{"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":"979590b9-1f63-52db-a3c4-cb8a59b03528","task_key":"train--Challenge~5f19~5fmain","task_revision_id":"1","upstream_id":"Challenge_19_main","short_description":"Consider two cascaded optical parametric amplifiers (OPAs) pumped by lasers at…","config":"","split":"train","body":"{\"code_template\":\"import sympy as sp\\n\\nr_1, r_2 = sp.symbols('r_1 r_2')\\nmu, eta = sp.symbols('mu eta')\\nphi_1, phi_2 = sp.symbols('phi_1 phi_2')\\ntheta, nu, Omega = sp.symbols('theta nu Omega')\\n\\ndef answer(r_1, r_2, mu, eta, phi_1, phi_2, theta, nu, Omega):\\n    r\\\"\\\"\\\"\\n    Return the expression of $\\\\left\\\\langle {{{\\\\left| {I_{\\\\theta}\\\\left( \\\\nu  \\\\right)} \\\\right|}^2}} \\\\right\\\\rangle$\\n    and its maximum squeezed and anti-squeezed values in Sympy format.\\n\\n    Inputs\\n    ----------\\n    r_1, r_2 : sympy.Symbol, gain parameters of the first and second OPAs\\n    mu       : sympy.Symbol, transmission coefficient ($\\\\mu_s=\\\\mu_i=\\\\mu$)\\n    eta      : sympy.Symbol, detection efficiency ($\\\\eta_s=\\\\eta_i=\\\\eta$)\\n    phi_1    : sympy.Symbol, pump phase of the first OPA\\n    phi_2    : sympy.Symbol, pump phase of the second OPA\\n    theta    : sympy.Symbol, as defined in $I_{\\\\theta}(\\\\nu)$\\n    nu       : sympy.Symbol, modulation frequency\\n    Omega    : sympy.Symbol, half of the frequency of the pump laser\\n\\n    Outputs\\n    ----------\\n    original : sympy.Expr\\n        Sympy expression for the original $\\\\left\\\\langle {{{\\\\left| {I_{\\\\theta}\\\\left( \\\\nu  \\\\right)} \\\\right|}^2}} \\\\right\\\\rangle$\\n    max_squeezed, max_antisqueezed : sympy.Expr\\n        Sympy expressions for the maximum squeezed and anti-squeezed\\n        $\\\\left\\\\langle {{{\\\\left| {I_{\\\\theta}\\\\left( \\\\nu  \\\\right)} \\\\right|}^2}} \\\\right\\\\rangle$\\n        when $\\\\phi_2-\\\\phi_1=\\\\pi$.\\n    \\\"\\\"\\\"\\n\\n    # ------------------ FILL IN YOUR RESULTS BELOW ------------------\\n    original = ...\\n    max_squeezed = ...\\n    max_antisqueezed = ...\\n    # ---------------------------------------------------------------\\n\\n    return original, max_squeezed, max_antisqueezed\",\"problem_description\":\"# Problem setup:\\nConsider two cascaded optical parametric amplifiers (OPAs) pumped by lasers at frequency $2\\\\Omega$. The first OPA is pumped with a phase of $\\\\phi_1$, while the second OPA is pumped with a phase of $\\\\phi_2$. The signal and idler beams generated by the first OPA undergo on-chip loss before entering the second OPA, with transmission coefficients $\\\\mu_s=\\\\mu_i=\\\\mu$. After passing through the second OPA, there are additional detection efficiencies $\\\\eta_s=\\\\eta_i=\\\\eta$. Suppose the gain parameters of the two OPAs are $r_1$ and $r_2$, respectively. Define the mean squared power of the photocurrent's sideband $\\\\left\\\\langle {{{\\\\left| {I_{\\\\theta}\\\\left( \\\\nu  \\\\right)} \\\\right|}^2}} \\\\right\\\\rangle$. Here $I_{\\\\theta}(\\\\nu ) = {a_{\\\\Omega  + \\\\nu }}{e^{ - i\\\\theta }} + a_{\\\\Omega  - \\\\nu }^\\\\dagger {e^{i\\\\theta }}$ and $a_\\\\omega$ is the annihilation operator of the photon mode of optical frequency $\\\\omega$.\\n\\n# Main problem:\\nDerive $\\\\left\\\\langle {{{\\\\left| {I_{\\\\theta}\\\\left( \\\\nu  \\\\right)} \\\\right|}^2}} \\\\right\\\\rangle$, then find out the maximum squeezed and anti-squeezed $\\\\left\\\\langle {{{\\\\left| {I_{\\\\theta}\\\\left( \\\\nu  \\\\right)} \\\\right|}^2}} \\\\right\\\\rangle$ if $\\\\phi_2 - \\\\phi_1 = \\\\pi$.\"}","display_format":"text","language":"","answer_status":"published","assets":[],"source_url":"https://critpt.com/","history":"initial import","indexing_mode":"noindex","subproblems":[],"grids":[]}