General immittance-only classification of compact and extended directions

Establish a complete immittance-only classification of compact and extended periodic port-flux directions for general galvanic and multiport circuit environments.

Background

The paper uses the pole structure of the impedance or admittance response to give sufficient criteria for deciding whether periodic port-flux directions are extended or compact. These criteria are particularly subtle in galvanic and multiport settings, where static inductive shunts and collective environmental translations can alter the global topology of the phase space.

The authors do not claim that the stated criteria provide a complete characterization and explicitly leave the general immittance-only classification unresolved.

References

A complete immittance-only classification of general galvanic and multiport cases is left for future work.

— Immittance formulas for exact blackbox quantization and divergence-free effective models in circuit QED  (2608.26027 - Gigon et al., 26 Aug 2026) in Section 2, subsection “Quantization of dressed-port degrees of freedom,” immediately after Eq. (ZExtendedDirectionCriterion)

In this manuscript, we mainly restrict ourselves to the case where all port canonical pairs are extended, with $[\hat{\Phi}p,\hat{Q}{Cp'}]=i\hbar\delta_{pp'}$. The systematic treatment of nontrivial compact sectors, connected to recent work on dissipative quantum phase transitions, is left for future work.

— Immittance formulas for exact blackbox quantization and divergence-free effective models in circuit QED  (2608.26027 - Gigon et al., 26 Aug 2026) in Section 2, subsection “Quantization of dressed-port degrees of freedom,” paragraph following Table I

More general time-dependent magnetic fluxes threading loops involving the black-box linear environment require a consistent flux allocation and are left for future work.

— Immittance formulas for exact blackbox quantization and divergence-free effective models in circuit QED  (2608.26027 - Gigon et al., 26 Aug 2026) in Section 2, subsection “Quantization of dressed-port degrees of freedom”