absorption can be treated additively. We analyze the frequency shift and the localization length of the modes. We also calculate the distribution of the frequency-dependent impedance of the chain. The distribution is Gaussian if the localization length is long compared to the absorption length; it has a power law tail in the opposite limit.

Phase sticking in one-dimensional Josephson junction chains

We studied current-voltage characteristics of long one-dimensional Josephson junction chains with Josephson energy much larger than charging energy, ${E}_{J}\ensuremath{\gg}{E}_{C}$. In this regime, typical I-V curves of the samples consist of a supercurrent-like branch at low-bias voltages followed by a voltage-independent chain current branch, ${I}_{\mathrm{chain}}$ at high bias. Our experiments showed that ${I}_{\mathrm{chain}}$ is not only voltage-independent but it is also practically temperature-independent up to $T=0.7{T}_{C}$. We have successfully model the transport properties in these chains using a capacitively shunted junction model with nonlinear damping.

Localizing quantum phase slips in one-dimensional Josephson junction chains

We studied quantum phase-slip (QPS) phenomena in long one-dimensional Josephson junction series arrays with tunable Josephson coupling. These chains were fabricated with as many as 2888 junctions, where one sample had a separately tunable link in the middle of the chain. Measurements were made of the zero-bias resistance, R0, as well as current–voltage characteristics (IVC). The finite R0 is explained by QPS and shows an exponential dependence on with a distinct change in the exponent at R0 = RQ = h/4e2. When R0 > RQ, the IVC clearly shows a remnant of the Coulomb blockade, which evolves to a zero-current state with a sharp critical voltage as EJ is tuned to a smaller value. The zero-current state below the critical voltage is due to coherent QPSs and we show that these are enhanced when the central link is weaker than all other links. Above the critical voltage, a negative, differential resistance is observed, which nearly restores the zero-current state.

Realization of a two-channel Kondo model with Josephson junction networks

We show that —in the quantum regime— a Josephson junction rhombi chain (i.e. a Josephson junction chain made by rhombi formed by joining 4 Josephson junctions) may be effectively mapped onto a quantum Hamiltonian describing Ising spins in a transverse magnetic field with open boundary conditions. Then, we elucidate how a Y-shaped network fabricated with 3 Josephson junction rhombi chains may be used as a quantum device realizing the two-channel Kondo model recently proposed by Tsvelik in Phys. Rev. Lett., 110 (2013) 147202. We point out that the emergence of a 2-channel Kondo effect in this superconducting network may be probed through the measurement of a pertinent Josephson current.

Experimental demonstration of Aharonov-Casher interference in a Josephson junction circuit

A neutral quantum particle with magnetic moment encircling a static electric charge acquires a quantum mechanical phase (Aharonov-Casher effect). In superconducting electronics the neutral particle becomes a fluxon that moves around superconducting islands connected by Josephson junctions. The full understanding of this effect in systems of many junctions is crucial for the design of novel quantum circuits. Here we present measurements and quantitative analysis of fluxon interference patterns in a six Josephson junction chain. In this multi-junction circuit the fluxon can encircle any combination of charges on five superconducting islands, resulting in a complex pattern. We compare the experimental results with predictions of a simplified model that treats fluxons as independent excitations and with the results of the full diagonalization of the quantum problem. Our results demonstrate the accuracy of the fluxon interference description and the quantum coherence of these arrays.

Enhanced coherence of a quantum doublet coupled to Tomonaga–Luttinger liquid leads

We use boundary field theory to describe the phases accessible to a tetrahedral qubit coupled to Josephson junction chains acting as Tomonaga–Luttinger liquid leads. We prove that, in a pertinent range of the fabrication and control parameters, an attractive finite coupling fixed point emerges due to the geometry of the composite Josephson junction network. We show that this new stable phase is characterized by the emergence of a quantum doublet which is robust not only against the noise in the external control parameters (magnetic flux, gate voltage) but also against the decoherence induced by the coupling of the tetrahedral qubit with the superconducting leads. We provide protocols allowing to read and to manipulate the state of the emerging quantum doublet and argue that a tetrahedral Josephson junction network operating near the new finite coupling fixed point may be fabricated with todayʼs technologies.

Josephson junction chains with long-range interactions: Phase slip proliferation versus Kosterlitz-Thouless transition

The role of long-range badly screened Coulomb interactions in a one-dimensional chain of Josephson junctions is studied. Correlation functions for the phase correlator are obtained as a function of the Josephson coupling energy, the short-range part of the Coulomb repulsion, and its long-range component. Although quasi-long-range order is no longer possible and the usual Kosterlitz-Thouless transition no longer exists, there are remnants of it. As an application, we calculate the $I\ensuremath{-}V$ curves for Andreev reflection when a normal metal is placed in contact with the chain. Formally, there is always an offset voltage ${V}_{0}$ below which no current can flow; however, in some regimes ${V}_{0}$ can be negligible. Contrary to what happens without long-range interactions, the Andreev current, as a function of applied voltage, increases faster than any power law. Signatures of long-range interactions and phase slips appear in the $I\ensuremath{-}V$ curves. A possible application for quasi-one-dimensional thin superconducting wires is outlined.

Measurement of the effect of quantum phase slips in a Josephson junction chain

Coulomb interactions can cause a rapid change in the phase of the wavefunction along a very narrow superconducting system. Such a phase slip is now measured in a chain of Josephson junctions. The interplay between superconductivity and Coulomb interactions has been studied for more than 20 years now1,2,3,4,5,6,7,8,9,10,11,12,13. In low-dimensional systems, superconductivity degrades in the presence of Coulomb repulsion: interactions tend to suppress fluctuations of charge, thereby increasing fluctuations of phase. This can lead to the occurrence of a superconducting–insulator transition, as has been observed in thin superconducting films5,6, wires7 and also in Josephson junction arrays4,9,11,12,13. The last of these are very attractive systems, as they enable a relatively easy control of the relevant energies involved in the competition between superconductivity and Coulomb interactions. Josephson junction chains have been successfully used to create particular electromagnetic environments for the reduction of charge fluctuations14,15,16. Recently, they have attracted interest as they could provide the basis for the realization of a new type of topologically protected qubit17,18 or for the implementation of a new current standard19. Here we present quantitative measurements of quantum phase slips in the ground state of a Josephson junction chain. We tune in situ the strength of quantum phase fluctuations and obtain an excellent agreement with the tight-binding model initially proposed by Matveev and colleagues8.

Minimization of thermal jitter in a balanced comparator single flux quantum cell

The process of reproduction of single flux quantum pulses in a Josephson junction chain composed of a pulse generator, a balanced comparator, and a single Josephson junction is investigated. It is shown that the reproduction of single flux quantum pulses in such a chain is more stable and the mean and the standard deviation of switching time are minimal for the McCumber-Stewart parameter value from 0.5 to 1 and for a normalized clock generator current around 1.5. The influence of input and output inductances on the characteristics of a balanced comparator is investigated. The recommended value of the output inductance leading to smooth pulses and relatively small jitter is around unity.

Superconductor-insulator duality for an array of Josephson wires

We propose novel model system for the studies of superconductor-insulator transitions, which is a regular lattice, whose each link consists of Josephson-junction chain of $N \gg 1$ junctions in sequence. The theory of such an array is developed for the case of semiclassical junctions with the Josephson energy $E_J$ large compared to the junctions's Coulomb energy $E_C$. Exact duality transformation is derived, which transforms the Hamiltonian of the proposed model into a standard Hamiltonian of JJ array. The nature of the ground state is controlled (in the absence of random offset charges) by the parameter $q \approx N^2 \exp(-\sqrt{8E_J/E_C})$, with superconductive state corresponding to small $q < q_c $. The values of $q_c$ are calculated for magnetic frustrations $f= 0$ and $f= \frac12$. Temperature of superconductive transition $T_c(q)$ and $q < q_c$ is estimated for the same values of $f$. In presence of strong random offset charges, the T=0 phase diagram is controlled by the parameter $\bar{q} = q/\sqrt{N}$; we estimated critical value $\bar{q}_c$.

Effective boundary field theory for a Josephson junction chain with a weak link

We show that a finite Josephson junction (JJ) chain, ending with two bulk superconductors, and with a weak link at its center, may be regarded as a condensed matter realization of a two-boundary sine-Gordon model. Computing the partition function yields a remarkable analytic expression for the DC Josephson current as a function of the phase difference across the chain. We show that, in a suitable range of the chain parameters, there is a crossover of the DC Josephson current from a sinusoidal to a sawtooth behavior, which signals a transition from a regime where the boundary term is an irrelevant operator to a regime where it becomes relevant.

Analysis of zero-bias resistance in overdamped mesoscopic Josephson junction chains

We measured the temperature dependence and Josephson-coupling-energy ( E J) dependence of the zero-bias resistance of one-dimensional Josephson junction chains. Each junction in the chains was shunted by an ohmic resistance of several kΩ, and the charging energy was more than an order of magnitude larger than E J. The results for overdamped junction chains are in agreement with a theory considering phase fluctuation in the presence of dissipation.

Pairing of Cooper pairs in a fully frustrated Josephson-junction chain.

We study a one-dimensional Josephson-junction chain embedded in a magnetic field. We show that, when the magnetic flux per elementary loop equals half the superconducting flux quantum phi0 = h/2e, a local Z2 symmetry arises. This symmetry is responsible for a nematic Luttinger liquid state associated with bound states of Cooper pairs. We analyze the phase diagram and discuss some experimental possibilities to observe this exotic phase.

Renormalization-group study of gate charge effects in Josephson-junction chains

We study the quantum phase transition in a chain of superconducting grains, coupled by Josephson junctions, with emphasis on the effects of gate charges induced on the grains. At zero temperature the system is mapped onto a two-dimensional classical Coulomb gas, where the gate charge plays the role of an imaginary electric field. Such a field is found relevant in the renormalization-group transformation and to change the nature of the superconductor-insulator transition present in the system, tending to suppress quantum fluctuations and helping establish superconductivity. On the basis of this observation, we propose the zero-temperature phase diagram on the plane of the gate charge and the energy ratio.