Resonator Decay, Linewidth, and Quality Factor
Node role: Concept.
Reader task: Translate one resonator loss process among time decay, complex-pole width, spectral linewidth, and quality factor.
Why it matters
The same resonator loss is described as a time-domain decay rate, a complex- pole imaginary part, a spectral linewidth, or a quality factor. Those numbers agree only when amplitude versus energy decay and angular frequency versus hertz are kept explicit.
This node defines that conversion once. Individual fits may choose symbols such as \(\kappa\), \(\gamma\), or \(\Gamma\), but they must state which physical rate each symbol owns.
One-mode contract
The starting damped-mode equation follows input–output theory (Gardiner and Collett 1985). This page derives the exponential, Lorentzian half-power width, and \(Q\) conversions from it. Its assumptions are one linear isolated mode and constant Markov damping; the later multimode discussion limits when a measured trace shares this Lorentzian width.
For a freely decaying field amplitude,
\[ \dot a(t)= -i\omega_0a(t)-\frac{\kappa}{2}a(t), \]
so
\[ a(t)=a(0)e^{-i\omega_0t}e^{-\kappa t/2}, \qquad n(t)=|a(t)|^2=n(0)e^{-\kappa t}. \]
Therefore \(\kappa\) is the energy or photon-number decay rate, while \(\kappa/2\) is the field-amplitude decay rate.
This \(\kappa/2\) is a decay-law factor. It is unrelated both to the classical energy factors in \(Q^2/(2C)+\Phi^2/(2L)\) and to the quantum zero-point \(+1/2\) in \(\hbar\omega(\hat a^\dagger\hat a+1/2)\).
The corresponding susceptibility has the form
\[ \chi(\omega)= \frac{1}{\kappa/2+i(\omega_0-\omega)}. \]
Its power response is Lorentzian:
\[ |\chi(\omega)|^2 \propto \frac{1}{(\kappa/2)^2+(\omega_0-\omega)^2}. \]
The half-power points are separated by \(\kappa\) in angular frequency. Thus
\[ \Delta\omega_{\mathrm{FWHM}}=\kappa, \qquad \Delta f_{\mathrm{FWHM}}=\frac{\kappa}{2\pi}. \]
| Quantity | Meaning | Value under this convention |
|---|---|---|
| Field-amplitude e-fold time | \(|a|\) falls by \(e^{-1}\) | \(2/\kappa\) |
| Energy or photon-number lifetime | \(|a|^2\) falls by \(e^{-1}\) | \(1/\kappa\) |
| Angular power FWHM | Width in \(\omega\) | \(\kappa\) |
| Ordinary-frequency power FWHM | Width in \(f\) | \(\kappa/2\pi\) |
| Loaded quality factor | Stored energy relative to loss per cycle | \(Q_l=\omega_0/\kappa=f_0/\Delta f_{\mathrm{FWHM}}\) |
Internal, external, and loaded loss
Here independent channels contribute additive energy flow. The loss-channel interpretation follows the damped-mode model (Gardiner and Collett 1985); inverse-\(Q\) addition below is this page’s algebra using the same resonance frequency. Shared baths and interfering outputs require more than independent scalar rates.
For independent Markovian loss channels,
\[ \kappa_l=\kappa_i+\sum_j\kappa_{c,j}, \]
and therefore
\[ \frac{1}{Q_l} =\frac{1}{Q_i}+\sum_j\frac{1}{Q_{c,j}}. \]
\(Q_i\) describes internal loss, while each \(Q_{c,j}\) describes coupling to one declared external channel. A two-sided feedline can use separate left- and right-going rates; their sum contributes to the loaded linewidth.
Boundary-conditioned linewidth versus one-port flow
For one mode coupled to several external continua under a declared boundary \(\mathcal B\),
\[ \dot a =-i\omega_a a -\frac{\kappa_{a,\mathrm{ext}}^{\mathcal B}+\gamma_a}{2}a, \qquad \kappa_{a,\mathrm{ext}}^{\mathcal B} =\sum_j\kappa_{a\to j}^{\mathcal B}. \]
\(\kappa_{a\to j}^{\mathcal B}\) is the partial energy flow into channel \(j\). The sum \(\kappa_{a,\mathrm{ext}}^{\mathcal B}\) is the boundary-conditioned external modal linewidth. A voltage measured at one output has the form
\[ V_j^{(+)}(t)\propto \sqrt{\kappa_{a\to j}^{\mathcal B}}\,a(t). \]
The partial rate therefore sets that port’s output amplitude, power, and collection fraction. The ring-down envelope at that port still follows the system pole: field amplitude decays with \((\kappa_{a,\mathrm{ext}}^{\mathcal B}+\gamma_a)/2\), and stored energy decays with \(\kappa_{a,\mathrm{ext}}^{\mathcal B}+\gamma_a\). Observing one port does not turn the pole linewidth into that port’s partial rate.
Direct open-operator linewidth
This section is this page’s open-circuit derivation: an attached matched boundary adds damping, and a complex root gives the free time dependence. The physical prefactors and boundary assumptions are derived in the node/line-flux bridge. No finite linewidth follows from conservative matrices alone.
Closed real capacitance and stiffness matrices contain stored energy and conservative frequencies, but no linewidth. A direct calculation must also attach the declared environment. For memoryless matched ports,
\[ \mathbf G_P =\mathbf B_P\operatorname{diag}(1/Z_{0,j})\mathbf B_P^{\mathsf T}, \qquad \mathbf D_{\mathrm{open}}(\omega) =\mathbf K-\omega^2\mathbf C-i\omega\mathbf G_P. \]
A general causal boundary instead contributes a frequency-dependent \(\boldsymbol\Sigma_{\mathrm{env}}(\omega)\). With the \(e^{-i\Omega t}\) convention, a simple passive root \(\Omega_\mu=\omega_\mu-i\kappa_\mu/2\) gives
\[ \kappa_\mu=-2\operatorname{Im}\Omega_\mu. \]
No residue factor changes this exact root linewidth. A nonzero left–right root slope is still required to establish a simple, conditioned root and to normalize residues. If a linewidth is approximated from a real-axis self-energy instead of an exact complex root, the derivative renormalization must be included.
For a retained coordinate set \(R\) and eliminated set \(E\), the Schur view
\[ \mathbf D_{R,\mathrm{eff}} =\mathbf D_{RR} -\mathbf D_{RE}\mathbf D_{EE}^{-1}\mathbf D_{ER} \]
represents the same open system wherever \(\mathbf D_{EE}\) is nonsingular. Roots of \(\det\mathbf D_{R,\mathrm{eff}}\) can represent full-system poles when they close against the full determinant. A root of one diagonal entry is instead a coordinate-effective or diabatic root while the other retained coordinates remain present; it is not automatically a hybrid pole. A frequency-dependent Schur operator may contain more roots and memory than its retained coordinate count suggests, so no selected pole sum inherits a trace identity without a separately qualified complete Markov witness.
A topology-constrained complex-response fit may reconstruct an open operator and derive fitted versions of these same roots. Direct and fitted quantities must be compared by identical operator, boundary, root, and branch identity. Fitting a free linewidth, using a visual \(-3\) dB trace width, or substituting a different pole sum changes the observable.
Appendix C of Rapid high-fidelity multiplexed readout uses \(\kappa_a\) for the reference filter linewidth (Heinsoo et al. 2018, Appendix C) in the \(\Gamma\to1\) single-port limit. After eliminating the input-capacitor boundary, it defines
\[ \widetilde\kappa_a =\kappa_P =\kappa_a\frac{1+\operatorname{Re}\Gamma}{2}. \]
The following output-power calculation is a reorganized explanation of the paper’s Appendix-C boundary: with no incident waves, the two port-resolved rates are
\[ \kappa_{a\to c} =\frac{\kappa_a}{4}|1-\Gamma|^2, \qquad \kappa_{a\to r} =\frac{\kappa_a}{4}|1+\Gamma|^2. \]
For the paper’s capacitor reflection coefficient, \(\operatorname{Re}\Gamma=|\Gamma|^2\), so their sum is exactly \(\widetilde\kappa_a\). At \(\Gamma=0\), each channel is \(\kappa_a/4\) and the total is \(\widetilde\kappa_a=\kappa_a/2\). At \(\Gamma\to1\), the \(c\) channel vanishes, the \(r\) channel becomes \(\kappa_a\), and the total becomes \(\widetilde\kappa_a=\kappa_a\).
Thus “effective” here means the total filter-mode linewidth after applying the altered feedline boundary. It does not mean a single observed channel. The paper’s \(\kappa_R\) is separately the readout-like hybrid-pole linewidth; qualify that paper symbol because this Knowledge Base also uses right-going channel notation. These limits describe the paper’s particular T-junction model; the absence of a named input capacitor in another topology does not assign that topology a value of \(\Gamma\).
Appendix C (Heinsoo et al. 2018, Appendix C) takes the propagation between the T-junction and input capacitor to be negligible. Its equations therefore define a zero-distance boundary, not a finite-distance feedback law. For separated reference planes, use the complete series-capacitor and propagation network. Do not extend all Appendix-C channel or linewidth formulas by merely replacing \(\Gamma\) with a phase-rotated reflection coefficient.
Finite-distance feedback and linewidth observables
Under the \(e^{-i\Omega t}\) convention, write a stable complex angular pole and its ordinary-frequency form as
\[ \Omega_p=2\pi\widetilde f_p, \qquad \kappa_p=-2\operatorname{Im}\Omega_p, \qquad \boxed{\frac{\kappa_p}{2\pi} =-2\operatorname{Im}\widetilde f_p}. \]
Finite-distance phase feedback makes the quantity label and extraction method part of the result:
| Quantity | What it measures | What it is not |
|---|---|---|
| Anchored-bare DirectSolve filter linewidth | Imaginary part of the identity-continued filter-like root of the declared Direct open network. | An HB fit parameter or a hybridized pole selected from a different operator. |
| HB fit linewidth | Linewidth inferred by fitting the declared pump-off HB response model. | Automatically identical to the Direct root linewidth. |
| Hybridized linewidth | Imaginary part of a pole of the complete coupled system. | A bare-coordinate linewidth. |
| \(S_{21}\) \(-3\) dB width | A trace width relative to a declared baseline and threshold. | Generically a pole linewidth when interference or multiple modes are present. |
Comparisons must retain the same boundary, reference planes, branch identity, and frequency units. Agreement of Direct and pump-off HB complex scattering is model evidence, not permission to rename these observables or a tolerance Gate.
Do not infer left-right symmetry from a local-hanger drawing. A useful parameterization is
\[ \kappa_L=\frac{\kappa_{\mathrm{ext}}}{2}(1+\eta), \qquad \kappa_R=\frac{\kappa_{\mathrm{ext}}}{2}(1-\eta), \qquad -1<\eta<1. \]
For scalar \(S_{21}\), a topology-specific physical line-shape fit may estimate \(\kappa_{\mathrm{ext}}\). A bounded \(\eta\) is diagnostic: scalar transmission is invariant under \(L\leftrightarrow R\), and a free complex residue can absorb the product \(\sqrt{\kappa_L\kappa_R}\). Promote separately labeled \(\kappa_L\) and \(\kappa_R\) only with \(S_{11}/S_{22}\), a complete calibrated two-port matrix, or equivalent port-calibrated evidence.
For the complex coupling convention used in resonator line-shape analysis (Probst et al. 2015), when mismatch or asymmetry is represented by a complex coupling quality factor \(\widehat Q_c\), the dissipative relation is
\[ \frac{1}{Q_i} = \frac{1}{Q_l} - \operatorname{Re}\!\left(\frac{1}{\widehat Q_c}\right). \]
Using \(1/|\widehat Q_c|\) silently turns the mismatch phase into dissipation and can bias \(Q_i\).
Complex poles and phasor convention
Following Dynamics Conventions and the Flux-to-Mode Symbol Bridge, the stable one-mode pole is
\[ \Omega=\omega_0-i\frac{\kappa}{2}. \]
The equally valid \(e^{+i\omega t}\) convention conjugates the pole location. Fit reports must therefore record the phasor convention before the sign of a pole imaginary part is interpreted. Positive physical linewidth and lifetime must be invariant after one convention is used consistently.
Naming contract for implementations
This is a retained engineering naming choice, not a further derivation or a universal eligibility rule. Its detailed promotion vocabulary describes the prior workflow; current executable artifact contracts belong to their package. The generic vector-fitting method no longer prescribes those project gates.
Every model or artifact should state:
- whether its rate is internal, external, or total loaded loss;
- whether the number is an energy-decay rate or amplitude-decay rate;
- whether it is expressed in angular units or hertz;
- whether a reported linewidth is power FWHM, power HWHM, or another width;
- which phasor convention defines complex-pole signs; and
- which system and coupling state the rate belongs to; and
- which mode layer—bare, off-reference, coupling-on diagonal, or hybridized—the rate belongs to.
Within the retained naming convention, an unqualified bare \(\kappa\) does not identify the intended artifact quantity. Use explicit layer and channel labels, for example \(\kappa_{x,\mathrm{ext,LB}}^{\mathcal T}\) or \(\kappa_{\mu,\mathrm{tot,H}}^{\mathcal T}\), and state whether the number is in rad/s or hertz. For a topology-isolated scalar response, a rational pole can estimate \(f_r\) and \(\kappa_{\mathrm{tot}}\) under the mapping above. Resolution, refinement, stability, continuation, physical interpretation, and complex residuals are evidence to examine, not universal thresholds supplied by vector fitting. A topology-specific complex-\(S_{21}\) fit may own the same pole and separately interpret complex \(Q_c\). Internal/external decomposition is promoted only when complex \(Q_c\), reference planes, background, and coupling topology are all identified; Vector Fitting alone does not provide that decomposition.
The reusable through-line rate notation is
\[ \underbrace{\gamma_p,\gamma_r}_{\text{internal energy-decay rates}}, \qquad \underbrace{\kappa_L,\kappa_R}_{\text{external channel rates}}, \qquad \underbrace{\Gamma}_{\text{total loaded energy-decay rate}}. \]
All are angular rates unless explicitly divided by \(2\pi\).
Failure modes
- Calling \(\kappa/2\) the energy linewidth introduces a factor-of-two error.
- Comparing \(\kappa\) in rad/s with a linewidth in Hz introduces a factor of \(2\pi\).
- Treating the FWHM of field magnitude as the Lorentzian power FWHM uses a different threshold.
- Combining rates from bare and hybridized modes mixes physical layers.
- Using a complex-pole sign without its phasor convention can label a stable pole as unstable.
- Reporting one \(Q_c\) without identifying the external channel hides the loading topology.
- Calling a boundary-conditioned modal linewidth a one-port partial rate confuses a pole’s decay constant with its output residue.
- Treating an isolated pole linewidth as the generic \(-3\) dB width of a coupled or interfering \(S_{21}\) trace changes the observable.
Connections
- Notch Resonator Complex S21 Fit extracts \(Q_l\), complex coupling \(\widehat Q_c\), and \(Q_i\) for one isolated resonance.
- Multimode Input–Output Scattering shows where the decay matrix enters the general response operator.
- From Node and Line Flux to Input–Output Scattering derives \(\kappa_{\mathrm{ext}}\) from a physical terminal admittance, mode normalization, and zero-point voltage before this page names its linewidth and quality-factor consequences.
- Poles, Zeros, and Residues uses the pole imaginary part as decay evidence only after the convention is declared.
References
- C. W. Gardiner and M. J. Collett, “Input and output in damped quantum systems,” Physical Review A 31, 3761 (1985), doi:10.1103/PhysRevA.31.3761.
- S. Probst et al., “Efficient and robust analysis of complex scattering data under noise in microwave resonators,” Review of Scientific Instruments 86, 024706 (2015), arXiv:1410.3365, doi:10.1063/1.4907935.
- J. Heinsoo et al., “Rapid high-fidelity multiplexed readout of superconducting qubits,” Physical Review Applied 10, 034040 (2018), arXiv:1801.07904v1, Appendix C.
| Field | Value |
|---|---|
| Status | Source-backed |
| Primary source | Complex scattering data under noise, Rapid high-fidelity multiplexed readout, and the primary input-output reference above. |