Final fixes

docs/source/programming.md

@@ -226,7 +226,7 @@ We have successfully defined the [Hamiltonian](programming.md#2-hamiltonian-evol
 - Selecting the `Channels` of the `Device` to use, which define the energy levels of the atoms to use - this step completely defines the [interaction Hamiltonian](programming.md#22-interaction-hamiltonian). The addressing property of each `Channel` also dictates the atoms that will be targeted by the `Pulse`.
 - Adding `Pulse` and delays to the `Channel`s defines the [driving Hamiltonian](programming.md#21-driving-hamiltonian) of each atom along time.    
 
-You can now simulate your first Hamiltonian by programming your first `Sequence` ! [In this tutorial](tutorials/creating.nblink), you will simulate the evolution of the state of an atom initialized in $\left|g\right>$ under a Hamiltonian $H(t)=\frac{\Omega(t)}{2} |g\rangle \langle r|+\frac{\Omega(t)}{2} |r\rangle\langle g|$, with $\Omega$ chosen such that the final state of the atom is the excited state $\left|r\right>$.
+You can now simulate your first Hamiltonian by programming your first `Sequence`! [In this tutorial](tutorials/creating.nblink), you will simulate the evolution of the state of an atom initialized in $\left|g\right>$ under a Hamiltonian $H(t)=\frac{\Omega(t)}{2} |g\rangle \langle r|+\frac{\Omega(t)}{2} |r\rangle\langle g|$, with $\Omega$ chosen such that the final state of the atom is the excited state $\left|r\right>$.
 
 Many concepts have been introduced here and you might want further explanations.
 - The `Device` object contains all the constraints and physical quantities that are defined in a QPU. [This section in the fundamentals](apidoc/core.rst) details these and provides examples of `Devices`. The `VirtualDevices` were also mentioned in this document ([here](programming.md#1-pick-a-device)), which is a more advanced feature described [here](tutorials/virtual_devices.nblink).

tutorials/creating_sequences.ipynb

@@ -83,7 +83,9 @@
     "\n",
     "    The final state will be the Rydberg state $\\left|r\\right>$ if $\\frac{\\Omega}{2} \\Delta t = \\frac{\\pi}{2}$. From this condition, we choose $\\Delta t = 1000\\ ns$ and $\\Omega=\\pi\\ rad/\\mu s$.\n",
     "\n",
-    "</details>"
+    "</details>\n",
+    "\n",
+    "<br>"
    ]
   },
   {
@@ -276,10 +278,9 @@
     "\n",
     "This final Hamiltonian should simultaneously favor having the largest number of atoms in the $\\left|r\\right>$ state (by having $\\delta > 0$) and discourage nearest neighbors from being both in $\\left|r\\right>$ (via the  [interaction Hamiltonian](../programming.md)). When these contributions are appropriately balanced, we get an Hamiltonian with $\\left|rgrgrgrgr\\right>$ as its ground state.\n",
     "\n",
-    "Let's follow the protocol from [this paper](https://journals.aps.org/prx/abstract/10.1103/PhysRevX.8.021070), where we define the parameters with respect to the interaction strength $U$ (see Table 1 of the paper):\n",
+    "Let's follow the protocol from [this paper](https://journals.aps.org/prx/abstract/10.1103/PhysRevX.8.021070), where we define the parameters with respect to the interaction strength between nearest neighbours, $U$ (see Table 1 of the paper):\n",
     "\n",
     "$$\n",
-    "\n",
     "U = 2\\pi\\ rad/\\mu s\\\\\n",
     "\\Omega_{max} = 2 U\\\\\n",
     "\\delta_0 = -6 U\\\\\n",
@@ -291,7 +292,13 @@
     "and define $\\Omega(t)$ and $\\delta(t)$ over time as (see Figure 1 (b)): \n",
     "<center>\n",
     "<img src=\"../files/AF_Ising_program.png\" alt=\"AF Pulse Sequence\" width=\"300\">\n",
-    "</center>"
+    "</center>\n",
+    "\n",
+    "The Hamiltonian we are implementing is (the phase is constant and equal to $0$ over time):\n",
+    "\n",
+    "$$H = \\hbar \\sum_i \\left (\\frac{\\Omega(t)}{2} \\left(|g\\rangle\\langle r| + |r\\rangle\\langle g|\\right) - \\delta(t) |r\\rangle\\langle r| + \\sum_{j<i}U_{ij}\\hat{n}_i \\hat{n}_j \\right)$$\n",
+    "\n",
+    "where $U_{ij} = \\frac{C_6}{\\hbar R_{ij}^6}$."
    ]
   },
   {