Stochastic dynamics of the resistively shunted superconducting tunnel junction system under the impact of thermal fluctuations

Abstract

In this work, a Josephson junction consisting of two superconducting layers sandwiching an insulating layer is explored, which is subject to the effects of thermal fluctuations. The precise expressions for the evolution of Josephson phase and the supercurrent through the junction are derived. A clockwise hysteresis cycle in the current-voltage characteristic curve is demonstrated mathematically. Additionally, the bifurcation of a planar limit cycle is established. The numerous stochastic thermodynamic properties of the resistively shunted superconducting tunnel junction system are described, considering the influence of three specific parameters: the conductance, the current bias and the noise intensity. Moreover, the probability density is characterized using the Fokker-Planck equation.

Figures (11)

• Figure 1: A SIS junction consists of two bulk superconductors separated by a thin insulator layer.
• Figure 2: (a) Full scheme of the experiment. The junction comprises three parallel lumped elements: a capacitor representing the internal capacitance of the junction, a shunt resistor, and a Josephson element having a current according to \ref{['supercurrent']}. The total junction current $I_S$ is a standard bias current coming from a source in parallel to the Josephson junction; (b) When $\phi\ll1$, the linear approximation $\sin(\phi)\approx\phi$ is valid, which means that $\sin(\phi)$ and $\phi$ are very close in this range. The system of motion, which consists of a single Josephson junction embedded in a superconducting loop, can be described by the equivalent circuit where the self inductance is $L_J$.
• Figure 3: (a) There exists a resistor $R$ in the absence of thermal noise when the temperature $T$ of the circuit is set to zero; (b) At any non-zero temperature, the resistor is replaced by a parallel combination of resistor $R$ and current noise source $N$, as dictated by the fluctuation-dissipation theorem.
• Figure 4: (a) The energy surface $H(x,v)$; (b) A phase portrait of a Hamiltonian system is plotted in the $(x,v)$ plane, where the vertical axis represents $v$, and the horizontal axis shows $x$.
• Figure 5: (a) A typical I-V characteristic curve is simulated for a superconducting tunnel Josephson junction. It exhibits a clockwise hysteresis. The Cooper pair tunneling current appears when $\langle\xi\rangle=0$, while the quasiparticle tunneling current is detected when $\langle\xi\rangle>2\Delta/e$. Note that $\kappa=I_S/I_c$, where $I_S$ represents the current through the Josephson junction, and $I_c$ is a parameter of the junction known as the critical current; (b) An animation of the bifurcating limit cycle.

Theorems & Definitions (1)

• Remark 2.1