The Magnus Expansion for Non-Hermitian Hamiltonians

The Magnus expansion offers a powerful, structure-preserving geometric method to express a time-ordered exponential operator solution—typical of time-dependent linear differential equations—as a single, ordinary operatorial exponential. This elegant representation possesses vital theoretical advantages, such as preserving the underlying Lie algebraic structure of the system while remaining an exact solution to the original differential equation. For any finite-dimensional Hermitian Hamiltonian governing a closed quantum system, the standard Magnus expansion inherently guarantees a manifestly unitary representation at every finite order of truncation, enforcing essential geometric conservation laws. However, this foundational symmetry completely breaks down when the underlying Hamiltonian is non-Hermitian, a regime increasingly vital for modeling open, dissipative, or topologically complex quantum networks. Under non-Hermitian configurations, standard Magnus generators cease to be self-adjoint, introducing unphysical norm inflation, numerical instabilities, and artificial exponential runaways that mask true phase dynamics.

In this work, we resolve these limitations by deriving a generalized, unitarized Magnus expansion framework that systematically restores and maintains the property of geometric unitarity for all bounded, finite-dimensional non-Hermitian Hamiltonians. Utilizing a dynamical metric normalization mapping inspired by right polar matrix decomposition, our formulation isolates coherent phase information from state-decay artifacts by consistently projecting the evolution manifold back onto a stable unitary subspace. We benchmark this framework against traditional expansions using two distinct numerical and analytical paradigms: a periodically driven Hatano–Nelson lattice exhibiting the non-Hermitian skin effect, and a driven $\mathcal{PT}$-symmetric two-level channel. The results demonstrate that our framework eliminates non-unitary drift down to machine precision, offering a robust tool for non-Hermitian quantum dynamics and topology.