A fluxon is a collective behavior which exists in a circular, parallel array of Josephson junctions.
If the array is cooled in the presence of a magnetic field, magnetic flux can be trapped inside. The “fluxon” is the solution of the coupled equations in this case. It corresponds to a static change of 2π in the phases of sequential junctions moving down the array. It has with it an associated magnetic flux and circulating supercurrent, which is why it is also referred to as a vortex. It behaves in many ways like a single-particle: its center-of-mass has a well-defined position and energy, which is periodic with the junctions in the array. If no current is applied to the array the fluxon will be pinned in a potential energy minimum; if the energy provided by the current exceeds the pinning potential, the fluxon will be depinned and move through the array, causing a voltage across each junction as it moves by. This voltage can be measured and is proportional to the velocity of the fluxon.

We have studied the dynamics of a fluxon trapped in a Josephson ring. Our work has focused on the prediction and observation of a new mode of transport. We dubbed this mode “fluxon diffusion.” It consists of a noise-activated, hopping mode where the fluxon moves from one potential well to the next under a constant current. This hopping mode causes a small voltage (a “diffusion voltage”) to appear across the array. It occurs in underdamped arrays, where there is hysteresis in the I-V curves. The combination of diffusion and hysteresis is very unusual; in single junctions, in fact, it cannot occur without the addition of colored noise or frequency-dependent damping.

In the first paper [1],
we showed the existence of Fluxon diffusion through numerical simulations of the equations of motion in the ring.
Together with our colleague Juan Mazo, we showed the conditions under which you would expect both a diffusion voltage
and hysteresis. In the second paper

[2], we showed the first experimental evidence for Fluxon diffusion, both through direct observation of the I-V curves and also in how the presence of fluxon diffusion affects the switching current measurements.

Our work will continue with different devices, where we will test different temperatures and different numbers of fluxons trapped in the array. With these experiments, we should be able to fully test our predictions regarding fluxon diffusion. In addition, we also have arrays in our possession which have been purposely made asymmetric or “ratchet” Our work should then be able to move in the direction of ratchets or “fluctuation induced transport,” which aims to study how noise can move a system in a preferred direction if the system is driven out of equilibrium.

One type of collective mode in a Josephson array is called a discrete breather. Physically, a discrete breather corresponds to a localized whirling mode: some junctions in the array have their phases rotating while others are static, despite the entire array being driven by a uniform current. This mode can only exist with nonlinear oscillators or pendulums, not with linear oscillators. The rotating junctions have a voltage across them, while the static junctions do not; thus, a discrete breather is
straightforward to measure. Discrete breathers are easiest to observe in a Josephson Ladder.

To date, much of the study of collective modes in nonlinear systems has been on individual modes separately. An exciting possibility is to think about how these modes interact with each other. Toward that end, we are working on the experimental
demonstration of the collision between a moving fluxon (or vortex, as they are called in Josephson ladders) and a discrete breather. In 2001, Trias et al. proposed such an experiment in a Josephson ladder.

Parameters for a ladder which could both support discrete breathers and moving vortices were calculated. Numerical simulations under different values of the applied current were performed. It was found that, under certain situations, the breather could “pin” the vortex, preventing further motion down the array. In other situations, the breather was destroyed by
the vortex, and the whole ladder was put into the high-voltage whirling mode. We are working to realize such an interaction in an
experiment.

The human brain contains tens of billions of neurons, each connected to thousands of its neighbors. Researchers today are trying to understand the interactions of limited numbers of these neurons as a stepping-stone to understanding larger regions of the brain.

One challenge for these studies is the sheer computational power necessary to simulate large networks of neurons over long timescales. In traditional computer simulations, each individual neuron must be simulated taking into account the behavior of all its neighbors. Each neuron might require 10-10000 floating point operations (FLOPs) to simulate a typical single pulse (action potential), depending on the model. Along with two colleagues at Colgate (Patrick Crotty, a computational neuroscientist, and Dan Schult, an applied mathematician), we recently proposed to alleviate this computational bottleneck by using circuits of Josephson junctions as direct, analog models of neurons. Not only do the spiking times occur on the scale of picoseconds (about a million times faster than a neuron spike in the human brain and thousands of times faster than the fastest digital simulations) but all the circuits would “compute” in parallel with live communication between neurons. The resulting speed-up in simulation time could have an enormous impact on our ability to understand how networks of thousands of neurons would behave (a realistic upper limit on the number of model neuron circuits one could realistically put on a chip).

In our first publication, we showed how a circuit of two Josephson junctions, a JJ Neuron, mimics several of the key behaviors that an analog neuron model should have. It produces action potentials, has a threshold current below which no action potentials are created, and has a refractory period which puts an upper limit on its firing frequency. We showed the mathematical correspondence of our model to other accepted models. We also showed numerical demonstrations of how these neurons could couple to one another. Finally, we showed how fast it could potentially simulate large numbers of neurons over long timescales.