{ "cells": [ { "cell_type": "markdown", "id": "aab9ba3c", "metadata": {}, "source": [ "# Fibonacci States" ] }, { "cell_type": "markdown", "source": [ "ZPGenerator is designed to simulate sources of light that require complicated pulse sequences to produce the photonic state. To demonstrate this capability, this example explores constructing and simulating a source of entangled photonic Fibonacci states that were first proposed and demonstrated in a Quandela research paper [[S. C. Wein et al., Nature Photonics 16, 374–379 (2022)](https://www.nature.com/articles/s41566-022-00979-z)]." ], "metadata": { "collapsed": false } }, { "cell_type": "markdown", "id": "f32ab5ac", "metadata": {}, "source": [ "## Theory background" ] }, { "cell_type": "markdown", "id": "4671b23b", "metadata": {}, "source": [ "Two-level atoms, or artificial atoms such as quantum dots, have been widely studied as a source of single photons. To deterministically generate entangled states of light, usually we must turn to sources of light based on multi-level systems, such as charged exciton containing the trion state (see the [Quantum Dots](quantum_dots.ipynb) and [Entanglement Generation](entanglement_generation.ipynb) tutorials). Surprisingly, a two-level system composed of a ground state $|g\\rangle$ and an excited state $|e\\rangle$ _can_ be used to deterministically generate entangled states of light encoded in the photon-number basis. This relies on the fact that the interaction between a two-level system and the electromagnetic field is essentially an entangling gate that preserves quantum coherence [[J. C. Loredo et al., Nature Photonics 13, 803–808 (2019)](https://www.nature.com/articles/s41566-019-0506-3)].\n", "\n", "The basic idea is to apply a sequence of well-timed excitation pulses to re-excite the atom before it has had time to fully decay. These excitation pulses play the role of single-qubit bit-flip gates on an ancillary qubit (the source) that is used to entangled the electromagnetic field. The photonic qubits are then the occupation of time bin modes defined between each subsequent pulse.\n", "\n", "As an example, the state of the source after a single $\\pi$ pulse is $|e\\rangle$. After a delay exactly equal to the atomic half-life $\\tau = T_1\\ln(2)$ where $T_1$ is the source lifetime, the atom may or may not have emitted a photon. Since the light-matter interaction is coherent, the total source-field system enters into an entangled state $|\\Psi\\rangle = (|e\\rangle|0\\rangle + |g\\rangle|1\\rangle)/\\sqrt{2}$. If, at this moment, we apply a second $\\pi$ pulse, the state of the source is flipped and the total light-matter state becomes $|\\Psi\\rangle = (|g\\rangle|0\\rangle + |e\\rangle|1\\rangle)/\\sqrt{2}$. Now, if we let the atom finish decaying to the ground state, we are left with the separable state $|\\Psi\\rangle = |g\\rangle|\\psi\\rangle$ where the state of the light is a maximally entangled Bell state $|\\psi\\rangle = (|00\\rangle + |11\\rangle)/\\sqrt{2}$.\n", "\n", "This idea can be extended to $N$ $\\pi$ pulses where each subsequent pulse adds a qubit to the state of light. It turns out that the delay between each pulse that maximises the entanglement in the state of light are determined by the Fibonacci sequence:\n", "\\begin{equation}\n", "\\tau_i = T_1\\ln\\left(\\frac{F_{N+2-i}}{F_{N-i}}\\right)\n", "\\end{equation}\n", "where $F_i = F_{i-1}+F_{i-2}$, $F_{0} = 0$, and $F_{1}=1$. Applying a pulse sequence based on this pattern will produce a sequence of entangled states that are expressed as an equally-weighted superposition of $F_N$ basis states:\n", " \n", "$$\n", "\\begin{aligned}\n", " |\\psi_0\\rangle &= |0\\rangle\\\\\n", " |\\psi_1\\rangle &= |1\\rangle\\\\\n", " |\\psi_2\\rangle &= \\left(|00\\rangle+|11\\rangle\\right)/\\sqrt{2}\\\\\n", " |\\psi_3\\rangle &= \\left(|001\\rangle+|100\\rangle+|111\\rangle\\right)/\\sqrt{3}\\\\\n", " |\\psi_4\\rangle &= \\left(|0000\\rangle+|0011\\rangle+|1001\\rangle+|1100\\rangle+|1111\\rangle\\right)/\\sqrt{5}\\\\\n", " &\\vdots\n", "\\end{aligned}\n", "$$\n", " \n", "We can also notice that $|\\psi_N\\rangle$ is a superposition state of photon number states with the same parity as $N$. Either $N$ is odd and $|\\psi_N\\rangle$ is a superposition of odd-numbers of photons or $N$ is even and $|\\psi_N\\rangle$ is a superposition of even-numbers of photons. Furthermore, the number of states composing $|\\psi_N\\rangle$ with the same total number of photons will follow Pascal's triangle:\n", "\n", "