Dynamics Conventions and the Flux-to-Mode Symbol Bridge
Node role: Concept.
Reader task: Translate a derivation, solver, or fit into one consistent sign, time-evolution, and symbol convention before combining equations or artifacts.
Reading map
Use the four-convention table for a quick translation. Follow phasor → Fourier → rotating frame → quantum picture for the sign derivations, then use the flux/charge-to-amplitude section when changing state variables. The final example shows which observables survive conjugating the representation.
What this page owns
Engineering choice. The project’s phasor, Fourier kernel, detuning sign and qualifier vocabulary are selected conventions, not uniquely mandated by circuit-physics literature.
This page is the single convention source of truth for linear circuit, input–output, and open-mode derivations in SCQ_Design. It fixes four choices that are individually valid but dangerous when mixed:
- the time-harmonic or phasor convention;
- the Fourier-transform convention;
- the rotating-frame convention; and
- Schrödinger/Heisenberg quantum time evolution.
It also owns the symbol bridge from dimensional generalized flux coordinates to canonical oscillator amplitudes. Other Knowledge pages should link here and state only local exceptions or source-specific translations.
The separate Symbol Conventions page owns topology, coupling-state, parameter-layer, basis, channel, processing, and unit qualifiers. This page does not redefine those semantic labels.
SCQ_Design uses
\[ e^{-i\omega t}, \qquad X(\omega)=\int_{-\infty}^{\infty}x(t)e^{+i\omega t}\,dt, \qquad a_{\rm lab}(t)=a_{\rm rot}(t)e^{-i\omega_dt}, \qquad U(t)=e^{-iHt/\hbar}. \]
A paper, solver, or library may use the conjugate convention. Its output is not wrong, but it must be translated at the boundary; one sign may never be changed in isolation.
Four conventions that answer different questions
| Convention | What is being chosen? | Canonical SCQ_Design choice |
|---|---|---|
| Time-harmonic / phasor | Which complex exponential represents a real single-frequency oscillation? | \(x_{\rm real}(t)=\operatorname{Re}[X(\omega)e^{-i\omega t}]\) |
| Fourier transform | Which kernel maps a complete time signal to frequency space? | Forward \(e^{+i\omega t}\), inverse \(e^{-i\omega t}\) |
| Rotating frame | Which known carrier phase is removed from a dynamical amplitude? | \(a_{\rm rot}=e^{+i\omega_dt}a_{\rm lab}\) |
| Quantum evolution / picture | Whether time dependence is assigned to states, operators, or both | \(U=e^{-iHt/\hbar}\); Schrödinger and Heisenberg pictures related by \(U\) |
The rotating frame is not another Fourier convention, and the Heisenberg picture is not another phasor convention. They are different transformations that must be composed consistently.
1 — Time-harmonic and phasor convention
This-page derivation. Differentiate the selected harmonic ansatz and apply the result to constitutive laws and time delay. The signs are consequences of the convention, not independent physical assumptions.
For a real classical quantity,
\[ \boxed{ x_{\rm real}(t) =\operatorname{Re}\!\left[X(\omega)e^{-i\omega t}\right]. } \]
Therefore
\[ \boxed{ \frac{d}{dt}\longrightarrow-i\omega, \qquad \frac{d^2}{dt^2}\longrightarrow-\omega^2. } \]
For example, the canonical capacitor and inductor admittances are
\[ Y_C=-i\omega C, \qquad Y_L=\frac{1}{-i\omega L}=\frac{i}{\omega L}. \]
A wave traveling toward increasing \(z\) is written
\[ V^+(z,t)=V_0^+e^{+ikz}e^{-i\omega t}. \]
A physical delay satisfies
\[ x_{\rm out}(t)=x_{\rm in}(t-\tau) \quad\Longleftrightarrow\quad X_{\rm out}(\omega)=e^{+i\omega\tau}X_{\rm in}(\omega). \]
Thus, under this convention, positive propagation phase and positive delay appear with \(+i\). This sign reverses under the conjugate phasor convention.
Engineering example / this-page derivation. If the selected JosephsonCircuits.jl HB output uses the conjugate \(e^{+i\omega t}\) convention, the translation below follows for consistently defined waves. Establish the actual convention from the owning selected-version package documentation and adapter; the documentation entrypoints (JosephsonCircuits.jl, n.d.-a, n.d.-b) do not certify every version or the current Workbench adapter. Under that source-specific assumption, compare with an SCQ_Design analytical trace by conjugating the complete complex scattering matrix at each positive real frequency:
\[ \boxed{ S_{ij}^{\rm SCQ}(\omega) = \left[S_{ij}^{\rm JosephsonCircuits,HB}(\omega)\right]^*. } \]
This rule assumes identical port order, wave directions, reference impedances, and reference planes. Conjugate once at the solver-adapter boundary; do not transpose the matrix, exchange port indices, or conjugate it again downstream. The adapter must record both the source and target phasor conventions and selected source version.
2 — Fourier-transform convention
Engineering choice / this-page derivation. Choose the transform pair below; integration by parts gives its derivative rule when the boundary terms vanish.
The project Fourier pair is
\[ \boxed{ X(\omega) =\int_{-\infty}^{\infty}x(t)e^{+i\omega t}\,dt, \qquad x(t) =\frac{1}{2\pi}\int_{-\infty}^{\infty} X(\omega)e^{-i\omega t}\,d\omega. } \]
When boundary terms vanish,
\[ \mathcal F\!\left[\dot x\right]=-i\omega X(\omega). \]
For a real time-domain signal,
\[ X(-\omega)=X(\omega)^*. \]
The single-frequency phasor in the previous section is one component of this Fourier representation. A page that writes only a harmonic ansatz must not silently imply a different full Fourier kernel.
Control and Vector-Fitting libraries often use a Laplace variable \(s\) with stable poles in \(\operatorname{Re}s<0\). Under the project convention,
\[ \boxed{s=-i\widetilde\omega.} \]
Hence the positive-frequency physical pole
\[ \widetilde\omega=\omega_0-i\frac\kappa2 \]
maps to
\[ s=-\frac\kappa2-i\omega_0. \]
Many real-rational fitters report the conjugate partner \(-\kappa/2+i\omega_0\). A fit artifact must publish which member is mapped to positive physical frequency instead of inferring it from the sign of \(\operatorname{Im}s\) alone.
3 — Rotating-frame convention
This-page derivation. Substitute the declared carrier removal into the stated damped lab-frame equation. This yields the project’s detuning sign.
Remove the carrier from the lab-frame oscillator amplitude with
\[ \boxed{ a_{\rm lab}(t)=a_{\rm rot}(t)e^{-i\omega_dt}, \qquad a_{\rm rot}(t)=e^{+i\omega_dt}a_{\rm lab}(t). } \]
If
\[ \dot a_{\rm lab} =-\left(i\omega_0+\frac\kappa2\right)a_{\rm lab}+F_{\rm lab}, \]
then
\[ \boxed{ \dot a_{\rm rot} =-\left[i(\omega_0-\omega_d)+\frac\kappa2\right]a_{\rm rot} +F_{\rm rot}. } \]
SCQ_Design therefore uses
\[ \Delta=\omega_0-\omega_d \]
when an unqualified rotating-frame detuning is unavoidable. A source using \(\Delta'=\omega_d-\omega_0\) is equivalent after \(\Delta'=-\Delta\) and a consistent rewrite of every EOM term.
The rotating transformation changes coordinates, not the measured resonance, linewidth, or scattering probability.
4 — Quantum-state and operator time evolution
Reorganized explanation. Canonical circuit operators follow Hamiltonian time evolution (Vool and Devoret 2017, sec. 3.1). This-page derivation: the harmonic commutator below gives the annihilation-operator carrier phase; a classical phasor declaration is not its physical origin.
In the Schrödinger picture,
\[ \boxed{ i\hbar\frac{d}{dt}|\psi_S(t)\rangle =H_S|\psi_S(t)\rangle, \qquad |\psi_S(t)\rangle=U(t)|\psi_S(0)\rangle, \qquad U(t)=e^{-iH_St/\hbar} } \]
for a time-independent Hamiltonian. An energy eigenstate coefficient evolves as \(e^{-iE_nt/\hbar}\).
In the Heisenberg picture,
\[ \hat O_H(t)=U^\dagger(t)\hat O_SU(t), \]
and
\[ \boxed{ \frac{d\hat O_H}{dt} =\frac{i}{\hbar}[H_H,\hat O_H] +\left(\frac{\partial\hat O}{\partial t}\right)_H. } \]
For \(H=\hbar\omega_0\hat a^\dagger\hat a\),
\[ \dot{\hat a}=-i\omega_0\hat a, \qquad \hat a(t)=\hat a(0)e^{-i\omega_0t}. \]
This agrees with the project phasor convention, but it follows from the Hamiltonian commutator rather than from declaring a classical phasor. Schrödinger and Heisenberg pictures give identical expectation values when states, operators, and observables are transformed together.
Flux coordinates, voltage, and oscillator amplitudes
Physics path: Previous — Knowledge overview · Next — One LC in Every Language
For this variable change, the canonical quantization construction supplies the conjugate pair, while the normalization comparison distinguishes this canonical scaling from local response scaling.
The symbol must continue to identify the physical kind of quantity:
| Symbol | Meaning | Typical units |
|---|---|---|
| \(\vec\Phi_{\rm node}\) | Physical node-flux coordinates | Wb |
| \(\vec\Phi_{\rm local}\) | Device-aligned generalized flux coordinates before all cyclic reductions | Wb |
| \(\vec\Phi_{\rm bare}\) | Complete retained bare generalized-flux basis | Wb |
| \(\vec Q_{\rm bare}\) | Canonical charges dual to \(\vec\Phi_{\rm bare}\) | C |
| \(\vec\Phi_{\rm N}\) | Closed conservative normal-mode generalized-flux coordinates after diagonalization | Wb |
| \(\vec Q_{\rm N}\) | Canonical charges dual to \(\vec\Phi_{\rm N}\) | C |
| \(\dot{\vec\Phi}_{\rm bare}=\vec V_{\rm bare}\) | Coordinate voltages | V |
| \(\hat a_x\) | Dimensionless annihilation operator of bare coordinate \(x\) after canonical normalization | dimensionless |
| \(a_x=\langle\hat a_x\rangle\) | Coherent bare-coordinate amplitude | \(\sqrt{\text{occupation}}\) under photon normalization |
| \(\hat b_\mu\) | Canonically normalized annihilation operator corresponding to \(\Phi_{\mu,\rm N},Q_{\mu,\rm N}\) | dimensionless |
| \(s_{j,\rm in/out}\) | Port traveling-wave amplitude | \(\sqrt{\mathrm W}\) or \(\sqrt{\mathrm{s^{-1}}}\), according to declared normalization |
| \(\boldsymbol\chi_\Phi=\boldsymbol{\mathcal D}_{\rm open}^{-1}\) | Exact second-order generalized-flux susceptibility | Generalized flux per generalized current/force |
| \(\boldsymbol\chi_a=\mathbf M^{-1}\) | Reduced first-order oscillator susceptibility | s under canonical photon normalization |
| \(\zeta_x\equiv\chi_{x,0}^{-1}\) | Scalar inverse bare susceptibility | \(\mathrm{s^{-1}}\) |
Reorganized explanation. Canonical LC impedance scaling gives the zero-point flux/charge scales (Vool and Devoret 2017, sec. 3.1.4.1). This-page derivation: the invertible amplitude definition and voltage map below follow from these scales and the stated charge-energy matrix.
For diagonal bare-coordinate oscillator scales,
\[ \Phi_{x,\rm zpf}=\sqrt{\frac{\hbar Z_x}{2}}, \qquad Q_{x,\rm zpf}=\sqrt{\frac{\hbar}{2Z_x}}, \qquad \Phi_{x,\rm zpf}Q_{x,\rm zpf}=\frac\hbar2. \]
The symbol change from canonical flux/charge to an oscillator operator is the explicit invertible definition
\[ \boxed{ \hat a_x \equiv \frac{\hat\Phi_x}{2\Phi_{x,\rm zpf}} +i\frac{\hat Q_x}{2Q_{x,\rm zpf}}. } \]
Equivalently,
\[ \hat\Phi_x=\Phi_{x,\rm zpf}(\hat a_x+\hat a_x^\dagger), \qquad \hat Q_x=-iQ_{x,\rm zpf}(\hat a_x-\hat a_x^\dagger). \]
Therefore
\[ \boxed{a_x\not\equiv\dot\Phi_x.} \]
They do not even have the same units. The exact Hamilton equation is
\[ \dot{\vec\Phi}_{\rm bare} =\mathbf A_Q\vec Q_{\rm bare}, \qquad \mathbf A_Q=\mathbf C_{\rm bare}^{-1}. \]
Substituting the canonical charge definition gives the exact full voltage operator:
\[ \boxed{ \hat{\vec V}_{\rm bare} =-i\mathbf A_Q\mathbf Q_{\rm zpf} \left(\hat{\vec a}_{\rm bare}-\hat{\vec a}_{\rm bare}^{\dagger}\right). } \]
Taking first moments, the annihilation-sector contribution is
\[ \boxed{ \vec V_{\rm bare}^{(a)} =-i\mathbf A_Q\mathbf Q_{\rm zpf}\vec a_{\rm bare}, } \]
with
\[ \mathbf Q_{\rm zpf}:=\operatorname{diag}(Q_{x,\rm zpf}). \]
The annihilation sector is not necessarily the positive-frequency sector of the physical dynamics. For an isolated diagonal oscillator with positive frequency, it is the positive-frequency contribution and reduces to
\[ V_x^{(+)}=-i\omega_x\Phi_{x,\rm zpf}a_x. \]
In a kinetically coupled bare basis, one coordinate voltage can depend on several \(a_x\) amplitudes. Port coupling must therefore transform the complete voltage/current map, not merely reuse the label of the physically attached node.
From local oscillator amplitudes to modes
This-page derivation. A unitary transform applies to the stated number-conserving block. A pairing block requires the wider canonical transformation; these are different representations of the same starting quadratic model.
For a number-conserving diagonalization,
\[ \mathbf U^\dagger\mathbf h_{\rm bare}\mathbf U =\operatorname{diag}(\omega_\mu), \qquad \vec b=\mathbf U^\dagger\vec a_{\rm bare}. \]
A full quadratic Hamiltonian with a nonzero pairing block requires a Bogoliubov/symplectic transformation that mixes \(\vec a\) and \(\vec a^\dagger\); it is not generally one unitary rotation. In that realization, a bare annihilation operator can contain both signs of normal-mode frequency. Transform the full voltage operator above, including both annihilation and creation terms, before selecting its positive-frequency normal-mode contribution or constructing a port observable.
The distinction between linear first moments and quantum fluctuations follows the quantum LC example (Vool and Devoret 2017, sec. 1.1); input–output and noise theory add bath and noise operators (Gardiner and Collett 1985; Clerk et al. 2010).
For a linear or quadratic system under coherent drive, the quantum first moments \(\langle\hat a\rangle\) obey the same linear response equations as the corresponding classical complex amplitudes. Quantum theory additionally supplies commutators, zero-point fluctuations, noise, and covariance. That statement does not extend an unrestricted classical/mean-field equivalence to nonlinear dynamics.
One physical example under both phasor conventions
This-page derivation. Compare a decaying amplitude and a pure delay under both selected exponential signs, conjugating wave definitions together.
Under the canonical \(e^{-i\omega t}\) convention, a freely decaying amplitude and a delayed through path are represented by
\[ a(t)=a(0)e^{-i\omega_0t}e^{-\kappa t/2}, \qquad \widetilde\omega_-=\omega_0-i\frac\kappa2, \qquad t_-(\omega)=e^{+i\omega\tau}. \]
Under the conjugate \(e^{+i\omega t}\) convention, the same physical waveform and delay use
\[ \widetilde\omega_+=\omega_0+i\frac\kappa2, \qquad t_+(\omega)=e^{-i\omega\tau}. \]
For consistently conjugated port amplitudes,
\[ S_+(\omega)=S_-(\omega)^*. \]
The following do not change:
- the real time-domain voltage and current;
- \(|S_{ij}|^2\) and delivered power;
- the positive resonance frequency \(\omega_0\);
- the positive linewidth \(\kappa\), lifetime, and quality factor; and
- whether the system is causal, stable, reciprocal, or passive.
What changes are representation-dependent signs: susceptance, pole imaginary part, propagation phase, delay phase, and detuning terms. This is why a convention change is physically harmless only when the entire derivation and all input/output definitions change together.
Every frequency-domain derivation or fit artifact must declare:
- phasor sign and derivative map;
- Fourier forward and inverse kernels;
- angular frequency or ordinary frequency units;
- rotating-frame carrier and detuning sign, if any;
- Schrödinger or Heisenberg picture when operator evolution is shown;
- port current directions, wave normalization, impedances, and reference planes; and
- the exact basis attached to every vector and matrix.
Connections
- Circuit Models to Bare Coordinates, Open EOM, and Normal Modes owns the end-to-end modeling procedure.
- Circuit Lagrangian, Hamiltonian, and Canonical Quantization owns the Legendre transform and operator domain.
- From Node and Line Flux to Input–Output Scattering derives terminal-voltage and traveling-wave prefactors.
- Multimode Input–Output Scattering uses the canonical \(a_x\), \(\boldsymbol\chi\), and port symbols.
- Resonator Decay, Linewidth, and Quality Factor owns positive linewidth and lifetime semantics.
- Introduction to Quantum Electromagnetic Circuits records the local source trail for node flux, canonical charge, and circuit quantization.
References
- U. Vool and M. H. Devoret, “Introduction to Quantum Electromagnetic Circuits,” arXiv:1610.03438.
- C. W. Gardiner and M. J. Collett, “Input and output in damped quantum systems,” doi:10.1103/PhysRevA.31.3761.
- A. A. Clerk et al., “Introduction to quantum noise, measurement, and amplification,” arXiv:0810.4729.
| Field | Value |
|---|---|
| Status | Source-backed |
| Used by | Circuit Models to Bare Coordinates, Open EOM, and Normal Modes and all downstream frequency-domain, input–output, pole, linewidth, and worked-system derivations. |
| Open review question | Can a reader translate one damped delayed response between \(e^{-i\omega t}\) and \(e^{+i\omega t}\) without changing any physical observable, and explain why \(a_x\ne\dot\Phi_x\)? |