Resistively and Capacitively Shunted Junction (RCSJ) model simulator with thermal noise

Code repository accompanying our paper titled "AC Josephson effect in a gate-tunable Cd₃As₂ nanowire superconducting weak link" [1,2].

Outline

Our program simulates the phase dynamics of a Josephson junction under the external bias current $I(t)$, influenced by the thermal noise current $I_{F}(t)$. It solves the differential equation

$$ \frac{\hbar}{2e} C \frac{d^2 \varphi}{dt^2} + \frac{\hbar}{2e} \frac{1}{R} \frac{d\varphi}{dt} + I_c f(\varphi) + I_{F}(t) = I(t) $$

for the junction phase $\varphi(t)$. In the presence of the random current fluctuation term $I_{F}(t)$ this is a stochastic differential equation (SDE). We use Heun's method for the numerical integration, for details see the supplementary material of our paper.

The current-phase relation is represented by $f(\varphi) = a \sin(\varphi) + b \sin(\varphi/2)$, where $a$ and $b$ are the weights of the conventional and topological supercurrent term, respectively. Thus, the simulation of purely conventional, purely topological and mixed junctions is possible.

Requirements

• python >= 3.7
• numpy
• scipy
• matplotlib
• numba
• tqdm

Installation

Clone the repository:

git clone https://github.com/donfuge/rcsj_sde.git

And install in editable mode with pip:

pip install -e rcsj_sde

Basic usage

The functions simu.simulation_full() and simu.simulation_voltage() run an experiment with ramped DC current bias. For each value of the DC bias, they solve the SDE, resulting in $\varphi(\tau)$ and $\dot\varphi(\tau)$.

1. simu.simulation_voltage() only returns the time-averaged junction voltage as a function of the bias current. It enables the investigation of the V-I curve of the Josephson junction. This function is recommended to simulate Shapiro experiments, where both the DC current and the AC amplitude are swept, resulting in a potentially large dataset.

2. simu.simulation_full() returns the full solution, $\varphi(\tau)$ and $\dot\varphi(\tau)$ for each bias value, in the form of a SimuResult object. This enables the investigation of the phase dynamics at specific bias current values. It calculates the power spectral density of the Josephson radiation as well.

Examples

1. See examples/basic_example.ipynb for a V-I curve and Josephson radiation simulation.

2. See examples/shapiro_example.ipynb for a Shapiro simulation.

3. In examples/shapiro_topo_sweep.ipynb you can find Shapiro maps for different current-phase relations, with varying $a, b$ parameters.

Citation

If you use this code for a scientific publication, please cite our paper:

@article{PhysRevB.108.094514,
  title = {ac Josephson effect in a gate-tunable ${\mathrm{Cd}}_{3}{\mathrm{As}}_{2}$ nanowire superconducting weak link},
  author = {Haller, R. and Osterwalder, M. and F\"ul\"op, G. and Ridderbos, J. and Jung, M. and Sch\"onenberger, C.},
  journal = {Phys. Rev. B},
  volume = {108},
  issue = {9},
  pages = {094514},
  numpages = {8},
  year = {2023},
  month = {Sep},
  publisher = {American Physical Society},
  doi = {10.1103/PhysRevB.108.094514},
  url = {https://link.aps.org/doi/10.1103/PhysRevB.108.094514}
}

References

[1] Our paper on arXiv

[2] Phys. Rev. B 108, 094514 (2023)