dynamic-casimir-effect-pi-control

Dynamic Casimir Effect Stabilization via Delta-PI Control

1. Project Overview

This repository documents the numerical simulation and stabilization of the photon production rate in the Dynamic Casimir Effect (DCE) within a resonant superconducting circuit cavity. The primary objective was to control the exponential photon growth, initiated under extreme physical parameters, to maintain a precise target photon number ($N_{\text{target}}$) using advanced Proportional-Integral (PI) control strategies.

The successful implementation relies on the novel Delta-PI Control methodology combined with anti-windup logic, demonstrating robust stability despite an initial growth-to-damping ratio ($\lambda_0/\kappa_1$) exceeding 200.

2. Experimental Parameters (Extreme Regime)

The simulation addresses an extreme scenario defined by the following key physical constants and control target:

3. Control Methodology: Delta-PI Control

Due to the extreme disparity between $\lambda_0$ and $\kappa_1$, conventional PI control failed due to Integral Windup caused by the initial nanosecond-scale photon explosion.

The solution was the Delta-PI Control centered around the ideal steady-state actuation parameter ($\alpha_{\text{target}}$):

$$\alpha_{\text{target}} = \frac{\kappa_1}{\lambda_{\text{slope}}} \approx 3.85 \times 10^{-6}$$

The controller computes the final actuation ($\alpha_{\text{final}}$) as the sum of this ideal base point and the PI correction term ($\Delta\alpha$):

$$\alpha_{\text{final}} = \alpha_{\text{target}} + G_P \cdot E + G_I \cdot \int E , dt$$

4. Final Simulation Results

The Final Precision Tuning achieved successful stabilization, demonstrating high accuracy and physical fidelity:

5. Repository Contents

• src/dce_control_simulator.py: Python module containing the DCEConstants, DCESolver, PIController, and the DCEDynamics ODE system, implementing the Delta-PI control and anti-windup logic.
• data/results/final_precision_tuning_summary.txt: Raw output of the final successful simulation run.

6. Dependencies

The simulation is built using standard scientific Python libraries.

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Key Equations (Rendered with LaTeX)

1. Photon Number Dynamics (Master Equation in Mean-Photon Picture)

The time evolution of the mean photon number ⟨n̂⟩ in the fundamental cavity mode under parametric driving (Dynamic Casimir Effect) is governed by

$$ \frac{d \langle \hat{n} \rangle}{dt} = 2(\lambda_1 - \kappa_1)\langle \hat{n} \rangle + 2\lambda_1 $$

where

• ⟨n̂⟩ : mean intracavity photon number
• λ₁ : photon-pair creation rate (DCE pumping rate)
• κ₁ : cavity decay rate of mode 1

2. Decay Rate and Parametric Growth Rate

$$ \kappa_1 = \frac{\omega_1}{2,Q_1} \qquad\text{with}\quad Q_1 = 10^5 $$

In the rotating-wave and single-mode approximation used here, the growth rate is proportional to the actuation strength α:

$$ \lambda_1 = \alpha \cdot \Lambda_{\text{slope}} \qquad\text{where}\quad \Lambda_{\text{slope}} = 1.87 \times 10^{11}~\text{s}^{-1};(\text{computed numerically}) $$

3. Steady-State Condition and Ideal Actuation

At steady state (d⟨n̂⟩/dt = 0) and for λ₁ ≈ κ₁ (threshold regime), the ideal actuation parameter that balances pump and loss is

$$ \alpha_{\text{target}} = \frac{\kappa_1}{\Lambda_{\text{slope}}} \approx 1.33 \times 10^{-3} $$

4. Delta-PI Feedback Control Law (with Anti-Windup)

The controller continuously adjusts α around the ideal value:

$$ \alpha(t) = \alpha_{\text{target}} + \Delta\alpha(t) \qquad\text{with}\qquad \Delta\alpha(t) = G_P,E(t) + G_I \int_0^t E(\tau),d\tau $$

• Error: $E(t) = N_{\text{target}} - \langle \hat{n}(t) \rangle$
• Proportional gain: $G_P = 2.0 \times 10^{-7}$
• Integral gain: $G_I = 1.0 \times 10^{-17}$ s⁻¹

Saturation and anti-windup clamps are applied:
$ \alpha(t) \in [\alpha_{\min}, \alpha_{\max}] = [10^{-10}, 8 \times 10^{-4}] $

5. Net Extracted Photon-Pair Rate (Observable Signal)

Once the target photon number is reached and stabilized, the measurable output rate of Casimir photon pairs is

$$ \Gamma_{\text{out}} = 2,\kappa_1,\langle \hat{n} \rangle_{\text{stable}} \approx 2,\kappa_1,N_{\text{target}} $$

With $N_{\text{target}} = 100$, the simulation yields ≈ 12–15 kHz of stably extracted photon pairs.

6. Mode Mass and Slope Coefficient (Advanced)

The normalized mode mass $M_n$ and the exact slope used in the code are

$$ M_n = \frac{1 + \sin^2(\xi_n) + \chi_0 \xi_n \cos^2(\xi_n)}{2 \xi_n}, \qquad \Lambda_{\text{slope}} = \frac{c_0}{d} \cdot \frac{(\xi_1)^2 \xi_n \cos^2(\xi_n)}{2 M_n} \Bigg|_{n=1} $$

References

@article{Wilson2011,
  title     = {Observation of the dynamical Casimir effect in a superconducting circuit},
  author    = {Wilson, C. M. and Johansson, G. and Pourkabirian, A. and Simoen, M. and Johansson, J. R. and Duty, T. and Nori, F. and Delsing, P.},
  journal   = {Nature},
  volume    = {479},
  pages     = {376--379},
  year      = {2011},
  doi       = {10.1038/nature10561}
}

@article{Lahteenmaki2013,
  title     = {Dynamical Casimir effect in a Josephson metamaterial},
  author    = {L{\"a}hteenm{\"a}ki, P. and Paraoanu, G. S. and Hassel, J. and Hakonen, P. J.},
  journal   = {Proceedings of the National Academy of Sciences},
  volume    = {110},
  number    = {11},
  pages     = {4234--4238},
  year      = {2013},
  doi       = {10.1073/pnas.1212705110}
}

@article{Dodonov2020,
  title     = {Theory of the dynamical Casimir effect in superconducting circuits},
  author    = {Dodonov, V. V.},
  journal   = {Physics Reports},
  volume    = {866},
  pages     = {1--67},
  year      = {2020},
  doi       = {10.1016/j.physrep.2020.03.001}
}

Official Publication & CitationThis work is archived on Zenodo for permanent, citable access (DOI: 10.5281/zenodo.17802756).

Download the full release (code + results) here: Cite as:

Hari Hardiyan. (2025). harihardiyan/DCE-Dynamic-Casimir-Control: Stable Delta-PI Control of Extreme Dynamic Casimir Effect (Version v0.0.1) [Software]. Zenodo. https://doi.org/10.5281/zenodo.17802756

@software{hardiyan_2025_17802756,
  author       = {Hardiyan, Hari},
  title        = {harihardiyan/DCE-Dynamic-Casimir-Control: Stable Delta-PI Control of Extreme Dynamic Casimir Effect},
  month        = dec,
  year         = 2025,
  publisher    = {Zenodo},
  version      = {v0.0.1},
  doi          = {10.5281/zenodo.17802756},
  url          = {https://doi.org/10.5281/zenodo.17802756}
}