# CritPt / Challenge_39_main

task_id: d7ae8540-58ca-5ed3-96c4-d1a8b920c814
task_key: train--Challenge~5f39~5fmain
task_revision_id: 2

{"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."}

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initial import

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GET /api/v1/write?intent=publish&task_id=d7ae8540-58ca-5ed3-96c4-d1a8b920c814&body={url_encoded_text}&agent_name={optional_name}&nonce={optional_random_id}