Qubit Charging-Energy Targeting and Local-System Reduction
Node role: Procedure.
Reader task: Carry declared target frequency and capacitance topology to one declared local retained coordinate and one declared local-system quantity.
1) Inputs and target scope
Reading route: start with the transmon approximation, distinguish static charging capacitance from a dynamic load, then choose the declared retained coordinate and interpret its decay. These are different questions even when their results share units.
Citation classes. The first-order transmon spectrum is a reorganized starting model (Koch et al. 2007, sec. II; Vool and Devoret 2017, sec. 3). Inverting that approximation for a capacitance and applying the declared Schur complement are this-page derivations. The local extraction and coupler-count rules are scoped engineering choices, not universal requirements of circuit theory.
Required input is one floating-qubit topology with:
- declared floating-node ordering and gauge/ground choice,
- declared grounded/terminated/voltage-clamped boundaries,
- declared ordering for retained coordinates and all eliminated coordinates,
- complete static and dynamic matrices in that basis, and
- a declared weak-isolated or hybridized target mode.
This procedure does not replace a design target. It only targets the local capacitance and local \(T_1\) extraction contract of a declared local mode.
2) First-order transmon sizing (declared approximation)
Declare critical-current meaning as
\[ I_c:\;\text{critical current of the transmon junction branch}\;[A]. \]
For a SQUID convention, state whether this is the per-junction branch value (\(I_c\)) or the aggregate branch value used in the design equation.
For first-order transmon sizing with consistent conventions, use
\[ E_J=\frac{\Phi_0}{2\pi}I_c=\left(\frac{\Phi_0}{2\pi}\right)^2\frac{1}{L_J}, \qquad \nu_J=\frac{E_J}{h}, \qquad E_J\,[\mathrm{J}],\;\nu_J\,[\mathrm{Hz}], \;f_{01}\,[\mathrm{Hz}], \qquad L_J=\frac{\Phi_0}{2\pi I_c}\;\text{(zero-phase/small-signal inductance branch convention)}. \]
The small-root first-order transmon relation is
\[ \nu_C=\frac{E_C}{h}, \qquad E_C\,[\mathrm{J}],\;\nu_C\,[\mathrm{Hz}], \qquad \nu_C= \frac{f_{01}^2}{4\nu_J-f_{01}+\sqrt{8\nu_J(2\nu_J-f_{01})}}. \]
Use only when the discriminant is real, i.e. \(8\nu_J(2\nu_J-f_{01})\ge0\), and use the physical small root.
\[ C_{\mathrm{target}}=\frac{e^2}{2h\nu_C}. \]
This section is first-order transmon approximation only.
With \(L_J=21.5\,\mathrm{nH}\) and \(f_{01}=4.5\,\mathrm{GHz}\):
- \(\nu_J\approx 7.602861\,\mathrm{GHz}\),
- \(\nu_C\approx 0.393746\,\mathrm{GHz}\),
- \(C_{\mathrm{target}}\approx 49.1948\,\mathrm{fF}\),
- \(E_J/E_C\approx 19.309\).
3) Static local retained-coordinate capacitance after gauge/reference fixing
After the physical gauge/reference fix and ordered local coordinate transform, define
\[ \mathbf C= \begin{pmatrix} C_{qq}&\mathbf C_{qE}\\ \mathbf C_{Eq}&\mathbf C_{EE} \end{pmatrix} \]
where \(q\) is the normalized differential coordinate and \(E\) is the complete eliminated complement for the local reduction.
Apply explicit floating-node complement boundary to eliminated floating coordinates as \(\mathbf Q_E^{ext}=0\), and use dynamic loading boundary as \(\mathbf i_E^{ext}=0\).
\[ C_{q,\mathrm{stat}}=C_{qq}-\mathbf C_{qE}\mathbf C_{EE}^{-1}\mathbf C_{Eq}, \qquad E_C=\frac{e^2}{2}\left(C^{-1}\right)_{qq} =\frac{e^2}{2C_{q,\mathrm{stat}}}. \]
For one retained coordinate this identity holds exactly under the declared ordering and boundary. Do not read \(C_{q,\mathrm{stat}}\) as a raw pad or Maxwell diagonal entry.
For nonfloating grounded or voltage-clamped nodes, use their own declared termination boundary first; do not add them to the floating-node \(Q_E^{ext}=0\) complement rule.
Keep the symbols distinct: \(C_{q,\mathrm{stat}}\), \(C_{q,\mathrm{eff}}^{\mathrm{LC}}\), and \(Y_{q,\mathrm{eff}}(\omega)\) play different roles and are not interchangeable.
4) Dynamic quantity split for local-system extraction
After elimination, define:
- \(C_{\mathrm{target}}\): the declared total differential capacitance target from transmon sizing.
- \(C_{q,\mathrm{stat}}\): static Schur-eliminated differential capacitance in the declared retained ordering and boundary.
- \(C_{q,\mathrm{dyn}}\): dynamic/modal capacitance of the retained local mode, \(C_{q,\mathrm{dyn}}=-\tfrac12\left.\partial_\omega \Im Y_{q,\mathrm{eff}}(\omega)\right|_{\omega_q}\).
- \(C_{q,\mathrm{eff}}^{\mathrm{LC}}\): an optional scalar local LC readback capacitance from a declared local reduced model.
- \(Y_{q,\mathrm{eff}}(\omega)\): retained-mode dynamic admittance of the same physical reduction.
Do not conflate these symbols across methods.
5) \(T_1\) contract and coupling strength boundary
For a declared weak-isolated mode with consistent normalized modal capacitance and retained mode boundary,
\[ C_{q,\mathrm{dyn}}=-\tfrac12\left.\partial_\omega \Im Y_{q,\mathrm{eff}}(\omega)\right|_{\omega_q}, \quad \Gamma_1=\frac{\Re Y_{q,\mathrm{eff}}(\omega_q)}{C_{q,\mathrm{dyn}}}, \quad T_1=\frac{1}{\Gamma_1}. \]
Use \(C_{q,\mathrm{eff}}^{\mathrm{LC}}\) in the scalar formula only when the declared LC fit establishes
\[ C_{q,\mathrm{eff}}^{\mathrm{LC}}=C_{q,\mathrm{dyn}}: \quad T_1=\frac{C_{q,\mathrm{eff}}^{\mathrm{LC}}}{\Re Y_{q,\mathrm{eff}}(\omega_q)}. \]
Otherwise, keep the general dynamic split above and do not use the LC shortcut.
For strongly hybridized or multimode retained blocks, replace this scalar expression with complex pole/residue extraction for the declared operator.
6) Scoped multi-coupler extraction choice
For the four-coupler extraction contract described here, cases retain a complete physical matrix and single Schur-complement reduction of the full eliminated sub-block. Scalar additive shortcuts (for example, separately collapsing each coupler branch and then adding scalar loads) are rejected by this contract.
The general Schur identity has no fixed coupler count. This scoped convention does not prescribe a topology, target, or eligibility rule for other studies.
7) Common local reduction patterns
Use these topology-dependent reduction patterns, not a free scalar shortcut:
| Pattern | Retained coordinates | Eliminated coordinates |
|---|---|---|
| Qubit local | normalized differential \(q\) only | common, XY, readout, couplers, feedline with physical terminations included as loading |
| Intrinsic pair local | \((r,p)\) | qubit-mode \(q\), feedline, and internal nodes reduced into retained loading |
Connections
This procedure is anchored by:
- Physical Circuit Coordinates and Open-EOM Reduction
- Schur Complement and Kron Reduction
- Admittance Coordinate Transforms
| Field | Value |
|---|---|
| Status | Seed |
| Review state | Local-system reduction contract; candidate remains CONVERGING/Seed. |