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European Journal of Physics

Feynman's and Ohta's models of a Josephson junction

Roberto De Luca

Published Sep 6, 2012• 27 citations
Publisher

Abstract

The Josephson equations are derived by means of the weakly coupled two-level quantum system model given by Feynman. Adopting a simplified version of Ohta's model, starting from Feynman's model, the strict voltage–frequency Josephson relation is derived. The contribution of Ohta's approach to the comprehension of the additional term given by the classical external system is recognized. Extension of the model to double-barrier Josephson junctions and multilayered systems is finally briefly commented on.

DOI
10.1088/0143-0807/33/6/1547

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The Josephson equations are derived by means of the weakly coupled two-level quantum system model given by Feynman. Adopting a simplified version of Ohta's model, starting from Feynman's model, the strict voltage–frequency Josephson relation is derived. The contribution of Ohta's approach to the comprehension of the additional term given by the classical external system is recognized. Extension of the model to double-barrier Josephson junctions and multilayered systems is finally briefly commented on.
By starting from a microscopic quantum mechanical description of Josephson dynamics of a one-dimensional array of N coupled superconductors, we obtain a set of linear differential equations for the system order parameter and for additional macroscopic physical quantities. With opportune considerations, we adapt this description to two coupled superconductors, obtaining the celebrated Feynman model for Josephson junctions. These results confirm the correspondence between the microscopic picture and the semi-classical Ohta’s model adopted in describing the superconducting phase dynamics in multi-barrier Josephson junctions.
The Feynman model is a famous theoretical model establishing the Josephson equations. Feynman et al. based on the Schrodinger equations describing the two states system of superconductor junction, have derived the differential equations of the amplitudes and phases of the two wave functions versus time, and solved them with a slightly rough method, and established the Josephson equation. It is worth considering carefully that mathematical treatment in Feynman model was not exact. In this paper we have solved the differential equations in Feynman model exactly, the results we have gotten show that A. C. Josephson equations can be established only under certain conditions..
Double-barrier Josephson junctions, with outer electrodes of superconducting Nb and a central thin layer of non-superconducting Al, are being developed as self-shunted junctions for high-frequency superconducting integrated circuits. This paper presents a simplified phenomenological model of the behaviour of such junctions (often called SINIS, for superconductor-insulator-normal-insulator-superconductor) based on the resistively shunted junction model, using Josephson coupling between the layers and the time-dependent Ginzburg-Landau equation to describe the dynamics of the weak induced energy gap in the Al layer. This analysis shows how non-hysteretic I - V characteristics can be obtained but suggests some limitations in their high-frequency performance. Predictions for critical current generally agree with prior experiments. The technological implications of this model are discussed.
The strong-coupling version of the BCS theory for superconductors is used to derive microscopic models for all types of small Josephson junctions--charge qubit, flux qubit and phase qubit. Applied to Josephson qubits it yields a more complicated structure of the lowest-lying energy levels than that obtained from phenomenological models based on quantization of the Kirchhoff equations. In particular, highly degenerate levels emerge, which act as probability sinks for the qubit. The alternative formulae concerning spectra of superconducting qubits are presented and compared with the experimental data. In contrast to the existing theories those formulae contain microscopic parameters of the model. In particular, for the first time, the density of Cooper pairs at zero temperature is estimated for an Al-based flux qubit. Finally, the question whether small Josephson junctions can be treated as macroscopic quantum systems is briefly discussed.
The quantized Josephson junction equation that underpins the behaviour of charge qubits and other tunnel devices is usually derived through canonical quantization of the classical macroscopic Josephson relations. However, this approach may neglect effects due to the fact that the charge qubit consists of a superconducting island of finite size connected to a large superconductor. We show that the well-known quantized Josephson equation can be derived directly and simply from a microscopic many-body Hamiltonian. By choosing the appropriate strong-coupling limit we produce a highly simplified Hamiltonian that nevertheless allows us to go beyond the mean-field limit and predict further finite-size terms in addition to the basic equation.
Based on the entangled state representation and Feynman's idea that ‘electron pairs are bosons, …, a bound pair acts as a Bose particle’, we present a cooper-pair number-phase quantization scheme for the mesoscopic LC circuit including two coupled Josephson junctions (JJs). Then we use the Heisenberg equation of motion to obtain the modified current equation and voltage equation across each JJ, as well as the equation for realizing quantum control. Besides, we investigate how the phases in two JJs are affected mutually through the capacitor coupling and the coupled JJs.