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We develop a systematic procedure to quantise canonically Hamiltonians oflight-matter models of transmission lines coupled through lumped linearlossless ideal nonreciprocal elements, that break time-reversal symmetry, in acircuit QED set-up. This is achieved through a description of the distributedsubsystems in terms of both flux and charge fields. We prove that this apparentredundancy is required for the general derivation of the Hamiltonian for awider class of networks. By making use of the electromagnetic duality symmetryin transmission lines (waveguides), we provide unambiguous identification ofthe physical degrees of freedom, separating out the nondynamical parts. Thisdoubled description can also treat the case of other extended lumpedinteractions in a regular manner that presents no spurious divergences, as weshow explicitly in the example of a circulator connected to a Josephsonjunction through a transmission line. This theory enhances the quantumengineering toolbox to design complex networks with nonreciprocal elements.

Canonical quantisation of telegrapher's equations coupled by ideal nonreciprocal elements