The dynamics of SQUIDs and coupled pendula

Josephson [8] predicted in 1962 that a DC tunnel current would flow between two superconductors connected by a thin insulating layer of thickness less than about 20°A in the absence of a voltage difference, an effect now called the DC Josephson effect. The quantum-mechanical current, called the superconducting current, arises from the tunneling of Cooper pairs of electrons of opposite spin and momenta and is given by Is = Ic sin θ, where Ic is the critical current and φ is the difference of the phases of the wave functions of the two superconductors. This gives the ideal current through a junction, but in real circuits there are resistive and capacitive currents as well. One of the standard models of a more realistic circuit is the so-called Stewart- McCumber resistively-shunted-junction (or RSJ) model, which is described by the following equation for the current [6, 9]: (chemical equation presented) Here h is Planck's constant, e is the charge on an electron, h/2e is the flux quantum, C is the capacitance, R is the resistance, and I is the imposed bias current. To simplify (5.2) define the frequency = p2eIc/hC and the scaled time τ = t; then (5.2) becomes (chemical equation presented) where e = (RC)-1, i = I/Ic, and the dot denotes derivation with respect to the rescaled time τ . A very useful correspondence of this system to a pendulum provides insight into the dynamics studied later. In fact, the pendulum will serve as the basic physical model; see also [1]. Suppose that a pendulum consists of a bob of mass m that is attached to a (weightless) rod of length L. Then the equation of motion is (chemical equation presented) where ∩ = mL2 is the moment of inertia of the pendulum, g is the gravitational acceleration, n is the damping, φ is the angle between the bob and vertical measured from the downward position, and T is the applied torque. After non-dimensionalization this leads to (5.3). When a ring of superconducting material contains two Josephson junctions, the result is a superconducting quantum interference device (SQUID), so called because the wave functions of the Cooper pairs at each junction interfere. SQUIDS are among the most sensitive devices for detecting magnetic fields - a SQUID is capable of detecting magnetic fields of around 2 picotesla, i.e., at the quantum flux level. The coupling between phases across the junctions is proportional to the difference of phases, and therefore, the system of equations governing a SQUID is (chemical equation presented) Here, Y is the coupling coefficient, and the dimensionless bias current I is assumed to be the same for both junctions. An identical pair of equations governs the motion of two pendula coupled by a linear torsional spring or bar, and forced with an applied torque I; see Fig. 5.1. We use this system as the paradigm in this chapter and we attempt to synthesize the results of [3, 5] and the unpublished study [2], which are all written in collaboration with Eusebius Doedel. The work involves extensive numerical studies that were carried out using DsTool, MatLab, and primarily, Auto. In the next section we analyze the equilibria of (5.5). Section 5.2 considers the undamped undriven case, which is part of the unpublished results in [2]. We analyze both equilibria and periodic orbits for this case, and also discuss the computation of heteroclinic connections. Finally, Sect. 5.3 shows the existence of so-called rotations, periodic solutions with a period that is an integer multiple of the forcing frequency. We discuss their stability in Sect. 5.4 and draw some conclusions in Sect. 5.5.