Nonlinear dynamics of two-dimensional Josephson junction arrays

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Physics, 1996.Includes bibliographical references (p. 187-196).by Mauricio Barahona García.Ph.D

Localised excitations in long Josephson junctions with phase-shifts with time-varying drive

In this project, we consider a variety of ac-driven, inhomogeneous sine-Gordon equations describing an infinitely long Josephson junctions with phase shifts, driven by a microwave field. First, the case of a small driving amplitude and a driving frequency close to the natural (defect) frequency is considered. We construct a perturbative expansion for the breathing mode to obtain equations for the slow time evolution of the oscillation amplitude. We show that, in the absence of an ac-drive, a breathing mode oscillation decays with a rate of at least \mathcal{O}(t^{-1/4}) and \mathcal{O}(t^{-1/2}) for 0-\pi-0 and 0-\kappa junctions, respectively. Multiple scale expansions are used to determine whether, e.g., an external drive can excite the defect mode of a junction (a breathing mode), to switch the junction into a resistive state. Next, we extend the study to the case of large oscillation amplitude with a high frequency drive. Considering the external driving force to be rapidly oscillating, we apply an asymptotic procedure to derive an averaged nonlinear equation, which describes the slowly varying dynamics of the sine-Gordon field. We discuss the threshold distance of 0-\pi-0 junctions and the critical bias current in $0-\kappa$ junctions in the presence of ac drives. Then, we consider a spatially inhomogeneous sine-Gordon equation with two regions in which there is a \pi-phase shift, and a time periodic drive, modelling 0-\pi-0-\pi-0 long Josephson junctions. We discuss the interactions of symmetric and antisymmetric defect modes in long Josephson junctions. We show that the amplitude of the modes decay in time. In particular, exciting the two modes at the same time will increase the decay rate. The decay is due to the energy transfer from the discrete to the continuous spectrum. For a small drive amplitude, there is an energy balance between the energy input given by the external drive and the energy output due to radiative damping experience by the coupled mode. Finally, we consider spatially inhomogeneous coupled sine-Gordon equations with a time periodic drive, modelling stacked long Josephson junctions with a phase shift. We derive coupled amplitude equations considering weak coupling and strong coupling in the absence of ac-drive. Next, by considering the strong coupling with time periodic drive, we expect that the amplitude of oscillation tends to constant for long times

Localised excitations in long Josephson junctions with phase-shifts with time-varying drive

In this project, we consider a variety of ac-driven, inhomogeneous sine-Gordon equations describing an infinitely long Josephson junctions with phase shifts, driven by a microwave field. First, the case of a small driving amplitude and a driving frequency close to the natural (defect) frequency is considered. We construct a perturbative expansion for the breathing mode to obtain equations for the slow time evolution of the oscillation amplitude. We show that, in the absence of an ac-drive, a breathing mode oscillation decays with a rate of at least \mathcal{O}(t^{-1/4}) and \mathcal{O}(t^{-1/2}) for 0-\pi-0 and 0-\kappa junctions, respectively. Multiple scale expansions are used to determine whether, e.g., an external drive can excite the defect mode of a junction (a breathing mode), to switch the junction into a resistive state. Next, we extend the study to the case of large oscillation amplitude with a high frequency drive. Considering the external driving force to be rapidly oscillating, we apply an asymptotic procedure to derive an averaged nonlinear equation, which describes the slowly varying dynamics of the sine-Gordon field. We discuss the threshold distance of 0-\pi-0 junctions and the critical bias current in $0-\kappa$ junctions in the presence of ac drives. Then, we consider a spatially inhomogeneous sine-Gordon equation with two regions in which there is a \pi-phase shift, and a time periodic drive, modelling 0-\pi-0-\pi-0 long Josephson junctions. We discuss the interactions of symmetric and antisymmetric defect modes in long Josephson junctions. We show that the amplitude of the modes decay in time. In particular, exciting the two modes at the same time will increase the decay rate. The decay is due to the energy transfer from the discrete to the continuous spectrum. For a small drive amplitude, there is an energy balance between the energy input given by the external drive and the energy output due to radiative damping experience by the coupled mode. Finally, we consider spatially inhomogeneous coupled sine-Gordon equations with a time periodic drive, modelling stacked long Josephson junctions with a phase shift. We derive coupled amplitude equations considering weak coupling and strong coupling in the absence of ac-drive. Next, by considering the strong coupling with time periodic drive, we expect that the amplitude of oscillation tends to constant for long times

Parametric Interaction in Josephson Junction Circuits and Transmission Lines

This research investigates the realization of parametric amplification in superconducting circuits and structures where nonlinearity is provided by Josephson junction (JJ) elements. We aim to develop a systematic analysis over JJ-based devices toward design of novel traveling-wave Josephson parametric amplifiers (TW-JPA). Chapters of this thesis fall into three categories: lumped JPA, superconducting periodic structures and discrete Josephson transmission lines (DJTL). The unbiased Josephson junction (JJ) is a nonlinear element suitable for parametric amplification through a four-photon process. Two circuit topologies are introduced to capture the unique property of the JJ in order to efficiently mix signal, pump and idler signals for the purpose of signal amplification. Closed-form expressions are derived for gain characteristics, bandwidth determination, noise properties and impedance for this kind of parametric power amplifier. The concept of negative resistance in the gain formulation is observed. A design process is also introduced to find the regimes of operation for gain achievement. Two regimes of operation, oscillation and amplification, are highlighted and distinguished in the result section. Optimization of the circuits to enhance the bandwidth is also carried out. Moving toward TW-JPA, the second part is devoted to modelling the linear wave propagation in a periodic superconducting structure. We derive closed-form equations for dispersion and s-parameters of infinite and finite periodic structures, respectively. Band gap formation is highlighted and its potential applications in the design of passive filters and resonators are discussed. The superconducting structures are fabricated using YBCO and measured, illustrating a good correlation with the numerical results. A novel superconducting Transmission Line (TL), which is periodically loaded by Josephson junctions (JJ) and assisted by open stubs, is proposed as a platform to realize a traveling-wave parametric device. Using the TL model, this structure is modeled by a system of nonlinear partial differential equations (PDE) with a driving source and mixed-boundary conditions at the input and output terminals, respectively. This model successfully emulates parametric and nonlinear microwave propagation when long-wave approximation is applicable. The influence of dispersion to sustain three non-degenerate phased-locked waves through the TL is highlighted. A rigorous and robust Finite Difference Time Domain (FDTD) solver based on the explicit Lax-Wendroff and implicit Crank-Nicolson schemes has been developed to investigate the device responses under various excitations. Linearization of the wave equation, under small-amplitude assumption, dispersion and impedance analysis is performed to explore more aspects of the device for the purpose of efficient design of a traveling-wave parametric amplifier. Knowing all microwave characteristics and identifying different regimes of operation, which include impedance properties, cut-off propagation, dispersive behaviour and shock-wave formation, we exploit perturbation theory accompanied by the method of multiple scale to derive the three nonlinear coupled amplitude equations to describe the parametric interaction. A graphical technique is suggested to find three waves on the dispersion diagram satisfying the phase-matching conditions. Both cases of perfect phase-matching and slight mismatching are addressed in this work. The incorporation of two numerical techniques, spectral method in space and multistep Adams-Bashforth in time domain, is employed to monitor the unilateral gain, superior stability and bandwidth of this structure. Two types of functionality, mixing and amplification, with their requirements are described. These properties make this structure desirable for applications ranging from superconducting optoelectronics to dispersive readout of superconducting qubits where high sensitivity and ultra-low noise operation is required.1 yea

Microwave frequency power dependence in high-Tc thin films, grain boundaries, and Josephson junctions

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Physics, 1999.Includes bibliographical references (p. 106-113).by Youssef M. Habib.Ph.D

Scalable and high-sensitivity readout of silicon quantum devices

Quantum computing is predicted to provide unprecedented enhancements in computational power. A quantum computer requires implementation of a well-defined and controlled quantum system of many interconnected qubits, each defined using fragile quantum states. The interest in a spin-based quantum computer in silicon stems from demonstrations of very long spin-coherence times, high-fidelity single spin control and compatibility with industrial mass-fabrication. Industrial scale fabrication of the silicon platform offers a clear route towards a large-scale quantum computer, however, some of the processes and techniques employed in qubit demonstrators are incompatible with a dense and foundry-fabricated architecture. In particular, spin-readout utilises external sensors that require nearly the same footprint as qubit devices. In this thesis, improved readout techniques for silicon quantum devices are presented and routes towards implementation of a scalable and high-sensitivity readout architecture are investigated. Firstly, readout sensitivity of compact gate-based sensors is improved using a high-quality factor resonator and Josephson parametric amplifier that are fabricated separately from quantum dots. Secondly, an integrated transistor-based control circuit is presented using which sequential readout of two quantum dot devices using the same gate-based sensor is achieved. Finally, a large-scale readout architecture based on random-access and frequency multiplexing is introduced. The impact of readout circuit footprint on readout sensitivity is determined, showing routes towards integration of conventional circuits with quantum devices in a dense architecture, and a fault-tolerant architecture based on mediated exchange is introduced, capable of relaxing the limitations on available control circuit footprint per qubit. Demonstrations are based on foundry-fabricated transistors and few-electron quantum dots, showing that industry fabrication is a viable route towards quantum computation at a scale large enough to begin addressing the most challenging computational problems

Novel excitations in driven vortex channels in a superconductor, and solitary waves of light and atoms in photonic crystal fibres

This is a thesis in two parts. In Part I, we will study the shear response of confined vortices. In Part 2, we will study light and matter interactions in photonic crystal fibres. Whilst the approaches of each are completely different, they both have the same central theme: solitons. In the first part of this thesis we study the static and dynamic properties of vortices within a Type-II superconductor, confined within a channel. The channel comprises a collection of pinned vortices, which form the perfect triangular lattice in the boundary, and rows of “free” particles which are driven via an external force. We provide two main results within this system. First we calculate the potential stemming from the boundary, and derive (under certain approximations) the phenomenologically accepted result for the critical shear dependence on the system width. We then study a novel system in which a defect is placed in a deformable potential; specifically a system comprised of two channels where one or both channels have a defect. This system provides a mechanism for the proliferation of kink/kink and anti-kink/anti-kink pairs as the defect binds to a local excitation in the form of a “breather”. We observe and explain what appears to be an action at a distance style interaction between excitations. In Part II, we will utilise the nonlinear effects of a Bose condensate and the unique optical properties of a photonic crystal fibre to demonstrate there are nonlinearly stable configurations which exist in the vicinity of an optical mode with a cut-off. These are solitary waves, whose relative composition of atoms and photons may be changed via altering the detuning of light from an atomic transition and Feshbach resonances