Existence and stability analysis of finite<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mn>0</mml:mn><mml:mtext>−</mml:mtext><mml:mi>π</mml:mi><mml:mtext>−</mml:mtext><mml:mn>0</mml:mn></mml:mrow></mml:math>Josephson junctions

Ahmad, Saeed; Susanto, Hadi; Wattis, Jonathan A. D.

doi:10.1103/physrevb.80.064515

Cited by 3 publications

(2 citation statements)

References 31 publications

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“…It was reported that the microwave power needed to switch the junction into a resistive state depends on the magnitude of the eigen frequency to be measured. The eigen frequency of the ground state in Josephson junctions with one and two phase-shifts has been calculated theoretically in [43][44][45][46][47][48] and confirmed experimentally in [49,50]. The experimental measurements were made by applying microwave radiation of fixed frequency and power to the Josephson junction.…”

Section: Introductionmentioning

confidence: 73%

Localized modes in parametrically driven long Josephson junctions with a double-well potential

Gul

Ali

Ullah³

2018

J. Phys. A: Math. Theor.

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In this paper, we study, both analytically and numerically, the localized modes in long Josephson junctions in the presence of a variety of parametric drives. The phase-shift applied acts as a double-well potential which is known as the junction. The system is described by an inhomogeneous sine-Gordon equation that depicts the dynamics of long Josephson junctions with phase-shift. Using asymptotic analysis together with multiple scale expansion, we obtain the oscillation amplitudes for the junction, which show the decay behaviour with time due to radiative damping and the emission of high harmonic radiations. We found that the energy taken away from the internal modes by radiative waves can be balanced by some external drives. We also discuss in detail the excitation by parametric drives with frequencies to be either in the vicinity or double the natural frequency of the system. It is shown that the presence of externally applied drives stabilizes the nonlinear damping, producing stable breather modes in the junction. It is observed that, in the presence of external drives, the driving effect is stronger for a case of driving frequency nearly equal to the system’s natural frequency as compared to that of the driving frequency nearly equal to twice of the natural frequency of the oscillatory mode. In the numerical section, we calculate the Josephson voltage for the sine-Gordon equation with external drives. We observe that, an infinitely long Josephson junction with double well potential cannot be switched to a resistive state by an external drive with a frequency close to and double the systems eigenfrequency, provided that the driven amplitude is small enough. It is due to the fact that the higher harmonics with frequencies in the phonon band are excited in the form of continuous wave radiation.

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Section: Introductionmentioning

confidence: 73%

Localized modes in parametrically driven long Josephson junctions with a double-well potential

Gul

Ali

Ullah³

2018

J. Phys. A: Math. Theor.

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“…The eigenfrequency of the ground state in the simplest case of Josephson junctions with one and two phase-shifts has been theoretically calculated in [7,8,9,10,11,12]. More importantly, the eigenfrequency calculation in the former case has been confirmed experimentally recently in [13,14].…”

Section: Introductionmentioning

confidence: 91%

Breathing Modes of Long Josephson Junctions with Phase-Shifts

Ali¹,

Susanto²,

Wattis³

2011

SIAM J. Appl. Math.

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Abstract. We consider a spatially inhomogeneous sine-Gordon equation with a time-periodic drive, modeling a microwave driven long Josephson junction with phase-shifts. Under appropriate conditions, Josephson junctions with phase-shifts can have a spatially nonuniform ground state. In recent reports [Phys. Rev. Lett. 98, 117006 (2007), arXiv:0903.1046, it is experimentally shown that a microwave drive can be used to measure the eigenfrequency of a junction's ground state. Such a microwave spectroscopy is based on the observation that when the frequency of the applied microwave is in the vicinity of the natural frequency of the ground state, the junction can switch to a resistive state, characterized by a non-zero junction voltage. It was conjectured that the process is analogous to the resonant phenomenon in a simple pendulum motion driven by a time periodic external force. In the case of long junctions with phase-shifts, it would be a resonance between the internal breathing mode of the ground state and the microwave field. Nonetheless, it was also reported that the microwave power needed to switch the junction into a resistive state depends on the magnitude of the eigenfrequency to be measured. Using multiple scale expansions, we show here that an infinitely long Josephson junction with phase-shifts cannot be switched to a resistive state by microwave field with frequency close to the system's eigenfrequency, provided that the applied microwave amplitude is small enough, which confirms the experimental observations. It is because higher harmonics with frequencies in the continuous spectrum are excited, in the form of continuous wave radiation. The breathing mode thus experiences radiative damping. In the absence of driving, the breathing mode decays at rates of at most O(t −1/4 ) and O(t −1/2 ) for junctions with a uniform and nonuniform ground state, respectively. The presence of applied microwaves balances the nonlinear damping, creating a stable breather mode oscillation. As a particular example, we consider the so-called 0 − π − 0 and 0 − κ Josephson junctions, respectively representing the two cases. We confirm our analytical results numerically. Using our numerical computations, we also show that there is a critical microwave amplitude at which the junction switches to the resistive state. Yet, it appears that the switching process is not necessarily caused by the breathing mode. We show a case where a junction switches to a resistive state due to the continuous wave background becoming modulationally unstable.

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Multiple-scale analysis of the parametric-driven sine-Gordon equation with phase shifts

et al. 2022

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In this article, we model the current and voltage across the weak link between two superconductors. This gives us a nonhomogeneous, nonlinear parametric-driven sine-Gordon equation with phase shifts. This model equation cannot be solved directly but can be approximated. For the approximations, we use two methods, and analytic perturbation method and the numerical approximation method known as the Runge–Kutta method. For the analytic method, we construct a perturbation expansion method with multiple-scale expansion. We discuss the parametric-driven in the sine-Gordon equation with phase shifts for the 0–π–0 junction. Further, we also describe the breathing modes for various order of perturbation. At the end, we compare the solutions obtained via perturbation and numerical methods of parametric-driven sine-Gordon equation with phase shifts. Finally, we concluded that the modes of the breathing decay to a constant in both cases. Also we found a good agreement between both approximate methods.

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Existence and stability analysis of finite0−π−0Josephson junctions

Cited by 3 publications

References 31 publications

Localized modes in parametrically driven long Josephson junctions with a double-well potential

Localized modes in parametrically driven long Josephson junctions with a double-well potential

Breathing Modes of Long Josephson Junctions with Phase-Shifts

Multiple-scale analysis of the parametric-driven sine-Gordon equation with phase shifts

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