Magnus Expansion

The Magnus expansion is a technique for solving time-dependent Schrödinger equations and computing effective Hamiltonians for periodically driven (Floquet) systems.

Theory

The Problem

Consider a time-dependent Schrödinger equation:

\[i\frac{\partial U}{\partial t} = H(t) U(t)\]

where $H(t)$ is periodic with period $T = 2\pi/\omega$. We want to find an effective time-independent Hamiltonian $H_{\text{eff}}$ such that after one period:

\[U(T) = e^{-i H_{\text{eff}} T}\]

Fourier Representation

A periodic Hamiltonian can be written in Fourier form:

\[H(t) = \sum_n H_n e^{in\omega t}\]

where the Hermiticity condition requires $H_{-n} = H_n^\dagger$.

The Magnus Series

The solution to the Schrödinger equation can be written as:

\[U(t) = e^{\Omega(t)}\]

where $\Omega(t)$ is the Magnus series:

\[\Omega(t) = \Omega_1(t) + \Omega_2(t) + \Omega_3(t) + \cdots\]

The effective Hamiltonian is:

\[H_{\text{eff}} = i\Omega(T)/T = \sum_{k=1}^\infty \Omega_k\]

Orders of the Expansion

First order (k=1):

\[\Omega_1 = H_0\]

The leading term is simply the time-averaged Hamiltonian.

Second order (k=2):

\[\Omega_2 = \sum_{n>0} \frac{-[H_n, H_{-n}]}{n\omega}\]

This captures effects like the Bloch-Siegert shift from counter-rotating drive terms.

Higher orders (k≥3):

For order $k$, the Magnus term involves nested commutators with $k$ Fourier components:

\[\Omega_k = \sum_{\substack{n_1,\ldots,n_k \\ \sum_j n_j = 0}} C(n_1,\ldots,n_k) \, [[\cdots[[H_{n_1}, H_{n_2}], H_{n_3}],\ldots], H_{n_k}]\]

The coefficients are:

\[C(n_1,\ldots,n_k) = \frac{1}{\omega^{k-1} \prod_{j=1}^{k-1} s_j}\]

where $s_j = n_1 + \cdots + n_j$ are partial sums.

Resonance Condition

The sum $\sum_j n_j = 0$ is the resonance condition. It ensures that only certain combinations of Fourier modes contribute to the effective Hamiltonian.

Reducible Terms

A term is reducible if any intermediate partial sum $s_j = 0$. Such terms are excluded because they factorize into products of lower-order terms.

Convergence

The Magnus expansion converges when:

\[\int_0^T \|H(t)\| \, dt < \pi\]

For high-frequency driving ($\omega \gg \|H\|$), convergence is typically rapid.

Physical Applications

Comparison with Floquet Theory

The Magnus expansion provides a high-frequency expansion of Floquet theory. While exact Floquet diagonalization gives the full quasienergy spectrum, the Magnus expansion produces analytical expressions valid in the $\omega \gg \|V\|$ regime.

How-To Guide

This section walks through using the Magnus expansion step by step.

Step 1: Define Fourier Modes

The Magnus expansion works with Hamiltonians in Fourier representation:

\[H(t) = \sum_n H_n e^{in\omega t}\]

Provide the Fourier modes as a dictionary mapping mode index to operator:

using UnitaryTransformations
using QuantumAlgebra
using Symbolics

QuantumAlgebra.use_σpm(true)

@variables Δ Ω ω

# H(t) = (Δ/2)σz + (Ω/2)(e^{iωt}σ⁺ + e^{-iωt}σ⁻)
modes = Dict(
    0  => Δ/2 * σz(),      # Time-independent part
    1  => Ω/2 * σp(),      # e^{iωt} coefficient
    -1 => Ω/2 * σm()       # e^{-iωt} coefficient
)

Step 2: Compute the Expansion

Call magnus_expansion with the Fourier modes and driving frequency:

result = magnus_expansion(modes, ω; order=4)

This returns a named tuple with:

• result.H_eff - The total effective Hamiltonian
• result.Ω1, result.Ω2, ... - Individual order contributions
• result.orders - Dictionary of all computed orders

Step 3: Analyze Results

println("H_eff = ", result.H_eff)
println("Ω₁ (time-averaged) = ", result.Ω1)
println("Ω₂ (Bloch-Siegert) = ", result.Ω2)
println("Ω₃ = ", result.Ω3)

Hermiticity Check

The Magnus expansion verifies that your Hamiltonian is Hermitian:

# This will check H₋ₙ = Hₙ† automatically
result = magnus_expansion(modes, ω; check_hermitian=true)

# Skip the check if you know what you're doing
result = magnus_expansion(modes, ω; check_hermitian=false)

Examples

Circularly Driven Two-Level System

A qubit driven by a circularly polarized field:

\[H(t) = \frac{\Delta}{2}\sigma_z + \frac{\Omega}{2}\left(e^{i\omega t}\sigma^+ + e^{-i\omega t}\sigma^-\right)\]

using UnitaryTransformations
using QuantumAlgebra
using Symbolics

QuantumAlgebra.use_σpm(true)

@variables Δ Ω ω

# Fourier modes: H(t) = Σₙ Hₙ exp(inωt)
modes = Dict(
    0  => Δ/2 * σz(),
    1  => Ω/2 * σp(),
    -1 => Ω/2 * σm()
)

# Compute Magnus expansion to 4th order
result = magnus_expansion(modes, ω; order=4)

println("H_eff = ", result.H_eff)
println("Ω₂ (Bloch-Siegert) = ", result.Ω2)
println("Ω₃ = ", result.Ω3)

Output (order 2):

\[\Omega_2 = -\frac{\Omega^2}{4\omega}\sigma_z\]

This is the Bloch-Siegert shift — a correction to the qubit frequency due to counter-rotating drive terms.

Higher-Order Corrections

The Magnus expansion can be computed to arbitrary order:

# High-order expansion
result = magnus_expansion(modes, ω; order=6)

# Access individual orders
println("Order 3: ", result.Ω3)  # ~ -ΔΩ²/(4ω²)
println("Order 4: ", result.Ω4)  # ~ +Δ²Ω²/(4ω³)
println("Order 5: ", result.Ω5)  # ~ -Δ³Ω²/(4ω⁴)

Pattern: For k ≥ 2, each order contributes:

\[\Omega_k = (-1)^{k-1} \frac{\Delta^{k-2} \Omega^2}{4\omega^{k-1}} \sigma_z\]

This geometric series sums to the exact Bloch-Siegert result:

\[H_{\text{eff}} = \frac{\Delta}{2}\left(1 - \frac{\Omega^2}{2\omega(\omega + \Delta)}\right)\sigma_z\]

Multiple Driving Frequencies

For Hamiltonians with multiple Fourier components:

@variables Δ Ω₁ Ω₂ ω

# Bichromatic driving
modes = Dict(
    0  => Δ/2 * σz(),
    1  => Ω₁/2 * σp(),
    -1 => Ω₁/2 * σm(),
    2  => Ω₂/2 * σp(),
    -2 => Ω₂/2 * σm()
)

result = magnus_expansion(modes, ω; order=3)

Example: AC Stark Shift

When a qubit is driven off-resonantly, the Magnus expansion captures the AC Stark shift:

@variables δ Ω ω  # δ = detuning from qubit frequency

modes = Dict(
    0  => δ/2 * σz(),
    1  => Ω/2 * σp(),
    -1 => Ω/2 * σm()
)

result = magnus_expansion(modes, ω; order=2)

# The AC Stark shift appears in Ω₂
println("AC Stark shift: ", result.Ω2)

Running the Examples

Complete example files are in the examples/ directory:

julia --project examples/driven_qubit.jl  # Magnus expansion

Key Points

1. Provide Fourier modes as a Dict{Int, Operator} mapping $n \to H_n$
2. Hermiticity requires $H_{-n} = H_n^\dagger$ — the package checks this by default
3. Access individual orders via result.Ω2, result.Ω3, etc.
4. High-frequency limit: The expansion is most accurate when $\omega \gg \|H\|$
5. Use order parameter to control the number of terms computed

References

1. W. Magnus, "On the exponential solution of differential equations for a linear operator," Comm. Pure Appl. Math. 7, 649 (1954).

2. S. Blanes et al., "The Magnus expansion and some of its applications," Physics Reports 470, 151 (2009).

3. A. Eckardt and E. Anisimovas, "High-frequency approximation for periodically driven quantum systems from a Floquet-space perspective," New J. Phys. 17, 093039 (2015).

4. M. Bukov, L. D'Alessio, and A. Polkovnikov, "Universal high-frequency behavior of periodically driven systems: from dynamical stabilization to Floquet engineering," Adv. Phys. 64, 139 (2015).