Circuit Lagrangian, Hamiltonian, and Canonical Quantization
Node role: Method.
Reader task: Start from an already reviewed set of independent conservative flux coordinates and derive its canonical Hamiltonian without hiding a singularity, changing basis, or mixing energy and frequency units.
This page owns only the downstream quantum/quantization mapping. It does not define classical anchored-bare open-EOM roots, a response fit, or a design objective. Those quantities are defined before Hamiltonian rewriting.
Reading map
The construction has three connected stages: assemble energy in independent flux coordinates; exchange velocities for their canonical charges; specify the operator domain and output units. The optional projection diagnostic is a separate basis-dependent calculation, not part of the Legendre transform.
Why it matters
An equivalent circuit becomes a quantum model only after its independent coordinates, conservative stored energies, Legendre transform, operator domains, and units are defined. A fitted curve or a schematic alone is not a Hamiltonian.
This node gives the shortest auditable path for a finite, conservative \(C\)/\(L\)/Josephson circuit with static external flux. Lossy environments, frequency-dependent fitted elements, time-dependent flux, and constrained or nearly singular limits require additional models.
Circuit Models to Bare Coordinates, Open EOM, and Normal Modes owns the definition of node flux, branch orientation, reference/gauge removal, incidence matrices, canonical charge as the dual coordinate, and the rule that all matrices and port maps must change basis together. This page assumes that coordinate contract before taking the Legendre transform. Time evolution and the \(\Phi/Q\leftrightarrow a\) symbol change follow Dynamics Conventions and the Flux-to-Mode Symbol Bridge.
These are requirements of the finite conservative construction below, not a new numerical eligibility test. No tolerance or parasitic cutoff is supplied.
1. Topology
Pass: Branch orientation, grounding, and static flux offsets are explicit.
Stop: A drawn connection has no owned electrical meaning.
2. Coordinates
Pass: The selected node fluxes are independent dynamical variables.
Stop: A reference, passive, or constrained coordinate is unresolved.
3. Energy
Pass: Every retained branch has a conservative stored-energy model.
Stop: A lossy or arbitrary frequency-dependent fit is treated as stored energy.
4. Legendre transform
Pass: The capacitance matrix is full rank on the selected subspace.
Stop: The velocity-to-charge map is singular or repaired with an unreviewed pseudoinverse.
5. Quantum domain
Pass: Each coordinate’s compact or extended domain and commutator are declared.
Stop: A matrix basis is chosen without physical boundary conditions.
6. Output units
Pass: Hamiltonian energy, hertz, and angular frequency are distinguished.
Stop: \(E\), \(E/h\), and \(E/\hbar\) are mixed.
Every card is a model boundary. Skipping the rank or domain distinction can produce a numerically diagonalizable matrix that is not the intended circuit.
Notation contract
| Symbol | Meaning | Units |
|---|---|---|
| \(\Phi_i(t)=\int_{t_0}^t V_i(t')dt'\) | Dimensional node-flux coordinate relative to a chosen reference node | Wb |
| \(\Phi_b\) | Oriented dimensional branch flux assembled from node coordinates and static offsets | Wb |
| \(\varphi_b=\Phi_b/\varphi_0\) | Dimensionless reduced Josephson phase under the branch convention | dimensionless |
| \(Q_i=\partial\mathcal L/\partial\dot\Phi_i\) | Canonical node charge | C |
| \(n_i=Q_i/(2e)\) | Dimensionless canonical charge under the declared positive-\(2e\) orientation | dimensionless |
| \(\Phi_0=h/(2e)\), \(\varphi_0=\Phi_0/(2\pi)\) | Flux quantum and reduced flux quantum | Wb |
\(\Phi\) and \(\varphi\) are deliberately different symbols. A project artifact may choose another font, but it must not use one unqualified phi for both a weber-valued coordinate and a dimensionless phase.
1. Accept the independent circuit coordinates
Reorganized explanation. The oriented branch and node-flux construction follows conservative circuit mechanics (Vool and Devoret 2017, sec. 2.1.7). The static offset allocation names the chosen loop convention.
Begin with the reviewed independent coordinate set from Circuit Models to Bare Coordinates, Open EOM, and Normal Modes. Its reduced branch-incidence matrix maps node coordinates to oriented branch fluxes:
\[ \boldsymbol\Phi_b =B\boldsymbol\Phi+\boldsymbol\Phi_{\mathrm{ext}}. \]
The static offset vector records the selected spanning-tree and loop-flux convention. At this stage the reference, passive, constrained, and gauge coordinates have already been resolved; this page does not repair an invalid coordinate set during matrix inversion.
2. Assemble stored energy
Reorganized explanation. Capacitive, inductive and Josephson stored energies supply the Lagrangian (Vool and Devoret 2017, sec. 2.1.8 and 4.1). The offset-charge sign below is explicitly part of this page’s starting model.
For common conservative branches,
\[ T_C=\frac{C_b}{2}\dot\Phi_b^2, \qquad U_L=\frac{\Phi_b^2}{2L_b}, \qquad U_J=-E_J\cos\left(\frac{\Phi_b}{\varphi_0}\right). \]
After branch contributions are assembled in the chosen coordinates, a common matrix form is
\[ \boxed{ \mathcal L(\boldsymbol\Phi,\dot{\boldsymbol\Phi}) =\frac12\dot{\boldsymbol\Phi}^{T} C\dot{\boldsymbol\Phi} +\mathbf Q_{\mathrm{off}}^{T}\dot{\boldsymbol\Phi} -U(\boldsymbol\Phi) }. \]
Capacitive energy plays the kinetic role; inductive and Josephson energies enter \(U\). The offset-charge term is written explicitly so its sign can be traced into the canonical charge.
3. Derive the classical equation of motion
This-page derivation. Apply Euler–Lagrange to the stated quadratic energy. A generalized drive is an added model term, not energy stored by the conservative circuit.
Before quantization, the Euler–Lagrange equation already fixes the circuit’s classical dynamics. With a generalized drive \(\mathbf f_\Phi(t)\),
\[ \frac{d}{dt} \frac{\partial\mathcal L}{\partial\dot{\boldsymbol\Phi}} -\frac{\partial\mathcal L}{\partial\boldsymbol\Phi} =\mathbf f_\Phi(t). \]
For the linearized conservative energy
\[ \mathcal L_2 =\frac12\dot{\boldsymbol\Phi}^T\mathbf C \dot{\boldsymbol\Phi} -\frac12\boldsymbol\Phi^T\mathbf K_\Phi\boldsymbol\Phi, \]
this becomes
\[ \boxed{ \mathbf C\ddot{\boldsymbol\Phi} +\mathbf K_\Phi\boldsymbol\Phi =\mathbf f_\Phi(t). } \]
This is the same physical model later written as a Hamiltonian. The Legendre transform changes the mathematical description from velocities to canonical charges; it does not change this dynamics or diagonalize the circuit.
4. Take the Legendre transform only when it exists
This-page derivation. Differentiate the stated Lagrangian and eliminate the velocities. This is the canonical construction of circuit mechanics (Vool and Devoret 2017, secs. 2.1.9–2.1.10), expressed here with the explicit offset charge.
The canonical charge is
\[ \mathbf Q =\frac{\partial\mathcal L} {\partial\dot{\boldsymbol\Phi}} =C\dot{\boldsymbol\Phi}+\mathbf Q_{\mathrm{off}}. \]
Only when \(C\) is invertible on the already-selected independent dynamical subspace may one solve for velocity and write
\[ \dot{\boldsymbol\Phi} =C^{-1}(\mathbf Q-\mathbf Q_{\mathrm{off}}), \]
\[ \boxed{ H =\mathbf Q^T\dot{\boldsymbol\Phi}-\mathcal L =\frac12 (\mathbf Q-\mathbf Q_{\mathrm{off}})^T C^{-1}(\mathbf Q-\mathbf Q_{\mathrm{off}}) +U(\boldsymbol\Phi) }. \]
Numerically, implementations should solve linear systems rather than form a dense inverse. Conceptually, however, the invertibility requirement remains: a solver cannot repair an undefined Legendre transform.
Stop before changing basis
The Legendre transform preserves the retained flux-coordinate directions. It adds their conjugate charges; it does not diagonalize the Hamiltonian or create normal modes. Closed conservative diagonalization, coordinate-bare oscillator normalization, open-system response, and projection diagnostics are downstream tasks described by Physical Circuit Coordinates and Open-EOM Reduction, Poles, Zeros, and Residues, and Resonator Decay, Linewidth, and Quality Factor. These distinguish anchored diagonal roots, hybridized poles, locally normalized response couplings, and channel-qualified decay from Hamiltonian coefficients.
A fitted pole, eigenfrequency, or projected coupling therefore cannot repair an undefined Legendre transform or retroactively define the coordinate basis used above.
This-page derivation / engineering choice. Start from the declared coupling-off energy-normalized basis and physical-on quadratic matrices below. These projected block formulas are an internal diagnostic, not a result attributed wholesale to the circuit-mechanics source. Their use does not supply a comparison tolerance or make the reference basis physically bare.
An already normalized set of coupling-off reference modes may be used as a declared projection basis for diagnostics. It does not become the coupling-on bare basis. Its normalization and frequency matrix are
\[ \mathbf V^T\mathbf C_0\mathbf V=\mathbf I, \qquad \mathbf W=\operatorname{diag}(\omega_1,\ldots,\omega_m), \]
The physical-on quadratic circuit uses \(\mathbf C_1\) and \(\mathbf K_{\Phi,1}\) in that same underlying coordinate basis. The projected blocks are
\[ \mathbf A =\mathbf V^T\mathbf C_0\mathbf C_1^{-1}\mathbf C_0\mathbf V, \qquad \mathbf B =\mathbf V^T\mathbf K_{\Phi,1}\mathbf V, \]
\[ \mathbf X=\mathbf W^{1/2}\mathbf A\mathbf W^{1/2}, \qquad \mathbf Y=\mathbf W^{-1/2}\mathbf B\mathbf W^{-1/2}. \]
The number-conserving and pairing blocks in angular-frequency units are
\[ \boxed{ \frac{\mathbf H_{\mathrm{nc,proj}}}{\hbar} =\frac{\mathbf X+\mathbf Y}{2} }, \qquad \boxed{ \frac{\boldsymbol\Delta_{\mathrm{proj}}}{\hbar} =\frac{\mathbf Y-\mathbf X}{2} }. \]
Implementations solve systems with \(\mathbf C_1\) rather than forming its inverse. The projected non-RWA frequencies are \(\sqrt{\operatorname{eig}(\mathbf A\mathbf B)}\); the projected RWA frequencies are the eigenvalues of \(\mathbf H_{\mathrm{nc,proj}}/\hbar\). Compare both with the complete physical-on generalized eigenproblem. The off-diagonal entries remain \(g_{\mathrm{proj}}\) or \(J_{\mathrm{proj}}\): agreement is a closure test, not permission to relabel them as coupling-on bare-coordinate parameters.
5. Promote canonical variables to operators
Reorganized explanation. Promote canonical charge and flux to operators with their commutator and physically specified domain (Vool and Devoret 2017, secs. 3.1.1–3.1.4).
Canonical quantization imposes
\[ [\hat\Phi_i,\hat Q_j] =i\hbar\delta_{ij}. \]
With \(\hat\varphi_i=\hat\Phi_i/\varphi_0\) and \(\hat n_i=\hat Q_i/(2e)\),
\[ [\hat\varphi_i,\hat n_j]=i\delta_{ij}. \]
The commutator does not select the Hilbert space by itself. Each independent coordinate must be classified from circuit topology and boundary conditions: a phase may be compact and have integer charge states, or an extended flux may live on the real line. Do not impose one global compact/extended default merely because the same symbol appears in two circuits.
Minimal Legendre-transform check
This-page derivation. The scalar starting Lagrangian below retains its offset-charge term. The four substitutions explicitly recover the familiar Cooper-pair-box Hamiltonian (Vool and Devoret 2017, sec. 4.2.2).
For one Josephson mode with total capacitance \(C_\Sigma\) and offset charge,
\[ \mathcal L =\frac{C_\Sigma}{2}\dot\Phi^2 +Q_{\mathrm{off}}\dot\Phi +E_J\cos(\Phi/\varphi_0). \]
Then
\[ Q=C_\Sigma\dot\Phi+Q_{\mathrm{off}} \]
and
\[ H =\frac{(Q-Q_{\mathrm{off}})^2}{2C_\Sigma} -E_J\cos(\Phi/\varphi_0). \]
Defining \(n=Q/(2e)\), \(n_g=Q_{\mathrm{off}}/(2e)\), and \(E_C=e^2/(2C_\Sigma)\) gives
\[ H=4E_C(n-n_g)^2-E_J\cos\varphi. \]
This compact example is not permission to collapse a multimode, lossy, or frequency-dependent equivalent circuit into the same one-coordinate model.
Energy and frequency units
The Hamiltonian above has energy units:
| Reported quantity | Correct conversion |
|---|---|
| Energy | \(E\) in joules |
| Ordinary frequency | \(E/h\) in hertz |
| Angular frequency | \(E/\hbar\) in radians per second |
A field named E_J should contain energy. A field stored in hertz should be named or documented as E_J / h; a value in angular units is E_J / ħ.
Rank-deficient capacitance matrix
The nearly singular versus exactly constrained distinction is a substantive quantization issue, not a numerical inconvenience (Rymarz and DiVincenzo 2023).
A proven global reference or gauge null coordinate may be removed. Any other rank deficiency means the velocity-to-charge map is not invertible as written. Do not silently apply a pseudoinverse or add a tiny ground capacitor to obtain a plausible spectrum. Exact constraints and nearly singular parasitic limits need reviewed specialist treatment; the exact-zero model need not equal the limit of a small physical parasitic.
Ineligible equivalent models
A resistor, lossy port response, arbitrary rational fit, or frequency-dependent element does not by itself define a unique finite conservative Hamiltonian. It first needs a physical energy-storing realization and, for loss, an environment or open-system contract. The same warning applies to an admittance fit whose topology and internal coordinates are not identified.
Scope not covered by this v1 node
- time-dependent external flux and the induced electromotive-force gauge;
- nonlinear or nonreciprocal capacitive elements;
- distributed continua without a declared modal truncation;
- constrained and nearly singular circuits beyond recognition and stop; and
- dissipation, drive, measurement, and master-equation construction.
Audit checklist
- Are the independent coordinates and reference node explicit?
- Are branch orientations and external-flux offsets reconstructible?
- Does every term in \(\mathcal L\) have energy units?
- Is \(C\) full rank on the selected dynamical subspace?
- Was a linear solve used without hiding a structural singularity?
- Are \(\Phi\), \(\varphi\), \(Q\), and \(n\) unambiguous?
- Is each quantum coordinate’s compact or extended domain declared?
- Is the reported output in joules, hertz, or radians per second?
- Is every fitted or lossy element eligible for the claimed Hamiltonian?
Connections
- Circuit Models to Bare Coordinates, Open EOM, and Normal Modes owns the physical coordinate construction that precedes this page’s Lagrangian and Legendre transform.
- From Node and Line Flux to Input–Output Scattering continues from finite conservative zero-point operators to a physical transmission-line boundary, derived radiative linewidth, and passive input–output matrices.
- Superconducting Order Parameter and Gauge-Invariant Phase separates branch phase from absolute condensate phase.
- Josephson Current, Phase, Energy, and Inductance supplies the Josephson branch energy.
- Josephson Cosine and Quantum Anharmonicity uses the quantized Hamiltonian to define spectral anharmonicity.
- Introduction to Quantum Electromagnetic Circuits is the primary paper page for the standard finite-circuit construction.
- Energy-participation quantization is a downstream weak-anharmonic distributed-mode method.
References
- U. Vool and M. H. Devoret, “Introduction to Quantum Electromagnetic Circuits,” arXiv:1610.03438, doi:10.1002/cta.2359.
- S. E. Rasmussen et al., “The Superconducting Circuit Companion—an Introduction with Worked Examples,” arXiv:2103.01225.
- M. Rymarz and D. P. DiVincenzo, “Consistent Quantization of Nearly Singular Superconducting Circuits,” Physical Review X 13, 021017 (2023), arXiv:2208.11767, doi:10.1103/PhysRevX.13.021017.
| Field | Value |
|---|---|
| Status | Source-backed |
| Semantic role | Derive a canonical Hamiltonian from a valid finite conservative coordinate model. |
| Used by | Josephson Cosine and Quantum Anharmonicity, Energy-participation quantization, and downstream circuit-to-Hamiltonian implementations. |
| Open review question | Can a reader identify the independent coordinate domain, prove that the Legendre transform exists, and distinguish \(H\), \(H/h\), and \(H/\hbar\)? |