Project Description I. Introduction: Josephson junction ...departments.colgate.edu/physics/faculty/KenSegall/Project... · I. Introduction: Josephson junction networks ... considered

Project Description: page 1

Project Description

I. Introduction: Josephson junction networks

Over the past 25 years, superconducting Josephson junctions have gradually become one of the major topics of research in low temperature physics. The Josephson community is wide and varied, spanning areas as fundamental as quantum measurement and coherence and as applied as ultra-sensitive detectors and voltage standards. Our research uses Josephson junctions as model systems for problems in nonlinear and neural dynamics.

A. Josephson junctions and nonlinearity At its heart, a single Josephson junction is simply two superconducting electrodes separated by a thin insulating tunnel barrier. A scanning electron micrograph of a Josephson junction is shown later in figure 4a. The electrons in each of the individual superconducting electrodes can be described by a single macroscopic wavefunction with a well-defined phase [1]. This phase undergoes an abrupt change in moving across the tunnel barrier. If the phase of the wavefunction is equal θ1 on one other side of the barrier and equal to θ2 on the other side of the barrier, then the change or phase difference is defined as φ = θ2 − θ1. This degree of freedom φ turns out to completely describe the junction’s electrical behavior; knowing the dynamics of φ one can predict all measurable properties such as the junction’s voltage and current. In most experiments φ is considered to be a classical variable, in that one can know its exact value and rate of change simultaneously. (In experiments on quantum computing, however, φ is usually considered to obey an uncertainty relationship, depending on the type of qubit.)

Brian Josephson’s prediction was that if φ was nonzero, a supercurrent will flow across the junction equal to Icsinφ, where Ic is a junction-dependent parameter called the critical current [2]. In addition, he predicted that the voltage V across the junction is proportional to dφ/dt. Besides the supercurrent, a classical displacement current and a resistive leakage current can also flow through the junction. The resulting statement of current conservation combined with the Josephson relationships gives the following:

φφπ

φπ

sin22

02

20

cJ Idt

d

Rdt

dCI +

Φ+

Φ= (1)

Here IJ is the total current flowing through the junction, Φ0 is the flux quantum (equal to Planck’s constant divided by twice the electron charge), C is the capacitance of the junction, and R is a resistance which characterizes the leakage through the junction. Equation (1) is a nonlinear, pendulum-like equation which describes the dynamics of φ. The current IJ can be supplied externally, or it can come from other junctions if more than one junction is in the circuit. Connecting multiple junctions together and forcing them to share current causes the phase difference from different junctions to be coupled together. Since equation (1) is nonlinear, the coupling between different φ’s is also nonlinear. If there are N junctions in a circuit, that circuit’s behavior represents the dynamics of N nonlinear differential equations, each of the


Figure 1: Coupling topologies. The junctions are represented by an “X”, and input current is represented by arrows. (a) SQUID; (b) Parallel Array; (c) 3-junction detector; (d) Josephson ladder; (e) Josephson Junction Neuron; (f) SQUID-ladder Array. For the cases (b), (d) and (f), the edges can be connected to each other in a circular geometry. form of equation (1). Our research focuses on the use of Josephson junctions connected in networks to address topics in nonlinear dynamics. B. Coupling Topologies Josephson junctions can be connected into networks in many different ways. Figure 1 shows the topologies relevant for this proposal. First and foremost is two junctions connected in a loop, driven by a constant current; this is commonly known as a SQUID (Superconducting Quantum Interference Device), shown in figure 1a. SQUIDs are used for magnetic field detection but can also be used in place of a single junction, where the critical current of that junction can be adjusted by varying the magnetic flux through the loop. Two particular structures that we have developed in our research are a 3-junction detector, shown in figure 1c, where one junction is put in parallel with two others, and a Josephson junction neuron (JJ neuron), shown in figure 1e, which is similar to a SQUID except that it is driven by two currents (an input current and a bias current). If several junctions are put in parallel it is called a parallel array, shown in figure 1b. Parallel arrays can be open-ended or wrapped in a ring to give periodic boundary conditions, which leads to a structure known as a Josephson ring. If we take the parallel array and replace the wires with additional junctions, we have a structure known as a Josephson ladder, shown in figure 1d. In the Josephson ladder the vertical junctions receive the external current while the horizontal junctions are the coupling junctions. If we now replace the horizontal junctions with SQUIDs, we have the SQUID ladder array (figure 1e). This structure


Figure 2: Scaling of computational time with number of junctions. The red dots show the computational time for 1 µs of real-time dynamics computed by our group. The two lines indicate scaling by N2 and N3. allows the coupling between vertical junctions to be tuned by the external flux. This structure can also be wrapped in a circle to give periodic boundary conditions. The equations of motion for a large network of Josephson junctions are determined by two sets of constraints. The first is charge conservation or Kirchhoff’s current law, which states that the sum of the currents entering a node is equal to the sum of the currents leaving a node. The second would appear to be Kirchhoff’s voltage law, but in superconducting circuits that is replaced with a stronger constraint, fluxoid quantization [1]. Fluxoid quantization ensures that the total phase change of the superconducting wavefunction going around any closed loop is an integer multiple of 2π. The phase can change abruptly going across a junction and can also change continuously with a magnetic flux perpendicular to the loop. Fluxoid quantization thus relates the junction phases to the magnetic flux. C. Speed and Computational Power One type of network we have studied in our lab has nine junctions coupled together (N = 9) in a Josephson ring. To compute the dynamics of nine junctions over the time for a typical lab experiment at different temperatures with sufficient averaging takes close to a week on a desktop computer. We make use of a high-powered workstation and sometimes borrow time on a computing cluster at Colgate, which reduces the time to a day or two and makes comparison of theory with experiment moderately feasible. However, as N gets much larger than nine, we quickly run into intractable computational times. Figure 2 shows the time to compute just 1 µs of real-time dynamics in a Josephson circuit versus the number of junctions N in the circuit. These timing experiments were done in our group. The scaling is roughly N2 to N3, depending on the properties of the coupling and the initial conditions. An interesting aspect of figure 2 is that although the dynamics of a large Josephson circuit would be essentially impossible to compute, one can straightforwardly measure the properties of a large circuit and, if the circuit is designed appropriately, infer the dynamics from those


Figure 3: Collective modes in Josephson arrays. (a) Fluxon trapped in a parallel array. The phase changes by 2π going down the array. Because a non-zero phase implies a supercurrent through the junction, there is circulating supercurrent and a magnetic flux coming out of the page. (b) Moving fluxon. Under the application of a current larger than the depinning current, the fluxon will move down the array. (c) Phase-synchronized whirling mode in a Josephson ladder. The phases of all vertical junctions rotate, and all the junctions are at the same phase. (d) Discrete Breather. Only one vertical junction rotates, despite all the junctions being driven by a uniform current. The horizontal junctions in the adjacent cells must rotate in the opposite direction and at half the voltage to maintain voltage conservation. measurements. In thinking about future research, Josephson junctions have a unique possibility: the analog simulation of the dynamics of a large, coupled, nonlinear network. Contributing to the plot in figure 2 is the fact that the characteristic times in Josephson junctions are extremely fast. A typical time for a 2π rotation of the phase (a single “whirl” of the pendulum) is on the order of 50 ps. This is much faster than the switching time for transistors (on the order of ns). When this is combined with the advantage of coupling, as discussed above, the result can be an enormous difference in the speed of a Josephson network as compared with simulation on a computer. This speed advantage could potentially be exploited to explore neural networks with the JJ neuron, a point that was discussed in our recent paper [3]. It also attracted some attention in the press [4-6]. D. Emergent Behavior: Collective Modes The dynamics of a single Josephson junction, expressed by equation (1), are exactly that of a damped, driven pendulum. Although they include interesting phenomena such as hysteresis and chaos, they are fairly straightforward and have been fully studied to date. However, when multiple junctions are coupled together, more complex and interesting behavior can result, including several emergent behaviors. These emergent behaviors can have a coherent structure and act in a collective, organized fashion. We are very interested in these behaviors, which we call collective modes. Figure 3 shows a schematic of each of the different collective modes we study. In these schematics we used the pendulum analogy to represent the state of each junction. If the phase of the pendulum is static and at zero degrees deflection, the current and


voltage across the junction are zero. If the phase is “tilted” to a non-zero angle but not rotating, there is a supercurrent but no voltage. If the phase of the junction is rotating, there is a voltage, and the size of the voltage is proportional to the speed of rotation. The first collective mode which we study is called a fluxon or vortex [8]. It can exist in either a parallel array or a Josephson ladder with similar properties. Here we discuss only the parallel array, as the fluxon in a Josephson ladder behaves the same. 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 sometimes 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. A second collective mode is called a whirling mode. It corresponds to the phases of the junctions rotating around at constant angular velocity. It can exist in either a parallel array or a Josephson ladder; in a Josephson ladder, it is the vertical junctions which rotate. If all junctions rotate at the same frequency, and hence have the same voltage, the mode is said to be frequency synchronized; if, in addition, the junctions all have exactly the same phase at all times, the mode is said to be phase synchronized. A third collective mode is called a Discrete Breather [7]. Physically, a Discrete Breather corresponds to a localized whirling mode: some junctions in the array are rotating while some 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. A Discrete Breather can only exist in a Josephson ladder1, since in a parallel array all junctions must have the same voltage across each junction. Note that in the ladder, the vertical junctions receive the current and therefore rotate; in order to maintain voltage conservation, some of the horizontal junctions must rotate as well. In some families of breathers [7], the horizontal junctions rotate at different speeds that the vertical ones. This 1 The kind of breather that can exist in a Josephson Ladder is called a “Rotobreather.” In a parallel array, technically another kind of breather can exist called an “Oscillobreather,” but these have yet to be observed.


(a) (b)

(c) (d) Figure 4: SEM micrographs of Josephson arrays tested in our group. All devices were designed at Colgate by undergraduate students and fabricated at Hypres. (a) Single junction, about 2 µm on a side. (b) Josephson Ring. The current comes in one set of nine leads (those attached to the dark grey areas) and leaves out of another nine leads (those attached to the big leads in the center). This new design eliminated the asymmetry in the I-V curves. (c) Josephson ladder. The parameters are designed to support both a discrete breather and a moving vortex. (d) Circular SQUID-ladder array. The horizontal junctions are replaced by SQUIDs to allow for variable coupling in the synchronization experiment.

means that some of the horizontal junction voltages will be different than some of the vertical junction voltages. If the ladder is shrunk down to a single cell, this can lead to a situation with only 3 junctions around the loop where two of them have voltages which adds up to the voltage of the third. It was these ideas that led to the 3-junction detector work [9-10]. E. Devices and Measurements Our Josephson arrays are designed on a computer here at Colgate by undergraduate students and fabricated at Hypres, Inc. SEM Micrographs of a single junction, a Josephson Ring, a Josephson ladder, and a SQUID-ladder Array are shown in figure 4. We test our devices at temperatures from 0.25 K to about 9K (the transition temperature of niobium) in a pumped 3He system. We do straightforward measurements of the DC I-V (Current-Voltage) curves, varying temperature and the perpendicular external magnetic field


(a) (b) (c) Figure 5: (a) Measured I-V curve for a circular parallel array. The current increases from zero along a supercurrent branch, with a small amplifier offset. It then switches to about 3 mV; the value of current at this point is known as the switching current, Isw. The measurement is repeated for current of both signs, giving different values of Isw (called Isw+ and Isw- for positive and negative current). (b) Histogram of 5000 values of Isw. (c) Using information from the histogram and the rate at which the current is swept, the probability per unit time for the fluxon to switch from the pinned state to the running state can be calculated at each current. applied to the chip. We can measure the voltage of several junctions simultaneously and apply current to two or three different points in the circuit. The measurements are effectively DC (0-100 kHz), so we heavily filter our leads with powder/Pi filters [11] and RC filters. Typical experiments proceed with a sequence of current pulses to certain parts of the array while measuring voltages at certain junctions; these sequences are then repeated at different temperatures and magnetic fields. The I-V curves contain a lot of information about the dynamics of the array. As an example, figure 5(a) shows a typical experimental I-V curve of a Josephson Ring with a fluxon trapped inside. The I-V curve contains the dynamics of the fluxon. The current first increases along a supercurrent branch where V = 0; in the figure there is a small amplifier offset. Along this supercurrent branch the fluxon is pinned in the array, unable to move. Applying a current puts a force on the fluxon. When the current has reached a certain value, the fluxon becomes depinned and moves to a running state, where the fluxon continuously moves through the array. This causes a DC voltage to appear. The value of current at the point where the array switches from zero voltage is called the switching current, Isw. The exact value of Isw for a given measurement is affected by fluctuations in the system; thus Isw is a random variable. Measuring the I-V curve multiple times to obtain the probability distribution of Isw is called a “switching current measurement.” Switching current measurements have a long history in the Josephson community [ref], and we have made use of them as well. Figure 5(b) shows a histogram of 5000 values of Isw. In the data shown in figure 5(a), the magnitude of the switching current in the negative direction, which we call Isw-, is larger than in the positive direction, Isw+. Since the applied current is proportional to the force on the fluxon, it thus takes a larger force to move the fluxon in the negative direction (i.e. clockwise) than in the positive direction (counter-clockwise). This indicates that the fluxon is traveling in an asymmetric (or “ratchet”) potential.


In order to compare the measurements with theory, we calculate the average and standard deviation for a given switching current distribution. The information in the distribution of Isw+ or Isw-, along with knowledge of how fast the current was ramped, can be used to calculate the switching rate, the probability per unit time for the fluxon to switch to the running state [12]. This rate is shown in figure 5(c). The rate is exponential in current and follows Kramer’s law. The small scatter in the points in figure 5(c) is indicative of the high quality and low noise of our measurement technique. II. Previous Results A. Fluxon diffusion in a Josephson Ring Our prior NSF-sponsored work (DMR-0804865, “RUI: Classical and Quantum Ratchets in Josephson arrays,” 7/2008 – 6/2011, $175,000) was to study the dynamics of a fluxon trapped in a Josephson ring. This work yielded two publications [13-14]. They focused on the prediction and observation of a new mode of transport for a fluxon in a Josephson Ring. 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 [15]. In the first paper [13], we showed the existence of Fluxon diffusion through numerical simulations of the equations of motion in the ring. Together with my colleague Juan Mazo, we showed the conditions under which you would expect both a diffusion voltage and hysteresis. In the second paper [14], 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. In this experimental study [14], we were hampered in our efforts to compare experiment to theory because of a flaw in the design of the Josephson ring. There was an asymmetry in the way the current was applied and extracted from the ring. It had a separate lead going to each of the junctions to apply the current, but all the current returned through a single lead. This return lead had nine times the current of one of the entering leads (since there were nine junctions), and the magnetic field from this one return lead affected the motion of the fluxon. We addressed this with a new set of devices where the current both comes in and leaves through nine leads. An example of this new device was shown in figure 4d. As shown later in the proposal, the new design alleviated this problem.


B. Subgap biasing of a superconducting detector using a 3-junction circuit One of the active areas of Josephson junction research is the use of Josephson tunnel junctions as single photon detectors. In this context they are usually called “Superconducting Tunnel Junctions,” or STJs. STJs are a promising technology, but they require a parallel magnetic field to achieve the necessary biasing in the subgap voltage region, away from both the supercurrent and the full voltage state. Application of a parallel magnetic field is straightforward but could be a limiting factor for extension to large arrays of detectors. Also, it also prohibits use of these detectors in conjunction with a scanning electron microscope, a common application for these types of detectors. We recently proposed [9] a method to bias the junctions without a magnetic field. This method uses a circuit of three junctions in place of a single junction. The nonlinear interaction between different junctions in the circuit holds one of the junctions at a subgap voltage, which can then function as the detector junction. This nonlinear interaction is similar to the kind of interaction in a Discrete Breather. In a recent publication [10] we demonstrated this idea experimentally. A 3-junction detector was studied and its operating voltage was measured as a function of temperature. We compared with numerical simulations and found a very good match. C. Analog simulation of neuron dynamics 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. I, along with two colleagues at Colgate (Patrick Crotty, a computational neuroscientist, and Dan Schult, an applied mathematician), 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 [3], 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. III. Proposed Research The objective of this proposal is to continue the study of the dynamics of Josephson networks through the investigation of collective modes, their interaction, and the applicability to neural dynamics. In particular, we will focus on four areas: (a) the continued study of fluxon diffusion in a Josephson Ring; (b) the study of the interaction of a Discrete Breather and a moving vortex in a Josephson ladder; (c) the Kuramoto-type synchronization of a disordered SQUID-ladder; and (d) the first experimental demonstration of the Josephson Junction Neuron. We discuss (a)-(d) below. In addition, the testing of a 3-junction detector remains on the table, if time permits. This work will have relevance for the Josephson community, the nonlinear dynamics community, and the field of computational neuroscience. In the Josephson community, largely because of the success of the quantum computing field, there is interest in the collective behavior of large coupled Josephson circuits. Although we focus on the classical dynamics, the modes that we observe will still be present in smaller, quantum circuits. Our work will thus inform the quantum computing field, either to avoid or take advantage of the classical, collective behaviors we study. In addition, it is also relevant to the superconducting detector community, and to those studying Josephson arrays for RF sources. In the nonlinear dynamics community, the move toward collective behavior of large, coupled systems has been going on for years. Josephson systems have been an important contributor to that community, because of their flexibility in design, ability to reach different parameter regimes, and straightforward measurements. Finally, this work will reach across disciplinary boundaries to the computational neuroscience community. Our work on the Josephson junction neuron, although only computational at this point, has already attracted a lot of attention [4-6]. A simple experimental demonstration will be extremely important if this work is to move forward. If we can reach that community, there is a very high ceiling for the impact of the JJ neuron. To accomplish these goals we will perform low-temperature electrical measurements at Colgate with Josephson arrays that have been or will be fabricated elsewhere, primarily at Hypres, Inc. The measurements will be performed by the P.I., a post-doctoral assistant, and undergraduate students. The low-temperature experimental setup at Colgate is complete and is producing high quality data, and we have in our possession devices for all the studies except for (d), which we plan to design over the next year. Our experimental data will be compared with numerical simulations of the equations of motion. Our past NSF funds have purchased a quad-core, dual processor machine on which we run simulations in our group at Colgate. In addition, we will continue our collaboration with Juan Mazo from the University of Zaragoza for computational and theoretical assistance. Finally, we will continue to work with Colgate professors Dan Schult and Patrick Crotty. Crotty is a computational neuroscientist, and Schult is an applied mathematician who specializes in nonlinear dynamics. In working together to devise the


(a) (b) Figure 6: Data from new Josephson Ring shown in figure 4(b). (a) Full I-V curve, showing symmetric switching points under positive and negative current. (b) Blow-up of the boxed region under positive current showing the fluxon diffusion voltage. We compare to numerical simulation of the equations of motion for the array. Josephson Junction Neuron, the three of us have formed a tight collaboration. We meet regularly to discuss research and teach together in various classes. The broader impact of this proposal will be as part of Colgate’s strong tradition in undergraduate research. The P.I. will continue working closely with undergraduates to take measurements and understand the data, exposing the students to different aspects of the research. The undergraduates will attack difficult questions in a cutting-edge field, preparing them for careers in science and technology. In the P.I.’s first seven years as an assistant professor, 24 undergraduate students worked with the P.I. either during the school year or over the summer. Of those 24, 18 went on to graduate school in either physics or engineering, and two more are applying this year. The schools attended by these students include the following: Yale University, the University of California at Santa Barbara, Cornell University, Cambridge University (England), SUNY Stony Brook, R.P.I., the University of California at Davis, Columbia University, Boston University, University of Delaware, Virginia Tech, University of Maryland, Carnegie Melon and Washington University. One former student, a woman, won a national fellowship. A full table of the P.I.’s graduates and their career paths is given in the RUI impact statement. A. Fluxon dynamics in a Josephson Ring Our previous experimental study [14] on a Josephson ring suffered from a design flaw which was fixed in a new set of devices, one of which is shown in figure 4b. The flaw led to an asymmetry in the switching current for the two directions in applying the current, similar to that seen in the data shown in figure 5. This asymmetry should not be there, since the arrays were fabricated with all the junctions having the same size and all the cells having the same inductance. Figure 6a shows the I-V curves for the new devices. The I-V curves are now indeed symmetric. The positive and negative switching currents are the same to a few percent.


In addition, the small fluxon diffusion voltage at currents before the switching occurs is clearly visible. Figure 6b shows a blow-up of the low-voltage range in 6a. We have now normalized the current to (NIC), where N=9 is the number of junctions and IC is the critical current of each junction, and normalized the voltage to (ICR), where R is the normal state resistance of each junction. Since the I-V curves are now symmetric, we can compare these results with theory. A numerical I-V curve is also plotted in figure 6b. The parameters for this curve are the same as in the experiment, except for the rate at which the current is ramped. Because the simulation is so much slower than the experiment, the simulation was ramped 100 times faster than the data was taken. If the ramp is slower, the system tends to switch to non-zero voltage much faster, which is why the data switch earlier than the simulation. We have attempted to correct for this (dotted line in 6b), but more work needs to be done. In the meantime one can see that the agreement is still pretty reasonable, as the diffusion voltage in the data and the simulation are almost identical. To continue this work, we will simply run the same devices and test different temperatures and different numbers of fluxons trapped in the array, as well as testing different arrays with different values of λ, the fluxon wavelength. 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” [16]. Now that our regular arrays are behaving as they should, we can test those structures as well and have confidence that the asymmetry is real. 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 [17-18]. B. Interaction of a breather and a vortex in a Josephson ladder To date, much of the study of collective modes such as shown in figure 3 has been on each mode 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 [19]. 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. We have designed and received arrays that have parameters that match the work of Trias et al. The procedure for experimentally exciting a breather has been documented in past work, and we have been able to excite breathers in past experiments. The problem that we face is to be able to experimentally


(b) (c) Figure 7: Simulations of breather-vortex collisions. (a) Schematic of the ladder array, showing the injection current in red and the array current in grey. The injection current puts flux into one cell of the ladder, exciting a vortex which then travels down the ladder toward the breather. The simulation calculates the voltage of the junction before the breather, (b), and after the breather, (c). The voltage color scale is “rainbow”, with blue representing lower voltages and red representing higher voltages. The boxed region shows a set of currents where the junction before the breather has switched to the voltage state while the junction after the breather has not. Thus, at those currents, the breather has stopped or “pinned” the vortex. At other places which are at large voltage in the second plot the breather has been destroyed. The other differences in the two plots, around injection currents of 0 and 100 and array currents of 120, is due to another effect which is not related to the breather-vortex collision. generate a vortex on demand. In the numerical work by Trias et al., the vortex was put into the ladder by forcing a kink in the phases numerically; there is no conventional way to do this in an experiment as of yet. Juan Mazo and I have worked on this problem and have come up with the idea of using a flux injection line which puts flux in just one of the cells of the array. We have tried this in numerical simulations, and this seems to create a vortex in the ladder. With this as the method of injecting the vortex, we have done simulations which now simply adjust the amount of injection flux into one cell in the array, increase the current in the array, and calculate the voltages of all the junctions. The results are shown in figure 7, where we simulate the dynamics of the ladder using the parameters of the array we have and use an injection current to start a vortex moving down the ladder. We focus on the junction before the breather, figure 7b, and the junction after the breather, figure 7c. In the figure we can clearly see


where the breather has pinned the vortex (boxed region) and where the breath has been destroyed (other purple regions in the second plot). With this device in our possession, we are ready to take the data.

C. Kuramoto-Type Synchronization of a disordered SQUID-ladder array The problem of coupled oscillators is ubiquitous in nature, from fireflies to heart cells to neurons in the brain [20-21]. One particularly important collective behavior which can occur in coupled oscillators is synchronization. Synchronization occurs when oscillators that have different natural frequencies begin to oscillate together because they are coupled to each other. For example, in the brain many neurons have their own natural frequency, the frequency they would like to fire at when they are not coupled to each other. Since these frequencies vary, in general there is no coherence in the firing times of different neurons. However, during certain periods of sleep or deep thought some neurons become strongly coupled to each other and start to fire in unison. These synchronized states can be measured in EEG (Electroencephalography) scans and typically represent thousands of neurons firing synchronously. The problem of how synchronization occurs in a group of coupled oscillators was first taken on by Kuramoto [22]. He assumed a certain form of the coupling between oscillators: the phase change of a given oscillator is proportional to the sine of the phase difference between itself and each neighbor:

( )∑ −

+==

N

jijii N

K

1sin θθωθ& . (2)

Here ωι is the natural frequency of the ith oscillator, K is the coupling strength and N is the total number of oscillators. With these assumptions, one can derive how the order parameter, a measure of how synchronized the system is at a given time, changes with the coupling K. As one varies the coupling from weak to strong, an abrupt onset of synchronization is seen. Because the specific form of the coupling is a sine function, this kind of synchronization lends itself particularly well to Josephson arrays; the supercurrent through a Josephson junction is also proportional to the sine of the phase difference. The Josephson ladder is the geometry of choice, because the vertical junctions can be thought of as the oscillators while the horizontal junctions act to couple them together. Daniels et al. [23] have worked out the theory for this case. They assumed different areas for each of the vertical junctions, giving them different natural frequencies when they are in the voltage state. By varying the size of the horizontal junctions, they were able to vary the coupling between the vertical junctions. As the coupling increased, they observed a transition to the synchronized state. We have started to try to realize the work of Daniels et al. in an experiment. To use a Josephson ladder to test this idea is difficult, because the coupling cannot be varied during the experiment; one would need a different array with different sized horizontal junctions for each value of the coupling. Instead, we decided to study the SQUID-ladder array, replacing each horizontal junction with a SQUID. Now, the flux can be varied in the experiment to get different values of the coupling. We have simulated this geometry and it does indeed show a varying amount of


synchronization. We have designed an array and now have it in our possession, as shown in figure 4(d). Our first cool-down with this device did indeed show that one of the junctions changed the size of its voltage (or rotation frequency) in response to a magnetic field (data not shown). We are thus excited that this project is on its way toward yielding results. D. Preliminary Testing of Josephson junction neurons Our initial publication [3] on the Josephson junction neuron was purely computational. In that work we demonstrated that a JJ Neuron could produce action potentials and also has a threshold current and a refractory period. The form of the action potential was consistent with being produced by activating and restoring forces, the same way that action potentials are produced in real neurons. Synapses could be represented by transmission line RLC filters. Both inhibitory and excitatory coupling from one neuron to the next could be achieved. Our goal in the first experimental work is to fabricate the first JJ Neurons and show that all of these properties exist. We will design circuits exactly as described in [3]. To detect the action potentials, we will use a simple RSFQ circuit, the so-called SFQ-to-DC converter [24]. This will give us a simple DC voltage that is proportional to the firing rate of each JJ Neuron. First, we will excite a single neuron and attempt to measure its threshold current and its refractory period. For the case of the threshold current, we should find that the output of the SFQ-to-DC converter goes from zero to some finite value when we reach the threshold. For the refractory period, we should find that as we increase the input current, we reach a maximum firing rate. Next, we will couple one neuron to another in an excitatory fashion, and show by exciting the first neuron, that the second neuron fires at a similar rate to the first. Finally, we will couple one neuron to another in an inhibitory fashion, and show that by exciting the first neuron, the second neuron has its firing rate decrease. These designs will be a little more difficult than previous ones because, following other RSFQ work, we need to work over a ground plane and tune our inductances very carefully. We will use freeware by Whiteley research to estimate the inductance of all of our structures. Buffer junctions [25] will be used for starters to make sure that our elements connect to one another without any impedance mismatch. We will be careful in the design, probably spending the first few months of the funding period making sure that it is solid. After sending out the design, we should have devices to test in about nine months into the funding period. ____________________________________________________________________________

In conclusion, this proposal aims to continue the work of our laboratory in studying nonlinear and neural dynamics in Josephson networks. We will build on our previous results from NSF funding and answer the open questions regarding the dynamics of a fluxon in a 1-D Josephson array. Hopefully we will attain the first results on breather-vortex collisions, the synchronization of SQUID-ladder arrays, and the demonstration of the Josephson junction neuron. Undergraduates will be heavily involved in this work, preparing them for careers in the technical sciences.


References

[1] T.P. Orlando and K.A. Delin, Foundations of Applied Superconductivity, Addison-Wesley Publishing, 1991.

[2] B.D. Josephson, Physics Letters 1, 251 (1962). [3] P. Crotty, D. Schult and K. Segall, “Josephson junction simulation of neurons,” Physical

Review E 82, 011914 (2010) [4] http://physicsworld.com/cws/article/news/41916, “Superconductors could simulate the

brain.” [5] http://physicsandcake.wordpress.com/2010/02/23/josephson-junction-neurons, “Josephson

junction neurons.” [6] “CU Claims Superconductors Simulate Neuronal Activity,” Superconductor Week, Vol. 24,

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Experimental Detection in a Josephson Array,“ Physical Review Letters 84, 741 (2000); P. Binder, D. Abriamov, A.V. Ustinov, S. Flach and Y. Zolotaryuk,“Observation of Breathers in Josephson Ladders,“ Physical Review Letters 84, 745 (2000).

[8] K.K. Likharev, Dynamics of Josephson junction circuits, Gordon and Breach, 1986. [9] K. Segall, J.J. Mazo and T.P. Orlando, “Multiple-junction biasing of superconducting

tunnel junction detectors,” Applied Physics Letters 86, 153507 (2005). [10] K. Segall, J. Moyer and J.J. Mazo, “Subgap biasing of Superconducting Tunnel Junctions

without a Magnetic Field,” Journal of Applied Physics 104, 043920 (2008). [11] A. Lukashenko and A.V. Ustinov, “Improved Powder Filters for Qubit Measurements,”

Review of Scientific Instruments 79, 014710 (2008) [12] T. Fulton and L.N. Dunkleberger, “Lifetime of the zero-voltage state in Josephson tunnel

junctions,” Physical Review B9, 4760-4768 (1974). [13] J.J. Mazo, F. Naranjo and K. Segall, “Thermal depinning of Josephson Fluxons in

superconducting rings,” Physical Review B78, 174510 (2008). [14] K. Segall, A. Dioguardi, N. Fernandes and J.J. Mazo, “Experimental observation of Fluxon

Diffusion in Josephson Rings,” Journal of Low Temperature Physics 154, 41-54 (2009). [15] R.L. Kautz and J.M. Martinis, “Noise-affected I-V curves in small hysteretic Josephson

junctions,” Physical Review B42, 9903-37 (1990). [16] F. Falo, J.J. Mazo, K. Segall, T.P. Orlando, and E. Trias, “Fluxon ratchet potentials in

superconducting circuits,” Applied Physics A75, 263-269 (2002). [17] R.D. Astumian, “Making molecules into motors,” Scientific American 285, 57-64 (2001). [18] P. Reimann, “Brownian motors: Noisy transport far from equilibrium,” Physics Reports 361,

57-265 (2002). [19] E. Trias, J.J. Mazo and T.P. Orlando, “Interactions between Josephson vortices and

breathers,” Physical Review B 65, 054517 (2002). [20] S.H. Strogatz, Sync: The emerging science of spontaneous order, Hyperion, (2003). [21] S.H. Strogatz, "From Kuramoto to Crawford: exploring the onset of synchronization in

populations of coupled oscillators", Physica D, 143:1-20 (2000).


[22] Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, Springer-Verlag, 1984. [23] B.C. Daniels, S.T.M. Dissanayake, and B.R. Trees, “Synchronization of coupled rotators:

Josephson junction ladders and the locally coupled Kuramoto Model,” Physical Review E67, 026216 (2003).

[24] P. Bunyk, K. Likharev, D. Zinoviev, “RSFQ Technology: Physics and Devices”, International Journal on High Speed Electronics and Systems 11:257-306 (2001).

[25] Kadin, Alan, M., Introduction to Superconducting Circuits, Wiley, USA, (1999).

Documents

Project Description I. Introduction: Josephson junction ...departments.colgate.edu/physics/faculty/KenSegall/Project... · I. Introduction: Josephson junction networks ... considered