We present an analytical and computational study of resonances and transient responses in a classical Josephson junction system. A theoretical basis for resonances in a superconducting loop with three junctions is presented, outlining both the direct relationship between the dynamics of single- and multijunction systems and the direct relationships between observations of the classical counterparts to Rabi oscillations, Ramsey fringes, and spin-echo oscillations in this class of systems. We show simulation data along with analytical analyses of the classical model, and the results are related to previously reported experiments conducted on three junction loops. We further investigate the effect of off-resonant microwave perturbations to, e.g., the Rabi-type response of the Josephson system, and we relate this response back to the nonlinear and multivalued resonance behavior previously reported for a single Josephson junction. The close relationships between single and multijunction behaviors demonstrate the underlying dynamical mechanism for a whole class of classical counterparts to expected quantum-mechanical observations in a variety of systems, namely, the resonant and transient behavior of a particle in an anharmonic potential well with subsequent escape.
Nonlinear effects in mesoscopic devices can have both quantum and classical origins. We show that a three-Josephson-junction (3JJ) flux qubit in the classical regime can produce low-frequency oscillations in the presence of an external field in resonance with the (high-frequency) harmonic mode $\ensuremath{\omega}$ of the system. Like in the case of quantum Rabi oscillations, the frequency of these pseudo-Rabi oscillations is much smaller than $\ensuremath{\omega}$ and scales approximately linearly with the amplitude of the external field. This classical effect can be reliably distinguished from its quantum counterpart because it can be produced by the external drive not only at the resonance frequency $\ensuremath{\omega}$ and its subharmonics $(\ensuremath{\omega}/n)$, but also at its overtones, $n\ensuremath{\omega}$.
An electrical circuit consisting of two capacitively coupled inductive loops, each interrupted by a Josephson junction, is analyzed through the classical resistively and capacitively shunted junction (RCSJ) model. The same circuit has recently been studied experimentally and the results were used to demonstrate quantum-mechanical entanglement in the system by observing the correlated states of the two inductive loops after initial microwave perturbations. Our classical analysis shows that the observed phenomenon exists entirely within the classical RCSJ model, and we provide a detailed intuitive description of the transient dynamics responsible for the observations.
We present a classical analysis of the transient response of Josephson junctions perturbed by microwaves and thermal fluctuations. The results include a specific low-frequency modulation in phase and amplitude behavior of a junction in its zero-voltage state. This transient modulation frequency is linked directly to an observed variation in the probability for the system to switch to its nonzero voltage state. Complementing previous work on linking classical analysis to the experimental observations of Rabi oscillations, this expanded perturbation method also provides closed-form analytical results for attenuation of the modulations and the Rabi-type oscillation frequency. Results of perturbation analysis are compared directly (and quantitatively) to numerical simulations of the classical model as well as published experimental data, suggesting that transients to phase-locking are closely related to the observed oscillations.
We study analytically and numerically the phase-modulation properties of a classical Josephson tunnel junction biased in the zero-voltage state and phase locked to an external ac field. We show that the phase-locked state is being modulated in the transients, or in response to perturbations, and the modulation frequency is calculated as a function of relevant system parameters, such as microwave field amplitude. Our analysis demonstrates that the modulation of a phase-locked state in an entirely classical Josephson junction produces oscillations analogous to quantum mechanical Rabi oscillations, expected to be observed under the same conditions.
The properties of phase escape in a dc superconducting quantum interference device (SQUID) at 25 mK, which is well below quantum-to-classical crossover temperature
Tcr, in the presence of strong resonant ac driving have been investigated. The SQUID contains two
Nb/Al-AlOx/Nb tunnel junctions with Josephson inductance much larger than the loop inductance so it can be viewed as a single junction having adjustable critical current. We find that with increasing microwave power
W and at certain frequencies $\ensuremath{\nu}$ and $\ensuremath{\nu}/2$, the single primary peak in the switching current distribution, which is the result of macroscopic quantum tunneling of the phase across the junction, first shifts toward lower bias current
I and then a resonant peak develops. These results are explained by quantum resonant phase escape involving single and two photons with microwave-suppressed potential barrier. As
W further increases, the primary peak gradually disappears and the resonant peak grows into a single one while shifting further to lower
I. At certain
W, a second resonant peak appears, which can locate at very low
I depending on the value of $\ensuremath{\nu}$. Analysis based on the classical equation of motion shows that such resonant peak can arise from the resonant escape of the phase particle with extremely large oscillation amplitude resulting from bifurcation of the nonlinear system. Our experimental result and theoretical analysis demonstrate that at
T⪡Tcr, escape of the phase particle could be dominated by classical process, such as dynamical bifurcation of nonlinear systems under strong ac driving.
We have investigated numerically a system consisting of a long Josephson junction embedded into a microstrip resonator. For such a configuration the Josephson vortex dynamics in the junction is driven by the oscillating currents in the resonator. We have calculated the complex impedance of the junction at the resonant frequency. The Q factor of the resonator and the change of the resonant frequency of the whole system can then be easily calculated knowing the parameters of the resonator without Josephson junction. The fluxon dynamics and the complex impedance of the junction are simulated for different values of the junction length, damping, dc bias current, and dc magnetic field. This allows us to explain the behavior of the impedance of the junction and different loss mechanisms. In particular, we predict a nonmonotonic dependence of the both real and imaginary parts of the impedance on the amplitude of the oscillating current in the resonator for some range of parameters. Our results show that such a combination of active and passive superconducting elements forms a rather interesting nonlinear physical system which might be useful for development of tunable/switchable superconducting resonators and suggests an experimental technique for detecting trapped vortices in long junctions.