{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Cat vs coherent states in a Kerr resonator, and the role of measurement\n", "$\\newcommand{\\ket}[1]{| #1 \\rangle}$\n", "$\\newcommand{\\bra}[1]{\\langle #1 |}$\n", "$\\newcommand{\\braket}[1]{\\langle #1 \\rangle}$\n", "$\\newcommand{\\CC}{\\mathcal{C}}$\n", "Author: F. Minganti (minganti@riken.jp)\n", "\n", "In this notebook we show how the same system can produce extremely different results according to the way an observer collects the emitted field of a resonator. This notebook closely follows the results obtained in Refs. [1-3]." ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "import matplotlib.pyplot as plt\n", "import numpy as np\n", "from qutip import *\n", "from IPython.display import display, Math, Latex" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "## The two-photon Kerr Resontator\n", "\n", "Let us consider a single nonlinear Kerr resonator subject \n", "to a parametric two-photon driving.\n", "In a frame rotating at the pump frequency, the Hamiltonian reads\n", "\$$\\label{Eq:Hamiltonian}\n", "\\hat{H}\n", "=\\frac{U}{2}\\,\\hat{a}^\\dagger\\hat{a}^\\dagger\\hat{a}\\hat{a}\n", "+\\frac{G}{2}\\left(\\hat{a}^\\dagger\\hat{a}^\\dagger+\\hat{a}\\hat{a}\\right),\n", "\$$\n", "where $U$ is the Kerr photon-photon interaction strength, $G$ is the two-photon driving amplitude, and $\\hat{a}^\\dagger$ ($\\hat{a}$) is the bosonic creation (annihilation) operator.\n", "\n", "![cavity-1.png](./images/cavity-1.png \"The system under consideration is a single Kerr resonator, with parametric drive and one- and two-photon dissipation.\")\n", "\n", "The time dynamics of the density matrix $\\hat{\\rho}$ of this sytem is given by a Lindblad master equation $i \\partial_t \\hat{\\rho} = \\mathcal{L} \\hat{\\rho}$, where $\\mathcal{L}$ is the Liouvillian superoperator.\n", "The superoperator $\\mathcal{L}$ is made of an Hamiltonian part and \n", "a non-hermitian contribution, which describe the dissipation of energy, particle and information into the environment, as detailed \n", "in e.g. [5].\n", "\n", "Given the parametric drive, the dissipation processes include one- and two-photon dissipation, and the Lindblad superoperator become\n", "\$$\\label{Eq:Lindblad}\n", "\\mathcal{L} \\hat{\\rho} = - i \\left[\\hat{H},\\hat{\\rho}\\right]\n", "+\\frac{\\gamma}{2} \\left(2\\hat{a}\\hat{\\rho}\\hat{a}^\\dagger\n", "-\\hat{a}^\\dagger\\hat{a}\\hat{\\rho}\n", "-\\hat{\\rho}\\hat{a}^\\dagger\\hat{a}\\right)\n", "+ \\, \\frac{\\eta}{2} \\left(2\\hat{a}\\hat{a}\\hat{\\rho}\\hat{a}^\\dagger\\hat{a}^\\dagger\n", "-\\hat{a}^\\dagger\\hat{a}^\\dagger\\hat{a}\\hat{a}\\hat{\\rho}\n", "-\\hat{\\rho}\\hat{a}^\\dagger\\hat{a}^\\dagger\\hat{a}\\hat{a}\\right),\n", "\$$\n", "where $\\gamma$ and $\\eta$ are, respectively, the one- and two-photon dissipation rates.\n", "\n", "We define the system parameters in the following cells. " ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "font_size=20\n", "label_size=30\n", "title_font=35" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [], "source": [ "a=destroy(20)\n", "U=1\n", "G=4\n", "gamma=1\n", "eta=1\n", "H=U*a.dag()*a.dag()*a*a + G*(a*a + a.dag()*a.dag())\n", "c_ops=[np.sqrt(gamma)*a,np.sqrt(eta)*a*a]\n", "\n", "parity=1.j*np.pi*a.dag()*a\n", "parity=parity.expm()\n", "\n", "rho_ss=steadystate(H, c_ops)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This model can be solved exactly for its steady state [2,3].\n", "The corresponding density matrix $\\hat{\\rho}_{\\rm ss}$ is well approximated by the statistical mixture of two orthogonal states:\n", "\$$\\label{Eq:MixtureCats}\n", "\\hat{\\rho}_{\\rm ss}\\simeq\n", "p^+\\,\\ket{\\CC^+_\\alpha}\\!\\bra{\\CC^+_\\alpha}\n", "+p^-\\,\\ket{\\CC^-_\\alpha}\\!\\bra{\\CC^-_\\alpha},\n", "\$$\n", "where $\\ket{\\CC^\\pm_\\alpha}\\propto\\ket{\\alpha}\\pm\\ket{-\\alpha}$ are photonic Schr\\\"odinger cat states whose complex amplitude $\\alpha$ is determined by the system parameters [2-4].\n", "We recall that the coherent state $\\ket{\\alpha}$ is the eigenstate of the destruction operator: $\\hat{a} \\ket{\\alpha}=\\alpha \\ket{\\alpha}$.\n", "The state $\\ket{\\CC^+_\\alpha}$ is called the even cat, since it can be written as a superposition of solely even Fock states, while $\\ket{\\CC^-_\\alpha}$ is the odd cat. \n", "In the previous equation, the coefficients $p^\\pm$ can be interpreted as the probabilities of the system of being found in the corresponding cat state.\n", "\n", "Below, we demonstrate this feature by diagonalising the steady-state density matrix, and by plotting the photon-number probability for the two most probable states." ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "scrolled": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "The mean number of photon is 3.4606002041553974\n" ] }, { "data": { "image/png": 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\n", 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