Dynamics of Josephson Junctions and Circuits

Konstantin K. Likharev

• Publisher: Routledge
• Year: 2022
• Print ISBN: 9782881240423
• eBook ISBN: 9781351454186
• Topic: Physical Sciences
• Subtopic: Civil Engineering

This monograph offers a detailed description of the statistics, dynamics and statics of Josephson junctions. Particular emphasis is placed on the dynamics of new circuits and analog and digital devices using single quanta of magnetic flux.

Part IV

Microwave Properties of Josephson Junctions

For everything there is a season and a time for every matter under heaven…

Eccl. 3:1 [RSV]

Now, the time has come to study more complex properties of the Josephson junctions and circuits. For this, we should first discuss the behavior of a single Josephson junction under the action of external microwave signals (Chapters 10 and 11). This would enable us to discuss the junction interactions with external microwave circuits (Chapter 12) and the microwave interactions of several junctions (Chapter 13).

CHAPTER 10

Small Microwave Signals

10.1. LINEAR EFFECTS

We will start our consideration of the microwave signal influence upon the Josephson junction with the simplest case of a small signal. According to the discussion of Section 2.4, the signal is represented by an ac current Ĩ(t) with sufficiently small amplitude I_ω (the exact conditions on the amplitude will be specified later). For the small signal, one can expand its effects to the Taylor series with respect to I_ω and consider consequently the linear (∝ I_ω), quadratic $\left(\propto I_{\omega }^2\right)$ and higher-order effects. The first-order (linear) effects are, of course, the most significant ones at I_ω → 0.

1. Linear effects in the S state

If the junction is in its S state (ϕ = ϕ_n = const, V = 0), the application of a small signal Ĩ(t) results in small oscillations $\tilde{\varphi }(t)$ of the phase around the value ϕ_n :

$\begin{array}{cc}\varphi (t)={\varphi }_n+\tilde{\varphi }(t),& \left|\tilde{\varphi }\right|\ll 1.\end{array}$

(10.1)

Consequently, the signal also produces Small voltage oscillations $\tilde{V}(t)$. To find the main (linear) part of the oscillations, one should linearize the junction equation (1.50) with respect to $\tilde{\varphi }$. For the resulting linear equation, the superposition principle is valid, and enables one to limit oneself to a sinusoidal signal^†

$\begin{array}{cc}\tilde{I}\left(t\right)=\mathrm{Im}\left[I_{\omega }{e}^{j{\Theta }_{\omega }}\right],& {\dot{\Theta }}_{\omega }=\omega ,\end{array}$

(10.2)

because the junction response to a more complex signal can be found as a sum of its responses to the sinusoidal components of the signal.

At this point, the attentive reader should note that this problem has already been solved in Section 3.2 in the course of the fluctuation analysis: the settled solut...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Dedication
• Table of Contents
• Preface
• I Introduction
• II Basic Properties of a Single Junction
• III Quantum Interference in Josephson Junction Circuits
• IV Microwave Properties of Josephson Junctions
• V Supplementary Chapters of Josephson Dynamics
• Conclusion
• Appendix: Josephson Junction Analog Simulators
• Author Index
• Subject Index