A new theoretical framework for superconducting circuits sets out how to build Hamiltonians directly from a surrounding passive linear environment's electrical response, then derive reduced descriptions for selected regimes. Its clearest practical warning is that, at sufficiently large characteristic impedance or qubit frequency, the full circuit Hamiltonian should be retained instead of eliminating the continuum with a Markov approximation.
In ordinary language, a Hamiltonian is the equation set used to track a circuit's energies and interactions. The paper applies this approach to Josephson-junction superconducting qudits, or superconducting quantum circuit elements, coupled to passive linear environments through capacitive, inductive or galvanic connections. The central aim is to move from impedance or admittance responses to exact Hamiltonians and reduced qudit dynamics.
The study is made up of illustrative circuit models rather than a sampled population. The cases span discrete resonator filters, finite-band metamaterial environments, nonreciprocal waveguide-QED systems and superconducting giant atoms.
From electrical response to a circuit model
The exact construction begins with an immittance response, meaning the impedance or admittance description of the linear environment. It extracts zero- and infinite-frequency poles - terms that capture the response at those limits - and diagonalizes a Hermitian matrix whose size is set by the number of ports, or coupling points. The resulting construction provides the exact Hamiltonian for the circuit.
The formulas cover capacitive, inductive and galvanic connections between Josephson-junction superconducting qudits and passive linear environments.
For spectrally resolved environments, where distinct features can be kept as separate modes, the paper uses a Schrieffer-Wolff reduction to remove weak, off-resonant effects from the explicit description. For smooth dissipative continua, it uses weak-coupling GKSL dynamics, a master-equation description of open-system evolution, with secular or partial-secular approximations. The reduced results are divergence-free dispersive Hamiltonians in the first case and weak-coupling master equations in the second.
Examples expose the trade-offs
In the strongly anharmonic resonator example, the dispersive Hamiltonian accurately reproduced the full dynamics of the circuit. The comparison was illustrative and came without a reported error metric or statistical uncertainty.
The paper also compares the Y and Z-tilde constructions, two formulations of the coupling. They converge in weak-coupling and effectively harmonic or vanishing-coupling limits, but their decay rates differ increasingly as adiabatic elimination becomes less controlled. The text leaves the optimal gauge or system-bath partition unresolved.
In the finite-band metamaterial model, exact capacitive loading regularized continuum coupling and transferred part of the spectral weight to a discrete pole below the band. With weak loading, the model instead showed stronger coupling near the bare band edges.
The nonreciprocal waveguide-QED example produced another contrast. In the chiral configuration, waveguide-induced dissipation vanished at the decoherence-free interaction point while the coherent interaction remained finite. The coherent coupling was twice that of the bidirectional configuration.
The shortcut has a boundary
The reduced equations are not universal replacements for the exact model. Their use depends on weak coupling, dispersive or smooth-spectrum conditions, and Markovian, secular or partial-secular approximations. Those conditions define when the simplified description is intended to apply.
At larger characteristic impedance or qubit frequency, the authors advise retaining the full circuit Hamiltonian rather than eliminating the continuum with a Markov approximation. It is a direct limit on the reduced-model shortcut.
A preprint with model-based evidence
Taken together, the examples present a route between exact circuit descriptions and reduced models across resonator, metamaterial and waveguide-QED environments. The framework's reach is broad in type, but its reduced equations remain tied to the assumptions used to derive them.
The manuscript is an arXiv preprint dated 26 Aug 2026. Its acknowledgments disclose ChatGPT assistance with code development, proofs and language editing, and state that the authors reviewed and verified the content, claims and conclusions.
Funding acknowledgments list Swiss National Science Foundation support to P. G. through Project CR-SII 222812/1, European Union Marie Sklodowska-Curie Actions support to A. P.-R. under grant agreement 101204967 for FTM-cQED, and Bavarian state-government support for Munich Quantum Valley through Hightech Agenda Bayern Plus.
Paper data and sources
Original title: Immittance formulas for exact blackbox quantization and divergence-free effective models in circuit QED
Authors: Philippe Gigon, Peter Rabl, Adrian Parra-Rodriguez
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text