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domain_expansion(term, structure={}, subspaces=None)

The domain_expansion function is designed to convert a term—represented as a dictionary containing the keys 'other', 'finite', and 'infinite'—into a corresponding MulGroup object.

Parameters

  • term (dict):
    A dictionary representing the term. It must include the following keys:
    • 'other': A list of factors that do not belong to the finite or infinite groups.
    • 'finite': A list of finite operators that will later be converted into a finite matrix.
    • 'infinite': A dictionary mapping keys (typically number operators) to lists of infinite operators.

Note: The term dictionary is typically obtained from the output of group_by_order.

  • structure (dict, optional):
    A dictionary that maps bosonic subspaces to a positional index. This contains the ordering structure in case of multiple bosonic subspaces are present. This is the result of count_bosonic_subspaces.

    • Default: An empty dictionary.

  • subspaces (list, optional):
    An ordered list of RDBasis objects to consider when forming the tensor products of the finite part of the operators.

    • Default: None.

Returns

The function returns a tuple containing:

  • A MulGroup object

  • A boolean flag: Indicates whether the resulting operator is diagonal.

Example Usage

Below is an example demonstrating the use of domain_expansion:

from sympt import RDBasis, BosonOp, Dagger, group_by_order, domain_expansion


spin = RDBasis("sigma", 2)  # Define a basis with two states
s0, sx, sy, sz = spin.basis

# Define multiple bosonic operators for a composite system
a_1 = BosonOp("a_1")
ad_1 = Dagger(a_1)
a_2 = BosonOp("a_2")
ad_2 = Dagger(a_2)

# Construct a Hamiltonian-like expression
H = ad_1 * a_1 * sz + ad_2 * a_2 * sy + ad_1 * sx  # Includes creation/annihilation terms

# Determine the structure of the bosonic subspaces
subspace_structure = count_bosonic_subspaces(H)
print("\nBosonic Subspace Structure:")
print(subspace_structure)

# Group terms in H by perturbative order and extract bosonic factors
grouped_terms = group_by_order(H)

first_term = grouped_terms[0][0]  # Extract a term dictionary

print("\nExample Term from Order 0:")
print(first_term)

mul_group_first_term, is_diagonal = domain_expansion(first_term, structure=subspace_structure, subspaces=[spin])

print("\nExpanded Term:")
display(mul_group_first_term)

print("\nIs Diagonal: ", is_diagonal)

Bosonic Subspace Structure:
{Dagger(a_2)*a_2: 0, Dagger(a_1)*a_1: 1}

Example Term from Order 0:
{'other': [], 'finite': [sigma_3], 'infinite': {Dagger(a_1)*a_1: [Dagger(a_1), a_1]}}

Expanded Term:
\(\displaystyle \left[\begin{matrix}1 & 0\\0 & -1\end{matrix}\right] \cdot {a_1^\dagger} a_1\)

Is Diagonal:  True

License

SymPT is licensed under the MIT License. See the LICENSE file for details.

Citation

If you use SymPT in your research, please cite the following paper:

BibTeX Entry:

@misc{diotallevi2024symptcomprehensivetoolautomating,
      title={SymPT: a comprehensive tool for automating effective Hamiltonian derivations}, 
      author={Giovanni Francesco Diotallevi and Leander Reascos and Mónica Benito},
      year={2024},
      eprint={2412.10240},
      archivePrefix={arXiv},
      primaryClass={quant-ph},
      url={https://arxiv.org/abs/2412.10240}, 
}

APA Citation:

Diotallevi, G. F., Reascos, L., & Benito, M. (2024). SymPT: a comprehensive tool for automating effective Hamiltonian derivations. arXiv preprint arXiv:2412.10240.

IEEE Citation:

G. F. Diotallevi, L. Reascos, and M. Benito, "SymPT: a comprehensive tool for automating effective Hamiltonian derivations," arXiv preprint arXiv:2412.10240, 2024.

References