Driven Josephson junction

A relatively unexplored extension of the Kramers theory is the escape of a Brownian particle out of a potential well in the presence of an external periodic force. Processes such as multiphoton dissociation and isomerization of molecules in high-pressure gas or in condensed phases/ laser-assisted desorption/ and transitions in current-driven Josephson junctions under the influence of microwaves " may be described with such a model, where the pieriodic force results from the radiation field. 

ZENO AND ANTI-ZENO EFFECTS IN DRIVEN JOSEPHSON JUNCTIONS CONTROL OF MACROSCOPIC QUANTUM TUNNELING... 

Zeno and anti-Zeno effects in driven Josephson junctions... 

The PT model is frequently used as a minimalistic approximation for more complex models. For instance, it is the mean-field version of the Frenkel Kontorova (FK) model as stressed by D. S. Fisher [29,83] in the context of the motion of charge-density waves. The (mean-field) description of driven, coupled Josephson junctions is also mathematically equivalent to the PT model. This equivalence has been exploited by Baumberger and Carol for a model that, however, was termed the lumped junction model [84] and that attempts to... 

Condensed-matter physics (Josephson junction, charge-density waves) Mechanics (Overdamped pendulum driven by a constant torque)... 

We now give a more quantitative discussion of the Josephson effect. Suppose that a Josephson junction is connected to a de current source (Figure 4.6.2), so that a constant current I >0 is driven through the junction. Using quantum mechanics, one can show that if this current is less than a certain critical current, no voltage will be developed across the junction that is, the junction acts as if it had zero resistance However, the phases of the two superconductors will be driven apart to a constant phase difference 0 = 0 - 0, where 0 satis-... 

N junctions, resistive load) Generalize Exercise 4.6.4 as follows. Instead of the two Josephson junctions in Figure 1, consider an array of N junctions in series. As before, assume the array is in parallel with a resistive load R, and that the junctions are identical, overdamped, and driven by a constant bias current 7. Show that the governing equations can be written in dimensionless form as... 

Pendulum driven by constant torque) The equation 0-t-sin0 = y describes the dynamics of an undamped pendulum driven by a constant torque, or an undamped Josephson junction driven by a constant bias current. 

This chapter is organized as follows for each bifurcation, we start with a simple prototypical example, and then graduate to more challenging examples, either briefly or in separate sections. Models of genetic switches, chemical oscillators, driven pendula and Josephson junctions are used to illustrate the theory. 

This section deals with a physical problem in which both homoclinic and infinite-period bifurcations arise. The problem was introduced back in Sections 4.4 and 4.6. At that time we were studying the dynamics of a damped pendulum driven by a constant torque, or equivalently, its high-tech analog, a superconducting Josephson junction driven by a constant current. Because we weren t ready for two-dimensional systems, we reduced both problems to vector fields on the circle by looking at the heavily overdamped limit of negligible mass (for the pendulum) or negligible capacitance (for the Josephson junction). 

In Section 8.5 we used a Poincare map to prove the existence of a periodic orbit for the driven pendulum and Josephson junction. Now we discuss Poincare maps more generally. 

As tools for analyzing differential equations. We have already encountered maps in this role. For instance, Poincare maps allowed us to prove the existence of a periodic solution for the driven pendulum and Josephson junction (Section 8.5), and to analyze the stability of periodic solutions in general (Section 8.7). The Lorenz map (Section 9.4) provided strong evidence that the Lorenz attractor is truly strange, and is not just a long-period limit cycle. 

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