Phase-locked Regimes Research Articles

Shapiro steps for skyrmion motion on a washboard potential with longitudinal and transverse ac drives

We numerically study the behavior of two-dimensional skyrmions in the presence of a quasi-one-dimensional sinusoidal substrate under the influence of externally applied dc and ac drives. In the overdamped limit, when both dc and ac drives are aligned in the longitudinal direction parallel to the direction of the substrate modulation, the velocity-force curves exhibit classic Shapiro step features when the frequency of the ac drive matches the washboard frequency that is dynamically generated by the motion of the skyrmions over the substrate, similar to previous observations in superconducting vortex systems. In the case of skyrmions, the additional contribution to the skyrmion motion from a non-dissipative Magnus force shifts the location of the locking steps to higher dc drives, and we find that the skyrmions move at an angle with respect to the direction of the dc drive. For a longitudinal dc drive and a perpendicular or transverse ac drive, the overdamped system exhibits no Shapiro steps; however, when a finite Magnus force is present we find pronounced transverse Shapiro steps along with complex two-dimensional periodic orbits of the skyrmions in the phase-locked regimes. Both the longitudinal and transverse ac drives produce locking steps whose widths oscillate with increasing ac drive amplitude. We examine the role of collective skyrmion interactions and find that additional fractional locking steps occur for both longitudinal and transverse ac drives. At higher skyrmion densities, the system undergoes a series of dynamical order-disorder transitions, with the skyrmions forming a moving solid on the phase locking steps and a fluctuating dynamical liquid in regimes between the steps.

Dynamic trajectory analysis of superparamagnetic beads driven by on-chip micromagnets

We investigate the non-linear dynamics of superparamagnetic beads moving around the periphery of patterned magnetic disks in the presence of an in-plane rotating magnetic field. Three different dynamical regimes are observed in experiments, including (1) phase-locked motion at low driving frequencies, (2) phase-slipping motion above the first critical frequency fc1, and (3) phase-insulated motion above the second critical frequency fc2. Experiments with Janus particles were used to confirm that the beads move by sliding rather than rolling. The rest of the experiments were conducted on spherical, isotropic magnetic beads, in which automated particle position tracking algorithms were used to analyze the bead dynamics. Experimental results in the phase-locked and phase-slipping regimes correlate well with numerical simulations. Additional assumptions are required to predict the onset of the phase-insulated regime, in which the beads are trapped in closed orbits; however, the origin of the phase-insulated state appears to result from local magnetization defects. These results indicate that these three dynamical states are universal properties of bead motion in non-uniform oscillators.

Effects of periodic forcing on the temporally correlated spikes of a semiconductor laser with feedback.

Optical excitable devices that mimic neuronal behavior can be building-blocks of novel, brain-inspired information processing systems. A relevant issue is to understand how such systems represent, via correlated spikes, the information of a weak external input. Semiconductor lasers with optical feedback operating in the low frequency fluctuations regime have been shown to display optical spikes with intrinsic temporal correlations similar to those of biological neurons. Here we investigate how the spiking laser output represents a weak periodic input that is implemented via direct modulation of the laser pump current. We focus on understanding the influence of the modulation frequency. Experimental sequences of inter-spike-intervals (ISIs) are recorded and analyzed by using the ordinal symbolic methodology that identifies and characterizes serial correlations in datasets. The change in the statistics of the various symbols with the modulation frequency is empirically shown to be related to specific changes in the ISI distribution, which arise due to different phase-locking regimes. A good qualitative agreement is also found between simulations of the Lang and Kobayashi model and observations. This methodology is an efficient way to detect subtle changes in noisy correlated ISI sequences and may be applied to investigate other optical excitable devices.

Nonlinear cyclotron acceleration of massless Dirac charge carriers in graphene and topological insulators

Two-dimensional motion of quasiparticles with the Dirac dispersion relation in an external magnetic field in the presence of electromagnetic radiation has been investigated. It has been shown that the mechanisms of acceleration in the phase-locked (autoresonance) regime known for classical charged particles can also take place for massless charge carriers. In this case, the energy distribution function of electrons exhibits a gap, the magnitude of which is governed by the magnetic field.

Phase-locked regimes in delay-coupled oscillator networks.

For an ensemble of globally coupled oscillators with time-delayed interactions, an explicit relation for the frequency of synchronized dynamics corresponding to different phase behaviors is obtained. One class of solutions corresponds to globally synchronized in-phase oscillations. The other class of solutions have mixed phases, and these can be either randomly distributed or can be a splay state, namely with phases distributed uniformly on a circle. In the strong coupling limit and for larger networks, the in-phase synchronized configuration alone remains. Upon variation of the coupling strength or the size of the system, the frequency can change discontinuously, when there is a transition from one class of solutions to another. This can be from the in-phase state to a mixed-phase state, but can also occur between two in-phase configurations of different frequency. Analytical and numerical results are presented for coupled Landau-Stuart oscillators, while numerical results are shown for Rössler and FitzHugh-Nagumo systems.

Perfect and robust phase-locking of a spin transfer vortex nano-oscillator to an external microwave source

We study the synchronization of the auto-oscillation signal generated by the spin transfer driven dynamics of two coupled vortices in a spin-valve nanopillar to an external source. Phase-locking to the microwave field hrf occurs in a range larger than 10% of the oscillator frequency for drive amplitudes of only a few Oersteds. Using synchronization at the double frequency, the generation linewidth is found to decrease by more than five orders of magnitude in the phase-locked regime (down to 1 Hz, limited by the resolution bandwidth of the spectrum analyzer) in comparison to the free running regime (140 kHz). This perfect phase-locking holds for frequency detuning as large as 2 MHz, which proves its robustness. We also analyze how the free running spectral linewidth impacts the main characteristics of the synchronization regime.

The interplay between STDP rules and anticipated synchronization in the organization of neuronal networks

Neuronal networks are able to synchronize in phase-locking regimes. Depending on the structural connectivity it is possible to determine the sign of the phase difference between oscillatory nodes. Specially in unidirectionally coupled motifs the relative phase difference is usually positive. However it has been shown that in the presence of dynamical inhibitory loops, unidirectionally coupled neuron models may present both positive or negative phase difference [1]. The unusual regime in which the phase difference is negative is called anticipated synchronization (AS) [2]. It means that two neurons or two neuronal populations coupled in a master-slave configuration can oscillate in such a way that the slave leads the master. These results are important in light of the growing experimental evidence that the synaptic strength between neurons can undergo spike-timing-dependent plasticity (STDP) [3]. In the delayed synchronization regime the master (pre-synaptic) neuron fires a spike before the slave (post-synaptic) neuron, which under STDP rules would facilitate long term potentiation (LTP), whereas in the AS regime the slave neuron fires a spike before the master neuron, contributing to long term depression (LTD) [4]. Since it has been shown previously that a simple 3-neuron motif can undergo a continuous transition from positive to negative values of relative phase via changes in synaptic conductances [1], the interplay between these regimes and STDP mechanisms is likely to play a very significant role in the organization of the system dynamics. Considering that in certain situations the relative potentiation in strong synapses is less intense than in weak synapses, but the depression do not show this dependence [5], we compare the effects of additive, multiplicative and hybrid rules in our motifs. Together with AS regimes in neuronal populations, STDP rules give synaptic weight distributions that are comparable to experiments in cortex [6,7].

Heterogeneous Connections Induce Oscillations in Large-Scale Networks

Realistic large-scale networks display a heterogeneous distribution of connectivity weights that might also randomly vary in time. We show that, depending on the level of heterogeneity in the connectivity coefficients, different qualitative macroscopic and microscopic regimes emerge. We evidence, in particular, generic transitions from stationary to perfectly periodic phase-locked regimes as the disorder parameter is increased, both in a simple model treated analytically and in a biologically relevant network made of excitable cells.

Phase-locking regimes of photonic crystal nanocavity laser arrays

We model and analyze the dynamical properties of coupled photonic crystal nanocavity lasers. The model includes Purcell enhancement of the spontaneous emission and intercavity coupling. The coupling strength between neighboring cavities is an essential parameter, and by performing finite-difference time-domain calculations, the typical coupling strength is extracted for realistic structures. Phase-locking regimes are identified, and their stability with respect to parameter variation is investigated. The results suggest that quantum well devices are not well suited for phase-locked nanocavity laser array devices.

Nonlinear energy transfer during the transition to drift-interchange turbulence

The transition to drift-interchange turbulence is investigated by estimating the nonlinear spectral power transfer functions of density and potential fluctuations representative for the behavior of the polarization and E × B drift nonlinearities using the magnetic field strength as the control parameter. As the control parameter increases the system undergoes several changes from a quasi-periodic to a phase locked to a weakly turbulent regime. The polarization drift nonlinearity transfers kinetic energy to larger scales. The E × B drift nonlinearity provides access to the free energy by coupling density fluctuations at smaller scales to the larger scale potential structures, resulting in a formation of a very robust phase-locked regime and acting as a mechanism of self-sustainment in the weakly turbulent regime. Although the interplay between these nonlinearities is different in the different regimes, it results in a successive increase in the degrees of freedom in all regimes, which is the most important feature of a transition to turbulence.

Multiplexing superparamagnetic beads driven by multi-frequency ratchets

Here, we explore the single particle dynamics of superparamagnetic beads exposed to multifrequency ratchets. Through a combination of theory, simulation, and experiment, we determine the important tuning parameters that can be used to implement multiplexed separation of polydisperse colloidal mixtures. In particular, our results demonstrate that the ratio of driving frequencies controls the transition between open and closed trajectories that allow particles to be transported across a substrate. We also demonstrate that the phase difference between the two frequencies controls not only the direction of motion but also which particles are allowed to move within a polydisperse mixture. These results represent a fundamentally different approach to colloidal separation than the previous methods which are based on controlling transitions between phase-locked and phase-slipping regimes, and have a higher degree of multiplexing capabilities that can benefit the fields of biological separation and sensing as well as provide crucial insights into general ratchet behavior.

Shunt-capacitor-assisted synchronization of oscillations in intrinsic Josephson junctions stack

We show that a shunt capacitor, by coupling each Josephson junction to all the other junctions, stabilizes synchronized oscillations in an intrinsic Josephson junction stack biased by a dc current. This synchronization mechanism is similar to the previously discussed radiative coupling between junctions, however, it is not defined by the geometry of the stack. It is particularly important in crystals with smaller numbers of junctions (where the radiation coupling is weak), and is comparable with the effect of strong super-radiation in crystals with many junctions. The shunt also helps to enter the phase-locked regime in the beginning of the oscillations, after switching on the bias current. Furthermore, it may be used to tune radiation power, which drops as the shunt capacitance increases.

Regular and chaotic transport of discrete solitons in asymmetric potentials

Ratchet dynamics of topological solitons of the forced and damped discrete double sine-Gordon system are studied. Directed transport occurring both in regular and in chaotic regions of the phase space and its dependence on damping, amplitude, and frequency of the driving, asymmetry parameter, and coupling constant, has been extensively investigated. We show that the passage from ratchet phase-locked regime to chaotic ratchets occurs via a period doubling route to chaos and that, quite surprisingly, pinned states can exist inside phase locking and chaotic transport regions for intermediate values of the coupling constant. The possibility to control chaotic discrete soliton ratchets by means of both small subharmonic signals and more general periodic drivings, has also been investigated.

Phase-locking and frustration in an array of nonlinear spin-torque nano-oscillators

We demonstrate that the cooperative dynamics of an array of coupled spin-torque nano-oscillators (STNO) can be controlled by introduction of an additional external phase shift βc between microwave current, which couples STNOs, and microwave voltage on the array. When this external phase shift βc compensates the intrinsic phase shift β0, caused by the STNO nonlinearity, a phase-locking regime with increased output power and vanishing inhomogeneous linewidth broadening is achieved. In the opposite case, when external and intrinsic phase shifts are added, the STNO array demonstrates a frustration regime with low output power and wide and noisy frequency spectrum.

Intercellular Communication Via Intracellular Calcium Oscillations

In this letter, we present the results of a simple model for intercellular communication via calcium oscillations, motivated in part by a recent experimental study. The model describes two cells (a “donor” and “sensor”) whose intracellular dynamics involve a calcium-induced, calcium release process. The cells are coupled by assuming that the input of the sensor cell is proportional to the output of the donor cell. As one varies the frequency of calcium oscillations of the donor cell, the sensor cell passes through a sequence of N : M phase-locked regimes and exhibits a “Devil's staircase” behavior. Such a phase-locked response has been seen experimentally in pulsatile stimulation of single cells. We also study a stochastic version of the coupled two-cell model. We find that phase locking holds for realistic choices for the cell volume.

Decadal Variability of Two Oceans and an Atmosphere

A model of the midlatitude, large-scale interaction between the upper ocean and the troposphere is used to illustrate possible mechanisms of connection between the decadal variability in the North Atlantic and in the North Pacific. The two ocean basins are connected to each other only through their coupling to the common, zonally averaged atmosphere. The ocean–atmosphere coupling takes place via wind-driven torques and heat fluxes at the air–sea interface. In this formulation, the decadal variability in each ocean basin consists of ocean–atmosphere modes and arises from a delayed feedback of the upper-ocean heat content onto the wind-driven flow mediated by the atmosphere through the requirements of global heat and momentum balances. The presence of two ocean basins leads to three basic kinds of coupling-induced behavior: phase locking, oscillation death, and chaos. In the phase-locked regime, the western boundary currents of the two basins oscillate in synchrony, with the narrower basin following the wider basin by a small time lag. In the oscillation death solutions, a steady solution is reached, even though each ocean basin, when uncoupled, would have sustained oscillations. In the chaotic regime, the interbasin coupling induces aperiodic fluctuations in both basins characterized by variability at centennial, as well as decadal, timescales.

Active length control of two phase locked CO2 lasers with a digital signal processor

Two CO2 lasers are phase locked together by injection locking, where radiation between the two lasers is exchanged using a copper prism as a beam folding device in the resonators. Extraction of the output radiation is achieved by a common output coupler. As this method is only utilizing reflective optics (except for the output coupler), it is also practical as a technique for phase-locking lasers with powers in the multikilowatt range. Single frequency oscillation is only achieved if the lengths of both resonators are within a so-called locking range. This can only be fulfilled if the length of one resonator is actively controlled. The method of detection of the phase difference between the lasers and actively controlling the length of one resonator is presented as well as its implementation into a digital signal processor. Stable phase locked operation is achieved and proved through detection of twice the intensity in the far field in the phase-locked regime.

Mechanisms of optical confinement in phase-locked laser arrays

The nature of optical confinement in phase-locked laser arrays (PLLAs) with a mesa-stripe structure (MSS) has been studied. Two main mechanisms are distinguished, which are based on the variation of the waveguide effective refractive index due to MSS formation and on the refractive index modulation induced by the heating of the structure. Stable operation was achieved when either weak or strong optical coupling was realized in the PLLA. A phase-locked regime of radiation was obtained only for laser diodes with strong optical coupling. In the latter case the angle divergency was not greater than for the antisymmetric supermode emission from the PLLA.

Coherent flux-flow in vertically stacked long Josephson tunnel junctions

Vertical stacks of two Nb/Al-AlO/sub x//Nb long Josephson junctions with nearly identical parameters have been studied experimentally. The magnetic coupling between the junctions was provided by their common electrode of thickness d/spl sime//spl lambda//sub L/. The I-V characteristics displayed flux-flow resonances of the two junctions. Two different mutually phase-locked regimes were found: the lower velocity anti-phase mode, and the high velocity in-phase mode. These modes, while staying locked, could be tuned by the magnetic field in a broad voltage range which corresponds to the Josephson frequencies of 150-600 GHz for the lower mode and 280-400 GHz for the upper mode. Our observations suggest applications of the stacked long Josephson junctions as local oscillators for millimeter and sub-millimeter wave circuits. >

Uniform coupling of microwaves to nonlinear resonant modes in Josephson junctions.

Uniform coupling of ac fields to Fiske modes in sine-Gordon systems is considered. We demonstrate that a uniformly distributed ac force gives rise to linear and symmetric phase-locked regimes for the Fiske modes in the limit of small amplitude perturbations. In the general case of large amplitudes of the ac force we demonstrate that phase locking can be described much like the Shapiro step for a single pendulum. We give a comprehensive analytical treatment of the problem and compare the results to direct numerical simulations of the perturbed sine-Gordon equation. Excellent agreement is found.