sqzcomb

Squeezed light in Kerr microcombs, computed end to end: from the classical Lugiato-Lefever steady state, through the linearized quantum fluctuations around it, to the output quadrature-noise spectrum a homodyne detector would report. The package exists because intracavity squeezing is not the observable; what leaves the extraction port is, and the coupling that maximizes one does not maximize the other.

Detected entanglement (new in v0.8)

v0.6 answered "are these two comb lines entangled inside the cavity"; v0.8 answers the question the detector asks: output_covariance_xxpp is the full symmetrized covariance of the OUTPUT field at analysis frequency omega (vacuum exactly 0.5 I), output_entanglement the logarithmic negativity a two-line homodyne pair would certify there -- frequency-resolved, through the extraction port, the intrinsic loss, thermal baths and an optional detection-efficiency channel. Anchored to exact identities, not trusted: every quadrature read off the matrix equals the independent scalar spectra path to 1e-13; a passive cavity outputs exactly ((2 nbar + 1)/2) I at every frequency and coupling, hence zero entanglement; for the symmetric twin-beam process E_N(omega) equals -ln(2 V_EPR_min(omega)) computed by the independent joint-quadrature path; E_N vanishes at large omega and never increases under detection loss.

Entanglement of the comb (new in v0.6)

The entangle module quantifies twin-beam entanglement directly from the covariance matrices the package produces: the partial-transpose symplectic eigenvalue through the Simon invariants (necessary and sufficient for two-mode Gaussian states), the logarithmic negativity, and the symmetric Duan-Simon EPR sum with its separability bound. The invariant formula is cross-checked against an independent explicit-partial-transpose computation, against the closed-form two-mode squeezed vacuum (E_N = 2r exactly), and on the driven photonic molecule, whose two rings turn out to be PPT-entangled below threshold while the symmetric Duan sum misses it, a working demonstration of why the sharper criterion matters.

from sqzcomb import (photonic_molecule, intracavity_covariance,
                     covariance_xxpp, entanglement_report)
M, gammas = photonic_molecule(mu=0.8, J=1.0)
sigma = covariance_xxpp(intracavity_covariance(M, gammas))
print(entanglement_report(sigma, 0, 1))

Status

v0.8.0 (alpha). Implemented and tested (72 tests, Python 3.9-3.13):

• Lugiato-Lefever solver (Strang splitting; the Kerr step and the linear-plus-pump step are each exact)
• homogeneous steady states against the exact cubic
• linearized fluctuation (Bogoliubov) matrix around an arbitrary steady state, with a stability guard that refuses above-threshold states
• input-output quadrature spectra with an extraction port and intrinsic loss; single-mode and joint two-mode quadratures
• photonic molecule (new in v0.2): the two-ring coupled-mode model (photonic_molecule), per-mode-coupling output spectra (output_variance_ports), and the exact instability threshold (molecule_threshold)
• multimode comb molecule (new in v0.3): every retained comb line of the LLE fluctuation matrix coupled to a matching auxiliary-ring mode (molecule_fluctuation_matrix), with multi-line bus detection and joint twin-beam quadratures through the auxiliary ring
• Gaussian-state interop (new in v0.4): steady-state covariance matrices of the intracavity Gaussian state (intracavity_covariance, numpy-only Lyapunov solve), export in the standard xxpp quadrature ordering with an explicit hbar convention (covariance_xxpp, vacuum exactly the identity at hbar = 2), Williamson symplectic spectra (symplectic_eigenvalues), and a QuTiP adapter (drift_from_qutip) that turns any quadratic QuTiP Hamiltonian into a drift matrix this package's spectra machinery accepts, refusing non-quadratic Hamiltonians rather than silently linearizing them. Install sqzcomb[interop] for the adapter; its test asserts that the released two-ring molecule is reproduced exactly from a QuTiP Hamiltonian.

Verified against closed forms in the test-suite: vacuum passes a passive cavity, and a passive molecule, unchanged for every coupling, port and frequency; the degenerate parametric oscillator output spectrum is reproduced to 1e-10; and the textbook result that detectable squeezing saturates at 3 dB at critical coupling, while full extraction breaks that limit, emerges from the machinery rather than being asserted.

For the molecule, the test-suite additionally asserts: exact reduction to the single ring at zero coupling; passive supermodes split by exactly 2J; the resonant threshold mu = 1 + J^2/gamma (static branch, Hopf branch 1 + gamma beyond J = gamma); the quarter-turn quadrature rotation of the -iJ hop; the exact zero-frequency equivalence of the auxiliary-ring port to an effective single mode with escape efficiency (J^2/gamma)/(1 + J^2/gamma); and a J^2/gamma = 3 molecule reaching 6 dB detected squeezing through the auxiliary port although the Kerr ring itself has no extraction port, which is the molecule extraction mechanism in its simplest form.

For the multimode molecule, the asserts continue in the same spirit: at one retained line the builder equals the released two-ring matrix to machine precision; at zero coupling it reduces exactly to the plain fluctuation matrix and the v0.1 spectra, single-line and twin-beam; a passive multimode molecule returns exact vacuum through any bus; the resonant auxiliary ring at zero frequency is exactly the single ring with J^2/gamma_b extra loss per line and the quarter-turn rotation; and when J^2/gamma_b exceeds one, twin-beam squeezing detected through the auxiliary bus is strictly deeper than through the main bus of the same device. The stability guard is also asserted to refuse a flat state that is above a pair's modulational-instability threshold once the molecule's added loss is removed.

Later releases are documented in their own sections below: imperfect detection (v0.5), two-mode Gaussian entanglement of the comb (v0.6), and solitons, supermodes and thermal baths (v0.7).

Every item of the original roadmap is now implemented. Deliberate scope, stated plainly -- designed-out, not overlooked: quantum noise beyond the Gaussian linearization (no non-Gaussian states, no above-threshold dynamics -- unstable and marginal drift matrices are refused or must be explicitly acknowledged, never silently averaged); technical noise of the resonator itself (thermorefractive and Raman noise are material physics with their own modelling choices); and pulsed or synchronously pumped operation (the pump here is CW). The maximally informative multimode decomposition for pure states (Bloch-Messiah) is likewise out; principal_quadratures answers the question experiments ask -- the deepest collective squeezing and its supermode -- exactly, for pure and mixed states alike.

Install and use

pip install sqzcomb

For development, clone the repository and pip install -e .[test].

import numpy as np
from sqzcomb import (lle_evolve, fluctuation_matrix,
                     output_quadrature_variance, squeezing_db)

# steady state at pump F and detuning alpha, anomalous dispersion d2
psi = lle_evolve(np.full(256, 0.05 + 0j), F=1.2, alpha=0.8,
                 dispersion=(-0.02,), t_end=300.0)

M, modes = fluctuation_matrix(psi, alpha=0.8, dispersion=(-0.02,))
i0 = int(np.where(modes == 0)[0][0])
v = output_quadrature_variance(M, eta=0.5, omega=0.0,
                               mode_index=i0, n_modes=modes.size)
print(squeezing_db(v), "dB relative to vacuum")

Units are the standard normalized LLE units: time in photon lifetimes, eta = kappa_ex / kappa, vacuum variance 1/2.

Imperfect detection (new in v0.5)

Every spectrum above is the noise at the extraction port; the photodiodes report less. detection applies the standard beamsplitter model of optical loss and quantum efficiency plus additive electronic noise, in both languages the package speaks: as a scalar map on quadrature variances (detected_variance, detected_squeezing_db, with dark_from_clearance_db converting a receiver's dark clearance in dB into a variance) and as the lossy Gaussian channel on xxpp covariance matrices (lossy_channel_xxpp, per-mode efficiencies allowed). required_efficiency inverts the loss model into the number an experiment plans around: the minimum efficiency that still delivers a target squeezing from a given source. Loss stages compose by multiplying efficiencies, and the test suite asserts that composition exactly, along with the vacuum fixed point, physicality of the channel (symplectic eigenvalues never fall below hbar/2), and agreement of the scalar and matrix forms on a squeezed mode.

References: the beamsplitter model of detector inefficiency, U. Leonhardt, Measuring the Quantum State of Light (Cambridge, 1997); the Gaussian lossy channel, C. Weedbrook et al., Rev. Mod. Phys. 84, 621 (2012).

Solitons, supermodes and thermal baths (new in v0.7)

The soliton module finds localized steady states -- dissipative Kerr solitons and soliton crystals -- by Newton's method in Fourier space (newton_state, seeded by soliton_seed, swept by continuation). The Jacobian it inverts is the same linearization the quantum-noise machinery builds, in its exact discrete form, so the solver refuses to return anything whose stationary residual is not verified below tolerance. The test suite then closes the loop through independent code paths: the converged soliton barely moves under the split-step time evolver, its background matches the exact cubic root, its linearization carries the exact translation (Goldstone) zero mode with eigenvector d psi / d theta, and a two-pulse crystal equals, grid point for grid point, the single soliton of the equation with dispersion scaled by four -- the exact rescaling of the periodic domain. Because the Goldstone mode makes a soliton's drift matrix marginally rather than asymptotically stable, the spectra now distinguish the two: a marginal matrix is refused with an explanation unless allow_marginal=True says the translation mode is understood, while a genuinely unstable one stays refused regardless.

principal_quadratures adds the supermode decomposition of any multimode covariance matrix: the eigendecomposition of sigma, whose smallest eigenvalue is -- exactly, by linear algebra -- the deepest squeezing any generalized quadrature of the state attains, and whose eigenvector is the supermode carrying it. The two-mode squeezed vacuum pins it in closed form ((hbar/2) e^{-/+ 2r} with EPR supermodes), and it is deliberately distinct from the Williamson spectrum, which measures mixedness, not squeezing.

Thermal input noise completes the bath model: output_quadrature_variance takes Bose occupations for the extraction-port and loss baths, and intracavity_covariance for each mode's bath, with thermal_occupation supplying the physical number from the exact SI constants (h and k_B are exact by definition since the 2019 redefinition). The anchors are closed forms: a passive cavity with baths at n_bar emits exactly (2 n_bar + 1)/2 at every frequency, coupling and phase; the parametric oscillator's spectra scale by exactly (2 n_bar + 1); a passive mode holds exactly n_bar photons; and the hot-loss/cold-port mixture matches its hand-derived form.

Methodological basis

T. M. Mahim, M. M. Rahman and A. S. M. Mohsin, "Overcoming the 3 dB squeezing extraction limit in silicon carbide microcombs with a photonic molecule," Optics Express 34(18), 34822-34834 (2026), https://doi.org/10.1364/OE.612248 (open access); code for the paper: https://github.com/Tanvir-Mahmud-Mahim/sic-molecule-squeezer

This package is the general-purpose engine; the paper repository reproduces the specific published study. v0.2 adds the two-mode photonic molecule, the extraction mechanism in its simplest form; the paper's full multimode comb molecule remains in the paper repository.

Support and governance

The package is written and maintained by Tanvir Mahmud Mahim (Department of Electrical and Electronic Engineering, BRAC University), who reviews every change and takes the final decision on scope and releases. There is no separate governance body; design questions are discussed in the open in issues and pull requests, and the standing rule of CONTRIBUTING.md binds the maintainer exactly as it binds contributors: a change that touches physics arrives with a test, and a constant arrives with its source.

Support runs through the issue tracker at https://github.com/TaN-MM-Org/sqzcomb/issues. Usage questions are welcome there alongside bug reports; a docstring that left a unit or a sign convention unclear is treated as a documentation bug, not as user error. The maintainer aims to respond within a week.

While the version is below 1.0 the API may still move between minor versions; such changes are called out in the release notes. The normalized-unit conventions stated above are stable: any change to them would be a breaking change named in the release notes, never a quiet renormalization.

License

Apache-2.0