Physical Circuit Coordinates and Open-EOM Reduction
Node role: Method.
Reader task: Trace a layout-realizable circuit from physical nodes to the small retained coordinate set used for response calculations without losing the loading of eliminated nodes, ports, or environments.
Reading map
Follow the physical node model into branch stamps and anchored coordinates, then attach the open boundary before eliminating the complete complement. The last table distinguishes diagonal roots, hybridized poles, response couplings and port traces. None is defined merely by a matrix diagonal.
Core idea
Engineering choice. The G1–G7 sequence below is this site’s modeling and handoff organization. Anchored subsystem coordinates and selected target operands are declared choices, not a universal basis supplied by physics.
A node-flux basis describes the circuit at the resolution of its chosen physical discretization. More CPW/MTL sections produce more coordinates and can approach the distributed system more closely. The design calculation then selects coordinates that represent the cared subsystems and eliminates the complete nonretained complement.
\[ \boxed{ \text{subsystem intent} \rightarrow \text{hybridized physical node-flux model} \rightarrow \text{anchored/local coordinates} \rightarrow \text{open-EOM Schur reduction}. } \]
An Equivalent Circuit is one possible reduced/local representation of this physical path. Its elements have design meaning only when the coordinate map and response close back to the hybridized model.
Design-target physical foundation
Before quantities or optimization are defined, a Design Target must bind:
| Item | Required declaration |
|---|---|
| Subsystems | Qubits, resonators, filters, feedlines, couplers, ports, grounds, and environments that matter to the decision. |
| Physical realization | Distributed/lumped topology, geometry/material inputs, localized extracted blocks, and branch orientations. |
| Node basis | Ordered nonreference node-flux vector and reference conductor or gauge convention. |
| Open boundary | Ports, reference planes, wave/phasor convention, loads, and terminations or causal self-energies. |
| Coordinate transforms | Every physical-node to local-coordinate map, including topology-specific common/differential transforms. |
| Reduction | Ordered retained coordinates \(R\), complete complement \(E\), and the exact operation applied to matrices and port maps. |
| Quantities | The root, pole, residue, cofactor, or calibrated network observable that owns each reported value. |
The foundation owns physical meaning. A later eigensolve, fit, or matrix factorization cannot create coordinate ownership that was not declared here.
G1 — Assemble the physical node-flux model
Reorganized explanation. Node flux and conservative electric/magnetic energy follow the method of nodes (Vool and Devoret 2017, secs. 2.1.7–2.1.8). Dissipation is kept outside the conservative stored-energy Lagrangian.
Choose a reference conductor or an explicit gauge treatment and order the physical node fluxes,
\[ \vec\Phi_{\mathrm{node}} =(\Phi_1,\ldots,\Phi_N)^T. \]
Stamp every capacitor, inductor, transmission-line section, conductance or memoryless port load into
\[ \mathcal L =\frac12\dot{\vec\Phi}^{T}\mathbf C\dot{\vec\Phi} -\frac12\vec\Phi^{T}\mathbf K\vec\Phi, \]
with the dissipative/open part represented by \(\mathbf G\) or a declared causal self-energy. The node order, branch signs, and units are part of the artifact.
G2 — Stamp branches before interpreting matrix entries
This-page derivation. Substitute an oriented branch difference \(\Phi_b=\mathbf b^T\vec\Phi\) into its scalar stored-energy term. The rank-one stamps below are the resulting quadratic coefficients.
For an oriented branch vector \(\mathbf b\) between two nodes,
\[ \Delta\mathbf C=C_b\mathbf b\mathbf b^T, \qquad \Delta\mathbf K=L_b^{-1}\mathbf b\mathbf b^T. \]
Distributed sections apply the same principle with their labeled per-unit- length matrices and section lengths. Interpret an off-diagonal entry only after the branch map, conductor order, reference, and sign convention are known. Matrix position alone does not identify a subsystem or coupling.
G3 — Choose anchored/local subsystem coordinates
Engineering choice / this-page derivation. Select physical coordinate columns before response extraction; substitution into quadratic energy gives the congruence transforms below. Choosing a subsystem coordinate is not choosing an eigenmode.
Let \(\mathbf T\) map physical nodes to an ordered local basis,
\[ \vec\Phi_{\mathrm{node}}=\mathbf T\vec\Phi_{\mathrm{local}}. \]
Select columns of \(\mathbf T\) because they correspond to declared physical subsystems: for example, a resonator open-end coordinate or a qubit island differential coordinate. Persist the selectors and ordering. These coordinates are anchored before solving the mutual response; they are not necessarily normal modes.
Transform every quadratic operator consistently,
\[ \mathbf C_{\mathrm{local}}=\mathbf T^T\mathbf C\mathbf T, \quad \mathbf K_{\mathrm{local}}=\mathbf T^T\mathbf K\mathbf T, \quad \mathbf G_{\mathrm{local}}=\mathbf T^T\mathbf G\mathbf T, \]
and transform the drive/observation maps with their corresponding dual rules.
G4 — Apply topology-specific coordinate reductions
A floating two-island qubit illustrates, but does not universalize, this step. One may define differential and common coordinates,
\[ \Phi_q=\Phi_R-\Phi_L, \qquad \Phi_c=w_L\Phi_L+w_R\Phi_R, \]
with weights fixed by the declared energy/basis convention. If the common coordinate is not cared and its block is nonsingular in the use region, it may be eliminated by Schur complement. This operation must include its effect on all other coordinates and port maps.
Other topologies may instead require loop, differential-port, or common-mode coordinates. Use the transform that follows from the circuit, not a generic name copied from another design.
G5 — Complete the open operator
This-page derivation. Fourier-transform the stated second-order circuit EOM using the selected phasor convention. The memoryless \(\mathbf G\) term is the starting boundary model, not every possible environment.
Under \(e^{-i\omega t}\),
\[ \boxed{ \mathbf D(\omega)=\mathbf K-\omega^2\mathbf C-i\omega\mathbf G } \]
for a memoryless boundary. More general environments replace \(-i\omega\mathbf G\) with their causal self-energy. Attach the physical ports, reference planes, drive map, observation map, and direct path before reducing the operator. This keeps feedline, qubit, and environment loading in every retained response.
G6 — Reduce the complete complement
This-page derivation. Solve the eliminated block equation and substitute it into the retained one. The Schur result is a response identity where the eliminated block is invertible; it is not a quantum partial trace.
Partition the ordered local coordinates into cared coordinates \(R\) and every other coordinate \(E\):
\[ \mathbf D= \begin{pmatrix} \mathbf D_{RR}&\mathbf D_{RE}\\ \mathbf D_{ER}&\mathbf D_{EE} \end{pmatrix}. \]
When \(\mathbf D_{EE}\) is nonsingular,
\[ \boxed{ \mathbf D_{RR}^{\mathrm{eff}} =\mathbf D_{RR} -\mathbf D_{RE}\mathbf D_{EE}^{-1}\mathbf D_{ER}. } \]
This is exact at each frequency. It is not the same as deleting \(E\) or setting its couplings to zero. The eliminated coordinates remain as dynamic loading in \(\mathbf D_{RR}^{\mathrm{eff}}\).
Use solves, not an explicit inverse. A singular or nonfinite eliminated solve means the declared reduced observable is not evaluable at that point; do not substitute a stale or simplified model.
G7 — Extract target-owned quantities
Engineering choice. The table names this site’s selected operations and qualifiers. It does not impose one root or coupling extraction on all designs.
The reduced matrix is a dynamic matrix, not a Hamiltonian. Its diagonal values at an arbitrary frequency are not themselves resonance frequencies or linewidths. A Design Target must name the operation that owns each quantity:
| Quantity | Typical exact operation |
|---|---|
| Anchored local frequency/linewidth | Complex root of one declared diagonal entry of \(\mathbf D_{RR}^{\mathrm{eff}}\). |
| Hybridized pole | Root of \(\det\mathbf D_{RR}^{\mathrm{eff}}\). |
| Local coherent exchange | Exact open-EOM \(J_{\rm res}(\omega_*)\) from the residue-normalized retained off-diagonal at a declared evaluation frequency. |
| Interference zero | Transfer cofactor, generalized eigenzero, or calibrated \(S/Y/Z\) zero. |
| Port response | Consistently transformed drive/observation maps and \(\mathbf D^{-1}\), including the direct path. |
Different quantities may use different exact operations on the same physical operator. Compute only what the Design Target needs.
Optional downstream quantum mapping
Canonical quantization, oscillator normalization, Hamiltonian coefficients, pairing terms, Bogoliubov transformations, and RWA are downstream quantum representations. They may be constructed from the same physical foundation when a quantum-simulation task needs them. They do not define the current classical open-EOM coordinates or response quantities unless a Design Target explicitly says so.
See Circuit Lagrangian, Hamiltonian, and Canonical Quantization for the conservative construction, and Symbol Conventions for concise coupling labels tied to a declared canonical basis and normalization. The full quadratic exchange/pairing block derivation is not developed in this selected material; these links do not supply that missing derivation or make a response reduction a Hamiltonian.
Review checklist
- Is the physical node order explicit?
- Can every local coordinate be traced to a selector or transform?
- Are common/differential reductions topology-specific and rank-valid?
- Does \(E\) contain every nonretained open-system coordinate?
- Are ports and environments attached before Schur reduction?
- Is each reported quantity tied to an exact root, pole, residue, cofactor, or network observable?
- Is any Equivalent Circuit visibly mapped back to the hybridized model?
Review Links
| Field | Value |
|---|---|
| Status | Seed |
| Used by | Qubit Charging-Energy Targeting and Local-System Reduction. |
| Review state | Current physical-coordinate and open-EOM common language; documentation candidate remains CONVERGING. |