From Node and Line Flux to Input–Output Scattering

A prefactor-complete bridge from a physical node-flux circuit and transmission-line field through exact S/Y/Z response, mode normalization, radiative linewidth, and passive Markov input–output matrices.

Node role: Procedure.

Reader task: Derive an open response and passive input–output parameters from node flux, a transmission-line field, and a physical coupling boundary.

What this page owns

How can a capacitance in a circuit become a decay rate and a square-root drive coefficient? The answer is not a fit parameter inserted by hand: write the line’s energy, choose its traveling-wave boundary, solve the attached circuit, and only then normalize a retained mode. The circuit/line starting model follows (Vool and Devoret 2017; Clerk et al. 2010); the bath and passive coupled-mode starting relations follow (Gardiner and Collett 1985; Suh et al. 2004). This page derives the displayed direct-terminal and two-sided-capacitor specializations, rather than claiming that those sources print every equation below.

This page derives the boundary between a physical circuit and input–output theory. Its starting objects are node fluxes, a transmission-line flux field, and a complete physical coupling block. Section 8 derives the scalar \(C_{\mathrm{ext}}\) special case. Its outputs are:

  1. the exact open node-domain response and its \(S/Y/Z\) observables;
  2. the terminal participation and zero-point voltage of a declared reduced coordinate;
  3. the radiative rate \(\kappa\) derived from that physical boundary; and
  4. the passive Markov matrices \(\mathbf D_{\rm port}\), \(\mathbf K_{\rm port}\), and \(\boldsymbol\Gamma_{\mathrm{ext}}\) used by multimode input–output theory.

The general finite-circuit construction before this boundary belongs to Circuit Models to Bare Coordinates, Open EOM, and Normal Modes. The general scattering algebra after this boundary belongs to Multimode Input–Output Scattering.

Reading route. Sections 1–7 derive a directly matched terminal and its exact response. Section 8 is an alternative physical specialization in which an internal node couples through \(C_{\mathrm{ext}}\) to a two-sided feedline; do not stack it on top of the direct-terminal boundary. Continue through Sections 9–15 only when deriving line elimination, \(\kappa\), or a reduced Markov input–output model.

Stage Question and reading destination What is still not assumed
Physical variables and line energy Sections 1–3: what stores energy, and which current direction is used? No oscillator rate or phenomenological linewidth.
Direct-terminal response Sections 4–6: how do waves drive the exact finite circuit? No mode truncation or Markov reduction of a resonant mode.
Alternative capacitive boundary Sections 8–10: what does eliminating the coupling node load and radiate? Not an additional stamp on the direct-terminal model.
Mode and terminal normalization Section 11: which physical voltage is attached to the bath? A bare-coordinate selector need not be a single oscillator amplitude.
Decay and reduced response Sections 12–15: when can the boundary be summarized by constant rates? Narrow-band, weak-damping, RWA/Markov assumptions remain visible here.

The stages are reading choices, not eligibility requirements. For the power-wave reference itself use Port Reference Impedance Semantics; for analytic response features use Poles, Zeros, and Residues.

ImportantThe order is part of the physics

\[ \boxed{ \text{node/line flux Lagrangian} \rightarrow \text{traveling-wave boundary} \rightarrow \text{exact }S/Y/Z \rightarrow \text{basis and energy normalization} \rightarrow \kappa,\mathbf D_{\rm port},\boldsymbol\Gamma \rightarrow \text{Markov input--output model}. } \]

\(\kappa\) is not inserted beside the physical capacitance block as a second primitive. It is a derived rate. A reduced input–output model is accepted only when its complex response closes against the exact circuit response on the declared frequency and parameter boundary.

There are two valid response lanes. The exact flux EOM produces \(S/Y/Z\) directly and is the closure authority. The reduced mode EOM first converts \((\vec\Phi_{\rm bare},\vec Q_{\rm bare})\) to \(\vec a_{\rm bare}\), then derives \(\kappa\), \(\mathbf K_{\rm port}\), and \(\mathbf D_{\rm port}\). The second lane may not redefine the basis or replace the first lane’s response without passing closure.

Conventions and dimensions

This page follows Dynamics Conventions and the Flux-to-Mode Symbol Bridge and therefore uses

\[ x(t)=x(\omega)e^{-i\omega t}, \qquad \dot x\mapsto-i\omega x. \]

Frequencies and decay rates are angular rates in \(\mathrm{rad\,s^{-1}}\). The physical line is lossless with real \(l',c'>0\) and real \(Z_0>0\). Complex or frequency-dependent wave-reference impedances require the separate conversion contract in Port Reference Impedance Semantics.

WarningDo not identify mode amplitude with voltage

The exact symbol change is

\[ \hat a_x =\frac{\hat\Phi_x}{2\Phi_{x,\rm zpf}} +i\frac{\hat Q_x}{2Q_{x,\rm zpf}}, \]

not \(a_x\equiv\dot\Phi_x\). The terminal voltage is obtained afterward from \(\dot{\vec\Phi}=\mathbf C^{-1}\vec Q\) and can mix several bare amplitudes. Section 11.2 derives that complete port-voltage map.

Symbol Units Meaning
\(\Phi_n\) \(\mathrm{Wb}\) Dimensional node flux, with \(\dot\Phi_n=V_n\).
\(\phi(x,t)\) \(\mathrm{Wb}\) Distributed transmission-line flux field.
\(C_0\) \(\mathrm F\) Internal capacitance matrix before the explicitly separated line-coupling branch.
\(K_\Phi\) \(\mathrm{H^{-1}}\) Linearized flux-stiffness matrix.
\(l'\) \(\mathrm{H\,m^{-1}}\) Line inductance per length.
\(c'\) \(\mathrm{F\,m^{-1}}\) Line capacitance per length.
\(s_{j,\mathrm{in/out}}\) \(\sqrt{\mathrm W}\) Classical RMS power-wave amplitude.
\(b_{j,\mathrm{in/out}}\) \(\mathrm{s^{-1/2}}\) Quantum traveling-wave amplitude; \(b^\dagger b\) is photon flux.
\(a_m\) dimensionless Energy-normalized mode amplitude; \(a_m^\dagger a_m\) is occupation.
\(\kappa\) \(\mathrm{rad\,s^{-1}}\) Energy-decay rate; amplitude decays at \(\kappa/2\).

Peak rather than RMS classical phasors move factors of two in the power formulas. The quantum positive-frequency convention below fixes those factors explicitly and must not be mixed with an undeclared classical convention.

1. Internal node-flux circuit

The node-flux Lagrangian reorganizes the conservative circuit-energy model (Vool and Devoret 2017). Selecting a terminal below is a physical declaration made here; it does not select a normal mode.

After resolving reference and gauge coordinates, the physical node-flux vector is

\[ \boldsymbol\Phi \equiv (\Phi_1,\ldots,\Phi_N)^T. \]

The quadratic internal Lagrangian is

\[ \boxed{ \mathcal L_{\mathrm{sys}} =\frac12\dot{\boldsymbol\Phi}^{T}C_0\dot{\boldsymbol\Phi} -\frac12\boldsymbol\Phi^{T}K_\Phi\boldsymbol\Phi. } \]

At this point there is no linewidth and no port-wave amplitude. The matrices describe stored electric and magnetic energy in the physical node basis. A port terminal \(P\) is selected by a node vector \(\mathbf b_P\):

\[ \Phi_P=\mathbf b_P^T\boldsymbol\Phi, \qquad V_P=\dot\Phi_P. \]

For a differential terminal, \(\mathbf b_P\) is the corresponding oriented incidence vector rather than a one-hot selector. This physical selector must later undergo the same basis changes as the circuit matrices.

2. Transmission line as a flux field

The distributed-flux model and its voltage/current identification follow (Vool and Devoret 2017; Clerk et al. 2010). The wave speed and impedance are derived below from the displayed energy densities, with the stated lossless-uniform assumptions.

For a uniform line coordinate \(x\), the continuum Lagrangian is

\[ \boxed{ \mathcal L_{\mathrm{TL}} =\int dx\left[ \frac{c'}{2}\dot\phi^2 -\frac{1}{2l'}(\partial_x\phi)^2 \right]. } \]

The field Euler–Lagrange equation gives

\[ c'\,\partial_t^2\phi -\frac1{l'}\partial_x^2\phi=0, \]

or

\[ \boxed{ \partial_t^2\phi-v^2\partial_x^2\phi=0, \qquad v=\frac1{\sqrt{l'c'}}. } \]

Voltage and current follow from the same field:

\[ \boxed{ V(x,t)=\partial_t\phi(x,t), \qquad I_{+x}(x,t)=-\frac1{l'}\partial_x\phi(x,t). } \]

For a harmonic traveling wave these equations give

\[ \boxed{ Z_0=\sqrt{\frac{l'}{c'}}, \qquad k=\frac{\omega}{v}=\omega\sqrt{l'c'}. } \]

Thus \(Z_0\) and propagation phase do not enter as empirical fit decorations. They descend from the two energy densities in the line Lagrangian.

3. Incoming and outgoing waves at a boundary

This page derives the current signs from the outward half-line coordinate. Semi-infinite-line input–output boundary reasoning is supported by (Clerk et al. 2010); reversing the arrow is an explicit convention change, not a different physical boundary.

Choose the line coordinate by

\[ x=0\quad\text{at the coupling point}, \qquad x>0\quad\text{outward into the semi-infinite line}. \]

The general solution is

\[ \boxed{ \phi(x,t) =\phi_{\mathrm{in}}(t+x/v) +\phi_{\mathrm{out}}(t-x/v). } \]

The first term travels toward the boundary and the second away from it. At \(x=0\),

\[ V=V_{\mathrm{in}}+V_{\mathrm{out}}, \]

and the current flowing outward from the boundary is

\[ \boxed{ I_{\mathrm{outward}} =\frac{V_{\mathrm{out}}-V_{\mathrm{in}}}{Z_0} =\frac{V-2V_{\mathrm{in}}}{Z_0}. } \]

The project-wide port convention instead takes current into the DUT. At this same boundary,

\[ \boxed{I_{\rm port}=-I_{\rm outward}.} \]

This arrow change is the only reason the two current formulas look different; all later signs use \(I_{\rm port}\) entering the finite circuit.

The minus sign on the incoming contribution is geometric: its power travels opposite to the outward current direction. Changing the current arrow changes this equation and every following port sign together.

Propagation phase

Under \(e^{-i\omega t}\), a wave traveling toward increasing \(z\) has

\[ V^+(z,t)=V_0^+e^{+ikz}e^{-i\omega t}. \]

Two reference planes separated by \(\ell\) therefore satisfy

\[ \boxed{ \frac{V^+(\ell,\omega)}{V^+(0,\omega)} =e^{+ik(\omega)\ell}. } \]

A matched lossless line is all-pass, not zero-phase. With weak distributed loss,

\[ t_0(\omega) =e^{-\alpha(\omega)\ell}e^{+ik(\omega)\ell}. \]

This factor belongs to the direct device path when the two reference planes enclose the line segment. Cable delay outside those planes belongs to the measurement error box. Moving a reference plane moves phase between these descriptions but cannot move a physical pole of the declared device.

4. Classical power-wave normalization

The real-positive power-wave definition is reorganized from (Kurokawa 1965). This page verifies its power identity with RMS phasors; peak phasors would require a different factor in the average-power expression.

For a real positive reference impedance and RMS phasors, the power waves on a chosen positive propagation axis are

\[ \boxed{ s_+ =\frac12\left(\frac{V}{\sqrt{Z_0}}+\sqrt{Z_0}I\right), \qquad s_- =\frac12\left(\frac{V}{\sqrt{Z_0}}-\sqrt{Z_0}I\right). } \]

Then

\[ |s_+|^2-|s_-|^2 =\operatorname{Re}(VI^*) \]

is the net time-averaged power in watts. On the local half-line boundary this reduces to

\[ s_{\mathrm{in}}=\frac{V_{\mathrm{in}}}{\sqrt{Z_0}}, \qquad s_{\mathrm{out}}=\frac{V_{\mathrm{out}}}{\sqrt{Z_0}}. \]

The scattering matrix maps these incident and outgoing waves,

\[ \boxed{\vec s_{\rm out}=\mathbf S\vec s_{\rm in}.} \]

It does not map total terminal voltage to total terminal voltage. The total voltage is reconstructed from both waves. Standard RF notation may call them \(\vec a_{\rm pw}\) and \(\vec b_{\rm pw}\); in this Knowledge Base \(\vec a_{\rm pw}\equiv\vec s_{\rm in}\) and \(\vec b_{\rm pw}\equiv\vec s_{\rm out}\) so they cannot be confused with internal oscillator operators \(a_x\).

5. Direct matched-line attachment

This is this page’s exact circuit solve for a selected topology. Use the wave/current relation just derived as the generalized force in KCL, then read outgoing waves from the solved terminal voltages. The lossless-line model (Vool and Devoret 2017) and real-positive wave definition (Kurokawa 1965) are starting principles, not a blanket source attribution for these matrix prefactors.

This section attaches each retained terminal directly to its own matched semi-infinite line. It is one complete port topology. The capacitive side-coupling topology in Section 8 is an alternative, not an additional stamp.

Let

\[ \mathbf B_P=(\mathbf b_1,\ldots,\mathbf b_P), \qquad \vec\Phi_P=\mathbf B_P^T\vec\Phi, \qquad \vec V_P=\mathbf B_P^T\dot{\vec\Phi}, \]

and let \(\mathbf Y_{\rm line}=\operatorname{diag}(1/Z_{{\rm line},j})\) contain the characteristic admittances of the physically attached semi-infinite matched lines. In this subsection the wave references are chosen equal to those line impedances, so \(\mathbf Y_0\equiv\mathbf Y_{\rm line}\). With every port current entering the finite circuit,

\[ \vec I_P =2\mathbf Y_0^{1/2}\vec s_{\rm in} -\mathbf Y_0\vec V_P, \qquad \vec s_{\rm out} =\mathbf Y_0^{1/2}\vec V_P-\vec s_{\rm in}. \]

Substitution into the Euler–Lagrange EOM gives

\[ \boxed{ \mathbf C\ddot{\vec\Phi} +\mathbf B_P\mathbf Y_0\mathbf B_P^T\dot{\vec\Phi} +\mathbf K_\Phi\vec\Phi =2\mathbf B_P\mathbf Y_0^{1/2}\vec s_{\rm in}. } \]

For \(e^{-i\omega t}\), define the intrinsic dynamic stiffness, the physical line self-energy, the open operator, and the exact flux susceptibility:

\[ \boxed{ \begin{aligned} \boldsymbol{\mathcal D}_\Phi(\omega) &=\mathbf K_\Phi-\omega^2\mathbf C,\\ \boldsymbol\Sigma_{\Phi,\rm env}(\omega) &=-i\omega\mathbf B_P\mathbf Y_0\mathbf B_P^T,\\ \boldsymbol{\mathcal D}_{\rm open}(\omega) &=\boldsymbol{\mathcal D}_\Phi(\omega) +\boldsymbol\Sigma_{\Phi,\rm env}(\omega),\\ \boldsymbol\chi_\Phi(\omega) &=\boldsymbol{\mathcal D}_{\rm open}^{-1}(\omega). \end{aligned} } \]

Then

\[ \boxed{ \mathbf S(\omega) =-\mathbf I -2i\omega\mathbf Y_0^{1/2}\mathbf B_P^T \boldsymbol\chi_\Phi(\omega) \mathbf B_P\mathbf Y_0^{1/2}. } \]

The \(-\mathbf I\) term is the terminal direct map for this boundary. If a finite through line or another direct network is retained between the reference planes, its propagation is part of the retained dynamic circuit or of a separately declared \(\mathbf S_{\rm dir}\); it is not an arbitrary Fano background.

Using \(\mathbf A_Q=\mathbf C^{-1}\) does not change the response. The exact same susceptibility is

\[ \boxed{ \boldsymbol\chi_\Phi =\left[ \mathbf A_Q\mathbf K_\Phi-\omega^2\mathbf I +\mathbf A_Q\boldsymbol\Sigma_{\Phi,\rm env}(\omega) \right]^{-1}\mathbf A_Q. } \]

The final \(\mathbf A_Q\) is required because the generalized drive has not been rescaled. Dropping it changes the meaning and units of susceptibility.

For an intrinsic terminal-current experiment before attaching the physical matched lines,

\[ \boldsymbol{\mathcal D}_\Phi\vec\Phi=\mathbf B_P\vec I_P, \qquad \boxed{ \mathbf Z_{\rm port}(\omega) =-i\omega\mathbf B_P^T \boldsymbol{\mathcal D}_\Phi^{-1}(\omega)\mathbf B_P. } \]

Choosing the wave references equal to the line impedances, the canonical real-\(Z_0\) conversion in Port Reference Impedance Semantics gives the same \(\mathbf S\). A later reference-only renormalization changes the numerical \(\mathbf S\) coordinates but not \(\mathbf B_P\mathbf Y_{\rm line}\mathbf B_P^T\), the physical self-energy, or the open poles. This is the general finite-circuit route; the physical \(C_{\rm ext}\) boundary below is one specialization.

6. Why an internal basis change cannot change \(S/Y/Z\)

This page’s proof is substitution, not an approximation: replace the flux coordinate in both the equations of motion and the output equation. The inverse transformation of the susceptibility cancels the transformations of the drive and observation maps. This explains why a changed internal plot can describe the same external experiment.

Write any linear drive and observation boundary as

\[ \boldsymbol{\mathcal D}_\Phi\vec\Phi =\mathbf F_\Phi\vec s_{\rm in}, \qquad \vec s_{\rm out} =\mathbf S_{\rm dir}\vec s_{\rm in}+\mathbf O_\Phi\vec\Phi, \]

where \(\mathbf F_\Phi\) maps port waves to generalized-force coordinates and \(\mathbf O_\Phi\) maps flux coordinates back to outgoing waves. Their exact prefactors are fixed by the chosen physical boundary, as in Section 5.

For a real invertible coordinate map \(\vec\Phi=\mathbf T\vec q\), every operator and port map must move together:

\[ \begin{aligned} \boldsymbol{\mathcal D}_q&=\mathbf T^T \boldsymbol{\mathcal D}_\Phi\mathbf T,& \boldsymbol\chi_q&=\mathbf T^{-1} \boldsymbol\chi_\Phi\mathbf T^{-T},\\ \mathbf B_{P,q}&=\mathbf T^T\mathbf B_P,& \mathbf F_q&=\mathbf T^T\mathbf F_\Phi,& \mathbf O_q&=\mathbf O_\Phi\mathbf T. \end{aligned} \]

Therefore

\[ \boxed{ \mathbf O_q\boldsymbol\chi_q\mathbf F_q =\mathbf O_\Phi\boldsymbol\chi_\Phi\mathbf F_\Phi, \qquad \mathbf B_{P,q}^T\boldsymbol\chi_q\mathbf B_{P,q} =\mathbf B_P^T\boldsymbol\chi_\Phi\mathbf B_P. } \]

Also

\[ \det\boldsymbol{\mathcal D}_q =\det(\mathbf T)^2\det\boldsymbol{\mathcal D}_\Phi, \]

so exact poles and terminal \(S/Y/Z\) are unchanged. Susceptibility entries, \(h_{ij}\), \(\Delta_{ij}\), and coordinate participation are not invariants; they must keep the basis that gives them physical meaning.

This invariance requires the physical ports, reference planes, reference impedances, wave normalization, drive map, and observation map to remain fixed and be transformed consistently. Changing an external port-wave basis, reference impedance, reference plane, termination, or calibration can change the reported \(\mathbf S\) representation and the entry called \(S_{21}\). Selecting ports does not uniquely choose an internal coordinate basis.

7. Quantum photon-flux normalization

Continuum photon-flux normalization and the narrow-band envelope picture are reorganized from quantum input–output/noise theory (Gardiner and Collett 1985; Clerk et al. 2010). The voltage crosswalk below uses the explicitly displayed commutator and Fourier normalization; its factors cannot be transferred to an unspecified solver-byte convention.

For a quantized line, choose continuum operators

\[ [b_j(\omega),b_k^\dagger(\omega')] =\delta_{jk}\delta(\omega-\omega'). \]

The incoming positive-frequency voltage is

\[ \boxed{ \hat V_{j,\mathrm{in}}^{(+)}(t) =\int_0^\infty d\omega \sqrt{\frac{\hbar\omega Z_0}{4\pi}} \,b_j(\omega)e^{-i\omega t}. } \]

In a narrow band around \(\omega_0\), the envelope operator is

\[ b_{j,\mathrm{in}}(t) =\frac1{\sqrt{2\pi}} \int d\omega\, b_j(\omega)e^{-i(\omega-\omega_0)t}. \]

Then

\[ \boxed{ \hat V_{j,\mathrm{in}}^{(+)}(t) \simeq \sqrt{\frac{\hbar\omega_0Z_0}{2}} \,b_{j,\mathrm{in}}(t)e^{-i\omega_0t}, } \]

and

\[ \boxed{ P_j=\hbar\omega_0 \left\langle b_{j,\mathrm{in}}^\dagger b_{j,\mathrm{in}}\right\rangle. } \]

Thus the classical and quantum power-wave normalizations are related locally by \(s_j^{(\mathrm W)}=\sqrt{\hbar\omega_0}\,b_j\). The square root in the input–output drive term is ultimately the amplitude form of this energy-flux normalization.

8. Capacitive side coupling to a two-sided feedline

This is the alternative topology, derived on this page by KCL at the coupling node. Keeping coupling capacitance in the physical circuit is important in multimode circuit analysis (Malekakhlagh and T"ureci 2016); that source does not supply the specific two-sided network algebra below.

The coupling topology is

\[ P\xleftrightarrow{\ C_{\mathrm{ext}}\ }F. \]

This section is the exact single cross-branch special case

\[ C_{PG}=C_{FG}=0. \]

A physical interdigitated component may also contribute terminal-to-ground branches. In that case retain its complete local Maxwell block in the physical node model, or rederive the eliminated boundary with those shunts present. They cannot be hidden inside this scalar \(C_{\mathrm{ext}}\). Geometry and region ownership follow Localized Electrostatic Components.

Its branch Lagrangian is

\[ \boxed{ \mathcal L_{\mathrm{ext}} =\frac{C_{\mathrm{ext}}}{2} (\dot\Phi_P-\dot\phi_F)^2. } \]

This one branch contributes both reactive loading and radiation. Write

\[ Y_c(\omega)=-i\omega C_{\mathrm{ext}}. \]

At \(F\), the two semi-infinite lines have incoming voltages \(V_{L,\mathrm{in}}\) and \(V_{R,\mathrm{in}}\). Applying the boundary-current equation to both lines and imposing KCL gives

\[ Y_c(V_F-V_P) +\frac{V_F-2V_{L,\mathrm{in}}}{Z_0} +\frac{V_F-2V_{R,\mathrm{in}}}{Z_0}=0. \]

Therefore

\[ \boxed{ V_F =\frac{ 2(V_{L,\mathrm{in}}+V_{R,\mathrm{in}}) +Z_0Y_cV_P }{2+Z_0Y_c}. } \]

The current leaving the internal terminal through the capacitor is

\[ \boxed{ I_{\mathrm{ext}} =Y_{\mathrm{ext}}(\omega) \left[V_P-V_{L,\mathrm{in}}-V_{R,\mathrm{in}}\right], \qquad Y_{\mathrm{ext}} =\frac{2Y_c}{2+Z_0Y_c}. } \]

The outgoing voltages obey \(V_{j,\mathrm{out}}=V_F-V_{j,\mathrm{in}}\), so

\[ \boxed{ \begin{pmatrix} V_{L,\mathrm{out}}\\ V_{R,\mathrm{out}} \end{pmatrix} = \frac1{2+Z_0Y_c} \begin{pmatrix} -Z_0Y_c&2\\ 2&-Z_0Y_c \end{pmatrix} \begin{pmatrix} V_{L,\mathrm{in}}\\ V_{R,\mathrm{in}} \end{pmatrix} +\frac{Z_0Y_c}{2+Z_0Y_c} \begin{pmatrix}1\\1\end{pmatrix}V_P. } \]

This is an exact classical boundary relation for the declared lumped \(C_{\mathrm{ext}}\) and two lossless half-lines. It already contains a direct feedline term, a resonant terminal-radiation term, their relative phase, and left/right symmetry. No Fano coefficient has been introduced.

9. Eliminate the line and obtain exact \(S/Y/Z\)

The substitution below retains both the reactive and dissipative parts of the same causal boundary. It is this page’s elimination of the already declared capacitor/line model, not an independent loss term added to a retained branch.

Here \(C_0\) is the internal capacitance matrix with the separated \(P\)–\(F\) branch omitted. Since \(V_P=-i\omega\mathbf b_P^T\boldsymbol\Phi\), the internal node equation becomes

\[ \boxed{ \left[ K_\Phi-\omega^2C_0 -i\omega\mathbf b_PY_{\mathrm{ext}}(\omega)\mathbf b_P^T \right]\boldsymbol\Phi =\mathbf b_PY_{\mathrm{ext}}(\omega) (V_{L,\mathrm{in}}+V_{R,\mathrm{in}}). } \]

The open dynamic operator is

\[ \boxed{ \boldsymbol{\mathcal D}_{\mathrm{open}}(\omega) =K_\Phi-\omega^2C_0 -i\omega\mathbf b_PY_{\mathrm{ext}}(\omega)\mathbf b_P^T. } \]

Then

\[ \boldsymbol\Phi =\boldsymbol{\mathcal D}_{\mathrm{open}}^{-1}\mathbf b_PY_{\mathrm{ext}} (V_{L,\mathrm{in}}+V_{R,\mathrm{in}}). \]

Substituting \(V_P=-i\omega\mathbf b_P^T\boldsymbol\Phi\) into the exact outgoing-voltage equation constructs the complete two-port \(S\) matrix. Port currents and voltages construct the corresponding \(Y\) and \(Z\) matrices. This is the response authority before modal truncation or Markov approximation.

WarningDo not count \(C_{\mathrm{ext}}\) twice

Two equivalent circuit descriptions are allowed:

  1. retain node \(F\), the branch \(C_{\mathrm{ext}}\), and the line boundary in the full physical matrix; or
  2. eliminate them and use \(Y_{\mathrm{ext}}\) in \(\boldsymbol{\mathcal D}_{\mathrm{open}}\).

Do not retain the branch contribution inside \(C_0\) and also add the eliminated \(Y_{\mathrm{ext}}\). That duplicates both reactive loading and radiation.

10. What the weak-coupling limit actually approximates

Expand the exact denominator from Section 8 rather than fitting the reactive and radiative terms separately. This page’s series expansion identifies which terms the small-capacitor limit keeps; the asymptotic inequality is a model assumption, not a numerical search bound.

When \(|\omega C_{\mathrm{ext}}Z_0|\ll1\),

\[ \boxed{ Y_{\mathrm{ext}}(\omega) =-i\omega C_{\mathrm{ext}} +\frac{\omega^2C_{\mathrm{ext}}^2Z_0}{2} +\mathcal O(\omega^3C_{\mathrm{ext}}^3Z_0^2). } \]

The imaginary term is the leading reactive capacitance. The positive real term is the leading radiation conductance into both line directions. They come from one causal admittance and must not be tuned independently without a declared reduced-model reason.

The exact model retains the full frequency dependence. Replacing it by a constant frequency shift and a constant \(\kappa\) is a local narrow-band Markov reduction, not a property of the capacitor itself.

11. Map physical flux coordinates to energy-normalized amplitudes

Canonical oscillator normalization follows the circuit quantization starting model (Vool and Devoret 2017). The terminal selector is then transformed on this page alongside the coordinate and charge maps. A normal-mode voltage and a bare- cluster voltage are different cases of that same accounting.

This step has two related but distinct cases. A conservative normal mode is an eigenvector of the declared closed conservative matrix pair. In the \(C_0\) specialization below, that is the internal reference with the explicitly separated line-coupling branch omitted; a full coupling-on normal mode instead uses the complete retained closed matrices. A physically anchored bare coordinate keeps its declared device label and generally remains coupled to the other retained coordinates. Their terminal-voltage maps must not be interchanged.

11.1 One conservative normal mode

A conservative mode vector satisfies

\[ K_\Phi\mathbf u_m =\omega_m^2C_0\mathbf u_m. \]

Its node-flux expansion is

\[ \boldsymbol\Phi=\mathbf u_m\phi_m. \]

The effective mode capacitance, stiffness, and terminal participation are

\[ \boxed{ C_m=\mathbf u_m^TC_0\mathbf u_m, \qquad K_m=\mathbf u_m^TK_\Phi\mathbf u_m=\omega_m^2C_m, \qquad \eta_m=\mathbf b_P^T\mathbf u_m. } \]

The scale of \(\mathbf u_m\) changes \(C_m\) and \(\eta_m\) together, so the physical ratios below are invariant. Canonical quantization gives

\[ \boxed{ \Phi_{m,\mathrm{zpf}} =\sqrt{\frac{\hbar}{2\omega_mC_m}}, \qquad |V_{P,m}^{\mathrm{zpf}}| =\omega_m|\eta_m|\Phi_{m,\mathrm{zpf}}. } \]

11.2 A physically anchored bare-coordinate cluster

For several retained anchored coordinates, first declare the coordinate map. If \(\boldsymbol\Phi\simeq T\boldsymbol z\), the physical terminal map is

\[ \boxed{ \mathbf c_P=T^T\mathbf b_P, \qquad \Phi_P=\mathbf c_P^T\boldsymbol z. } \]

Every later oscillator scaling, rotation, or Bogoliubov transformation must also transform this port vector. A visually convenient Hamiltonian written in one basis cannot use a p-only port vector borrowed from another basis.

There is a further distinction in a physically anchored but kinetically coupled basis. If

\[ H=\frac12\mathbf P_z^TA_Q\mathbf P_z +\frac12\boldsymbol z^TB_\Phi\boldsymbol z, \]

Hamilton’s equation gives

\[ \boxed{ V_P=\dot\Phi_P =\mathbf c_P^T\dot{\boldsymbol z} =\mathbf c_P^TA_Q\mathbf P_z. } \]

For the downstream canonical oscillator scales

\[ Z_x=\sqrt{\frac{(A_Q)_{xx}}{(B_\Phi)_{xx}}}, \qquad P_{x}^{(a)}=-i\sqrt{\frac{\hbar}{2Z_x}}\,a_x, \]

the annihilation-sector contribution to the terminal-voltage operator is

\[ \boxed{ V_P^{(a)} =-i\sum_x \left[\mathbf c_P^TA_Q\right]_x \sqrt{\frac{\hbar}{2Z_x}}\,a_x. } \]

Therefore a physical branch attached only to coordinate \(p\) establishes \(\mathbf c_P=(0,0,1)^T\) in that coordinate basis, but it does not by itself establish a p-only input–output vector when \(A_Q\) is dense. The latter is a statement about the transformed terminal voltage operator and must be tested after the complete canonical normalization. In a true conservative normal-mode basis, the corresponding result is recovered by transforming \(A_Q\), \(\mathbf c_P\), and the ladder operators together.

The bare annihilation sector is not generally the positive-frequency part: the full charge observable contains both \(a_x\) and \(a_x^\dagger\). With pairing, transform that full observable into the actual conservative normal modes before selecting its positive-frequency part. A Bogoliubov transformation of the Hamiltonian alone does not supply the transformed terminal observable.

In a stated number-conserving realization or appropriate RWA, for a narrow cluster around one carrier \(\omega_0\), the following vector gives the annihilation-sector terminal-voltage coefficients:

\[ v_{P,x} =\left[\mathbf c_P^TA_Q\right]_x \sqrt{\frac{\hbar}{2Z_x}}. \]

Within that number-conserving/RWA scope, the same bath produces the Markov decay matrix

\[ \boxed{ [\boldsymbol\Gamma_{\mathrm{ext}}]_{xy} =\frac{2\operatorname{Re}Y_{\mathrm{ext}}(\omega_0)} {\hbar\omega_0} v_{P,x}^*v_{P,y}. } \]

Its diagonal entries are direct coordinate rates and its off-diagonal entries are correlated decay through the shared terminal. When the mode frequencies span a range over which the admittance or rotating-wave factors vary materially, retain the frequency-dependent projected self-energy instead of using this common-carrier Markov matrix.

12. Derive the radiative linewidth

The numerator is power leaving the chosen terminal; the denominator is stored mode energy. Dividing the two yields this page’s weak-damping rate, with the same peak-phasor factor in both. Quantum noise/input–output theory gives the related bath interpretation (Clerk et al. 2010; Gardiner and Collett 1985), not a license to use the scalar expression for every multimode or delayed environment.

For one weakly damped mode with peak terminal phasor \(V_P=\eta_mV_m\), the time-averaged power delivered to the eliminated environment is

\[ \overline P_{\mathrm{rad}} =\frac12\operatorname{Re}Y_{\mathrm{ext}}(\omega_m) |\eta_mV_m|^2. \]

At resonance its total stored energy is

\[ \overline E_m=\frac12C_m|V_m|^2. \]

Because \(\dot E_m=-\kappa_{m,\mathrm{ext}}E_m\),

\[ \boxed{ \kappa_{m,\mathrm{ext}} =\frac{ \operatorname{Re}Y_{\mathrm{ext}}(\omega_m)|\eta_m|^2 }{C_m} =\frac{ 2\operatorname{Re}Y_{\mathrm{ext}}(\omega_m) |V_{P,m}^{\mathrm{zpf}}|^2 }{\hbar\omega_m}. } \]

For the symmetric two-sided boundary,

\[ \kappa_{m,L}=\kappa_{m,R} =\frac12\kappa_{m,\mathrm{ext}}. \]

In the weak-\(C_{\mathrm{ext}}\) limit,

\[ \boxed{ \kappa_{m,\mathrm{ext}} \simeq \frac{|\eta_m|^2\omega_m^2C_{\mathrm{ext}}^2Z_0}{2C_m}. } \]

This expression states exactly which assumptions permit a physical capacitor to be summarized by a rate: one isolated weakly damped coordinate, a local frequency window, a real matched line, and the declared terminal participation. Outside that boundary, use \(Y_{\mathrm{ext}}(\omega)\) or the full node/line response instead of forcing one constant \(\kappa\).

13. Why the EOM contains \(\kappa/2\) and \(\sqrt\kappa\)

This section reorganizes the flat-bath input–output reduction (Gardiner and Collett 1985; Clerk et al. 2010). The choice of continuum measure and channel phase is displayed rather than absorbed into an unnamed coupling constant.

The continuum line may equivalently be represented by bath operators. In a channel phase convention, a rotating-wave interaction has the form

\[ \frac{H_{\mathrm{int}}}{\hbar} =i\sum_j\int\frac{d\omega}{\sqrt{2\pi}} \left[ g_j(\omega)b_j^\dagger(\omega)a -g_j^*(\omega)a^\dagger b_j(\omega) \right]. \]

Solving the bath equation, substituting it back into the mode equation, and taking a flat narrow-band spectral density gives

\[ \kappa_j=|g_j(\omega_m)|^2, \]

up to the displayed continuum normalization. The resulting local equation is

\[ \dot a =-\left(i\omega_m+\frac12\sum_j\kappa_j\right)a +\sum_j e^{i\theta_j}\sqrt{\kappa_j}\,b_{j,\mathrm{in}}. \]

The factor \(1/2\) is required because \(|a|^2\) is energy or occupation: if \(|a|^2\propto e^{-\kappa t}\), then \(a\propto e^{-\kappa t/2}\). The \(\sqrt\kappa\) drive coefficient is required because \(b^\dagger b\) is an incident photon flux whereas \(a^\dagger a\) is a stored photon number. Channel phases \(e^{i\theta_j}\) are fixed by the physical boundary and reference-plane gauge, not by the rate magnitude alone.

14. Passive multimode input–output matrices

Energy-conserving direct/resonant-path constraints follow coupled-mode theory (Suh et al. 2004). This page rewrites them in the stated photon-normalized channel gauge and derives the scattering formula by the same substitution used above. A unitary direct path and a passive canonical metric are visible assumptions.

The energy-normalized internal amplitudes and photon-flux-normalized channels are

\[ \mathbf a\in\mathbb C^N, \qquad \mathbf b_{\mathrm{in/out}}\in\mathbb C^P. \]

A passive Markov realization is

\[ \boxed{ \dot{\mathbf a} =\left( -i\mathbf h -\frac12\boldsymbol\Gamma_{\mathrm{int}} -\frac12\mathbf D_{\rm port}^\dagger\mathbf D_{\rm port} \right)\mathbf a -\mathbf D_{\rm port}^\dagger\mathbf S_{\mathrm{dir}} \mathbf b_{\mathrm{in}}, } \]

\[ \boxed{ \mathbf b_{\mathrm{out}} =\mathbf S_{\mathrm{dir}}\mathbf b_{\mathrm{in}} +\mathbf D_{\rm port}\mathbf a. } \]

Consequently,

\[ \boxed{ \boldsymbol\Gamma_{\mathrm{ext}} =\mathbf D_{\rm port}^\dagger\mathbf D_{\rm port}, \qquad \mathbf K_{\rm port}=-\mathbf D_{\rm port}^\dagger\mathbf S_{\mathrm{dir}}. } \]

These relations follow from conservation of internal energy plus port power when \(\boldsymbol\Gamma_{\mathrm{int}}=0\) and \(\mathbf S_{\mathrm{dir}}\) is unitary. They also show why \(\mathbf K_{\rm port}\), \(\mathbf D_{\rm port}\), and external damping are not three independent fit objects in a passive model.

With

\[ \mathbf M(\omega) =\frac12\boldsymbol\Gamma_{\mathrm{int}} +\frac12\mathbf D_{\rm port}^\dagger\mathbf D_{\rm port} +i(\mathbf h-\omega\mathbf I), \]

the scattering matrix is

\[ \boxed{ \mathbf S(\omega) =\mathbf S_{\mathrm{dir}} -\mathbf D_{\rm port}\mathbf M^{-1}(\omega) \mathbf D_{\rm port}^\dagger\mathbf S_{\mathrm{dir}}. } \]

Equivalently,

\[ \mathbf S =\mathbf S_{\mathrm{dir}} +\mathbf D_{\rm port}\boldsymbol\chi\mathbf K_{\rm port}, \qquad \boldsymbol\chi=\mathbf M^{-1}. \]

For an ideal symmetric hanger, one convenient gauge is

\[ \mathbf S_{\mathrm{dir}} =\begin{pmatrix}0&1\\1&0\end{pmatrix}, \qquad \mathbf D_{\rm port} =-\sqrt{\frac\kappa2} \begin{pmatrix}1\\1\end{pmatrix}. \]

It gives \(\mathbf K_{\rm port}=\sqrt{\kappa/2}(1,1)\) and produces destructive interference between the direct through wave and resonator radiation. A finite line replaces the off-diagonal ones by the appropriate propagation factor; mismatch requires the complete direct two-port and may also create frequency-dependent feedback in \(\mathbf M\).

15. Exact, reduced, and fitted quantities

Use this summary to decide which approximation produced a reported quantity. The table is an engineering organization of this page’s derivation, not a source’s universal artifact schema. The exact line/network treatment has a broader applicability than the local Markov model (Parra-Rodriguez and Egusquiza 2025); no current package capability is claimed by listing either representation.

Layer Primitive inputs Outputs What may be fitted
Exact node/line \(\mathbf C_0,\mathbf K_\Phi,\mathbf C_{\rm local},l',c'\), topology, port planes Frequency-dependent \(\boldsymbol{\mathcal D}_{\mathrm{open}}\) and exact \(S/Y/Z\) Physical element or calibrated network parameters with provenance.
Reduced conservative One declared basis map and energy normalization \(\mathbf h\), counter-rotating block, terminal vector \(\mathbf c_P\) Structured diagonal and coherent entries only if response closure passes.
Open reduced Projected \(Y_{\mathrm{ext}}(\omega)\) or bath spectral density Shifts, \(\mathbf D_{\rm port}(\omega)\), correlated decay, complex poles Frequency-dependent self-energy or bounded Markov parameters.
Markov input–output Narrow-band constant \(\mathbf h,\mathbf D_{\rm port},\Gamma\) and \(S_{\mathrm{dir}}\) Structured complex \(S\) and derived hybridized poles One passive parameter set; \(\mathbf D_{\rm port}\), \(\mathbf K_{\rm port}\), and radiative damping remain constrained.
Rational response Calibrated complex trace Visible poles, total linewidths, residues Vector Fitting; no bare-coordinate decomposition.

The structured physical fit and Vector Fitting meet at the hybridized pole and complex-response layers. They do not own the same bare parameters.

The exact node/line row or either declared reduced row may be packaged as a circuit-derived finite-order port-response model. Its artifact must state the retained representation and carry the physical coordinate, drive, and output maps. See Finite-Order Port-Response Models and Chain Realizations.

Interpretation limits and diagnostic comparisons

The derivation makes the following interpretation limits explicit:

  • Node, branch, port-current, phasor, RMS/peak, and reference-plane conventions determine the signs, factors, and observable being described.
  • Counting a local capacitance block both as retained branches and as an eliminated admittance duplicates reactive loading and radiation.
  • A physical linewidth interpretation depends on the terminal selector, mode normalization, environment admittance, and angular-versus-ordinary-frequency convention.
  • A p-only port map borrowed from a different basis does not describe the Hamiltonian’s transformed terminal voltage.
  • In the stated passive Markov model, \(\mathbf D_{\rm port}\), \(\mathbf K_{\rm port}\), and radiative damping obey the coupled constraints derived above; treating them as independent fit objects does not establish that passive realization.
  • A constant Markov rate is a local approximation, not a replacement for arbitrary frequency-dependent environmental memory.
  • A scalar \(S_{21}\) projection does not independently identify left/right channel rates.

Comparing the reduced and exact node/line complex \(S/Y/Z\) responses under the same physical boundary can reveal the effects of mode truncation, normalization, or a Markov approximation. Differences are diagnostic evidence to interpret for the intended use, not an automatic publishability or promotion decision. This page prescribes no held-out frequencies, sweep points, sampling rule, numeric tolerance, or acceptance threshold.

Connections

References

Part of this page Primary support
Node/line-flux variables, distributed line Lagrangian, and semi-infinite-line Langevin boundary Vool & Devoret; Clerk et al.
Continuum bath elimination, amplitude damping, \(\sqrt\kappa\) drive, and input–output relation Gardiner & Collett; Clerk et al.
Coupling capacitor retained in the physical open-circuit Lagrangian and its effect on modes/normalization Malekakhlagh & Türeci
Passive multimode/multiport direct-path constraints Suh, Wang & Fan
Exact transmission-line/multiport quantization beyond the local Markov reduction Parra-Rodriguez & Egusquiza

No source is claimed to print every device specialization of this general boundary. The explicit two-sided \(C_{\mathrm{ext}}\) elimination on this page is derived from the cited line-field and boundary principles. Device-specific internal matrices and reduced response formulas belong in their worked-system pages, where they must close back to this general boundary.

Field Value
Status Seed
Used by Multimode Input–Output Scattering.
Evidence boundary The line-field, power-wave, bath-elimination, and passive-coupled-mode starting models have the source support identified in their sections. The explicit two-sided \(C_{\mathrm{ext}}\) boundary is derived here from the same Lagrangian and KCL conventions. Sign, limiting-case, and exact-circuit reconstruction comparisons can diagnose its algebra and physical interpretation; this row defines no promotion or acceptance condition.
Open review question Can a reader derive every factor of two and square root from node/line energy and power flow, then identify exactly where the Markov approximation begins?

Bibliography

Clerk, A. A. et al. 2010. “Introduction to Quantum Noise, Measurement, and Amplification.” Reviews of Modern Physics 82: 1155. https://doi.org/10.1103/RevModPhys.82.1155.
Gardiner, C. W., and M. J. Collett. 1985. “Input and Output in Damped Quantum Systems.” Physical Review A 31: 3761. https://doi.org/10.1103/PhysRevA.31.3761.
Kurokawa, K. 1965. “Power Waves and the Scattering Matrix.” IEEE Transactions on Microwave Theory and Techniques 13 (2): 194–202. https://doi.org/10.1109/TMTT.1965.1125964.
Malekakhlagh, M., and H. E. T"ureci. 2016. Lamb Shift of a Superconducting Qubit in a Multimode Cavity. https://doi.org/10.1103/PhysRevA.93.012120.
Parra-Rodriguez, A., and I. L. Egusquiza. 2025. Exact Quantization of Nonreciprocal Quasi-Lumped Electrical Networks with Transmission Lines. https://doi.org/10.1103/PhysRevX.15.011072.
Suh, W., Z. Wang, and S. Fan. 2004. Temporal Coupled-Mode Theory and the Presence of Non-Orthogonal Modes in Lossless Multimode Cavities. https://doi.org/10.1109/JQE.2004.834773.
Vool, U., and M. H. Devoret. 2017. “Introduction to Quantum Electromagnetic Circuits.” International Journal of Circuit Theory and Applications 45: 897–934. https://doi.org/10.1002/cta.2359.