examples/demo-jaynes-cummings.ipynb

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    "# scqubits example: Jaynes-Cummings model\n",
    "J. Koch and P. Groszkowski\n",
    "\n",
    "For further documentation of scqubits see https://scqubits.readthedocs.io/en/latest/.\n",
    "\n",
    "---"
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   "source": [
    "import numpy as np\n",
    "\n",
    "import scqubits as scq\n",
    "from scqubits import HilbertSpace, InteractionTerm, ParameterSweep"
   ]
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    "## The Jaynes-Cummings model\n",
    "\n",
    "The Jaynes-Cummings mode describes a system composed of a qubit (two-level system) coupled to a harmonic oscillator. The coupling is such that a qubit excitation can be \"converted\" into a photon and vice versa,\n",
    "\n",
    "$\\displaystyle H=-\\frac{\\omega_q}{2}\\sigma_z + \\omega_r a^\\dagger a + g (a\\sigma_+ + a^\\dagger \\sigma_-)$\n",
    "\n",
    "This model can be solved exactly, resulting in the following eigenenergies:\n",
    "\n",
    "$\\displaystyle E_0 = -\\frac{1}{2}\\omega_q$,\n",
    "\n",
    "$\\displaystyle E_{n\\pm} =  \\bigg(n-\\frac{1}{2}\\bigg) \\omega_r \\pm \\sqrt{\\frac{\\Delta^2}{4}+n g^2}$,\n",
    "\n",
    "where $\\Delta=\\omega_r-\\omega_q$ and $n=1,2,\\ldots$.\n"
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    "# exact eigenenergies for comparing with numerics\n",
    "\n",
    "def energies(frequency_r, frequency_q, g, n_cutoff):\n",
    "    delta = frequency_r - frequency_q\n",
    "    energies1 = (np.arange(1, n_cutoff) - 0.5) * frequency_r\n",
    "    energies2 = np.sqrt(delta**2/4 + np.arange(1, n_cutoff) * g**2)\n",
    "    energies_plus = energies1 + energies2\n",
    "    energies_minus = energies1 - energies2\n",
    "    energies_0 = np.asarray([[-0.5 * frequency_q]])\n",
    "    all_energies = np.append(energies_0, energies_minus)\n",
    "    all_energies = np.append(all_energies, energies_plus)\n",
    "    return np.sort(all_energies)"
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   "source": [
    "### Set up the subsystems"
   ]
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    "frequency_q = 1.0\n",
    "frequency_r = 0.8\n",
    "g = 0.1\n",
    "\n",
    "\n",
    "qubit = scq.GenericQubit(E=frequency_q)\n",
    "\n",
    "osc = scq.Oscillator(\n",
    "    E_osc=frequency_r,\n",
    "    truncated_dim=10  # up to 9 photons\n",
    ")"
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   "source": [
    "### Set up Hilbert space"
   ]
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    "# Form a list of all components making up the Hilbert space.\n",
    "hilbertspace = HilbertSpace([qubit, osc])\n",
    "\n",
    "hilbertspace.add_interaction(\n",
    "    g_strength = g,\n",
    "    op1 = qubit.sm_operator,\n",
    "    op2 = osc.creation_operator,\n",
    "    add_hc = True\n",
    ")"
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      "HilbertSpace:  subsystems\n",
      "-------------------------\n",
      "\n",
      "GenericQubit--------| [GenericQubit_1]\n",
      "                    | E: 1.0\n",
      "                    |\n",
      "                    | dim: 2\n",
      "\n",
      "\n",
      "Oscillator----------| [Oscillator_1]\n",
      "                    | E_osc: 0.8\n",
      "                    | l_osc: None\n",
      "                    | truncated_dim: 10\n",
      "                    |\n",
      "                    | dim: 10\n",
      "\n",
      "\n",
      "\n",
      "HilbertSpace:  interaction terms\n",
      "--------------------------------\n",
      "InteractionTerm----------| [Interaction_1]\n",
      "                         | g_strength: 0.1\n",
      "                         | operator_list: [(0, <bound method GenericQubit.sm_operator of Gen ...\n",
      "                         | add_hc: True\n",
      "\n",
      "\n"
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    "print(hilbertspace)"
   ]
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      "[-0.5         0.25857864  0.54142136  1.02679492  1.37320508]\n"
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    "evals = hilbertspace.eigenvals(evals_count=5)\n",
    "print(evals)"
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    "energies(frequency_r, frequency_q, g, 3)"
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