{"kind":"task","effective_mode":"full","benchmark":{"kind":"benchmark","effective_mode":"full","slug":"critpt","formal_name":"CritPt","introduction":"CritPt 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","profile":{"resources":[],"task_format":"","scoring":"","metric":"","size":"","answer_access":"","license":"","citation":"","maintainer":"","released":"","why_hard":"","related":[]}},"task_id":"d7ae8540-58ca-5ed3-96c4-d1a8b920c814","task_key":"train--Challenge~5f39~5fmain","task_revision_id":"2","upstream_id":"Challenge_39_main","short_description":"Consider a three-level atom with two ground states, $|b\\rangle$ and $|d\\rangle$,…","config":"","split":"train","body":"{\"code_template\":\"import sympy as sp\\n\\nn, np = sp.symbols('n n_prime', integer=True, nonnegative=True)\\ng, gamma = sp.symbols('g gamma', positive=True)\\nalpha = sp.symbols('alpha')\\n\\ndef answer(n, np, g, gamma, alpha):\\n    r\\\"\\\"\\\"\\n    Return the expression of the steady-state cavity coherence in SymPy format.\\n\\n    Inputs\\n    ----------\\n    n : sympy.Symbol\\n        Photon number in state $|n\\\\rangle$.\\n    np : sympy.Symbol\\n        Photon number in state $|n'\\\\rangle$.\\n    g : sympy.Symbol\\n        Atom-cavity coupling constant $g$.\\n    gamma : sympy.Symbol\\n        Spontaneous-emission rate from $|e\\\\rangle$ to $|d\\\\rangle$, $\\\\gamma$.\\n    alpha : sympy.Symbol\\n        Complex amplitude of the initial coherent state $|\\\\alpha\\\\rangle$.\\n\\n    Outputs\\n    ----------\\n    coherence : sympy.Expr\\n        Symbolic expression for the steady-state cavity coherence $\\\\langle n'| \\\\hat \\\\rho_{c,ss}|n\\\\rangle$.\\n    \\\"\\\"\\\"\\n\\n    # ------------------ FILL IN YOUR RESULTS BELOW ------------------\\n    coherence = ...  # a SymPy expression of the inputs\\n    # ---------------------------------------------------------------\\n\\n    return coherence\",\"problem_description\":\"# Problem setup:\\nConsider a three-level atom with two ground states, $|b\\\\rangle$ and $|d\\\\rangle$, and a single excited state $|e\\\\rangle$. The atom is placed in a lossless cavity that is prepared in a coherent state, i.e., $\\\\hat \\\\rho_c(t=0) = |\\\\alpha\\\\rangle\\\\langle\\\\alpha|$. The cavity is resonant with the ''bright'' $|b\\\\rangle \\\\leftrightarrow |e\\\\rangle$ atomic transition. Setting $\\\\hbar \\\\equiv 1$, the Hamiltonian describing the atom-cavity interaction is then\\n\\\\begin{equation}\\n\\\\hat H = \\\\frac{g}{2} \\\\Big(|b\\\\rangle\\\\langle e| \\\\hat a^\\\\dagger +  |e\\\\rangle\\\\langle b| \\\\hat a\\\\Big).\\n\\\\end{equation}\\nThe atom undergoes spontaneous emission into free space (but not into the cavity) from $|e\\\\rangle$ to $|d\\\\rangle$. But once this happens, this atom no longer interacts with the cavity and hence becomes ''dark.'' The spontaneous emission is described by the dissipator\\n\\\\begin{equation}\\n    \\\\mathcal{D} \\\\hat \\\\rho =\\n\\t\\t\\\\hat J \\\\hat \\\\rho \\\\hat J^\\\\dagger\\n\\t\\t- \\\\frac{1}{2} \\\\left(\\n\\t\\t\\\\hat J^\\\\dagger \\\\hat J \\\\hat \\\\rho + \\\\hat \\\\rho \\\\hat J^\\\\dagger \\\\hat J\\n\\t\\t\\\\right),\\n\\\\end{equation}\\nwhere $\\\\hat J = \\\\sqrt{\\\\gamma} |d\\\\rangle\\\\langle e|$, so that the evolution of the atom-cavity system is governed by the quantum master equation\\n\\\\begin{equation}\\n    \\\\frac{d\\\\hat \\\\rho}{dt} = -i\\\\left[ \\\\hat H, \\\\hat \\\\rho \\\\right] + \\\\mathcal{D} \\\\hat \\\\rho.\\n\\\\end{equation}\\nAssume the atom is prepared in the bright state so that the initial atom-cavity state is $\\\\hat \\\\rho_0 = |b\\\\rangle\\\\langle b| \\\\otimes  |\\\\alpha\\\\rangle\\\\langle\\\\alpha|$.\\n\\n# Main problem:\\n\\nDerive an expression for the cavity field coherences,  $\\\\langle n'| \\\\hat \\\\rho_{c,ss}|n\\\\rangle$, in steady state.\"}","display_format":"text","language":"","answer_status":"published","assets":[],"source_url":"https://critpt.com/","history":"initial import","indexing_mode":"noindex","subproblems":[],"grids":[]}