{ "cells": [ { "cell_type": "markdown", "metadata": { "collapsed": true }, "source": [ "# Spin Squeezing in presence of local and collective noise\n", "\n", "Notebook author: Nathan Shammah (nathan.shammah at gmail.com)\n", "\n", "Here we study the effect of collective and local processes on a spin squeezing Hamiltonian. \n", "\n", "We consider a system of $N$ two-level systems (TLSs) with identical frequency $\\omega_{0}$, which can de-excite incoherently or collectively at the rates $\\gamma_\\text{E}$ and $\\gamma_\\text{CE}$,\n", "\n", "\\begin{eqnarray}\n", "\\dot{\\rho} &=&-i\\lbrack -i\\Lambda\\left(J_{+}^2-J_{-}^2\\right),\\rho \\rbrack\n", "+\\frac{\\gamma_\\text {CE}}{2}\\mathcal{L}_{J_{-}}[\\rho]\n", "+\\frac{\\gamma_\\text{E}}{2}\\sum_{n=1}^{N}\\mathcal{L}_{J_{-,n}}[\\rho]\n", "\\end{eqnarray}\n", "\n", "We study the time evolution of the spin squeezing parameter [1-4]\n", "\\begin{eqnarray}\n", "\\xi^2 &=& N\\langle\\Delta J_y^2\\rangle/\\left(\\langle J_z\\rangle^2+\\langle J_x\\rangle^2\\right)\n", "\\end{eqnarray}\n", "\n", "We assess how different dynamical conditions and initial states can be explored to optimize the spin squeezing of a given Dicke state [5-7]. This study can be generalized to other types of local and collective incoherent processes. A table grouping this processes is given below, \n", "\n", "
Keyword | \n", "Rate $\\gamma_j$ | \n", "Lindbladian $\\mathcal{L}[\\rho]$ | \n", "
$\\texttt{emission}$ | \n", "$\\gamma_\\text{E}$ | \n", "\\begin{eqnarray}\\mathcal{L}[\\rho]&=&\\sum_n^N \\left(J_{-,n}\\rho J_{+,n} - \\frac{1}{2}J_{+,n}J_{-,n}\\rho - \\frac{1}{2}\\rho J_{+,n}J_{-,n} \\right)\\end{eqnarray} | \n", "
$\\texttt{pumping}$ | \n", "$\\gamma_\\text{P}$ | \n", "\\begin{eqnarray}\\mathcal{L}[\\rho]&=&\\sum_n^N \\left(J_{+,n}\\rho J_{-,n} - \\frac{1}{2}J_{-,n}J_{+,n}\\rho - \\frac{1}{2}\\rho J_{-,n}J_{+,n} \\right)\\end{eqnarray} | \n", "
$\\texttt{dephasing}$ | \n", "$\\gamma_\\text{D}$ | \n", "\\begin{eqnarray}\\mathcal{L}[\\rho]&=&\\sum_n^N \\left(J_{z,n}\\rho J_{z,n} - \\frac{1}{2}J_{z,n}J_{z,n}\\rho - \\frac{1}{2}\\rho J_{z,n}J_{z,n} \\right)\\end{eqnarray} | \n", "
$\\texttt{collective}\\_\\texttt{emission}$ | \n", "$\\gamma_\\text{CE}$ | \n", "\\begin{eqnarray}\\mathcal{L}[\\rho]&=& J_{-}\\rho J_{+} - \\frac{1}{2}J_{+}J_{-}\\rho - \\frac{1}{2}\\rho J_{+}J_{-} \\end{eqnarray} | \n", "
$\\texttt{collective}\\_\\texttt{pumping}$ | \n", "$\\gamma_\\text{CP}$ | \n", "\\begin{eqnarray}\\mathcal{L}[\\rho]&=& J_{+}\\rho J_{-} - \\frac{1}{2}J_{-}J_{+}\\rho - \\frac{1}{2}\\rho J_{-}J_{+} \\end{eqnarray} | \n", "
$\\texttt{collective}\\_\\texttt{dephasing}$ | \n", "$\\gamma_\\text{CD}$ | \n", "\\begin{eqnarray}\\mathcal{L}[\\rho]&=& J_{z}\\rho J_{z} - \\frac{1}{2}J_{z}^2\\rho - \\frac{1}{2}\\rho J_{z}^2 \\end{eqnarray} | \n", "