{"kind":"task","effective_mode":"full","benchmark":{"kind":"benchmark","effective_mode":"full","slug":"critpt","formal_name":"CritPt","introduction":"研究水準の物理問題で、科学的理解と多段階の推論・計算を評価するベンチマークです。公開データには70の課題があり、問題文とコード雛形を組み合わせて解答を構成します。\n\nCritPt evaluates scientific understanding and multi-step reasoning and computation on research-level physics problems. Its public dataset contains 70 challenges with problem descriptions and code templates.","introduction_ja":"","introduction_en":"","category":"Category not supplied","task_count":null,"acquisition_status":"Acquisition status not supplied","official_url":"https://critpt.com/","indexing_mode":"noindex"},"task_id":"00f3dda1-5be1-5b35-9710-4134f2eb7656","task_key":"train--Challenge~5f20~5fmain","task_revision_id":"1","upstream_id":"Challenge_20_main","short_description":"Two identical dielectric ellipsoids are confined in two Gaussian optical…","config":"","split":"train","body":"{\"code_template\":\"import numpy as np\\nfrom scipy.constants import c\\n\\ndef answer(a, b, rho, k, epsilon_r, P_0, w_0, R):\\n    r\\\"\\\"\\\"\\n    Return the values of $\\\\omega_t$ and $g$.\\n\\n    Inputs\\n    ----------\\n    a, b: float, The semi-major and semi-minor axes of the ellipsoids, in nm.\\n    rho: float, The mass density of the ellipsoids, in kg/m^3.\\n    k: float, The wave vector of the laser, in 1/m.\\n    epsilon_r: float, The relative permittivity of the ellipsoids, dimensionless.\\n    P_0: float, The power of the laser, in mW.\\n    w_0: float, The beam waist of the laser, in nm.\\n    R: float, The distance between the ellipsoids, in nm.\\n\\n    Outputs\\n    ----------\\n    omega_t : float\\n        The torsional frequency $\\\\omega_t$, in 1/s.\\n    g       : float\\n        The coupling rate $g$, in 1/s.\\n    \\\"\\\"\\\"\\n\\n    # ------------------ FILL IN YOUR RESULTS BELOW ------------------\\n    omega_t = ...\\n    g = ...\\n    # ---------------------------------------------------------------\\n\\n    return omega_t, g\",\"problem_description\":\"# Problem setup:\\nTwo identical dielectric ellipsoids are confined in two Gaussian optical tweezers that propagate along the $z$-axis, are separated by a distance $R$ along the $x$-axis, and are also linearly polarized along the $x$-axis. The trapping lasers have wave vector $k$, beam waist $w_0$ and power $P_0$. Each ellipsoid possesses a semi-major axis of length $a$, two equal semi-minor axes of length $b$, a relative permittivity $\\\\epsilon_r$, and a mass density $\\\\rho$. Assume that the long axes execute small torsional oscillations about the $x$-direction and that the second-quantized Hamiltonian governing this system is\\n$$H = \\\\hbar {\\\\omega _t}\\\\left( {a_1^\\\\dagger {a_1} + a_2^\\\\dagger {a_2}} \\\\right) + \\\\hbar g\\\\left( {a_1^\\\\dagger {a_2} + {a_1}a_2^\\\\dagger } \\\\right).$$\\n\\n# Main problem:\\n\\nDerive $\\\\omega_t$ and $g$ with the given parameters.\"}","display_format":"text","language":"","answer_status":"published","assets":[],"source_url":"https://critpt.com/","history":"initial import","indexing_mode":"noindex","subproblems":[],"grids":[]}