Breathing Modes of Long Josephson Junctions with Phase-Shifts

Abstract

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.

Soliton and Breather Splitting on Star Graphs from Tricrystal Josephson Junctions

“…When excited, the mode will vibrate, but eventually, it will fade away because of 'radiative' damping, i.e., a damping mechanism due to the excitation of higher harmonics with frequencies in the continuous spectrum. Even more, trapped kink oscillations decay asymptotically at a rate of O(t −1/2 ), see [34]. Here, even though our excitations are non-topological, large-amplitude breathers are close to an intertwining pair of kink and anti-kink.…”

Time-Fractional Klein–Gordon Equation with Solitary/Shock Waves Solutions

“…By considering u≃ sin u, the considered system becomes the Sine-Gordon equation. In future work, it will be interesting to investigate the Sine-Gordon model with nonlinear AC/DC drives with different fractional operators to study the solitonic behavior, localized modes in single and in stacked long Josephson junctions with a variety of potentials, parity time symmetry, the nonlinearity/dispersion effects, and evolution of the localized monotonic shocks [67][68][69][70][71][72][73].…”

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

“…where L is a linear self-adjoint operator (L = L † ) given by the left-hand side of the above system. Applying the Fredholm theorem [32,51], we obtain the solvability condition…”

“…Recently, the localized modes excitation has been studied in 0 − π − 0 long Josephson junctions for single-mode and two mode oscillations [30][31][32][33] by considering an inhomogeneous sine-Gordon equation with a variety of direct and parametric drives. It has been reported that in 0 − π − 0 long Josephson junctions, the amplitudes of the breathing modes decay with time due to radiative damping and emission of high harmonic radiations, which can be re-balanced by applying appropriate external drives.…”

“…The experimental measurements were made by applying microwave radiation of fixed frequency and power to the Josephson junction. It has been shown that, in the presence of ac-drive the breathing mode cannot be excited to switch the junction into a resistive state, provided that the microwave amplitude is small enough, which means that, the breathing mode experiences radiative damping [32]. It should be noted that such damping is not present in short junctions as the phase difference in that limit is effectively independent of spatial variable x.…”