https://github.com/oameye/vanvleckrecursion.jl

Symbolic calculation of the (van Vleck) Floquet recursion formula
https://github.com/oameye/vanvleckrecursion.jl

floquet floquet-theory hamiltonian perturbation perturbation-theory quantum symbolic time-dependent-hamiltonian

Last synced: about 2 months ago
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Symbolic calculation of the (van Vleck) Floquet recursion formula

• Host: GitHub
• URL: https://github.com/oameye/vanvleckrecursion.jl
• Owner: oameye
• License: mit
• Created: 2025-06-01T15:35:42.000Z (4 months ago)
• Default Branch: main
• Last Pushed: 2025-08-11T08:01:32.000Z (about 2 months ago)
• Last Synced: 2025-08-11T10:13:10.775Z (about 2 months ago)
• Topics: floquet, floquet-theory, hamiltonian, perturbation, perturbation-theory, quantum, symbolic, time-dependent-hamiltonian
• Language: Julia
• Homepage: https://oameye.github.io/VanVleckRecursion.jl/
• Size: 359 KB
• Stars: 3
• Watchers: 0
• Forks: 0
• Open Issues: 2
• Metadata Files:
  • Readme: README.md
  • License: LICENSE

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README

          
# VanVleckRecursion.jl

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A Julia package for computing the Van Vleck canonical transformations recursion formula symbolically of rapidly driven systems.

## What does this package do?

This package implements the Van Vleck recursion method to compute effective time-independent Hamiltonians for quantum systems under fast periodic driving. Given a time-dependent Hamiltonian `H(t) = H₀ + H₁cos(ωt) + ...`, the method derives the formula to systematically eliminates secular divergences that plague traditional perturbation theory and produces a static effective Hamiltonian `H_eff` valid for long times.

The recursive algorithm computes:

- **Generators** `S(n)`: Canonical transformation functions that eliminate time dependence

- **Kamiltonians** `K(n,k)`: Effective Hamiltonian contributions ordered by perturbation order `n` and frequency order `k`

## Example

```julia

using VanVleckRecursion

# Define a driven system: H(t) = H₀ + H₁cos(ωt) 

H = Terms([

    Term(rotating=0),      # Static term H₀

    Term(rotating=1)    # Oscillating term H₁cos(ωt)

])

# Set the Hamiltonian and compute effective theory

set_hamiltonian!(H)

# First-order effective Hamiltonian contributions

s1 = S(1)        # First-order generator

k1 = K(1)        # First-order Kamiltonian

# Higher orders

k2 = K(2)        # Second-order corrections

```

## Citation

This package implements the recursive formulas from:

**J. Venkatraman, X. Xiao, R. G. Cortiñas, A. Eickbusch, M. H. Devoret**  

*"On the static effective Hamiltonian of a rapidly driven nonlinear system"*  

Physical Review Letters **129**, 100601 (2022)  

[arXiv:2108.02861](https://arxiv.org/abs/2108.02861) | [DOI:10.1103/PhysRevLett.129.100601](https://doi.org/10.1103/PhysRevLett.129.100601)

```bibtex

@article{venkatraman2022static,

  title={On the static effective Hamiltonian of a rapidly driven nonlinear system},

  author={Venkatraman, Jayameenakshi and Xiao, Xu and Corti{\~n}as, Rodrigo G and Eickbusch, Alec and Devoret, Michel H},

  journal={Physical Review Letters},

  volume={129},

  number={10},

  pages={100601},

  year={2022},

  publisher={American Physical Society},

  doi={10.1103/PhysRevLett.129.100601},

  url={https://arxiv.org/abs/2108.02861}

}

```

Based on [the original Python implementation by xiaoxu (2021)](https://github.com/xiaoxuisaac/vanVleck-recursion), translated to Julia.