Positions_and_Modes.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "233e7df5",
   "metadata": {},
   "source": [
    "# Ion Positions, Vibrational Modes, and Crosstalk Visualization"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "4af6623b",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2023-08-10T00:23:13.850161Z",
     "start_time": "2023-08-10T00:23:12.444076Z"
    }
   },
   "outputs": [],
   "source": [
    "from tools.IonChainTools import *"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "beb7989b",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2023-08-10T00:23:13.857321Z",
     "start_time": "2023-08-10T00:23:13.853225Z"
    }
   },
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "from matplotlib import animation, rc\n",
    "from IPython.display import HTML\n",
    "import scipy.constants as con\n",
    "\n",
    "from math import sin, pi as π\n",
    "mCa40 = con.atomic_mass*39.9626"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "150822bf",
   "metadata": {},
   "source": [
    "## Ion Positions"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "60a736d9",
   "metadata": {},
   "source": [
    "The relative distances of the ions from each other in the trap only depends on the number ions. Using the function `calcPositions(N)` will give a vector representing the positions of `N` ions, with `0` being the center of the trap. These positions are given in units of the length scale for the trap. To convert these positions to meters, we must multiply the positions by a length scale that is determined by the specifications of the trap and the ion: the trap frequency, the mass of the ion, and the degree of ionization of the ion (its net charge). This length scale is calculated using `lengthScale(ν, M, Z)`. The units of `ν` are radians/sec and the units of `M` are kg."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "4407475a",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2023-08-10T00:23:13.867522Z",
     "start_time": "2023-08-10T00:23:13.860339Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-1.74290321e+00, -8.22100756e-01, -1.19733315e-12,  8.22100756e-01,\n",
       "        1.74290321e+00])"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "calcPositions(5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "975da811",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2023-08-10T00:23:13.874510Z",
     "start_time": "2023-08-10T00:23:13.870008Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4.449042362254299e-06"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# The following lines are all equivalent\n",
    "# Default mass is mass of Ca40\n",
    "# Default ionization level is 1\n",
    "lengthScale(2*π*1e6)\n",
    "lengthScale(2*π*1e6, M=mCa40)\n",
    "lengthScale(2*π*1e6, Z=1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "f1549953",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2023-08-10T00:23:13.881687Z",
     "start_time": "2023-08-10T00:23:13.876568Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-7.75425023e-06, -3.65756109e-06, -5.32698590e-18,  3.65756109e-06,\n",
       "        7.75425023e-06])"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "calcPositions(5)*lengthScale(2*π*1e6)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ed841ed9",
   "metadata": {},
   "source": [
    "We can see that the spacing between ions changes as the number of ions changes by plotting many chains simultaneously. The coloring in the below plot is such that corresponding ions are colored the same in each chain. For example, in chains with odd numbers of ions, the center ion is always the same color; and in chains with even numbers of ions the ion to the right of the center is always the same color. This allows you to see that when going from a chain of $N$ ions to a chain of $N+2$ ions, the $N$ original ions get pushed closer together symmetrically about the center."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "79cea524",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2023-08-10T00:23:14.577783Z",
     "start_time": "2023-08-10T00:23:13.883590Z"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "Nrange = range(2, 30+1)\n",
    "fig, ax = plt.subplots()\n",
    "for N in Nrange:\n",
    "    positions = calcPositions(N)\n",
    "    c = [(-1)**i for i in range(N)]\n",
    "    if N%2 == 1:\n",
    "        c = [(-1)**((N-1)/2-i) for i in range(N)]\n",
    "        ax.scatter(positions, np.ones(N)*N, c=c, cmap='Paired')\n",
    "    else:\n",
    "        c = [(-1)**((N)/2-i) for i in range(N)]\n",
    "        ax.scatter(positions, np.ones(N)*N, c=c, cmap='bwr')\n",
    "ax.set_title('Positions of Ions vs Number of Ions')\n",
    "ax.set_xlabel('Positions of Ions')\n",
    "ax.set_ylabel('Number of Ions');"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "26ebeb8b",
   "metadata": {},
   "source": [
    "We observe that the spacing between the ions decreases as more ions are added to the string. We can visualize this by plotting the minimum spacing of the ions as a function of the number of ions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "122e186c",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2023-08-10T00:23:15.141870Z",
     "start_time": "2023-08-10T00:23:14.579401Z"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "Nrange = range(2, 30+1)\n",
    "fig, ax = plt.subplots()\n",
    "minspacings = []\n",
    "for N in Nrange:\n",
    "    positions = calcPositions(N)\n",
    "    minspacings.append(min([positions[i+1]-positions[i] for i in range(N) if i!=N-1]))\n",
    "plt.scatter(Nrange, minspacings)\n",
    "ax.set_title('Minimum Spacing of Ions vs Number of Ions')\n",
    "ax.set_xlabel('Minimum Spacing of Ions')\n",
    "ax.set_ylabel('Number of Ions');"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6a152d88",
   "metadata": {},
   "source": [
    "## Crosstalk"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6dc87132",
   "metadata": {},
   "source": [
    "We model crosstalk by considering one or more lasers, each centered on a particular ion in the chain. Each laser has a Gaussian beam intensity profile specified by a beamwidth.\n",
    "\n",
    "The electric field strength at position $x$ due to a laser centered on position $x_c$ which a beamwidth $w$ is given by:\n",
    "\n",
    "$$\n",
    "E(x) = \\exp\\left(-\\left(\\frac{x-x_c}{w}\\right)^2\\right)\n",
    "$$\n",
    "\n",
    "If we define $E_L(x)$ to be the electric field strength at position $x$ from some laser $L$, then we can calculate the total intensity at position $x$ as:\n",
    "\n",
    "$$\n",
    "I(x) = \\left[ \\sum_L E_L(x) \\right]^2\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8a4a297d",
   "metadata": {},
   "source": [
    "Crosstalk can be calculated using the `calcCrosstalkIntensities(N, targets, beamwidth)` function, which returns a vector with the intensities at each ion position. The intensities are scaled so that an intensity of 1 corresponds to the intensity at the center of a single laser beam. `targets` is a vector of the indices of the target ions (which determine the centers of the beams), and `beamwidth` is the beamwidth of the lasers. `beamwidth` is in units of the length scale, which can be found using the `lengthScale` function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "1d869110",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2023-08-10T00:23:15.149701Z",
     "start_time": "2023-08-10T00:23:15.145023Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.18465251, 1.13844097, 1.03520042, 1.13844097, 0.18465251])"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "calcCrosstalkIntensities(5, [1,3], 1)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "402d1a02",
   "metadata": {},
   "source": [
    "If you want to find the intensity at arbitrary positions, you can use the `calcCrosstalkIntensitiesAtPositions(beamcenters, bw, positions)` function. `beamcenters` is a vector of the positions of the centers of the beams, given in units of the length scale. `bw` is the beamwidth of the lasers, also in units of the length scale. `positions` are the positions to calculate the intensities at, in units of the length scale. We can use this to define a function to help us visualize the positions of the ions and the crosstalk at each position:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "05b35629",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2023-08-10T00:23:15.333434Z",
     "start_time": "2023-08-10T00:23:15.153433Z"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig,ax = plt.subplots(1,1)\n",
    "\n",
    "plotPositionsAndCrosstalk(10, [1,5], 1)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f6a4da48",
   "metadata": {},
   "source": [
    "## Vibrational Modes"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2fca746d",
   "metadata": {},
   "source": [
    "The axial and radial vibrational modes of an $N$-ion chain can be calculated using the `calcAxialModes(N, masses)` function and the `calcRadialModes(N, masses, νratio)` function respectively. `N` is the number of ions and `masses` is an optional parameter that is a list of the masses of each ion. `νratio`, which is only necessary for calculating radial modes, is the ratio of the radial trap frequency to the axial trap frequency (typically somewhere between 3 and 10). Only the ratios between the masses matter to the vibrational modes, so they can be provided in any units. Both functions return a list of vibrational modes in order of increasing mode frequency. Each vibrational mode is represented by a tuple where the first element of the tuple is the vibrational mode frequency and the second element is the mode vector. The vibrational mode frequencies are given in units of the axial trap frequency. Vibrational mode vectors represent ratios of displacements of the ions, so they do not depend on the length scale. The vibrational mode vectors are normalized."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "97c38bc5",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2023-08-10T00:23:15.340650Z",
     "start_time": "2023-08-10T00:23:15.335019Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(1.0, array([0.70710678, 0.70710678])),\n",
       " (1.7320508075688772, array([ 0.70710678, -0.70710678]))]"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "calcAxialModes(2) # Default to equal masses for all ions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "1554a3e4",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2023-08-10T00:23:15.347378Z",
     "start_time": "2023-08-10T00:23:15.342259Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(1.0, array([0.70710678, 0.70710678])),\n",
       " (1.7320508075688772, array([ 0.70710678, -0.70710678]))]"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "calcAxialModes(2, masses=[1,1]) # Equivalent to above"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "164473b1",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2023-08-10T00:23:15.353333Z",
     "start_time": "2023-08-10T00:23:15.348818Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(0.7962252170181257, array([0.59069049, 0.80689822])),\n",
       " (1.5381890013208515, array([ 0.9390708 , -0.34372377]))]"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "calcAxialModes(2, masses=[1,2]) # Unequal masses change vibrational modes"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "dd69efc8",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2023-08-10T00:23:15.360172Z",
     "start_time": "2023-08-10T00:23:15.355197Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(2.8284271247461903, array([ 0.70710678, -0.70710678])),\n",
       " (3.0, array([0.70710678, 0.70710678]))]"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "calcRadialModes(2, νratio=3)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a01bb244",
   "metadata": {},
   "source": [
    "### Axial Mode Visualization"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cbb6aaeb",
   "metadata": {},
   "source": [
    "To help visualize the modes, we can create an animation of them. The following function `animateAxialMode` produces an animation that is playable within the notebook. The ions are represented by blue circles, and the red dots indicate their equilibrium positions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "248c3245",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2023-08-10T00:23:16.258748Z",
     "start_time": "2023-08-10T00:23:15.361829Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "<link rel=\"stylesheet\"\n",
       "href=\"https://maxcdn.bootstrapcdn.com/font-awesome/4.4.0/\n",
       "css/font-awesome.min.css\">\n",
       "<script language=\"javascript\">\n",
       "  function isInternetExplorer() {\n",
       "    ua = navigator.userAgent;\n",
       "    /* MSIE used to detect old browsers and Trident used to newer ones*/\n",
       "    return ua.indexOf(\"MSIE \") > -1 || ua.indexOf(\"Trident/\") > -1;\n",
       "  }\n",
       "\n",
       "  /* Define the Animation class */\n",
       "  function Animation(frames, img_id, slider_id, interval, loop_select_id){\n",
       "    this.img_id = img_id;\n",
       "    this.slider_id = slider_id;\n",
       "    this.loop_select_id = loop_select_id;\n",
       "    this.interval = interval;\n",
       "    this.current_frame = 0;\n",
       "    this.direction = 0;\n",
       "    this.timer = null;\n",
       "    this.frames = new Array(frames.length);\n",
       "\n",
       "    for (var i=0; i<frames.length; i++)\n",
       "    {\n",
       "     this.frames[i] = new Image();\n",
       "     this.frames[i].src = frames[i];\n",
       "    }\n",
       "    var slider = document.getElementById(this.slider_id);\n",
       "    slider.max = this.frames.length - 1;\n",
       "    if (isInternetExplorer()) {\n",
       "        // switch from oninput to onchange because IE <= 11 does not conform\n",
       "        // with W3C specification. It ignores oninput and onchange behaves\n",
       "        // like oninput. In contrast, Mircosoft Edge behaves correctly.\n",
       "        slider.setAttribute('onchange', slider.getAttribute('oninput'));\n",
       "        slider.setAttribute('oninput', null);\n",
       "    }\n",
       "    this.set_frame(this.current_frame);\n",
       "  }\n",
       "\n",
       "  Animation.prototype.get_loop_state = function(){\n",
       "    var button_group = document[this.loop_select_id].state;\n",
       "    for (var i = 0; i < button_group.length; i++) {\n",
       "        var button = button_group[i];\n",
       "        if (button.checked) {\n",
       "            return button.value;\n",
       "        }\n",
       "    }\n",
       "    return undefined;\n",
       "  }\n",
       "\n",
       "  Animation.prototype.set_frame = function(frame){\n",
       "    this.current_frame = frame;\n",
       "    document.getElementById(this.img_id).src =\n",
       "            this.frames[this.current_frame].src;\n",
       "    document.getElementById(this.slider_id).value = this.current_frame;\n",
       "  }\n",
       "\n",
       "  Animation.prototype.next_frame = function()\n",
       "  {\n",
       "    this.set_frame(Math.min(this.frames.length - 1, this.current_frame + 1));\n",
       "  }\n",
       "\n",
       "  Animation.prototype.previous_frame = function()\n",
       "  {\n",
       "    this.set_frame(Math.max(0, this.current_frame - 1));\n",
       "  }\n",
       "\n",
       "  Animation.prototype.first_frame = function()\n",
       "  {\n",
       "    this.set_frame(0);\n",
       "  }\n",
       "\n",
       "  Animation.prototype.last_frame = function()\n",
       "  {\n",
       "    this.set_frame(this.frames.length - 1);\n",
       "  }\n",
       "\n",
       "  Animation.prototype.slower = function()\n",
       "  {\n",
       "    this.interval /= 0.7;\n",
       "    if(this.direction > 0){this.play_animation();}\n",
       "    else if(this.direction < 0){this.reverse_animation();}\n",
       "  }\n",
       "\n",
       "  Animation.prototype.faster = function()\n",
       "  {\n",
       "    this.interval *= 0.7;\n",
       "    if(this.direction > 0){this.play_animation();}\n",
       "    else if(this.direction < 0){this.reverse_animation();}\n",
       "  }\n",
       "\n",
       "  Animation.prototype.anim_step_forward = function()\n",
       "  {\n",
       "    this.current_frame += 1;\n",
       "    if(this.current_frame < this.frames.length){\n",
       "      this.set_frame(this.current_frame);\n",
       "    }else{\n",
       "      var loop_state = this.get_loop_state();\n",
       "      if(loop_state == \"loop\"){\n",
       "        this.first_frame();\n",
       "      }else if(loop_state == \"reflect\"){\n",
       "        this.last_frame();\n",
       "        this.reverse_animation();\n",
       "      }else{\n",
       "        this.pause_animation();\n",
       "        this.last_frame();\n",
       "      }\n",
       "    }\n",
       "  }\n",
       "\n",
       "  Animation.prototype.anim_step_reverse = function()\n",
       "  {\n",
       "    this.current_frame -= 1;\n",
       "    if(this.current_frame >= 0){\n",
       "      this.set_frame(this.current_frame);\n",
       "    }else{\n",
       "      var loop_state = this.get_loop_state();\n",
       "      if(loop_state == \"loop\"){\n",
       "        this.last_frame();\n",
       "      }else if(loop_state == \"reflect\"){\n",
       "        this.first_frame();\n",
       "        this.play_animation();\n",
       "      }else{\n",
       "        this.pause_animation();\n",
       "        this.first_frame();\n",
       "      }\n",
       "    }\n",
       "  }\n",
       "\n",
       "  Animation.prototype.pause_animation = function()\n",
       "  {\n",
       "    this.direction = 0;\n",
       "    if (this.timer){\n",
       "      clearInterval(this.timer);\n",
       "      this.timer = null;\n",
       "    }\n",
       "  }\n",
       "\n",
       "  Animation.prototype.play_animation = function()\n",
       "  {\n",
       "    this.pause_animation();\n",
       "    this.direction = 1;\n",
       "    var t = this;\n",
       "    if (!this.timer) this.timer = setInterval(function() {\n",
       "        t.anim_step_forward();\n",
       "    }, this.interval);\n",
       "  }\n",
       "\n",
       "  Animation.prototype.reverse_animation = function()\n",
       "  {\n",
       "    this.pause_animation();\n",
       "    this.direction = -1;\n",
       "    var t = this;\n",
       "    if (!this.timer) this.timer = setInterval(function() {\n",
       "        t.anim_step_reverse();\n",
       "    }, this.interval);\n",
       "  }\n",
       "</script>\n",
       "\n",
       "<style>\n",
       ".animation {\n",
       "    display: inline-block;\n",
       "    text-align: center;\n",
       "}\n",
       "input[type=range].anim-slider {\n",
       "    width: 374px;\n",
       "    margin-left: auto;\n",
       "    margin-right: auto;\n",
       "}\n",
       ".anim-buttons {\n",
       "    margin: 8px 0px;\n",
       "}\n",
       ".anim-buttons button {\n",
       "    padding: 0;\n",
       "    width: 36px;\n",
       "}\n",
       ".anim-state label {\n",
       "    margin-right: 8px;\n",
       "}\n",
       ".anim-state input {\n",
       "    margin: 0;\n",
       "    vertical-align: middle;\n",
       "}\n",
       "</style>\n",
       "\n",
       "<div class=\"animation\">\n",
       "  <img id=\"_anim_imgdab40f9de8714347a262a31e90bc7a0f\">\n",
       "  <div class=\"anim-controls\">\n",
       "    <input id=\"_anim_sliderdab40f9de8714347a262a31e90bc7a0f\" type=\"range\" class=\"anim-slider\"\n",
       "           name=\"points\" min=\"0\" max=\"1\" step=\"1\" value=\"0\"\n",
       "           oninput=\"animdab40f9de8714347a262a31e90bc7a0f.set_frame(parseInt(this.value));\"></input>\n",
       "    <div class=\"anim-buttons\">\n",
       "      <button onclick=\"animdab40f9de8714347a262a31e90bc7a0f.slower()\"><i class=\"fa fa-minus\"></i></button>\n",
       "      <button onclick=\"animdab40f9de8714347a262a31e90bc7a0f.first_frame()\"><i class=\"fa fa-fast-backward\">\n",
       "          </i></button>\n",
       "      <button onclick=\"animdab40f9de8714347a262a31e90bc7a0f.previous_frame()\">\n",
       "          <i class=\"fa fa-step-backward\"></i></button>\n",
       "      <button onclick=\"animdab40f9de8714347a262a31e90bc7a0f.reverse_animation()\">\n",
       "          <i class=\"fa fa-play fa-flip-horizontal\"></i></button>\n",
       "      <button onclick=\"animdab40f9de8714347a262a31e90bc7a0f.pause_animation()\"><i class=\"fa fa-pause\">\n",
       "          </i></button>\n",
       "      <button onclick=\"animdab40f9de8714347a262a31e90bc7a0f.play_animation()\"><i class=\"fa fa-play\"></i>\n",
       "          </button>\n",
       "      <button onclick=\"animdab40f9de8714347a262a31e90bc7a0f.next_frame()\"><i class=\"fa fa-step-forward\">\n",
       "          </i></button>\n",
       "      <button onclick=\"animdab40f9de8714347a262a31e90bc7a0f.last_frame()\"><i class=\"fa fa-fast-forward\">\n",
       "          </i></button>\n",
       "      <button onclick=\"animdab40f9de8714347a262a31e90bc7a0f.faster()\"><i class=\"fa fa-plus\"></i></button>\n",
       "    </div>\n",
       "    <form action=\"#n\" name=\"_anim_loop_selectdab40f9de8714347a262a31e90bc7a0f\" class=\"anim-state\">\n",
       "      <input type=\"radio\" name=\"state\" value=\"once\" id=\"_anim_radio1_dab40f9de8714347a262a31e90bc7a0f\"\n",
       "             >\n",
       "      <label for=\"_anim_radio1_dab40f9de8714347a262a31e90bc7a0f\">Once</label>\n",
       "      <input type=\"radio\" name=\"state\" value=\"loop\" id=\"_anim_radio2_dab40f9de8714347a262a31e90bc7a0f\"\n",
       "             checked>\n",
       "      <label for=\"_anim_radio2_dab40f9de8714347a262a31e90bc7a0f\">Loop</label>\n",
       "      <input type=\"radio\" name=\"state\" value=\"reflect\" id=\"_anim_radio3_dab40f9de8714347a262a31e90bc7a0f\"\n",
       "             >\n",
       "      <label for=\"_anim_radio3_dab40f9de8714347a262a31e90bc7a0f\">Reflect</label>\n",
       "    </form>\n",
       "  </div>\n",
       "</div>\n",
       "\n",
       "\n",
       "<script language=\"javascript\">\n",
       "  /* Instantiate the Animation class. */\n",
       "  /* The IDs given should match those used in the template above. */\n",
       "  (function() {\n",
       "    var img_id = \"_anim_imgdab40f9de8714347a262a31e90bc7a0f\";\n",
       "    var slider_id = \"_anim_sliderdab40f9de8714347a262a31e90bc7a0f\";\n",
       "    var loop_select_id = \"_anim_loop_selectdab40f9de8714347a262a31e90bc7a0f\";\n",
       "    var frames = new Array(25);\n",
       "    \n",
       "  frames[0] = \"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAbAAAAEgCAYAAADVKCZpAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\\\n",
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       "\"\n",
       "\n",
       "\n",
       "    /* set a timeout to make sure all the above elements are created before\n",
       "       the object is initialized. */\n",
       "    setTimeout(function() {\n",
       "        animdab40f9de8714347a262a31e90bc7a0f = new Animation(frames, img_id, slider_id, 200.0,\n",
       "                                 loop_select_id);\n",
       "    }, 0);\n",
       "  })()\n",
       "</script>\n"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "animateAxialMode(5, 2)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8f78a11d",
   "metadata": {},
   "source": [
    "If we look at the frequencies of the axial modes, we see that the frequency spectrum is hardly changed by the addition of more ions to the chain. The first $N-1$ modes of an $N$ ion chain are very similar in frequency to the $N-1$ modes of an $N-1$ ion chain, there is simply one more mode added on the end."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "45342a19",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2023-08-10T00:23:16.511638Z",
     "start_time": "2023-08-10T00:23:16.260612Z"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots()\n",
    "for N in (5, 10, 15, 20):\n",
    "    freqs = [f for f, vec in calcAxialModes(N)]\n",
    "    plt.scatter(range(N), freqs, label=f\"{N}-ion chain\")\n",
    "ax.legend()\n",
    "ax.set_title('Frequencies of Axial Modes')\n",
    "ax.set_xlabel('Mode Index (COM is 0)')\n",
    "ax.set_ylabel('Mode Frequency')\n",
    "ax.set_xticks(np.arange(20));\n",
    "ax.grid(visible=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ffe2677f",
   "metadata": {},
   "source": [
    "We can also visualize the mode coupling statically:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "ea1c2724",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2023-08-10T00:23:16.626747Z",
     "start_time": "2023-08-10T00:23:16.513291Z"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig,ax = plt.subplots(1,1)\n",
    "visualizeAxialMode(5,2,ax,head_size = 0.005)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "86095554",
   "metadata": {},
   "source": [
    "### Radial Mode Visualization"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "b8d42ea1",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2023-08-10T00:23:16.979336Z",
     "start_time": "2023-08-10T00:23:16.628442Z"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x216 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, axs = plt.subplots(1,2,figsize=(6,3))\n",
    "ax,ax2 = axs\n",
    "for N in (5, 10, 15, 20):\n",
    "    freqs = [f for f, vec in calcRadialModes(N,νratio = 10)]\n",
    "    ax.plot(np.arange(N),freqs,'.', label=f\"{N}-ion chain\")\n",
    "    ax2.plot(np.arange(N)+(20-N), freqs,'.', label=f\"{N}-ion chain\")\n",
    "for a in axs:\n",
    "    a.legend()\n",
    "    a.set_title('Frequencies of Axial Modes')\n",
    "    a.set_xlabel('Mode Index (COM is 0)')\n",
    "    a.set_ylabel('Mode Frequency')\n",
    "    #a.set_xticks(np.arange(20));\n",
    "    a.grid(visible=True)\n",
    "    \n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "e148d130",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2023-08-10T00:23:17.902313Z",
     "start_time": "2023-08-10T00:23:16.981124Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "<link rel=\"stylesheet\"\n",
       "href=\"https://maxcdn.bootstrapcdn.com/font-awesome/4.4.0/\n",
       "css/font-awesome.min.css\">\n",
       "<script language=\"javascript\">\n",
       "  function isInternetExplorer() {\n",
       "    ua = navigator.userAgent;\n",
       "    /* MSIE used to detect old browsers and Trident used to newer ones*/\n",
       "    return ua.indexOf(\"MSIE \") > -1 || ua.indexOf(\"Trident/\") > -1;\n",
       "  }\n",
       "\n",
       "  /* Define the Animation class */\n",
       "  function Animation(frames, img_id, slider_id, interval, loop_select_id){\n",
       "    this.img_id = img_id;\n",
       "    this.slider_id = slider_id;\n",
       "    this.loop_select_id = loop_select_id;\n",
       "    this.interval = interval;\n",
       "    this.current_frame = 0;\n",
       "    this.direction = 0;\n",
       "    this.timer = null;\n",
       "    this.frames = new Array(frames.length);\n",
       "\n",
       "    for (var i=0; i<frames.length; i++)\n",
       "    {\n",
       "     this.frames[i] = new Image();\n",
       "     this.frames[i].src = frames[i];\n",
       "    }\n",
       "    var slider = document.getElementById(this.slider_id);\n",
       "    slider.max = this.frames.length - 1;\n",
       "    if (isInternetExplorer()) {\n",
       "        // switch from oninput to onchange because IE <= 11 does not conform\n",
       "        // with W3C specification. It ignores oninput and onchange behaves\n",
       "        // like oninput. In contrast, Mircosoft Edge behaves correctly.\n",
       "        slider.setAttribute('onchange', slider.getAttribute('oninput'));\n",
       "        slider.setAttribute('oninput', null);\n",
       "    }\n",
       "    this.set_frame(this.current_frame);\n",
       "  }\n",
       "\n",
       "  Animation.prototype.get_loop_state = function(){\n",
       "    var button_group = document[this.loop_select_id].state;\n",
       "    for (var i = 0; i < button_group.length; i++) {\n",
       "        var button = button_group[i];\n",
       "        if (button.checked) {\n",
       "            return button.value;\n",
       "        }\n",
       "    }\n",
       "    return undefined;\n",
       "  }\n",
       "\n",
       "  Animation.prototype.set_frame = function(frame){\n",
       "    this.current_frame = frame;\n",
       "    document.getElementById(this.img_id).src =\n",
       "            this.frames[this.current_frame].src;\n",
       "    document.getElementById(this.slider_id).value = this.current_frame;\n",
       "  }\n",
       "\n",
       "  Animation.prototype.next_frame = function()\n",
       "  {\n",
       "    this.set_frame(Math.min(this.frames.length - 1, this.current_frame + 1));\n",
       "  }\n",
       "\n",
       "  Animation.prototype.previous_frame = function()\n",
       "  {\n",
       "    this.set_frame(Math.max(0, this.current_frame - 1));\n",
       "  }\n",
       "\n",
       "  Animation.prototype.first_frame = function()\n",
       "  {\n",
       "    this.set_frame(0);\n",
       "  }\n",
       "\n",
       "  Animation.prototype.last_frame = function()\n",
       "  {\n",
       "    this.set_frame(this.frames.length - 1);\n",
       "  }\n",
       "\n",
       "  Animation.prototype.slower = function()\n",
       "  {\n",
       "    this.interval /= 0.7;\n",
       "    if(this.direction > 0){this.play_animation();}\n",
       "    else if(this.direction < 0){this.reverse_animation();}\n",
       "  }\n",
       "\n",
       "  Animation.prototype.faster = function()\n",
       "  {\n",
       "    this.interval *= 0.7;\n",
       "    if(this.direction > 0){this.play_animation();}\n",
       "    else if(this.direction < 0){this.reverse_animation();}\n",
       "  }\n",
       "\n",
       "  Animation.prototype.anim_step_forward = function()\n",
       "  {\n",
       "    this.current_frame += 1;\n",
       "    if(this.current_frame < this.frames.length){\n",
       "      this.set_frame(this.current_frame);\n",
       "    }else{\n",
       "      var loop_state = this.get_loop_state();\n",
       "      if(loop_state == \"loop\"){\n",
       "        this.first_frame();\n",
       "      }else if(loop_state == \"reflect\"){\n",
       "        this.last_frame();\n",
       "        this.reverse_animation();\n",
       "      }else{\n",
       "        this.pause_animation();\n",
       "        this.last_frame();\n",
       "      }\n",
       "    }\n",
       "  }\n",
       "\n",
       "  Animation.prototype.anim_step_reverse = function()\n",
       "  {\n",
       "    this.current_frame -= 1;\n",
       "    if(this.current_frame >= 0){\n",
       "      this.set_frame(this.current_frame);\n",
       "    }else{\n",
       "      var loop_state = this.get_loop_state();\n",
       "      if(loop_state == \"loop\"){\n",
       "        this.last_frame();\n",
       "      }else if(loop_state == \"reflect\"){\n",
       "        this.first_frame();\n",
       "        this.play_animation();\n",
       "      }else{\n",
       "        this.pause_animation();\n",
       "        this.first_frame();\n",
       "      }\n",
       "    }\n",
       "  }\n",
       "\n",
       "  Animation.prototype.pause_animation = function()\n",
       "  {\n",
       "    this.direction = 0;\n",
       "    if (this.timer){\n",
       "      clearInterval(this.timer);\n",
       "      this.timer = null;\n",
       "    }\n",
       "  }\n",
       "\n",
       "  Animation.prototype.play_animation = function()\n",
       "  {\n",
       "    this.pause_animation();\n",
       "    this.direction = 1;\n",
       "    var t = this;\n",
       "    if (!this.timer) this.timer = setInterval(function() {\n",
       "        t.anim_step_forward();\n",
       "    }, this.interval);\n",
       "  }\n",
       "\n",
       "  Animation.prototype.reverse_animation = function()\n",
       "  {\n",
       "    this.pause_animation();\n",
       "    this.direction = -1;\n",
       "    var t = this;\n",
       "    if (!this.timer) this.timer = setInterval(function() {\n",
       "        t.anim_step_reverse();\n",
       "    }, this.interval);\n",
       "  }\n",
       "</script>\n",
       "\n",
       "<style>\n",
       ".animation {\n",
       "    display: inline-block;\n",
       "    text-align: center;\n",
       "}\n",
       "input[type=range].anim-slider {\n",
       "    width: 374px;\n",
       "    margin-left: auto;\n",
       "    margin-right: auto;\n",
       "}\n",
       ".anim-buttons {\n",
       "    margin: 8px 0px;\n",
       "}\n",
       ".anim-buttons button {\n",
       "    padding: 0;\n",
       "    width: 36px;\n",
       "}\n",
       ".anim-state label {\n",
       "    margin-right: 8px;\n",
       "}\n",
       ".anim-state input {\n",
       "    margin: 0;\n",
       "    vertical-align: middle;\n",
       "}\n",
       "</style>\n",
       "\n",
       "<div class=\"animation\">\n",
       "  <img id=\"_anim_imgadd2a7b530c040d5a84ac93d781c547d\">\n",
       "  <div class=\"anim-controls\">\n",
       "    <input id=\"_anim_slideradd2a7b530c040d5a84ac93d781c547d\" type=\"range\" class=\"anim-slider\"\n",
       "           name=\"points\" min=\"0\" max=\"1\" step=\"1\" value=\"0\"\n",
       "           oninput=\"animadd2a7b530c040d5a84ac93d781c547d.set_frame(parseInt(this.value));\"></input>\n",
       "    <div class=\"anim-buttons\">\n",
       "      <button onclick=\"animadd2a7b530c040d5a84ac93d781c547d.slower()\"><i class=\"fa fa-minus\"></i></button>\n",
       "      <button onclick=\"animadd2a7b530c040d5a84ac93d781c547d.first_frame()\"><i class=\"fa fa-fast-backward\">\n",
       "          </i></button>\n",
       "      <button onclick=\"animadd2a7b530c040d5a84ac93d781c547d.previous_frame()\">\n",
       "          <i class=\"fa fa-step-backward\"></i></button>\n",
       "      <button onclick=\"animadd2a7b530c040d5a84ac93d781c547d.reverse_animation()\">\n",
       "          <i class=\"fa fa-play fa-flip-horizontal\"></i></button>\n",
       "      <button onclick=\"animadd2a7b530c040d5a84ac93d781c547d.pause_animation()\"><i class=\"fa fa-pause\">\n",
       "          </i></button>\n",
       "      <button onclick=\"animadd2a7b530c040d5a84ac93d781c547d.play_animation()\"><i class=\"fa fa-play\"></i>\n",
       "          </button>\n",
       "      <button onclick=\"animadd2a7b530c040d5a84ac93d781c547d.next_frame()\"><i class=\"fa fa-step-forward\">\n",
       "          </i></button>\n",
       "      <button onclick=\"animadd2a7b530c040d5a84ac93d781c547d.last_frame()\"><i class=\"fa fa-fast-forward\">\n",
       "          </i></button>\n",
       "      <button onclick=\"animadd2a7b530c040d5a84ac93d781c547d.faster()\"><i class=\"fa fa-plus\"></i></button>\n",
       "    </div>\n",
       "    <form action=\"#n\" name=\"_anim_loop_selectadd2a7b530c040d5a84ac93d781c547d\" class=\"anim-state\">\n",
       "      <input type=\"radio\" name=\"state\" value=\"once\" id=\"_anim_radio1_add2a7b530c040d5a84ac93d781c547d\"\n",
       "             >\n",
       "      <label for=\"_anim_radio1_add2a7b530c040d5a84ac93d781c547d\">Once</label>\n",
       "      <input type=\"radio\" name=\"state\" value=\"loop\" id=\"_anim_radio2_add2a7b530c040d5a84ac93d781c547d\"\n",
       "             checked>\n",
       "      <label for=\"_anim_radio2_add2a7b530c040d5a84ac93d781c547d\">Loop</label>\n",
       "      <input type=\"radio\" name=\"state\" value=\"reflect\" id=\"_anim_radio3_add2a7b530c040d5a84ac93d781c547d\"\n",
       "             >\n",
       "      <label for=\"_anim_radio3_add2a7b530c040d5a84ac93d781c547d\">Reflect</label>\n",
       "    </form>\n",
       "  </div>\n",
       "</div>\n",
       "\n",
       "\n",
       "<script language=\"javascript\">\n",
       "  /* Instantiate the Animation class. */\n",
       "  /* The IDs given should match those used in the template above. */\n",
       "  (function() {\n",
       "    var img_id = \"_anim_imgadd2a7b530c040d5a84ac93d781c547d\";\n",
       "    var slider_id = \"_anim_slideradd2a7b530c040d5a84ac93d781c547d\";\n",
       "    var loop_select_id = \"_anim_loop_selectadd2a7b530c040d5a84ac93d781c547d\";\n",
       "    var frames = new Array(25);\n",
       "    \n",
       "  frames[0] = \"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAbAAAAEgCAYAAADVKCZpAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\\\n",
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       "UCTOtOmMatAAAAAASUVORK5CYII=\\\n",
       "\"\n",
       "  frames[1] = \"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAbAAAAEgCAYAAADVKCZpAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\\\n",
       "AAALEgAACxIB0t1+/AAAADh0RVh0U29mdHdhcmUAbWF0cGxvdGxpYiB2ZXJzaW9uMy4xLjEsIGh0\\\n",
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       "EjAAUhIwAFISMABSEjAAUhIwAFL6f26h35XS511hAAAAAElFTkSuQmCC\\\n",
       "\"\n",
       "\n",
       "\n",
       "    /* set a timeout to make sure all the above elements are created before\n",
       "       the object is initialized. */\n",
       "    setTimeout(function() {\n",
       "        animadd2a7b530c040d5a84ac93d781c547d = new Animation(frames, img_id, slider_id, 200.0,\n",
       "                                 loop_select_id);\n",
       "    }, 0);\n",
       "  })()\n",
       "</script>\n"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "animateRadialMode(5, 0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "ff417954",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2023-08-10T00:23:42.688897Z",
     "start_time": "2023-08-10T00:23:42.562637Z"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig,ax = plt.subplots(1,1)\n",
    "visualizeRadialMode(5,2,ax,head_size = 0.25)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "64d02c3d",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.11.6"
  },
  "toc": {
   "base_numbering": 1,
   "nav_menu": {},
   "number_sections": false,
   "sideBar": true,
   "skip_h1_title": false,
   "title_cell": "Table of Contents",
   "title_sidebar": "Contents",
   "toc_cell": false,
   "toc_position": {},
   "toc_section_display": true,
   "toc_window_display": true
  },
  "varInspector": {
   "cols": {
    "lenName": 16,
    "lenType": 16,
    "lenVar": 40
   },
   "kernels_config": {
    "python": {
     "delete_cmd_postfix": "",
     "delete_cmd_prefix": "del ",
     "library": "var_list.py",
     "varRefreshCmd": "print(var_dic_list())"
    },
    "r": {
     "delete_cmd_postfix": ") ",
     "delete_cmd_prefix": "rm(",
     "library": "var_list.r",
     "varRefreshCmd": "cat(var_dic_list()) "
    }
   },
   "types_to_exclude": [
    "module",
    "function",
    "builtin_function_or_method",
    "instance",
    "_Feature"
   ],
   "window_display": false
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}