Cosmetic changes to the Hamiltonian in the intro, added a sentence to better describe relationship betweein r and D(t)

Author: DanPuzzuoli

qiskit/advanced/aer/8_pulse_simulator_duffing_model.ipynb

@@ -59,12 +59,12 @@
     "\n",
     "-  Each Duffing oscillator is specified by a frequency $\\nu$, anharmonicity $\\alpha$, and drive strength $r$, which result in the Hamiltonian terms:\n",
     "\\begin{equation}\n",
-    "    \\pi(2\\nu - \\alpha)a^\\dagger a + \\pi \\alpha (a^\\dagger a)^2 + 2 \\pi r (a + a^\\dagger) \\times D(t),\n",
+    "    2\\pi\\nu a^\\dagger a + \\pi \\alpha a^\\dagger a(a^\\dagger a - 1) + 2 \\pi r (a + a^\\dagger) \\times D(t),\n",
     "\\end{equation}\n",
-    "where $D(t)$ is the signal on the drive channel for the qubit, and $a^\\dagger$ and $a$ are, respectively, the creation and annihilation operators for the qubit.\n",
+    "where $D(t)$ is the signal on the drive channel for the qubit, and $a^\\dagger$ and $a$ are, respectively, the creation and annihilation operators for the qubit. Note that the drive strength $r$ sets the scaling of the control term, with $D(t)$ assumed to be a complex and unitless number satisfying $|D(t)| \\leq 1$. \n",
     "-  A coupling between a pair of oscillators $(l,k)$ is specified by the coupling strength $J$, resulting in an exchange coupling term:\n",
     "\\begin{equation}\n",
-    "    2 \\pi J (a_l \\otimes a_k^\\dagger + a_l^\\dagger \\otimes a_k),\n",
+    "    2 \\pi J (a_l a_k^\\dagger + a_l^\\dagger a_k),\n",
     "\\end{equation}\n",
     "where the subscript denotes which qubit the operators act on.\n",
     "- Additionally, for numerical simulation, it is necessary to specify a cutoff dimension; the Duffing oscillator model is *infinite dimensional*, and computer simulation requires restriction of the operators to a finite dimensional subspace.\n",