{ "finite_difference_convergence": { "analytic_alpha_over_hbar_k2": 0.1, "checks": { "error_decreases": true, "finest_step_accuracy": true, "fourth_order_regime": true }, "largest_error": 6.531111254064192e-05, "minimum_refinement_ratio": 15.948524835433782, "smallest_error": 6.249786799195078e-11 }, "friction": { "detuning_range_Delta_over_Gamma": [ -2.0, 2.0 ], "finite_difference": "five-point centered derivative", "samples": 801, "validation": { "analytic_weak_friction_optimum_Delta_over_Gamma": 0.2886751345948129, "checks": { "blue_negative_friction": true, "finite_difference_friction": true, "friction_optimum": true, "red_positive_friction": true }, "max_five_point_friction_error": 1.072114619304898e-11, "maximum_weak_alpha_over_hbar_k2": 0.12990381056766576, "numeric_weak_friction_optimum_Delta_over_Gamma": 0.2886751312424951, "weak_friction_optimum_error": 3.3523178122329966e-09 } }, "license": "MIT", "limitations": [ "closed two-level transition", "plane waves and balanced counterpropagating beams", "independent rates only in the weak-saturation benchmark", "local internal steady state at each velocity", "nonrelativistic classical center-of-mass motion", "dilute independent particles", "three-dimensional isotropic-emission diffusion ledger", "no hyperfine, Zeeman, polarization-gradient, or dark states", "no magnetic trap, spatial intensity profile, or gravity", "no branching, loss, reabsorption, collisions, or density effects", "no sub-Doppler, narrow-line, Raman, or sideband cooling", "no stochastic trajectory ensemble" ], "outputs": [ "laser-cooling-force-curves.csv", "laser-cooling-friction.csv", "laser-cooling-temperature.csv", "laser-cooling-intensity.csv", "laser-cooling-convergence.csv", "laser-cooling-relaxation.csv", "laser-cooling-rubidium.csv", "laser-cooling-metadata.json" ], "provenance": { "neutral_atom_cooling_proposal": { "authors": "T. W. Haensch and A. L. Schawlow", "doi": "10.1016/0030-4018(75)90159-5", "journal": "Optics Communications 13, 68-69 (1975)", "title": "Cooling of gases by laser radiation" }, "sub_Doppler_observation": { "authors": "P. D. Lett, R. N. Watts, C. I. Westbrook, W. D. Phillips, P. L. Gould, and H. J. Metcalf", "doi": "10.1103/PhysRevLett.61.169", "journal": "Physical Review Letters 61, 169-172 (1988)", "title": "Observation of atoms laser cooled below the Doppler limit" } }, "relaxation": { "equations": [ "d mean_velocity / d tau = -mean_velocity", "d (T/T_eq) / d tau = -2 (T/T_eq - 1)" ], "integrator": "classical explicit Runge-Kutta order 4", "max_internal_step": 0.005, "samples": 501, "time_range_tau": [ 0.0, 5.0 ], "validation": { "checks": { "cold_heats_monotonically": true, "cold_variance_relaxation": true, "hot_cools_monotonically": true, "hot_variance_relaxation": true, "mean_relaxation": true }, "max_cold_temperature_error": 1.5198509117908543e-11, "max_hot_temperature_error": 1.3678658206117689e-10, "max_mean_velocity_error": 7.349676423018536e-13 } }, "rubidium_benchmark": { "isotope": "rubidium-87", "transition": "D2 representative closed-two-level benchmark", "validation": { "Doppler_temperature_microkelvin": 145.53704611065731, "checks": { "Doppler_temperature_range": true, "damping_time_range": true, "recoil_temperature_range": true, "resonant_velocity_range": true }, "recoil_temperature_microkelvin": 0.18097823453839443, "resonant_velocity_m_per_s": 2.366081468372795, "velocity_damping_time_milliseconds": 0.21102627608949293 } }, "runtime": { "numpy": "2.3.5", "platform": "Windows-11-10.0.26200-SP0", "python": "3.12.13", "random_seed": null }, "scope": { "canonical_detuning": "Delta = omega_0 - omega_L", "claim": "Doppler-force and friction-diffusion benchmark, not a multilevel or apparatus-specific cooling prediction", "force_normalization": "F / (hbar k Gamma)", "model": "closed two-level atom in weak optical molasses with a declared representative shared-saturation extension", "red_detuning": "Delta > 0", "saturation": "s is the on-resonance value for one beam", "translated_detuning": "delta = omega_L - omega_0 = -Delta", "velocity_coordinate": "u = k v / Gamma" }, "shared_saturation_extension": { "denominator": "1 + s_total + 4 detuning^2", "intensity_samples": 241, "saturations_per_beam": [ 0.01, 0.1, 0.5, 2.0 ], "status": "representative two-level extension", "total_saturation": "s_total = 2 s_beam", "validation": { "checks": { "force_below_single_beam_ceiling": true, "force_grows_over_scan": true, "power_broadening_raises_minimum": true, "weak_temperature_normalization": true }, "highest_peak_force": 0.06519262969507308, "highest_shared_minimum_T_over_TD": 4.58257569495584, "lowest_peak_force": 0.0004020122786291433, "lowest_shared_minimum_T_over_TD": 1.000999500499376, "weak_intensity_independence_error": 0.0, "weak_temperature_at_Delta_half_over_TD": 1.0 } }, "temperature": { "Doppler_scale": "k_B T_D = hbar Gamma / 2", "definition": "k_B T = D_p / alpha", "detuning_range_Delta_over_Gamma": [ 0.02, 2.5 ], "diffusion_convention": "d Var(p_x) / dt = 2 D_p", "geometry": "three orthogonal weak beam pairs and isotropic spontaneous emission", "samples": 1001, "validation": { "checks": { "diffusion_friction_identity": true, "shared_optimum_locations": true, "shared_optimum_values": true, "weak_Doppler_location": true, "weak_Doppler_value": true }, "largest_diffusion_friction_identity_error": 1.7763568394002505e-15, "largest_shared_optimum_location_error": 2.131845722175285e-08, "largest_shared_optimum_value_error": 4.440892098500626e-16, "numeric_weak_minimum_T_over_TD": 1.0, "numeric_weak_temperature_optimum_Delta_over_Gamma": 0.49999999604270173, "shared_optima": { "s_0p01": { "analytic_Delta_over_Gamma": 0.5049752469181039, "analytic_T_over_TD": 1.0099504938362078, "numeric_Delta_over_Gamma": 0.5049752383715829, "numeric_T_over_TD": 1.009950493836208 }, "s_0p1": { "analytic_Delta_over_Gamma": 0.5477225575051661, "analytic_T_over_TD": 1.0954451150103321, "numeric_Delta_over_Gamma": 0.547722549073077, "numeric_T_over_TD": 1.0954451150103324 }, "s_0p5": { "analytic_Delta_over_Gamma": 0.7071067811865476, "analytic_T_over_TD": 1.4142135623730951, "numeric_Delta_over_Gamma": 0.7071067685330397, "numeric_T_over_TD": 1.4142135623730951 }, "s_2": { "analytic_Delta_over_Gamma": 1.118033988749895, "analytic_T_over_TD": 2.23606797749979, "numeric_Delta_over_Gamma": 1.1180339674314377, "numeric_T_over_TD": 2.2360679774997902 } }, "weak_temperature_location_error": 3.9572982668545365e-09, "weak_temperature_value_error": 0.0 } }, "validation": { "all_checks_passed": true, "sections": { "convergence": { "error_decreases": true, "finest_step_accuracy": true, "fourth_order_regime": true }, "force": { "balanced_zero_force": true, "blue_antidamps": true, "directional_sum": true, "odd_force": true, "red_damps": true }, "friction": { "blue_negative_friction": true, "finite_difference_friction": true, "friction_optimum": true, "red_positive_friction": true }, "intensity": { "force_below_single_beam_ceiling": true, "force_grows_over_scan": true, "power_broadening_raises_minimum": true, "weak_temperature_normalization": true }, "relaxation": { "cold_heats_monotonically": true, "cold_variance_relaxation": true, "hot_cools_monotonically": true, "hot_variance_relaxation": true, "mean_relaxation": true }, "rubidium": { "Doppler_temperature_range": true, "damping_time_range": true, "recoil_temperature_range": true, "resonant_velocity_range": true }, "temperature": { "diffusion_friction_identity": true, "shared_optimum_locations": true, "shared_optimum_values": true, "weak_Doppler_location": true, "weak_Doppler_value": true } } }, "weak_force": { "detunings_Delta_over_Gamma": [ -0.5, 0.25, 0.5, 1.0, 2.0 ], "samples": 1201, "saturation_per_beam": 0.1, "validation": { "checks": { "balanced_zero_force": true, "blue_antidamps": true, "directional_sum": true, "odd_force": true, "red_damps": true }, "largest_red_small_velocity_force": -0.0004999999987499893, "max_directional_sum_error": 6.938893903907228e-18, "max_force_oddness_error": 5.551115123125783e-17, "max_zero_velocity_force": 0.0, "smallest_blue_small_velocity_force": 0.0004999999987499893 }, "velocity_range_kv_over_Gamma": [ -3.0, 3.0 ] } }