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17 changes: 10 additions & 7 deletions docs/source/tutorials/ssh_model.ipynb
Original file line number Diff line number Diff line change
Expand Up @@ -16,18 +16,21 @@
"source": [
"Using a Microwave channel, the interaction Hamiltonian is given by\n",
"\n",
"```{math}\n",
"$$\n",
"H = \\sum_{i=1}^N \\sum_{j<i} C_{ij} (\\sigma_i^+ \\sigma_j^- + \\sigma_i^- \\sigma_j^+)\n",
"```\n",
"$$\n",
"where \n",
"{math}`C_{ij} = \\frac{C_3(1-3\\cos^2\\theta_{ij})}{R_{ij}^3}`.\n",
"Here {math}`R_{ij}` is the distance between qubits {math}`i` and {math}`j`, {math}`R_{ij}=|\\textbf{x}_i-\\textbf{x}_j|`, and {math}`\\theta_{ij}` is the angle between {math}`\\textbf{x}_i-\\textbf{x}_j` and the magnetic field.\n",
"$$\n",
"C_{ij} = \\frac{C_3(1-3\\cos^2\\theta_{ij})}{R_{ij}^3}\n",
"$$\n",
"\n",
"Here $R_{ij}$ is the distance between qubits $i$ and $j$, $R_{ij}=|\\textbf{x}_i-\\textbf{x}_j|$, and $\\theta_{ij}$ is the angle between $\\textbf{x}_i-\\textbf{x}_j$ and the magnetic field.\n",
"See the [Pulser docs](https://pulser.readthedocs.io/en/stable/tutorials/xy_spin_chain.html) for more information.\n",
"\n",
"By appropriate choice of the qubit geometry we can approximate the [SSH model](https://en.wikipedia.org/wiki/Su%E2%80%93Schrieffer%E2%80%93Heeger_model) for hard bosons.\n",
"For example, when \n",
"$$ \\theta_{ij} = \\cos^{-1}\\sqrt{1/3}$$\n",
"there will be no interaction between qubits {math}`i` and {math}`j`. We use this to model the SSH model below.\n"
"there will be no interaction between qubits $i$ and $j$. We use this to model the SSH model below.\n"
]
},
{
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],
"metadata": {
"kernelspec": {
"display_name": "qse",
"display_name": "env_qse",
"language": "python",
"name": "python3"
},
Expand All @@ -207,7 +210,7 @@
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.11.11"
"version": "3.12.11"
}
},
"nbformat": 4,
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