Noise-mediated dynamics in a two-dimensional oscillator: Exact solutions and numerical results

– We derive a Fokker-Planck equation (FPE) to analyze the oscillator equations describing a nonlinear amplifier, exemplified by a two-junction Superconducting Quantum Interference Device (SQUID), in the presence of thermal noise. We show that the FPE admits a unique stationary solution and obtain analytical results for several parameters ranges. To solve the FPE numerically, we develop an efficient spectral method which exploits the periodicity of the probability density. The numerical method, combined with the exact solutions, allow us to rapidly explore the noise-mediated dynamics as a function of the control parameters. The study of nonlinear dynamical behavior in systems that undergo bifurcations via changing a control parameter is of considerable interest. Tuning these systems to the onset of the bifurcation can lead to very large changes in the output in response to a very small external perturbation, resulting in a high gain and great sensitivity to a “target” input or forcing signal. Recent work has focused on the onset of spontaneous oscillations, via a saddle-node connection, in one realization out of many sharing such dynamics: the two-junction (or “dc”) SQUID [1]. The dc SQUID is characterized [2] by a two-dimensional (2D) set of dynamical equations for the junction Schrödinger phase differences. Our interest in the SQUID stems from its relevance as the most sensitive detector of magnetic fields. Experimental and numerical results [3, 4] have shown that the optimal response of the SQUID to an input signal occurs just beyond the bifurcation point. Noise, present from a variety of sources, can however change the dynamical response of the system and understanding the effect of noise on the SQUID is of obvious importance. In this letter, we examine the dynamics of a dc SQUID in the presence of thermal Johnson noise generated in the resistive shunts; this manifests itself as white voltage noise sources in the junctions. Previous calculations have been mostly numerical (e.g., [5]); instead, in this letter we use a Fokker-Planck equation (FPE) approach which yields the possibility of extracting analytic information. Our analytical results are compared against very efficient numerical simulations of the FPE. The SQUID dynamics are described by equations for the time-derivatives of the Schrödinger phase differences δi across the (assumed identical) Josephson junctions [2, 4]: τ δ̇i = Ib/2 + (−1)Is − I0 sin δi, i = 1, 2. (1) (∗) E-mail: [email protected] c © EDP Sciences J. A. Acebrón et al.: Noise-mediated dynamics in a two-dimensional etc. 355 The circulating current Is, induced in the loop by an external magnetic flux, is the experimental observable of interest and can be written in the form βIs/I0 = δ1 − δ2 − 2πΦe/Φ0. Here, τ = h̄/(2eR) is a characteristic time constant (R being the normal-state resistance of the junctions), β ≡ 2πLI0/Φ0 the nonlinearity parameter, L the loop inductance, I0 the junction critical current and Φ0 ≡ h/2e the flux quantum. The two natural experimental control parameters are the applied dc magnetic flux Φe and the dc bias current Ib, which we take to be symmetrically applied to the loop. It is convenient to rescale time by τ and introduce a scaled flux Φex ≡ Φe/Φ0 and bias current J ≡ Ib/(2I0). Inserting the Langevin noise sources in the dynamics yields the system δ̇1 = J − 1 β (δ1 − δ2 − 2πΦex)− sin δ1 + ξ1(t), δ̇2 = J + 1 β (δ1 − δ2 − 2πΦex)− sin δ2 + ξ2(t), (2) where ξi is white Gaussian noise having zero mean and correlation function 〈ξi(t)ξj(t′)〉 = 2Dδijδ(t− t′). As mentioned above, this system exhibits two (in the absence of noise, distinct) regimes of operation [4, 6]. For a fixed Φex, a saddle-node connection takes place when the bias current J exceeds a critical value Jc. For J < Jc, the noiseless system has two fixed points, one stable (a node) and one unstable (a saddle) [7]. This is the “superconducting regime” with the potential energy function admitting of stable minima corresponding to a current conservation 2J = sin δ1 +sin δ2. For J > Jc the fixed points disappear and we obtain oscillatory solutions whose frequency obeys the characteristic square-root scaling law [6]. This latter regime is the so-called “running regime.” The properties of the solutions near the bifurcation have recently been studied [6]. The coupled Langevin equations (2) lead to a Fokker-Planck equation for the probability density ρ(δ1, δ2, t): ∂ρ ∂t = D [ ∂ρ ∂δ2 1 + ∂ρ ∂δ2 2 ] − ∂ ∂δ1 (v1 ρ)− ∂ ∂δ2 (v2 ρ), (3) and v1(δ1, δ2, t) = J − 1 β (δ1 − δ2 − 2πn− 2πΦex)− sin δ1, v2(δ1, δ2, t) = J + 1 β (δ1 − δ2 − 2πn− 2πΦex)− sin δ2, (4) where v1,2 are the drift terms (with 2π-periodicity in δ1 and δ2), and the density function is normalized, ∫ 2π 0 ∫ 2π 0 ρ(δ1, δ2, t) dδ1 dδ2 = 1, with n chosen in order to obtain a periodic continuation of the coefficients. We are interested in finding solutions of the FPE for large time. This search is greatly facilitated by the fact that the FPE has a unique stationary solution. This can be seen by noting that the functional H(t) = ∫ ρ ln(ρ/ρ0) dδ1 dδ2, where ρ0 is the stationary solution, is a Lyapunov function (see [8] and references therein). It then follows that such a stationary solution is unique and globally stable. For J = 0, we have found an exact (stationary) solution: ρ0(δ1, δ2) = αe− 1 2βD (δ1−δ2−2πn−2πΦex)2e 1 D cos 1e 1 D cos δ2 , (5) 356 EUROPHYSICS LETTERS 0 20 40 60 80 100 t/τ -0.2 -0.1 0 0.1 < I s (t )> Langevin FPE Fig. 1 – Comparison between the numerical solution of the Langevin equations (averaged over 10000 realizations) and the solution of the Fokker-Planck equation by the spectral method with N = M = 20moments. Parameters are D = 0.1, J = 0.9, β = 1, and Φex = 0.2.