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We report on a comparative analysis of three different laboratory terahertz (THz) sources emitting into open space based on a backward wave oscillator with a frequency multiplier, a microwave frequency multiplier to large numbers of harmonics based on semiconductor superlattices, and a recently developed long Josephson junction-based oscillator. The weight‒size parameters, frequency responses, and radiation powers of the sources have been qualitatively and quantitatively compared, along with the complexity of their tuning and operation under laboratory conditions.

An electromagnetic source of terahertz radiation on the basis of a tunnel long Josephson junction was proposed and experimentally studied. Output radiation was transmitted to open space by means of a slot antenna, which was located on the same microcircuit with the junction and matched with the collecting lens. Several Oscillator structures designed for operation within the frequency ranges of 250–410, 330–530, and 390–700 GHz with the possibility of continuous frequency tuning were manufactured and tested. The amplitude-frequency characteristics of output radiation with a spectral resolution of nearly 0.1 MHz were studied with the use of a terahertz-spectrometer based on a superconducting receiver. The study of transmitting antenna characteristics throughout the entire bandwidth was performed with the use of a cooled highly sensitive silicon based bolometer. The experimental results correspond to numerical calculations.

Interaction of the fluxon with the finite size dipole impurity in the long Josephson junction is investigated. The impurity has polarity and will be referred to as a {\it dipole} impurity because it also has a direction and, consequently, changes its sign under the space inversion transform $x \to -x$. Such a model is used to describe the inductively coupled to the Josephson transmission line qubit and the misaligned Abrikosov vortex that penetrates into the long Josephson junction. We derive the approximate equations of motion for the fluxon center of mass and its velocity. With the help of these equations we demonstrate that pinning and scattering of the fluxon on the impurity differs significantly from the case of the point impurity which is modelled by the derivative of the Dirac's $\delta$-function, Comment: 7 figures; Phys. Lett. A (in press)

We have developed a theoretical model describing the dynamics of the Josephson vortices in the long planar Josephson junction under the influence of the alternating heterogeneous magnetic field created by the tip of magnetic force microscope (MFM), and external uniform, constant magnetic field. The fitting of the model parameters ensured the coincidence of the calculated dependencies of the junction critical current on the external magnetic field with the experimental data, with a standard deviation of less than 0.1 percent. We considered the range of values of the field corresponding to the penetration of more than 10 Josephson vortices at different positions of the MFM tip. From comparison with the experiment, we obtained the estimates of Josephson length, effective area of the junction, critical current distribution along the junction boundary, inductance of the lead wires of the experimental sample and the distribution of the magnetic flux inside the junction created by the MFM tip.

In the present work, the current-voltage (I-V) characteristics in a coupled long Josephson junction based on magnesium diboride are studied by establishing a system of equations of phase differences of various inter- and intra-band channels starting from the microscopic Hamiltonian of the junction system and simplifying it through the phenomenological procedures such as action, partition function, Hubbard-Stratonovich transformation (bosonization), Grassmann integral, saddle-point approximation, Goldstone mode, phase dependent effective Lagrangian and, finally, Euler-Lagrange equation of motion. The system of equations are solved using finite difference approximation for which the solution of unperturbed sine-Gordon equation is taken as the initial condition. Neumann boundary condition is maintained at both the ends so that the fluxon is capable of reflecting from the end of the system. The phase dependent current is calculated for different tunnel voltage and averaged out over space and time. The current-voltage characteristics are almost linear at low voltage and non-linear at higher voltage which indicates that the more complicated physical phenomena at this situation may occur. At some region of the characteristics, there exist a negative resistance which means that the junction system can be used in specific electronic devices such as oscillators, switches, memories etc. The non-linearity is also sensitive to the layer as well as to the junction thicknesses. Non-linearity occurs for lower voltage and for higher junction and layer thicknesses., Comment: 17 pages, 7 figures

A study of the output terahertz (THz) emission to open space from Nb/AlO x /Nb-based long Josephson junction (LJJ) was carried out. Both bolometric technique with a frequency resolution of about1 GHz and heterodyne technique with spectral resolution of about 0.1 MHz were used for detection of the THz emission. A comparative analysis with other types of THz sources was made, such as backward wave oscillator (BWO) and frequency multiplier based on a semiconductor superlattice GaAs/AlAs with large harmonic numbers. A qualitative and quantitative comparison with respect to weight and size characteristics, the spectral properties, the emission power, as well as a general usability in laboratory conditions, was made. The highest detected power was achieved with an LJJ-based oscillator, and the most convenience was demonstrated using a superlattice-based oscillator with large harmonic numbers.

A long Josephson junction comprising regions with phase discontinuities driven by an external ac-drive is studied. An inhomogeneous sine-Gordon equation is used, that depicts the dynamics of long Josephson junctions with phase discontinuities. Perturbation technique along with asymptotic analysis and the method of averaging are applied to obtain an average dynamics in the form of double sine-Gordon equations for both small and large driving amplitudes. From the obtained average dynamics, it is determined that, the external ac-drive may affect the presence of the ground state of the junction. Specifically, the cases for 0 − κ and 0 − π − 0 junctions are discussed. In the presence of an external ac-drive, the critical facet length bc is analyzed for 0 − π − 0 junction above which the ground state is non-uniform. The critical bias current γc for 0 − κ junction is investigated, at which the junction switches to a resistive state. Further, the interaction of localized defect modes and the fast oscillating drive is studied for both 0 − κ and 0 − π − 0 junctions, which is explained by using Lagrangian approach. Numerical simulations are performed to support our analytical calculations.

We demonstrate the double-peaked switching current distributions of a long Josephson junction from the zero to the finite voltage state in the case of asymmetrical bias feed profile. The considered asymmetry stands to be a natural feature of YBCO junctions fabricated on bicrystal substrates. We argue that such a picture is associated with the formation of solitons at the opposite junction edges for slightly different bias currents, arisen due to asymmetry. System parameters such as junction length, noise intensity and character of bias feed inhomogeneity leading to atypical distribution of switching currents are discussed.

We investigate analytically and numerically a long Josephson junction on an infnite domain, having arbitrary periodic phase shift of k, that is, the so-called 0-k long Josephson junction. The system is described by a one-dimensional sine-Gordon equation and has relatively recently been proposed as artificial atom lattices. We discuss the existence of periodic solutions of the system and investigate their stability both in the absence and presence of an applied bias current. We find critical values of the phase-discontinuity and the applied bias current beyond which a solution switches to its complementary counterpart. Due to the periodic discontinuity in the phase, the system admits regions of allowed and forbidden bands. We perturbatively investigate the Arnold tongues that separate the region of allowed and forbidden bands, and discuss the effect of an applied bias current on the band-gap structure. We present numerical simulations to support our analytical results.

We investigate the low-energy dynamics of two coupled anisotropic Bose-Einstein condensates forming a long Josephson junction. The theoretical study is performed in the framework of the two-dimensional Gross-Pitaevskii equation and the Bogoliubov--de Gennes formalism. We analyze the excitation spectrum of the coupled Bose condensates and show how low-energy excitations of the condensates lead to multiple-frequency oscillations of the atomic populations in the two wells. This analysis generalizes the standard bosonic Josephson equation approach. We also develop a one-dimensional hydrodynamic model of the coupled condensates, that is capable of reproducing the excitation spectrum and population dynamics of the system.

Compared to irreversible gate operations, reversible digital logic gates can provide a fundamental advantage in energy efficiency. Here we discuss previously discovered 1-bit ballistic reversible logic gates and new experimental plans to measure it. The gates consist of long Josephson junctions (LJJs) connected by a circuit interface. The gate dynamics evanescently extends into the LJJs to realize the physically resonant gate operation. It differs from known adiabatic superconducting gates because it does not work in the adiabatic limit. Depending on the gate type, that is the Identity or NOT gate, the polarity of the outgoing fluxon should be preserved or inverted, respectively. The dynamics of the scattering process, originally discovered in full numerical simulation, is understood using collective coordinate analysis. We plan to experimentally study how well a fluxon can travel ballistically towards the interface, and scatter with the designed gate operation. Here we present gate and SQUID-sensing layout designs for a NOT gate. The LJJ circuit layout uses niobium trilayer short junctions with connecting wiring for inductors. Part of the design includes the shunting capacitors at the gate interface which is key for the resonant dynamics in the gates.

A new implementation for a terahertz flux-flow oscillator (FFO) based on a long Josephson junction is presented. The oscillator is integrated with the lens antenna on a single chip providing the THz emission in the open space, and with a harmonic mixer (HM) based on a SIS junction for a frequency and phase locking. We used the double slot antenna to output the emission from the planar transmission line to the open space, this antenna is coupled to the elliptical silicon lens forming narrow output beam of THz radiation. Two designs of antenna and matching structures coupled to the oscillator are developed and fabricated, the operating ranges 0.33 - 0.57 THz and 0.42 - 0.73 THz were obtained experimentally for these designs. The emission to open space was measured by an external superconducting receiver located in the separate cryogenic system.

This article presents the phase dynamics of an inline long Josephson junction in voltage state under the influence of constant external magnetic field. Fractional calculus approach is used to model the evolution of the phase difference between the macroscopic wave functions of the two superconductors across the junction. The governing non-linear partial differential equation is then solved using finite difference-finite element schemes. Other quantities of interest like Josephson current density and voltage across the junction are also computed. The effects of various parameters in the model on phase difference,Josephson current density and voltage are analyzed graphically with the help of numerical simulations.

One of the crucial factors in achieving high performance of superconducting integrated circuits, such as analog-to-digital converters (ADCs), is sampling using a high-frequency clock source with a low cycle-to-cycle jitter. As the superconductor ADC technology matures toward more complex designs for higher dynamic range performance, the need for synchronous clocking of multiple comparators continues to grow. Since high-frequency external clock sources are expensive and make a significant contribution to the heat load of the system, a high-frequency and low-jitter on-chip clock source using long Josephson junction (LJJ) is considered the preferred long-term solution. Toward that end, we are working on improving an on-chip 110-GHz clock source based on an unshunted LJJ in annular geometry. Minimizing the additional jitter added by each fan-out of the clock signal is the effort's goal. For synchronous clocking of up to three comparators, we compare clock distribution using superconducting passive transmission lines and a new approach using novel active transmission lines. We also introduce a method of synchronizing LJJ clock source with an external stable oscillator by injection locking.

By applying a voltage to a Josephson junction, the charge in superconducting layers (S-layers) will oscillate. Wavelength of the charge oscillations in S-layers is related to external current in junction, by increasing the external current, the wavelength will decrease which cause in some currents the wavelength be incommensurate with width of junction, so the CVC shows Fiske like steps. External current throwing along junction has some components, resistive, capacitive and superconducting current, beside these currents there is a current in lateral direction of junction, (x direction). On the other hand, the emitted electromagnetic wave power in THz region is related to AC component of electric field in junction, which itself is related to charge density in S-layers, which is related to currents in the system. So we expect that features of variation of current components reflect the features of emitted THz power form junction. Here we study in detail the superconductive current in a long Josephson junction (JJ), the current voltage characteristics (CVC) of junction and emitted THz power from the system. Then we compare the results. Comparing the results we see that there is a good qualitative coincidence in features of emitted THz power and supercurrent in junction.

In the current work long Josephson junctions are being studied. Magnetic flux distribution is the physical measure for phase difference of the wave functions in the superconducting layers of the junction. The current phase relation, in most cases, can be considered as an odd strict 2π-periodic function, and hence, it can be presented in Fourier series of sinuses. It is well-known from the physical experiment that with a sufficient degree of precision, a number of physical systems are reliably described with the contribution of only first two harmonics. The adequate mathematical model for the distribution of the magnetic flux is then the double sine-Gordon equation with Neumann boundary conditions at the ends of the junction. Even in the stationary case, the boundary problem is highly nonlinear and the only tool for its comprehensive study is numerical methods. The aim of the present work is to show that in the case of zero external current, the stationary equation turns out to be a fully integrable model, derived from a variation principle with a cosine potential. In our work analytical solutions for the magnetic flux distributions described in the terms of Jacobi elliptic sinuses are derived. Analytical studies in this case serve to further numerically study of this multiparametric nonlinear boundary problem, which is so important in the applied nanophysics.In the current work long Josephson junctions are being studied. Magnetic flux distribution is the physical measure for phase difference of the wave functions in the superconducting layers of the junction. The current phase relation, in most cases, can be considered as an odd strict 2π-periodic function, and hence, it can be presented in Fourier series of sinuses. It is well-known from the physical experiment that with a sufficient degree of precision, a number of physical systems are reliably described with the contribution of only first two harmonics. The adequate mathematical model for the distribution of the magnetic flux is then the double sine-Gordon equation with Neumann boundary conditions at the ends of the junction. Even in the stationary case, the boundary problem is highly nonlinear and the only tool for its comprehensive study is numerical methods. The aim of the present work is to show that in the case of zero external current, the stationary equation turns out to be a fully integrable model,...

This report discusses the following topics on superconductivity: nonlinearities in hard superconductors such as surface impedance of a type II superconductimg half space and harmonic generation and intermodulation due to alternating transport currents; and nonlinearities in superconducting weak links such as harmonic generation by a long Josephson Junction in a superconducting slab.

Spin-torque-biased magnetic dynamics in an easy-plane ferromagnet (EPF) is theoretically studied in the presence of a weak in-plane anisotropy. While this anisotropy spoils U(1) symmetry, thereby quenching the conventional spin superfluidity, we show that the system instead realizes a close analog of a long Josephson junction (LJJ) model. The traditional magnetic-field and electric-current controls of the latter map, respectively, onto the symmetric and antisymmetric combinations of the out-of-plane spin torques applied at the ends of the magnetic strip. This suggests an alternative route towards realizations of superfluidlike transport phenomena in insulating magnetic systems. We study a spin-torque-biased phase diagram, providing an analytical solution for static multidomain phases in the EPF. We adapt an existing self-consistency method for the LJJ to develop an approximate solution for the EPF dynamics. The LJJ-EPF mapping has the potential for producing applications with superconductor-based circuit functionality at elevated temperatures. The results apply equally to antiferromagnets with suitable effective free energy in terms of the N\'eel order instead of in-plane magnetization.

on the PDFJournal of Institute of Science and Technology, 2015, 20(2): 115-121

We investigate both analytically and numerically an infinitely long Josephson junction with two π -discontinuities in the phase characterized by a jump of π in the phase difference of the junction, i.e. a 0– π –0 long Josephson junction on an infinite domain. The dynamics of the system is described by a modified one-dimensional perturbed sine-Gordon equation. We investigate an instability region of the trivial zero solution in which semifluxons are spontaneously generated. A perturbation technique is used to show that the existence of static semifluxons depends on the length of the junction, the facet length, and the applied bias current. Numerical simulations are presented accompanying our analytical results.

This paper describes current transport through а long Josephson junction with an alternating critical–current density. This alternating density can be achieved in experiments bу incorporating a magnetic layer to the weak link in а special manner. The prospects for the practical use of such structures are related to the possibility of obtaining bistable Josephson elements on their basis. Joint analysis of both current–phase relations and dynamic characteristics made it possible to optimize the operation mode for a fast superconducting memory cell based on bistable contact and to assess the energy dissipation for the read and write operations.

We investigate the coherent energy and thermal transport in a temperature-biased long Josephson tunnel junction, when a Josephson vortex, i.e., a soliton, steadily drifts driven by an electric bias current. We demonstrate that thermal transport through the junction can be controlled by the bias current, since it determines the steady-state velocity of the drifting soliton. We study the effects on thermal transport of the damping affecting the soliton dynamics. In fact, a soliton locally influences the power flowing through the junction and can cause the variation of the temperature of the device. When the soliton speed increases approaching its limiting value, i.e., the Swihart velocity, we demonstrate that the soliton-induces thermal effects significantly modify. Finally, we discuss how the appropriate material selection of the superconductors forming the junction is essential, since short quasiparticle relaxation times are required to observe fast thermal effects., Comment: 9 pages, 5 figures

New computing technologies are being sought near the end of CMOS transistor scaling, meanwhile superconducting digital, i.e., single-flux quantum (SFQ), logic allows incredibly efficient gates which are relevant to the impending transition. In this work we present a proposed reversible logic, including gate simulations and schematics under the name of Reversible Fluxon Logic (RFL). In the widest sense it is related to SFQ-logic, however it relies on (some approximately) reversible gate dynamics and promises higher efficiency than conventional SFQ which is logically irreversible. Our gates use fluxons, a type of SFQ which has topological-particle characteristics in an undamped Long Josephson junction (LJJ). The collective dynamics of the component Josephson junctions (JJs) enable ballistic fluxon motion within LJJs as well as good energy preservation of the fluxon for JJ-circuit gates. For state changes, the gates induce switching of fluxon polarity during resonant scattering at an interface between different LJJs. Related to the ballistic nature of fluxons in LJJ, the gates are powered, almost ideally, only by data fluxon momentum in stark contrast to conventionally damped logic gates which are powered continuously with a bias. At first the fundamental Identity and NOT gates are introduced. Then 2-bit gates are discussed, including the IDSN gate which actually allows low fluxon-number inputs for more than 4 input states. A digital CNOT, an important milestone for 2-bit reversible superconducting gates, is planned as a central result. It uses a store and launch gate to stop and then later route a fluxon. This use of the store and launch gate allows a clocked CNOT gate and synchronization within. The digital CNOT gate could enable high efficiency relative to conventional irreversible gates and shows the utility of the IDSN as a reversible gate primitive.

The magnetic field dependence of the Eck step voltage in long Josephson tunnel junctions (LJTJs) can be exploited for ultralow-noise magnetic sensing. The magnetic flux to be measured is captured by a non-interrupted superconducting pickup loop in which a narrow constriction, forming the LJTJ wiring layer, produces a local high density circulating currents which, in turn, induces a magnetic field large enough to set a biased LJTJ at a finite voltage proportional to the field strength (flux-flow state). On this basis, an innovative and ultralow-noise superconducting device has been realized having a highly linear voltage responsivity and an intrinsic voltage spectral density down to few pV/ $\surd $ Hz. Being that the intrinsic voltage fluctuations of these all-Niobium sensors are very low (at $T = 4.2$ K), a double transformer superconducting quantum interference device (SQUID) amplifier has been used to determine the amplitude of its voltage spectral density obtaining a value as low as 10 pV/ $\surd $ Hz. In addition, the 1/ ${f}$ noise does not affect the sensor performance down to a few hertz. The corresponding magnetic field noise, equal to 10 fT/ $\surd $ Hz, together with highly linear and broadband voltage responsivity over a wide magnetic flux range, features performances that are comparable with those of a SQUID magnetometers having a comparable flux capture area.

The kink solution in the long Josephson junction is studied. The perturbation scheme of constructing the fluxon solution in curved junction is formulated. The prediction from the perturbation scheme is compared with the prediction that follows from the numerical studies of the complete field model.

We present detailed numerical simulations of Josephson vortices in a long Josephson junction perpendicular to a thin film screen under strong dc and ac magnetic fields. By solving the sine-Gordon equation, we calculated the threshold magnetic field for penetration of fluxons as a function of frequency, and the power dissipated by oscillating fluxons as functions of the ac field amplitude and frequency. We considered the effects of superimposed ac and dc fields, and a bi-harmonic magnetic field resulting in a vortex ratchet dynamics. The results were used to evaluate the contribution of weak-linked grain boundaries to the nonlinear surface resistance of polycrystalline superconductors under strong electromagnetic fields, particularly thin film screens and resonator cavities., This paper has 23 figures and 11 pages in its pdf format. The paper has been published online and will be published in text soon in November in journal of Physica C

Modified sine-Gordon equation is solved for a soliton diode, which is a long Josephson junction in which the magnetic field of the control current reverses its direction at the center. Applying a properly directed bias current can start the spontaneous generation of soliton and antisolitons at the transition region of the diode and leads to a flux-flow voltage. The threshold bias current at which such a voltage generation starts is a function of the control current and the size of the transition region. Such phenomena can find applications in high-speed decision making electronic comparators.

The flux-flow oscillator (FFO) based on a long Josephson junction has been implemented as a broadband tunable terahertz (THz) source to open space. For this purpose, the transmitting slot antenna has been coupled to the oscillator. Additionally, an elliptical lens with a diameter of 10 mm has been matched to the antenna, forming a narrow output beam of the THz emission. Two designs for the antenna, integrated with the oscillator and developed for operation at different frequency ranges of 0.32-0.55 THz and 0.4-0.7 THz, have been investigated. The FFO has been phase locked to an external reference oscillator by utilizing a harmonic mixer. Its linewidth in the phase-locking regime is determined by the phase noise of the reference oscillator and the number of harmonics used and has been measured to be less than 0.1 MHz. A free-running FFO linewidth from about 2 MHz to several MHz, depending on the operating point, has been obtained. Output emission to open space has been measured by a superconducting integrated spectrometer located in a separate cryostat. The FFO operation as an external source with the achieved emission power and spectral characteristics has demonstrated its applicability for different tasks and purposes where tunable THz sources are required.

We investigate soliton–antisoliton interactions by numerical simulation of a nonlinear transmission line. Long Josephson junction acts as a nonlinear transmission line in which soliton solutions are obtained form Sine–Gordon equation. By proper initial and boundary conditions solitons and antisolitons are obtained inside the same junction, simulating the so called soliton diode. The role of loss in the annihilation of solitons and antisolitons injected to the transition region of the soliton diode is investigated by numerical simulation of their dynamics and phase revolution along the junction. It is shown that such loss can greatly reduce the speed of solitons and antisolitons, lowering the frequency response of the device. A so called “locally-lossy” soliton diode is proposed and it is shown that this configuration maintains annihilation process without much degradation of the device speed.

A long Josephson junction containing regions with phase shifts driven by a rapidly-oscillating ac-drive is considered. An average equation, based on multiple time scale analysis is derived, which is effectively a double sine-Gordon equation. In the presence of internal phase-shifts, it is analytically shown through the average equation that the ac-drive can influence the type of the ground state. In particular, the cases of 0-κ and 0-π-0 junctions are considered. For the 0-κ junction it is shown that an external ac-drive can reduce the critical bias current above which the junction switches to the resistive state. As for the 0-π-0 junction an ac-drive is shown to increase the critical length above which the ground state is nonuniform. Numerical comparisons are performed, and good agreement is obtained. © 2012 Elsevier B.V. All rights reserved.

© 2017 IOP Publishing Ltd.In this work we propose a hybrid device based on a long Josephson junction (JJ) coupled inductively to an external ferromagnetic (FM) layer. The long JJ in a zero-field operation mode induces a localized AC magnetic field in the FM layer and enables a synchronized magnetostatic standing wave. The magnetostatic wave induces additional dissipation for soliton propagation in the junction and also enables a phase locking (resonant soliton synchronization) at a frequency of natural ferromagnetic resonance. The later manifests itself as an additional constant voltage step on the current-voltage characteristics at the corresponding voltage. The proposed device allows to study magnetization dynamics of individual micro-scaled FM samples using just DC technique, and also it provides additional phase locking frequency in the junction, determined exclusively by characteristics of the ferromagnet.

We report on a "source-sink" algorithm which allows one to calculate time-resolved physical quantities from a general nanoelectronic quantum system (described by an arbitrary time-dependent quadratic Hamiltonian) connected to infinite electrodes. Although mathematically equivalent to the non equilibrium Green's function formalism, the approach is based on the scattering wave functions of the system. It amounts to solving a set of generalized Schr\"odinger equations which include an additional "source" term (coming from the time dependent perturbation) and an absorbing "sink" term (the electrodes). The algorithm execution time scales linearly with both system size and simulation time allowing one to simulate large systems (currently around $10^6$ degrees of freedom) and/or large times (currently around $10^5$ times the smallest time scale of the system). As an application we calculate the current-voltage characteristics of a Josephson junction for both short and long junctions, and recover the multiple Andreev reflexion (MAR) physics. We also discuss two intrinsically time-dependent situations: the relaxation time of a Josephson junction after a quench of the voltage bias, and the propagation of voltage pulses through a Josephson junction. In the case of a ballistic, long Josephson junction, we predict that a fast voltage pulse creates an oscillatory current whose frequency is controlled by the Thouless energy of the normal part. A similar effect is found for short junctions, a voltage pulse produces an oscillating current which, in the absence of electromagnetic environment, does not relax., Comment: 13 pages, 12 figures

Flux dynamics in an annular long Josephson junction is studied. Three main topics are covered. The first is chaotic flux dynamics and its prediction via Melnikov integrals. It turns out that DC current bias cannot induce chaotic flux dynamics, while AC current bias can. The existence of a common root to the Melnikov integrals is a necessary condition for the existence of chaotic flux dynamics. The second topic is on the components of the global attractor and the bifurcation in the perturbation parameter measuring the strength of loss, bias and irregularity of the junction. The global attractor can contain coexisting local attractors, e.g. a local chaotic attractor and a local regular attractor. In the infinite dimensional phase space setting, the bifurcation is very complicated. Chaotic attractors can appear and disappear in a random fashion. Three types of attractors (chaos, breather, spatially uniform and temporally periodic attractor) are identified. The third topic is ratchet effect. Ratchet effect can be achieved by a current bias field which corresponds to an asymmetric potential, in which case the flux dynamics is ever lasting chaotic. When the current bias field corresponds to a symmetric potentially, the flux dynamics is often transiently chaotic, in which case the ratchet effect disappears after sufficiently long time.

We present a new implementation for a terahertz flux-flow oscillator based on a long Josephson junction, in this concept the oscillator is integrated with the lens antenna providing the THz emission in the open space. The slot planar antenna is used for radiation and matched to the oscillator (by the input) and to the elliptical silicon lens with a width of 10 mm (by output). Three designs of antenna coupled with the oscillator by a microstrip line are developed and calculated, the obtained operating ranges are 250 – 410 GHz, 330 – 570 GHz and 420 - 700 GHz. The impedance and beam pattern of antenna designs are calculated.

The influence of curvature on kink creation during continuous phase transition in a long Josephson junction is considered. An example of a Josephson junction with constant curvature shows that the number density of kinks produced in this system is reduced in the curved regions of the junction.

We investigated the current–voltage characteristics and the critical current density (J c) of a YBa2Cu3O7−δ grain boundary (GB) long Josephson junction with respect to magnetic field (H) under different application directions. When H is rotated within the bc-plane of the film, a small peak of J c is observed when H is parallel to the c-axis, which denotes that there are some residual c-axis dislocations in the GB. When H is rotated within the ac-plane, some vortex-flow steps can be observed in the current–voltage curve at a high inclination angle of H. The vortex-flow step becomes smaller and wider with the decreasing angle between H and the c-axis. These results are a clear demonstration of vortex splitting at the GB plane in YBa2Cu3O7−δ GB long Josephson junctions.

We consider a long Josephson junction with a discontinuity point characterized by a gauge phase-shift. The system is described by a modified sine-Gordon equation. We study, in particular, the interactions between a fluxon and a fractional fluxon. A perturbation theory is developed in the small phase-shift limit to understand the characteristics of the interaction. Finally, numerical computations of the threshold bias current and the threshold velocity for a fluxon running over a fractional fluxon are presented.

A new superconducting component is studied: a cryogenic phase detector. It is based on a superconductor/ insulator/superconductor tunnel junction. The basic characteristics of the detector are studied. The optimum operation modes of the device are determined. A model of the output-signal formation is developed and tested. A model that qualitatively and quantitatively describes the relation between a cryogenic phase detector and a long Josephson junction is proposed. A cryogenic phase-lock loop system with a locking band of 27 MHz is realized.

Fluxon transmission through several impurities of different strength and type (i.e., microshorts and microresistors), placed in a long Josephson junction is investigated. Threshold pinning current on the impurities is computed as a function of the distance between them, their amplitudes and the dissipation parameter. It is shown that in the case of consequently placed microshorts or microresistors, the threshold pinning current exhibits a clear minimum as a function of the distance between the impurities. In the case of a microresistor, followed by a microshort, an opposite phenomenon is observed, namely the threshold pinning current exhibits maximum as a function of the distance between the impurities.

The magnetic field dependences of critical current are numerically constructed for a long Josephson junction with a shunt-or resistor-type microscopic inhomogeneities and compared to the critical curve of a junction with exponentially varying width. The numerical results show that it is adequate to replace the distributed inhomogeneity of a long Josephson junction by an inhomogeneity localized at one of its ends, which has certain technological advantages. It is also shown that the critical curves of junctions with exponentially varying width and inhomogeneities localized at the ends are unaffected by the mixed fluxon-antifluxon distributions of the magnetic flow. This fact may explain the improvement of the spectra of microwave radiation noted in the literature.

A zigzag boundary between a $d_{x^2-y^2}$ and an $s$-wave superconductor is believed to behave like a long Josephson junction with alternating sections of $0$ and $\pi$ symmetry. We calculate the field-dependent critical current of such a junction, using a simple model. The calculation involves discretizing the partial differential equation for the phase difference across a long $0$-$\pi$ junction. In this form, the equations describe a hybrid ladder of inductively coupled small $0$ and $\pi$ resistively and capacitively shunted Josephson junctions (RCSJ's). The calculated critical critical current density $J_c(H_a)$ is maximum at non-zero applied magnetic field $H_a$, and depends strongly on the ratio of Josephson penetration depth $\lambda_J$ to facet length $L_f$. If $\lambda_J/L_f \gg 1$ and the number of facets is large, there is a broad range of $H_a$ where $J_c(H_a)$ is less than $2\%$ of the maximum critical current density of a long $0$ junction. All of these features are in qualitative agreement with recent experiments. In the limit $\lambda_J/L_f \to \infty$, our model reduces to a previously-obtained analytical superposition result for $J_c(H_a)$. In the same limit, we also obtain an analytical expression for the effective field-dependent quality factor $Q_J(H_a)$, finding that $Q_J(H_a) \propto \sqrt{J_c(H_a)}$. We suggest that measuring the field-dependence of $Q_J(H_a)$ would provide further evidence that this RCSJ model applies to a long $0$-$\pi$ junction between a d-wave and an s-wave superconductor., Comment: 25 pages, 10 figures

We present the latest results of developing and implementing a fully superconducting integrated receiver of the submillimeter-wave range on the basis of the superconductor-insulator-superconductor tunnel junction. Characteristics of a new-generation unique device operated in the frequency range 500–600 GHz and intended for atmospheric research onboard a high-altitude balloon within the framework of the international TELIS project are given. This frequency range comprises spectral lines of many chlorine-containing compounds, the registration of which is a major problem of the environmental monitoring of the atmosphere. In the process of creating the spectrometer, we developed the technology for manufacturing Nb/AlN/NbN junctions, which made it possible to reduce the emission linewidth of the superconducting local oscillator based on the long Josephson junction and decrease the noise temperature of the receiver. At the same time, the possibility to continuously tune the local-oscillator frequency in the entire operating range was realized.

This Letter deals with a research subject in nonlinear mechanics and applied mathematics. It develops (i) accurate higher-order approximate analytical nonlinear oscillator system with negative dissipation, and (ii) analogy to long Josephson junction. Particular emphasis has been placed on the weakly damped nonlinear oscillating system with negative dissipation with respect to a transformed temporal variable derived from the weak link of the simplified Josephson junction model. Nevertheless, the system response is shown to be stable with positive dissipation with respect to the physical time at a specific location. The analysis forms an innovative extension of the harmonic balancing method commonly used in nonlinear oscillation and vibration systems such as the Duffing oscillator and van der Pol oscillator. Besides introducing coupling of linearized governing equation and harmonic balancing method, the method of averaging is also employed to obtain accurate higher-order analytical approximate solutions. Unlike the classical harmonic balance method without analytical solution, the approach not only considers energy dissipation but also presents simple linear algebraic approximate solutions. In addition, general approximate analytical expressions for the dispersion relations are also established. The presence of a small perturbed parameter is not required.

A stack of inductively coupled long Josephson junctions are modeled as a system of coupled sine-Gordon equations. One boundary of the stack is coupled electrically to a resonant cavity. With one fluxon in each Josephson junction, the inter-junction fluxon forces are repulsive. We look at a possible transition, induced by the cavity, to a bunched state.

Fluctuational dynamics in a long Josephson junction is studied via numerical solution of the sine-Gordon equation with allowance for the effect of white noise. It is shown that, in the case of a uniform distribution of the bias current and a constant density of the critical current, the lifetime of the superconducting state increases substantially with the junction length and tends toward a constant when a few Josephson lengths are attained.

We consider a long Josephson junction with alternating 0- and $\pi$-facets with different facet lengths between the 0- and the $\pi$-parts. Depending on the combinations between the 0- and the $\pi$-facet lengths, an antiferromagnetically ordered semifluxons array can be the ground state of the system. Due to the fact that in that case there are two independent ground states, an externally introduced 2$\pi$ fluxon will be splintered or fractionalized. The magnitude of the flux in the fractional fluxons is a function of the difference between the 0 and the $\pi$-facet lengths. Here, we present an analytical calculation of the flux of splintered Josephson fluxons for any combination of 0- and $\pi$-facet lengths. In the presence of an applied bias current, we show numerically that only one of the two fractional fluxons can be moved. We also consider the I–V characteristics of the ground state and the one of a 2$\pi$-fluxon in a zig-zag junction.

A YBa2Cu3O7-δ bicrystal Josephson junction was fabricated by electron beam lithography and Ar ion beam etching. The width of the junction (W) was 12 μm and the Josephson penetration length (λJ) was calculated to be 2.36 μm. As W/λJ is 5.1, this bicrystal junction is a long Josephson junction. In critical current - magnetic field characteristics, periodicity is not observed, but the maximum value and the minimum value of the critical current are different at a finite magnetic field. In the I-V curve with irradiating the rf signal, the rounded Shapiro step was observed. By applying the magnetic field, the rounded 1st Shapiro step across the zero current axis was observed.

In a well-known science fiction novel by Ray Bradbury, a butterfly in the pre-historic past is accidentally crushed by time travelers and this seemingly insignificant event radically changes history. We will call this the “Bradbury Butterfly” Effect (BBE). We know that such an effect occurs in Long Josephson Junctions as described by a time-dependent nonlinear sine-Gordon equation. This equation states that any alteration within the initial perturbation fundamentally changes the asymptotic state of the system. The actual manifestation of this effect is proven by a numerical simulation of the time-dependent sine-Gordon equation.

The FFO is a long Josephson junction in which the flow of magnetic flux quanta, and thus the oscillator frequency is controlled by the DC bias current IB, and the DC current ICL producing the magnetic field at the junction ends. We have experimentally studied the properties of the Nb/AlOx/Nb and Nb/AlN/Nb trilayer FFOs when IB also contributes to the magnetic field. A number of samples were fabricated, each containing an FFO integrated with a bias current loop designed so that the bias current can generate approximately the same positive or negative contribution to the magnetic fields at both ends of the FFO. The magnetic field is proportional to ICL + Lp*IB, where Lp is given by the loop design. We have made a study of both the DC parameters and the radiation linewidth for FFOs with different integrated loops and compared our results to that for the FFO without any loop. It was found that loops with positive contribution (Lp > 0) make the FFO linewidth smaller (although the differential resistance of the FFO on the bias current becomes larger), while loops with negative contribution (Lp < 0) result in broadening of the FFO radiation linewidth. The magnitude of this effect strongly depends on the critical current density JC ~ (Rn*S)−1. The Rn*S product was varied in our experiments from 13 to 35 Ohm*µm2. We propose a possible explanation of this effect.