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Results in Physics

From microscopic to macroscopic description of Josephson dynamics in one-dimensional arrays of weakly-coupled superconducting islands

A. Giordano, Roberto De Luca

Published Jan 1, 2015• 2 citations
Publisher

Abstract

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.

DOI
10.1016/j.rinp.2015.01.003

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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 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.
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.
There exists a large number of experimental and theoretical results supporting the picture of macroscopic qubits implemented by nanoscopic Josephson junctions of three different types -- charge qubit, flux qubit and phase qubit. The standard unified description of such systems is based on the formal quantization of the phenomenological Kirchhoff equations for the corresponding circuits. In this paper a simplified version of the BCS theory for superconductors is used to derive microscopic models for all types of small Josephson junctions. For these models the state-dependent individual tunneling of Cooper pairs couples ground pair states with excited pair states what leads to a more complicated structure of the lowest lying energy levels. In particular, the highly degenerate levels emerge, which act as probability sinks for the qubit. These models allow also for the coupling to phonons as an efficient mechanism of relaxation for all types of junctions. The alternative formulas concerning basic spectral parameters of superconducting qubits are presented and compared with the experimental data. Finally, the question whether small Josephson junctions can be treated as macroscopic quantum systems is briefly discussed.
We elaborate a theoretical description of large Josephson junctions which is based on Werthamer's microscopic tunneling theory. The model naturally incorporates coupling of electromagnetic radiation to the tunnel currents and, therefore, is particularly suitable for description of the self-coupling effect in Josephson junction. In our numerical calculations we treat the arising integro-differential equation, which describes temporal evolution of the superconducting phase difference coupled to the electromagnetic field, by the Odintsov-Semenov-Zorin algorithm. This allows us to avoid evaluation of the time integrals at each time step while taking into account all the memory effects. To validate the obtained microscopic model of large Josephson junction we focus our attention on the Josephson flux-flow oscillator. The proposed microscopic model of flux-flow oscillator does not involve the phenomenological damping parameter, rather the damping is taken into account naturally in the tunnel current amplitudes calculated at a given temperature. The theoretically calculated current-voltage characteristics is compared to our experimental results obtained for a set of fabricated flux-flow oscillators of different lengths.
Josephson junction arrays (JJAs) are engineered many-body quantum systems consisting of several superconducting islands connected by thin tunnel barriers - Josephson junctions (JJs). They are of interest both from a fundamental point of view (e.g. addressing questions of macroscopic quantum physics, providing model systems in which to study complex many-body phenomena) and from the prospect of technological applications (e.g. quantum computing, metrology, metamaterials). In this thesis we present various tools and techniques for the theoretical study of JJAs, and then put these tools to use in the modelling of two particular experiments on JJAs. In our presentation of the physics of JJAs, we emphasise phenomena unique to arrays (i.e. not seen in single junction devices such as qubits). We investigate charge transport through the the bilinear array, consisting of two linear chains of JJs, and show that it exhibits both Coulomb drag and depinning. We predict new phenomena arising from the interplay between Coulomb drag and depinning at voltages higher than have so far been experimentally reported. We then turn our attention to recent experiments in which planar JJAs were placed in 3-D microwave cavities, with vortices being injected via an externally applied magnetic eld. We calculate the impedance spectrum using linear response theory, which enables us to see when new vortices enter the system in the form of sharp changes in the resonance frequency.
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.