A Julia package for computing the Van Vleck canonical transformations recursion formula symbolically of rapidly driven systems.
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 ordernand frequency orderk
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 correctionsThis 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 | DOI:10.1103/PhysRevLett.129.100601
@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), translated to Julia.