cavsqueeze

Simulate spin squeezing -- the quantum trick that lets an ensemble of spins measure better than its atom number alone allows -- for spins coupled to a resonator, with the imperfections a real solid brings: frequency disorder, decay, thermal photons and uneven coupling. The solver's cost is set by the line shape, not the spin count, so a quadrillion spins are no harder than a thousand. It also works backward from experiment: measured shot records go in, squeezing parameters with honest error bars come out. Developed for 171Yb3+:CaWO4 and applicable to any spin ensemble coupled to a resonator.

cavsqueeze implements

• the adiabatically eliminated Tavis-Cummings model with collective emission, collective thermal absorption, single-spin dephasing and coupling inhomogeneity (resonator.py);
• discretization of Gaussian, Lorentzian and Voigt lines into frequency classes with tail resolution (ensemble.py);
• a class-resolved second-order cumulant expansion in connected variables, whose cost is set by the number of classes rather than by N, so that N = 1e15 is no harder than N = 1e3, and which loses no precision to cancellation at large N because the means are subtracted analytically (cumulant.py), together with the raw-moment form used as a test reference (cumulant_raw.py); the solver can condition the ensemble on a continuous measurement of J_z through the resonator (Rates.meas, Rates.meas_eta);
• exact references: QuTiP master equation for distinguishable spins and the permutation-invariant Dicke solver PIQS (exact.py);
• pulse sequences: echo twist, Ramsey, twist-untwist readout, plain squeezed readout (protocols.py);
• far-detuned spectator spins propagated analytically, which removes the stiffness of heavy-tailed lines (ensemble.tail_resolved_classes, cumulant.evolve);
• an independent solver for cross-checking: discrete truncated Wigner trajectories (dtwa.py), which truncate the equations of motion rather than the statistics.

Installation

From PyPI:

pip install cavsqueeze          # core solver (numpy, scipy, matplotlib)
pip install cavsqueeze[exact]   # adds QuTiP for the exact reference solvers

Or from source, to run the tests:

git clone https://github.com/TaN-MM-Org/cavsqueeze
cd cavsqueeze
pip install -e .[test]
pytest tests              # validation testbench (about 2 minutes)

The test suite validates the cumulant solver against exact QuTiP and PIQS references, against closed-form limits, and against the independent discrete truncated Wigner solver; it runs in CI on every push and pull request.

Minimal example

import numpy as np
from cavsqueeze import from_hz, homogeneous
from cavsqueeze.protocols import optimal_squeezing
N = 1e10
p = from_hz(g_hz=1e6/np.sqrt(N), kappa_hz=1e4, Delta_hz=30e6, T=0.02, T2=0.15)
best = optimal_squeezing(p, homogeneous(N), 1e-6, 1e-2)
print(10*np.log10(best["xi2"]), "dB at", best["t"], "s")

Metrology projections

The metrology module turns the solver's collective moments into the quantities an experiment is designed against, for any platform the solver can describe: Kitagawa-Ueda and Wineland squeezing parameters, single-shot phase sensitivity, the projection-noise-limited Allan deviation of a Ramsey clock, and the field sensitivity of a Ramsey magnetometer.

from cavsqueeze import (squeezing_parameters, clock_allan_deviation,
                        metrological_gain_db)
m = squeezing_parameters(state, ens.n)          # xi2_S, xi2_R, contrast, dphi
print(metrological_gain_db(m["xi2_R"]), "dB over the SQL")
print(clock_allan_deviation(m["dphi"], nu0=4.29e14, T_ramsey=0.1, tau=1.0))

The formulas are the standard ones (Kitagawa-Ueda 1993; Wineland 1992; Itano 1993; Ludlow RMP 2015) and are tested against the solver's own exact references and closed-form limits.

New in v1.13, the Dick effect is included: a clock's local oscillator is only sampled part of the time, and the noise it sneaks in during the dead time sets a stability floor that no amount of squeezing moves (Schulte et al., Nat. Commun. 2020). The floor is computed from your laser's measured noise spectrum through the exact closed-form Fourier coefficients of the Ramsey timing sequence, and total_clock_allan_deviation combines it with the projection noise and says plainly, via its dick_limited verdict, whether squeezing or the laser currently limits the clock.

oat_closed_form provides the exact unitary one-axis-twisting moments -- mean spin, extremal transverse variances, optimal angle and both squeezing parameters -- as the decoherence-free benchmark the dissipative solver is compared against; every returned quantity is asserted against brute-force exact evolution in the tests. The pulse-sequence layer (css_x, twist, twist_untwist, optimal_squeezing, plain_squeezed_readout, ...) is exported at the package root.

Squeezing from measured count data

Every other module predicts squeezing from a model; estimate_squeezing estimates it from an experiment. Feed it the standard Ramsey tomography record -- per-angle shot arrays of the measured population difference J_z = (N_up - N_down)/2 -- plus the atom number and the fringe contrast, and it returns Kitagawa-Ueda and Wineland parameters with honest uncertainties. The fit leans only on exact structure: the rotation law of a 2x2 covariance makes V(theta) = c + a cos 2theta

• b sin 2theta exactly (no Gaussian assumption), so the tomography fit is closed-form weighted linear least squares, and V_min = c - sqrt(a^2 + b^2) is an eigenvalue identity asserted against numpy.linalg.eigvalsh. The single Gaussian assumption (the exact 2 s^4/(M-1) sample-variance error bar) is stated, not hidden. A known detection variance can be subtracted per angle -- opt-in, recorded in the result, and refused with an explanation when it over-subtracts into unphysical territory.

from cavsqueeze import estimate_squeezing, metrological_gain_db
est = estimate_squeezing(angles, shots, N=480, contrast=0.92,
                         contrast_sigma=0.01, detection_variance=v_det)
print(est.xi2_R, "+/-", est.xi2_R_sigma,
      "->", metrological_gain_db(est.xi2_R), "dB")

Two helpers close the loop from raw files to the estimate:

• estimate_contrast fits the Ramsey fringe contrast -- the number estimate_squeezing previously asked you to bring from your own fringe fit -- from a phase-scan shot record, by the same exact linear-least-squares structure (the mean fringe is exactly A cos phi + B sin phi + d), with an uncertainty that feeds contrast_sigma directly. A fitted contrast significantly above 1 is refused with the likely cause (wrong N, or a J_z calibration problem) rather than clipped.
• save_shots_csv / load_shots_csv give both shot records (tomography and fringe scans -- the same data shape) a documented plain-text contract (angle_rad,jz, one row per shot) with an exact round trip and refusals for malformed files.

phases, fringe = cavsqueeze.load_shots_csv("fringe.csv")
cest = cavsqueeze.estimate_contrast(phases, fringe, N=480)
angles, shots = cavsqueeze.load_shots_csv("tomography.csv")
est = cavsqueeze.estimate_squeezing(angles, shots, N=480,
                                    contrast=cest.contrast,
                                    contrast_sigma=cest.contrast_sigma)

Anchors in the tests: exact covariance recovery with eigenvalues from an independent numpy path; the exact Kitagawa-Ueda one-axis-twisting closed form recovered from sampled shots; a coherent-spin-state sample estimating the standard quantum limit xi2_R = 1 through the complete file-to-estimate pipeline with no hand-entered contrast; Monte-Carlo scatter matching the reported sigmas in both fitters; exact fringe recovery on clean records; the detection-noise round trip; exact file round trips; and degenerate-design refusals in both fitters.

Planning the measurement before taking it

The estimator closes the loop after the experiment; plan_tomography and shots_for_squeezing close it before. Because the tomography fit is linear, its covariance depends only on the design -- the angles, the shot counts and the expected variance scale -- so the error bars of a planned measurement can be computed exactly before any data exists, from the same matrix the estimator will report:

import numpy as np
from cavsqueeze import plan_tomography, shots_for_squeezing

angles = np.linspace(0.0, np.pi, 9, endpoint=False)
plan = plan_tomography(angles, shots_per_angle=50, var_min=90.0,
                       var_max=160.0, N=480, contrast=0.92,
                       contrast_sigma=0.01)
print(plan["xi2_R_sigma"])          # the error bar you would get

m, plan = shots_for_squeezing(0.02, angles, 90.0, 160.0, N=480,
                              contrast=0.92, contrast_sigma=0.01)
print(m, "shots per angle")         # the cheapest design meeting it

Two exact facts do the work: equally spaced angles over half a turn make the three fitted parameters exactly independent (no angle wastes shots on redundancy), and the contrast uncertainty puts a floor under the Wineland error bar that no number of tomography shots removes -- a target below that floor is refused with the floor named, because only a better fringe measurement can buy it.

The Dick-free comparison

The record entanglement-enhanced clocks of 2024-2025 sidestep the Dick effect by interrogating two ensembles synchronously with one local oscillator, so its noise is common mode and the frequency DIFFERENCE averages down at the projection-noise floor alone (Robinson et al., Nat. Phys. 20, 208 (2024): 1.9(2) dB of stability enhancement at the 10^-17 level; Yang et al., PRL 135, 193202 (2025): 2.0(2) dB beyond the standard quantum limit at 1.1x10^-18). synchronized_comparison models exactly that statement -- combined projection noise, no Dick term in the difference, differential non-common-mode noise stated as out of scope -- and reports the conventional per-clock figure:

from cavsqueeze import phase_sensitivity, synchronized_comparison

out = synchronized_comparison(dphi_a, dphi_b, nu0=4.29e14,
                              T_ramsey=0.061, tau=1000.0, T_cycle=0.3)
print(out["differential"], out["per_clock"])

Into the bosonic quantum stack

The interop module extracts the Holstein-Primakoff mode of the collective spin from any solver state: the 2x2 Gaussian quadrature covariance (a coherent spin state maps to the vacuum), its symplectic eigenvalue, squeezing parameter, angle, thermal occupation and purity, and an export to a QuTiP density matrix that reproduces that covariance. The export verifies itself against the target covariance so Fock truncation can never silently corrupt it, and the QuTiP squeeze-phase convention is locked by a test rather than assumed.

from cavsqueeze import bosonic_mode, to_qutip
mode = bosonic_mode(state, ens.n)     # Sigma, nu, r, theta, n_th, purity
rho, mode = to_qutip(state, ens.n)    # QuTiP density matrix of the mode

How it is checked

68 tests (plus 3 that skip here because they compare against the paper's companion scripts, which live in that repository), on Python 3.10-3.13 plus a QuTiP-free 3.14 job, run in CI on every push. Every physics claim is pinned to an exact reference, never a stored number: the cumulant solver against exact QuTiP and PIQS references and closed-form limits; the independent discrete truncated Wigner solver as a cross-check; the one-axis-twisting closed form against brute-force exact evolution; the estimators recovering generating parameters with Monte-Carlo scatter matching their reported error bars; the Dick coefficients against independent numerical quadrature, with exact zero for continuous interrogation and the exact rectangular-window ratio; and exact file-contract round trips. The planner is held to the estimator itself: the planned covariance equals the fitted one to machine precision, the equal-spacing orthogonality is asserted as an exact identity, and seeded Monte-Carlo scatter matches the planned error bar.

Associated paper

The physics, the conventions and the validation of this package are described in: T. M. Mahim, M. M. Rahman, A. S. M. Mohsin, Synchronization sets the coherence and the squeezing limit of a spin ensemble in a cavity. The paper's companion repository, yb-cawo4-cavity-squeezing, contains the scripts, datasets and figures that reproduce the paper and is archived on Zenodo; this repository is the software's home for development, releases and support.

Contributing and support

Bug reports, questions and pull requests are welcome through GitHub issues; see CONTRIBUTING.md for the development setup and the testing requirements. Tagged releases are published to PyPI by CI.

License

Apache-2.0 (see LICENSE). Please cite the paper if you use this code; citation metadata is in CITATION.cff.