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arXiv:2309.15652 · IEEE Transactions on Applied Superconductivity

Influence of Noise on Josephson Junctions Dynamics With a BCS Theory-Based Model

Lucas Iwanikow, Pascal Febvre

Published Nov 28, 2023• 1 citation
arXivHTMLPDF

Abstract

We developed a weak-linked Josephson junction time-domain simulation tool based on the Bardeen-Cooper-Schrieffer (BCS) theory to account for the electrodynamics of Cooper pairs and quasiparticles in the presence of thermal noise. The model, based on Werthamer and Harris formalisms, allows us to describe the behavior of Josephson junctions for various current and/or voltage input waveforms, analog or digital, and for any operating temperature below the critical temperature of the superconducting materials. We show a good agreement between experimental and simulated IV curves of Josephson junctions, as well as a relative increase of the grey zone width of an SFQ balanced comparator between 4 and 20% due to the presence of quasiparticles, for McCumber parameters comprised between 0.1 and 1, respectively.

DOI
10.1109/tasc.2023.3334199

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We developed a weak-linked Josephson junction time-domain simulation tool based on the Bardeen-Cooper-Schrieffer (BCS) theory to account for the electrodynamics of Cooper pairs and quasiparticles in the presence of thermal noise. The model, based on Werthamer and Harris formalisms, allows us to describe the behavior of Josephson junctions for various current and/or voltage input waveforms, analog or digital, and for any operating temperature below the critical temperature of the superconducting materials. We show a good agreement between experimental and simulated IV curves of Josephson junctions, as well as a relative increase of the grey zone width of an SFQ balanced comparator between 4 and 20% due to the presence of quasiparticles, for McCumber parameters comprised between 0.1 and 1, respectively.
We use the Schrieffer-Wolff transformation (SWT) to analyze Josephson junctions between superconducting leads described by the charge-conserving BCS theory. Starting from the single-electron tunneling terms, we directly recover the conventional effective Hamiltonian, −EJcos⁡φ^-E_J\cos\hatφ−EJ​cosφ^​, with an operator-valued phase bias φ^\hatφφ^​. The SWT approach has the advantage that it can be systematically extended to more complex scenarios. We show that if a Bogoliubov quasiparticle is present its motion couples to that of Cooper pairs, introducing correlated dynamics that reshape the energy spectrum of the junction. Furthermore, higher-order terms in the SWT naturally describe Josephson harmonics, whose amplitudes are directly related to the microscopic properties of the superconducting leads and the junction. We derive expressions that could facilitate tuning the ratio between the different harmonics in a controlled way.
We investigate the noise-affected I-V curves of small-area Josephson junctions through experiment, simulation, and theory. In particular, we consider I-V curves in which two different states of finite voltage coexist at the same dc bias: a high-voltage state that corresponds to the usual quasiparticle branch and a low-voltage state that is characterized by thermally activated phase diffusion. The observed hysteresis between the phase-diffusion and quasiparticle branches cannot be explained within the context of the simple resistively and capacitively shunted junction (RCSJ) model but is explained by extended models in which the damping increases with frequency. Frequency-dependent damping is shown to produce a qualitative change in the attractors of the noise-free system which allows the two voltage states to be simultaneously stable. This picture is confirmed by Monte Carlo simulations which accurately reproduce the experimental I-V curves of two different samples over a wide range of temperatures. In addition we develop analytic expressions for three key parameters of the I-V curve of junctions displaying hysteresis between the phase-diffusion and quasiparticle branches: the initial slope of the phase-diffusion branch, the bias level at which the junction switches from the phase-diffusion branch to the quasiparticle branch, and the bias level at which it returns to the phase-diffusion branch.
Superconducting quantum circuits (SQCs) are being explored as model systems for scalable quantum computing architectures. Josephson junctions are extensively used in superconducting quantum interference devices (SQUIDs) and in persistent-current qubit systems. Noise excitations, however, have a critical influence on their dynamics. Thus, the primary focus of this research was to investigate the effects of thermal ac-tivation on the superconducting properties of Josephson junctions. Specifically, ther-mal noise tends to result in a range of switching currents, values less than the critical current at which a junction switches from the superconducting to the normal state. First, a general review of superconductivity concepts is given, including a treatment of the Josephson phenomena. Next, I describe some of my work on characterizing the current-voltage traces of Josephson junctions tested at 4 K with a Multi-Chip Probe (MCP). Then, I describe thermal activation theory and examine the equations useful for modeling switching current distributions. The Josephson junctions of a SQUID with a ramped bias current were tested for numerous temperatures T < 4.5 K (and
In the design of electronic applications (for example, rapid single flux quantum (RSFQ) circuits), the properties of Josephson junctions (JJs) provided by technology are the major requirement. The characteristic voltage IcRn an dt he dynamic parameter βc are non-negligibly important, bu tt he external shunted junctions are only completely described by including the parasitic inductance between the junction area and the shunt resistor. The first publications of the RSFQ circuit technique were closely followed by many studies on the dynamic influence of this inductance. It has been pointed out that no practical value of this parameter has to be taken into account in the circuit design and optimization process. The operation of RSFQ logic circuits made of high-temperature superconductors (HTS) are strongly influenced by thermal noise. Our paper contains the determination of the digital bit-error rate for all different types of JJs by using a Fokker–Planck equation for describing the influence of thermal fluctuations. Ou ra nalysis shows that the parasitic inductance cannot be neglected in terms of noise. The intrinsic shunted junctions like high current density-, SINIS- and all kinds of HTS-JJs provide the best noise immunity for a fixed temperature.
The full quantum theory of mixing has been widely developed by Tucker. From theoretical analysis it allows one to predict the optimal performances of a quasiparticle superconducting insulator-superconducting junction mixer from knowing the junction current-voltage characteristics. This theory, however, considers only quasiparticle processes and neglects pair effects. At frequencies close to the superconductor energy gap, the Josephson currents become unavoidable and are reputed to be the cause of strong instabilities and noise. We present here a theoretical calculation of the Josephson noise and losses, which is consistent with experimental results. The width of the Josephson noise peaks is discussed.
The use of high-T/sub c/ Josephson junctions for digital applications requires a careful study of the influence of thermal noise on circuit performance. We investigated a balanced current comparator, which contains a basic component of all RSFQ circuits: two overdamped Josephson junctions connected in series. The dependence of the dc-voltage across the Josephson junctions on signal current was measured to determine the threshold uncertainty of switching of the comparator. The experimental data obtained at different sampling pulse frequencies, in a temperature range from 10 to 50 K, were compared with a theoretical model, taking into account the influence of thermal noise as well as of sampling pulse frequency.