UnitaryTransformations.jl

A Julia package for performing symbolic unitary transformations on quantum Hamiltonians.

Summary of Schrieffer-Wolff transformations at different orders

Overview

Unitary transformations are essential tools in quantum mechanics for simplifying Hamiltonians, deriving effective low-energy theories, and block-diagonalizing systems with separated energy scales.

This package provides symbolic implementations that produce analytical expressions rather than numerical results. For example, the dispersive shift in circuit QED is computed as -g²/Δ, not as a floating-point number.

Available Transformations

Transformation	Purpose	Documentation
Schrieffer-Wolff	Block-diagonalize Hamiltonians, derive effective low-energy theories	Guide
Magnus Expansion	Effective Hamiltonians for periodically driven (Floquet) systems	Guide

Quick Start

using UnitaryTransformations
using QuantumAlgebra
using Symbolics

QuantumAlgebra.use_σpm(true)

@variables Δ g

# Jaynes-Cummings Hamiltonian
H = Δ/2 * σz() + g * (a'()*σm() + a()*σp())

# Define the low-energy subspace
P = Subspace(σz() => -1)

# Perform Schrieffer-Wolff transformation
result = schrieffer_wolff(H, P; order=2)

println(result.H_P)
# Output: -Δ/2 + (-g²/Δ) a†a

Supported Quantum Systems

The package works with quantum systems provided by QuantumAlgebra.jl:

System	Operators	Example
Two-level systems	`σx()`, `σy()`, `σz()`, `σp()`, `σm()`	Qubits, spin-1/2
Bosonic modes	`a()`, `a'()`	Cavities, phonons
N-level atoms	`nlevel_ops(N, :name)`	Multi-level atoms
SU(N) systems	`su_generators(N, :name)`	3-level Λ systems
Fermions	`f(:name)`, `f'(:name)`	Electrons
Symbolic sums	`SymSum(expr, i)`, `SymExpr`	Multi-atom systems (Tavis-Cummings)

Installation

using Pkg
Pkg.add(url="https://github.com/volkerkarle/UnitaryTransformations.jl")

Documentation

Citation

If you use this package in your research:

@software{UnitaryTransformations.jl,
  author = {Karle, Volker},
  title = {UnitaryTransformations.jl: Symbolic Unitary Transformations for Quantum Hamiltonians},
  url = {https://github.com/volkerkarle/UnitaryTransformations.jl},
  year = {2025}
}