Detection of signals in presence of noise through Josephson junction switching currents.

Josephson junctions can be employed to reveal a sinusoidal signal in presence of Gaussian noise. To mimic realistic setups, the detection is performed linearly ramping the bias current until a switch to the finite voltage occurs; the analysis of the resulting switching currents can be exploited to decide about the presence of the harmonic drive. The signal is applied in two conditions: with an unknown initial phase (incoherent strategy) and with a known initial phase (coherent strategy). In both conditions, the analysis of the efficiency of the detection, performed through the signal-to-noise ratio, as estimated by the Kumar-Carrol index, shows that the dependence upon the Josephson junction ramp rate is beneficial, especially for relatively fast speed. One can conclude that the collection of the switching currents is a robust technique, and thus it is possible to exploit the advantages of a predetermined finite time to collect the data.

Noise and disorder effects in a series of birhythmic Josephson junctions coupled to a resonator.

The effects of uncorrelated white noise on a series of Josephson junctions coupled to a linear RLC resonator are investigated. Both the cases of identical and nonidentical parameters are considered. The junctions are hysteretic, and hence can be considered birhythmic; that is, capable of oscillating at different frequencies for the same set of parameters. It is also found that, in the presence of noise, a uniform array behaves similarly to a single Josephson junction. However, the magnitude of the effective energy that characterizes the response to noise becomes smaller as the number of elements of the array increases, making the resonator less stable. Disorder in the parameters drastically changes the physics of the array. The disordered array of Josephson junctions misses the birhythmic properties for large values of the variance of the disorder parameter and remains birhythmic only for low values of the disorder parameter. Finally, disorder makes it difficult to locate the separatrix, hinting to a more complex structure of the effective energy landscape.

Noise effects on a birhythmic Josephson junction coupled to a resonator.

We study the effect of noise on a Josephson junction that, coupled to a linear RLC resonator, can oscillate at two frequencies. To establish the global stability of the attractors, we estimate the position of the separatrix, essential information to establish the stability of the attractor for this multidimensional system, from the analysis of the mean first passage time. We find that the frequency locked to the resonator is most stable at low bias and less stable at high bias, where the resonator exhibits the largest oscillations. The change in the birhythmic region is dramatic for the effective barrier changes of an order of magnitude and the corresponding lifetime of about seven decades.

Effective Fokker-Planck equation for birhythmic modified van der Pol oscillator.

We present an explicit solution based on the phase-amplitude approximation of the Fokker-Planck equation associated with the Langevin equation of the birhythmic modified van der Pol system. The solution enables us to derive probability distributions analytically as well as the activation energies associated with switching between the coexisting different attractors that characterize the birhythmic system. Comparing analytical and numerical results we find good agreement when the frequencies of both attractors are equal, while the predictions of the analytic estimates deteriorate when the two frequencies depart. Under the effect of noise, the two states that characterize the birhythmic system can merge, inasmuch as the parameter plane of the birhythmic solutions is found to shrink when the noise intensity increases. The solution of the Fokker-Planck equation shows that in the birhythmic region, the two attractors are characterized by very different probabilities of finding the system in such a state. The probability becomes comparable only for a narrow range of the control parameters, thus the two limit cycles have properties in close analogy with the thermodynamic phases.

Global stability analysis of birhythmicity in a self-sustained oscillator.

We analyze the global stability properties of birhythmicity in a self-sustained system with random excitations. The model is a multi-limit-cycle variation in the van der Pol oscillator introduced to analyze enzymatic substrate reactions in brain waves. We show that the two frequencies are strongly influenced by the nonlinear coefficients alpha and beta. With a random excitation, such as a Gaussian white noise, the attractor's global stability is measured by the mean escape time tau from one limit cycle. An effective activation energy barrier is obtained by the slope of the linear part of the variation in the escape time tau versus the inverse noise intensity 1/D. We find that the trapping barriers of the two frequencies can be very different, thus leaving the system on the same attractor for an overwhelming time. However, we also find that the system is nearly symmetric in a narrow range of the parameters.

Generalized coupling in the Kuramoto model.

We propose a modification of the Kuramoto model to account for the effective change in the coupling constant among the oscillators, as suggested by some experiments on Josephson junction, laser arrays, and mechanical systems, where the active elements are turned on one by one. The resulting model is analytically tractable and predicts that both first and second order phase transitions are possible, depending upon the value of a new parameter that tunes the coupling among the oscillators. Numerical simulations of the model are in accordance with the analytical estimates, and in qualitative agreement with the behavior of Josephson junctions coupled via a cavity.