Overdamped Josephson Junctions Research Articles

Development of an SIS Mixer-Based Low-Noise Amplifier Amenable to Josephson Oscillator Pumping

We propose a low-noise and low-power-consumption microwave amplifier using superconductor-insulator-superconductor (SIS) mixers as quasi-particle frequency up- and down-converters pumped by a Josephson array oscillator in the millimeter-wave band. A proof-of-concept study of an amplifier with two W-band waveguide mixers using Nb/Al-AlOx/Nb junctions in a cascade connection demonstrated an average gain of 6 dB and a noise temperature of 12 K from nearly DC to 5 GHz. We also designed and fabricated a Josephson array oscillator at 100 GHz, which consisted of 31 overdamped Josephson junctions placed at each half-wavelength in an Nb microstrip line. The estimated output power of the oscillator was 0.2 μW at 100 GHz, which can be increased by increasing the number of the oscillator junctions. These results suggest that the proposed SIS mixer-based amplifier is suitable for on-chip integrated circuits to be implemented in large-scale multipixel SIS receivers and quantum computers.

On Germs of Constriction Curves in Model of Overdamped Josephson Junction, Dynamical Isomonodromic Foliation and Painlevé 3 Equation

On Germs of Constriction Curves in Model of Overdamped Josephson Junction, Dynamical Isomonodromic Foliation and Painlevé 3 Equation

On families of constrictions in model of overdamped Josephson junction and Painlevé 3 equation* *Supported by RSF Grant 18-41-05003.

The tunnelling effect predicted by Josephson (Nobel Prize, 1973) concerns the Josephson junction: two superconductors separated by a narrow dielectric. It states existence of a supercurrent through it and equations governing it. The overdamped Josephson junction is modelled by a family of differential equations on two-torus depending on three parameters: B (abscissa), A (ordinate), ω (frequency). We study its rotation number ρ(B, A; ω) as a function of (B, A) with fixed ω. The phase-lock areas are the level sets L r ≔ {ρ = r} with non-empty interiors; they exist for (Buchstaber, Karpov, Tertychnyi). Each L r is an infinite chain of domains going vertically to infinity and separated by points. Those separating points for which A ≠ 0 are called constrictions. We show that: (1) all the constrictions in L r lie on the axis {B = ωr}; (2) each constriction is positive: this means that some its punctured neighbourhood on the axis {B = ωr} lies in Int(L r ). These results confirm experiments by physicists (1970ths) and two mathematical conjectures. We first prove deformability of each constriction to another one, with arbitrarily small ω and the same , using equivalent description of model by linear systems of differential equations on (Buchstaber, Karpov, Tertychnyi) and studying their isomonodromic deformations described by Painlevé 3 equations. Then non-existence of ghost constrictions (i.e., constrictions either with , or of non-positive type) with a given ℓ for small ω is proved by slow-fast methods.

Overdamped phase diffusion in hBN encapsulated graphene Josephson junctions

We investigate the zero-bias behavior of Josephson junctions made of encapsulated graphene boron nitride heterostructures in the long ballistic junction regime. For temperatures down to $2.7$ K, the junctions appear nonhysteretic with respect to the switching and retrapping currents ${I}_{C}$ and ${I}_{R}$. A small nonzero resistance is observed even around zero-bias current and scales with temperature as dictated by the phase diffusion mechanism. By varying the graphene carrier concentration we are able to confirm that the observed phase diffusion mechanism follows the trend for an overdamped Josephson junction. This is in contrast with the majority of graphene-based junctions which are underdamped and shorted by the environment at high frequencies.

Driving a low critical current Josephson junction array with a mode-locked laser

We report proof-of-concept experiments on an optically driven Josephson voltage standard based on a mode-locked laser (MLL), a time-division multiplexer, and a cryogenic ultrafast photodiode driving an overdamped Josephson junction array (JJA). Our optical pulse pattern generator (PPG) concept builds on the capability of MLLs to produce trains of picosecond-wide optical pulses with little amplitude and temporal spread. Our present setup enables multiplication of the original 2.3 GHz pulse repetition frequency by a factor of 8. A commercial photodiode converts the optical pulses into about 25 ps wide electrical pulses in liquid helium several cm from the JJA. Using a custom-made MLL, we can drive a JJA with a low critical current of 360 μA at multiple Shapiro steps. We have performed experiments with pulse pairs whose time interval can be set freely without distorting the shapes of individual pulses. Experimental results are in qualitative agreement with theoretical simulations, and they demonstrate, e.g., crossover in the Shapiro step pattern when the time interval between the pulses is approximately equal to the inverse of the characteristic frequency of the JJA. However, there are quantitative discrepancies, which motivate an improved integration of photodiodes and JJAs to improve both the understanding and fidelity of Josephson Arbitrary Waveform Synthesizers. Considering future quantum technologies in a wider perspective, our optical approach is a potential enabler for fast and energy-efficient pulse drive without an expensive high-bandwidth electrical PPG and without high-bandwidth electrical cables that yield too high thermal conductance between cryogenic and room temperatures.

On families of constrictions in the model of an overdamped Josephson junction

On families of constrictions in the model of an overdamped Josephson junction

On families of constrictions in the model of an overdamped Josephson junction

On families of constrictions in the model of an overdamped Josephson junction

Fabrication and characterization of YBa<sub>2</sub>Cu<sub>3</sub>O<sub>7–</sub><inline-formula><tex-math id="Z-20210129174424">\begin{document}$_{ \delta}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20201291_Z-20210129174424.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20201291_Z-20210129174424.png"/></alternatives></inline-formula> step-edge Josephson junctions on MgO substrate for high-temperature superconducting quantum interference devices

The YBa<sub>2</sub>Cu<sub>3</sub>O<sub>7–<i>δ</i></sub> (YBCO) step-edge Josephson junction on MgO substrate has recently been shown to have important applications in making advanced high-transition temperature (high-<i>T</i><sub>C</sub>) superconducting devices such as high-sensitivity superconducting quantum interference device (SQUID), superconducting quantum interference filter, and THz detector. In this paper, we investigate the fabrication and transport properties of YBCO step-edge junction on MgO substrate. By optimizing the two-stage ion beam etching process, steps on MgO (100) substrates are prepared with an edge angle <i>θ</i> of about 34°. The YBCO step-edge junctions are then fabricated by growing the YBCO thin films with a pulsed laser deposition technique and subsequent traditional photolithography. The resistive transition of the junction shows typical foot structure which is well described by the Ambegaokar-Halperin theory of thermally-activated phase slippage for overdamped Josephson junctions. The voltage-current curves with temperature dropping down to 77 K exhibit resistively shunted junction behavior, and the Josephson critical current density <i>J</i><sub>C</sub> is shown to follow the <inline-formula><tex-math id="Z-20210127094830-1">\begin{document}$(T_{\rm C}-T)^2$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20201291_Z-20210127094830-1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20201291_Z-20210127094830-1.png"/></alternatives></inline-formula> dependence. At 77 K, the <i>J</i><sub>C</sub> of the junction reaches 1.4 × 10<sup>5</sup> A/cm<sup>2</sup>, significantly higher than the range of 10<sup>3</sup>–10<sup>4</sup> A/cm<sup>2</sup> as presented by other investigators for YBCO step-edge junctions on MgO substrate with comparable <i>θ</i> of 35°–45°. This indicates a rather strong Josephson coupling of the junction, and by invoking the results of YBCO bicrystal junctions showing similar values of <i>J</i><sub>C</sub>, it is tentatively proposed that the presently fabricated junction might be described as an S-s′-S junction with s′ denoting the superconducting region of depressed <i>T</i><sub>C</sub> in the vicinity of the step edge or as an S-N-S junction with N denoting a very thin non-superconducting layer. By incorporating the MgO-based YBCO step-edge junction, high-<i>T</i><sub>C</sub> radio frequency (RF) SQUID is made. The device shows decent voltage-flux curve and magnetic flux sensitivity of 250 <inline-formula><tex-math id="Z-20210128093740-1">\begin{document}$ \text{μ}\Phi_0/{\rm Hz}^{1/2} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20201291_Z-20210128093740-1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20201291_Z-20210128093740-1.png"/></alternatives></inline-formula> at 1 kHz and 77 K, comparable to the values reported in the literature. To further improve the RF SQUID performance, efforts could be devoted to optimizing the junction parameters such as the junction <i>J</i><sub>C</sub>. By using the YBCO step-edge junction on MgO substrate, high-<i>T</i><sub>C</sub> direct current SQUID could also be developed, as reported recently by other investigators, to demonstrate the potential of MgO-based step-edge junction in making such a kind of device with superior magnetic flux sensitivity.

Group algebras acting on the space of solutions of a special double confluent Heun equation

We study properties of the space $$\boldsymbol{\Omega}$$ of solutions of a special double confluent Heun equation closely related to the model of a overdamped Josephson junction. We describe operators acting on $$\boldsymbol{\Omega}$$ and relations in the algebra $$\mathcal{A}$$ generated by them over the real number field. The structure of $$\mathcal{A}$$ depends on parameters. We give conditions under which $$\mathcal{A}$$ is isomorphic to a group algebra and describe two corresponding group structures.

Extending voltage range to 10 V rms in AC–DC difference measurements with AC programmable Josephson voltage standard

The larger-scale programmable Josephson voltage standard (PJVS) chip composed of 524 288 niobium nitride-based overdamped Josephson junctions enables us to generate stepwise-approximated AC waveforms with a root-mean-square (rms) amplitude of up to 10 V. In this study, we have extended the voltage range of the differential sampling system based on the AC-PJVS to an rms value of 10 V with the larger-scale PJVS chip. The sampling digital voltmeter (DVM) that is employed to measure differential voltages is modified to use a 10 MHz reference clock instead of an internal clock. This modification substantially improves the uncertainty of AC–DC difference measurements at low frequencies, which is caused by the internal clock error of the sampling DVM, by preventing the incoherent sampling of the differential voltages. The expanded uncertainties (k = 2) of the AC–DC difference measurements are reduced from 7.0 μV/V to 1.8 μV/V for a frequency of 62.5 Hz and from 8.0 μV/V to 4.4 μV/V for a frequency of 1 Hz. The improvements in the uncertainty are more significant compared with a conventional method at low frequencies, where the accuracy of a thermal voltage converter is reduced.

On Spectral Curves and Complexified Boundaries of the Phase-Lock Areas in a Model of Josephson Junction

The paper deals with a three-parameter family of special double confluent Heun equations that was introduced and studied by V. M. Buchstaber and S. I. Tertychnyi as an equivalent presentation of a model of overdamped Josephson junction in superconductivity. The parameters are \(l,\lambda ,\mu \in \mathbb {R}\). Buchstaber and Tertychnyi described those parameter values, for which the corresponding equation has a polynomial solution. They have shown that for μ ≠ 0 this happens exactly when \(l\in \mathbb {N}\) and the parameters (λ, μ) lie on an algebraic curve \({\Gamma }_{l}\subset \mathbb {C}^{2}_{(\lambda ,\mu )}\) called the l-spectral curve and defined as zero locus of determinant of a remarkable three-diagonal l × l-matrix. They studied the real part of the spectral curve and obtained important results with applications to model of Josephson junction, which is a family of dynamical systems on 2-torus depending on real parameters (B, A; ω); the parameter ω, called the frequency, is fixed. One of main problems on the above-mentioned model is to study the geometry of boundaries of its phase-lock areas in \(\mathbb {R}^{2}_{(B,A)}\) and their evolution, as ω decreases to 0. An approach to this problem suggested in the present paper is to study the complexified boundaries. We prove irreducibility of the complex spectral curve Γl for every \(l\in \mathbb {N}\). We also calculate its genus for \(l\leqslant 20\) and present a conjecture on general genus formula. We apply the irreducibility result to the complexified boundaries of the phase-lock areas of model of Josephson junction. The family of real boundaries taken for all ω > 0 yields a countable union of two-dimensional analytic surfaces in \(\mathbb {R}^{3}_{(B,A,\omega ^{-1})}\). We show that, unexpectedly, its complexification is a complex analytic subset consisting of just four two-dimensional irreducible components, and we describe them. This is done by using the representation of some special points of the boundaries (the so-called generalized simple intersections) as points of the real spectral curves and the above irreducibility result. We also prove that the spectral curve has no real ovals. We present a Monotonicity Conjecture on the evolution of the phase-lock area portraits, as ω decreases, and a partial positive result towards its confirmation.

Solution Space Monodromy of a Special Double Confluent Heun Equation and Its Applications

We consider three linear operators determining automorphisms of the solution space of a special double confluent Heun equation of positive integer order (L-operators). We propose a new method for describing properties of the solution space of this equation based on using eigenfunctions of one of the L-operators, called the universal L-operator. We construct composition laws for L-operators and establish their relation to the monodromy transformation of the solution space of the special double confluent Heun equation. We find four functionals quadratic in eigenfunctions of the universal automorphism; they have a property with respect to the considered equation analogous to the property of the first integral. Based on them, we construct matrix representations of the L-operators and also the monodromy operator. We give a method for extending solutions of the special double confluent Heun equation from the subset Re z > 0 of a complex plane to a maximum domain on which the solution exists. As an example of its application to the RSJ model theory of overdamped Josephson junctions, we give the explicit form of the transformation of the phase difference function induced by the monodromy of the solution space of the special double confluent Heun equation and propose a way to continue this function from a half-period interval to any given interval in the domain of the function using only algebraic transformations.

Stationary charge imbalance effect in a system of coupled Josephson junctions

We investigate the stationary charge imbalance effect in the system of coupled overdamped Josephson junctions. We show that the stationary charge imbalance in the resistive state brings to a decrease in the Josephson frequency in the junctions of the stack. The formed difference in frequencies leads to a nonuniform switch of junctions along the stack to the Shapiro step regime in the presence of external electromagnetic radiation. On the other hand, such nonuniform switching leads to the appearance of voltage spikes on the IV-characteristic of the stack. We show that the stationary charge imbalance is responsible for a slope in the Shapiro step due to the difference of the nonequilibrium potentials at the edges of the step. The obtained results have been compared with the voltage bias coupled Josephson junctions and experimental results.

On Constrictions of Phase-Lock Areas in Model of Overdamped Josephson Effect and Transition Matrix of the Double-Confluent Heun Equation

In 1973, B. Josephson received a Nobel Prize for discovering a new fundamental effect concerning a Josephson junction,—a system of two superconductors separated by a very narrow dielectric: there could exist a supercurrent tunneling through this junction. We will discuss the model of the overdamped Josephson junction, which is given by a family of first-order nonlinear ordinary differential equations on two-torus depending on three parameters: a fixed parameter ω (the frequency); a pair of variable parameters (B, A) that are called respectively the abscissa, and the ordinate. It is important to study the rotation number of the system as a function ρ = ρ(B, A) and to describe the phase-lock areas: its level sets Lr = {ρ = r} with non-empty interiors. They were studied by V.M.Buchstaber, O.V.Karpov, S.I.Tertychnyi, who observed in their joint paper in 2010 that the phase-lock areas exist only for integer values of the rotation number. It is known that each phase-lock area is a garland of infinitely many bounded domains going to infinity in the vertical direction; each two subsequent domains are separated by one point, which is called constriction (provided that it does not lie in the abscissa axis). Those points of intersection of the boundary ∂Lr of the phase-lock area Lr with the line Λr = {B = rω} (which is called its axis) that are not constrictions are called simple intersections. It is known that our family of dynamical systems is related to appropriate family of double–confluent Heun equations with the same parameters via Buchtaber–Tertychnyi construction. Simple intersections correspond to some of those parameter values for which the corresponding “conjugate” double-confluent Heun equation has a polynomial solution (follows from results of a joint paper of V.M. Buchstaber and S.I.Tertychnyi and a joint paper of V.M. Buchstaber and the author). There is a conjecture stating that all the constrictions of every phase-lock areaLrlie in its axis Λr. This conjecture was studied and partially proved in a joint paper of the author with V.A.Kleptsyn, D.A.Filimonov, and I.V.Schurov. Another conjecture states that for any two subsequent constrictions in Lr with positive ordinates, the interval between them also lies in Lr. In this paper, we present new results partially confirming both conjectures. The main result states that for every $r \in \mathbb {Z}\setminus \{0\}$ the phase-lock area Lr contains the infinite interval of the axis Λr issued upwards from the simple intersection in ∂Lr ∩Λr with the biggest possible ordinate. The proof is done by studying the complexification of the system under question, which is the projectivization of a family of systems of second-order linear equations with two irregular non-resonant singular points at zero and at infinity. We obtain new results on the transition matrix between appropriate canonical solution bases of the linear system; on its behavior as a function of parameters. A key result, which implies the main result of the paper, states that the off-diagonal terms of the transition matrix are both non-zero at each constriction. We also show that their ratio is real at the constrictions. We reduce the above conjectures on constrictions to the conjecture on negativity of the ratio of the latter off-diagonal terms at each constriction.

Superconducting and Dissipative Characteristics of Overdamped SNIS Josephson Junctions for Sensing Applications

Josephson-junction devices such as suitably designed nanoSQUIDs can provide a sensing element for micro and nanoelectronics. In this paper, the properties of such device must be tailored in order to have an optimal response. In particular, it relates its temperature stability, and a comparatively weak dependence of the critical supercurrent I c on temperature T in the working range. To realize it, a concave upward I c-versus- T curve is required. The aim of this paper was to study conditions when such temperature dependence can be realized in four-layered S/N-I-S Josephson junctions, mantaining the overdamped behavior. We show how the shape of the I c-versus- T dependence can provide important information about the strength of the S/N proximity coupling.

On the Synthesis of Stepwise Quantum Waves Using a SNIS Programmable Josephson Array in a Cryocooler

This paper presents recent advances in the application of binary-divided 1 V array, consisting of 8192 intrinsically shunted SNIS overdamped Josephson junctions (JJs), for the synthesis of stepwise waves with quantum accuracy. The maximum output voltage is ensured by opportunely driving the subsections of the SNIS array by means of three states bias-current setpoints to the Shapiro steps n = 0, ±1 or n = ±2, respectively. Reconfigurable digital modular electronics has been designed to bias individually each of the 13 subsections of the SNIS array. A two-stage closed cycle refrigerator equipped with LF and RF electrical lines is employed for cooling-down the SNIS array for temperatures ranging from 3.6 K to above 7 K. Stepwise sine waves with rms amplitude ranging from 1 to 2 V using the first ( n = 1) and second ( n = 2) Shapiro steps, different temperatures and bias-current setpoints have been synthesized up to the kHz range. The synthesized waves have been recorded and analyzed by a high-precision differential sampling system. We report the results of the first characterizations carried out with the new multibit current source and an improved version of the sample holder designed to optimize the heat dissipation of the SNIS array for operation in cryocooler setups.

Analytical derivation of DC SQUID response

We consider voltage and current response formation in DC superconducting quantum interference device (SQUID) with overdamped Josephson junctions in resistive and superconducting state in the context of a resistively shunted junction (RSJ) model. For simplicity we neglect the junction capacitance and the noise effect. Explicit expressions for the responses in resistive state were obtained for a SQUID which is symmetrical with respect to bias current injection point. Normalized SQUID inductance (where Ic is the critical current of Josephson junction, L is the SQUID inductance, e is the electron charge and ℏ is the Planck constant) was assumed to be within the range l ≤ 1, subsequently expanded up to using two fitting parameters. SQUID current response in the superconducting state was considered for arbitrary value of the inductance. The impact of small technological spread of parameters relevant to low-temperature superconductor (LTS) technology was studied, using a generalization of the developed analytical approach, for the case of a small difference of critical currents and shunt resistances of the Josephson junctions, and inequality of SQUID inductive shoulders for both resistive and superconducting states. Comparison with numerical calculation results shows that developed analytical expressions can be used in practical LTS SQUIDs and SQUID-based circuits design, e.g. large serial SQIF, drastically decreasing the time of simulation.

Automorphisms of the solution spaces of special double-confluent Heun equations

Two new linear operators determining automorphisms of the solution space of a special double-confluent Heun equation in the general case are obtained. This equation has two singular points, both of which are irregular. The obtained result is applied to solve the nonlinear equation of the resistively shunted junction model for an overdamped Josephson junction in superconductors. The new operators are explicitly expressed in terms of structural polynomials, for which recursive computational algorithms are constructed. Two functional equations for the solutions of the special double-confluent Heun equation are found.

Mechanical analog of an over-damped Josephson junction

An over-damped pendulum can be adopted as a mechanical analogue of an over-damped Josephson junction (JJ). The basic equations leading to the driving torque versus the time average of the angular frequency are studied. The mechanical analogue can be used to provide additional insight into the current–voltage characteristics of over-damped JJs.

Calibration System for Zener Voltage Standards Using a 10 V Programmable Josephson Voltage Standard at NMIJ

We have developed a calibration system for the Zener voltage standards using a programmable Josephson voltage standard (PJVS) for liquid-helium consumption reduction and improvement in the calibration performance. The PJVS array is composed of 524 288 niobium nitride (NbN)-based overdamped Josephson junctions and is cooled by a 10-K compact cooler. To verify the validity of the system, indirect and direct comparisons of the PJVS system and our conventional Josephson voltage standard system have been carried out. As a result, an agreement within a relative expanded uncertainty of $8.4\times 10^{-10}$ was obtained for a nominal voltage of 10 V. We also discuss the uncertainty sources in the calibration of the Zener voltage standards using our PJVS system.