Singular Perturbation Approximations for a Class of Linear Quantum Systems

This paper considers the use of singular perturbation approximations for a class of linear quantum systems arising in the area of linear quantum optics. The paper presents results on the physical realizability properties of the approximate system arising from singular perturbation model reduction.

💡 Research Summary

This paper investigates the use of singular perturbation techniques for model reduction of a class of linear quantum systems that arise naturally in quantum optics. The authors first formalize the class of systems under consideration as quantum stochastic differential equations (QSDEs) expressed in terms of annihilation operators. A central concept is “physical realizability,” which requires that the system matrices can be derived from a set of physically meaningful parameters: a positive‑definite commutation matrix Θ, a coupling matrix Λ, a Hermitian Hamiltonian matrix M, and a unitary scattering matrix S. The paper recalls that for square (equal numbers of inputs and outputs) linear quantum systems, physical realizability is equivalent to the lossless bounded‑real property: the state matrix F must be Hurwitz and the transfer function Φ(s)=H(sI−F)⁻¹G+K must be unitary on the imaginary axis.

The main contribution is the analysis of singularly perturbed quantum systems of the form

 da₁ = F₁₁ a₁ dt + F₁₂ a₂ dt + G₁ du,
 ε da₂ = F₂₁ a₁ dt + F₂₂ a₂ dt + G₂ du,
 dy = H₁ a₁ dt + H₂ a₂ dt + K du,

where ε>0 is a small parameter that scales the dynamics of the “fast” subsystem a₂. Assuming F₂₂ is nonsingular, the standard singular perturbation reduction sets ε→0 and yields a reduced‑order (slow) model

 da₁ = F₀ a₁ dt + G₀ du, dy = H₀ a₁ dt + K₀ du

with F₀ = F₁₁−F₁₂F₂₂⁻¹F₂₁, etc.

Theorem 3 states that if the original (full‑order) system is physically realizable for every ε>0 and F₂₂ is nonsingular, then the reduced system’s state matrix F₀ has all eigenvalues in the closed left half‑plane and its transfer function Φ₀(iω) is unitary for all real ω. In other words, the reduced model inherits the lossless bounded‑real property, which is a necessary condition for physical realizability. However, the theorem does not guarantee that F₀ is strictly Hurwitz nor that the reduced model is minimal. The authors illustrate this limitation with a concrete counter‑example where F₀ is not Hurwitz and the reduced model is non‑minimal, despite the full‑order system being physically realizable for all ε.

To address the gap, the paper examines a special class of singular perturbations that arise from scaling the Hamiltonian and coupling operators. Specifically, the authors consider Θ=I, Λ =