2D Gross-Pitaevskii Equation Solver: BEC Dynamics

Overview

This repository contains a modular Python framework for simulating Bose-Einstein Condensates (BECs) by solving the dimensionless time-dependent Gross-Pitaevskii Equation (GPE) in two dimensions.

The solver utilizes the Split-Step Fourier Method (SSFM), a spectral method that provides high accuracy and stability for non-linear Schrödinger-type equations. The simulation currently supports:

1. Imaginary Time Evolution: To find the energetic ground state of the system.
2. Real Time Evolution: To simulate dynamics, including breathing modes induced by coupling constant modulation and Gaussian pulse perturbations.

The Physics

The simulation solves the dimensionless GPE for a wavefunction $\psi(\mathbf{r}, t)$:

$$i \frac{\partial \psi}{\partial t} = \left[ -\frac{1}{2} \nabla^2 + V(\mathbf{r}) + g(t) |\psi|^2 \right] \psi$$

Where:

• $\nabla^2$ is the kinetic energy term.
• $V(\mathbf{r}) = \frac{1}{2}r^2$ is the harmonic trapping potential.
• $g(t)$ is the time-dependent non-linear interaction strength (representing s-wave scattering).

Numerical Method

The Split-Step Fourier Method decouples the linear (kinetic) and non-linear (potential + interaction) parts of the Hamiltonian over a small time step $\Delta t$:

$$\psi(t+\Delta t) \approx e^{-i \frac{\hat{T}}{2} \Delta t} e^{-i \hat{V}_{eff} \Delta t} e^{-i \frac{\hat{T}}{2} \Delta t} \psi(t)$$

By transforming the kinetic operator into momentum space using FFT, the evolution becomes computationally efficient ($O(N \log N)$).

Features

• Modular Architecture: Physics parameters, solver logic, and visualization are decoupled for easy experimentation.
• Variable Interaction Strength: Supports time-dependent modulation of $g(t)$ to study collective excitations (e.g., monopole/breathing modes).
• GPU Ready Structure: Vectorized NumPy operations allow for easy porting to CuPy for GPU acceleration.
• Automated Visualization: Generates phase and density evolution GIFs automatically.

Installation

1. Clone the repository:

   git clone [https://github.com/SomnathRoy123/GPE-Solver.git](https://github.com/SomnathRoy123/GPE-Solver.git)
   cd GPE-Solver

2. Install dependencies:

   pip install -r requirements.txt

Usage

The simulation is controlled via src/config.py and executed via main.py.

1. Configure Parameters: Open src/config.py to adjust grid size (NX, NY), time step (DT), or Trap Frequency (OMEGA).

2. Run the Simulation:

   python main.py

3. View Output: Results (frames and GIFs) are saved to the output/ directory.

Project Structure

BEC-dynamics-simulation/
├── src/
│   ├── config.py           # Physical constants and Grid generation
│   ├── solver.py           # SSFM implementation (Real & Imaginary time)
│   ├── visualization.py    # Matplotlib plotting and GIF generation
│   └── utils.py            # Normalization and helper math
├── main.py                 # Simulation entry point
├── requirements.txt        # Python dependencies
└── README.md               # Documentation