Shape_and_Tone_FFT.ipynb

{
  "nbformat": 4,
  "nbformat_minor": 0,
  "metadata": {
    "colab": {
      "name": "Shape_and_Tone_FFT.ipynb",
      "provenance": []
    },
    "kernelspec": {
      "name": "python3",
      "display_name": "Python 3"
    }
  },
  "cells": [
    {
      "cell_type": "code",
      "metadata": {
        "id": "JgrKmOc4V0k1",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "# %% Includes\n",
        "# ================================================\n",
        "\n",
        "import numpy as np\n",
        "import matplotlib.pyplot as plt\n"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "IfhTcfSOV62Q",
        "colab_type": "text"
      },
      "source": [
        "#Just the first Tone:"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "TGDDjtqyV54Y",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 283
        },
        "outputId": "dd0fc5ea-67d3-433e-e714-2154e2463cc1"
      },
      "source": [
        "\n",
        "M=1024\n",
        "\n",
        "a_m=np.zeros(M,dtype=complex)   # complex fft parameters\n",
        "\n",
        "a_m[1] = M*1j   #positive first sine\n",
        "\n",
        "STFT_block=np.fft.ifft(a_m).real  # take real part \n",
        "plt.plot(STFT_block)\n"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "[<matplotlib.lines.Line2D at 0x7fc094732ac8>]"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 4
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "NjOBXuJpWjU0",
        "colab_type": "text"
      },
      "source": [
        "# Just the (Hann) Shape:"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "P71TfH0gWafK",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 283
        },
        "outputId": "c8c260df-1300-4200-c3b9-592b8bd02032"
      },
      "source": [
        "\n",
        "M=1024\n",
        "\n",
        "a_m=np.zeros(M,dtype=complex)   # complex fft parameters\n",
        "\n",
        "a_m[0] = M/2\n",
        "a_m[-1] = -M/4\n",
        "a_m[1] = -M/4\n",
        "\n",
        "STFT_block=np.fft.ifft(a_m).real  # take real part \n",
        "plt.plot(STFT_block)\n"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "[<matplotlib.lines.Line2D at 0x7fc094619588>]"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 5
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "wAxq7eZMXOHV",
        "colab_type": "text"
      },
      "source": [
        "Multiplication in time domain corresponds to convolution in freq. domain\n",
        "# Convolved shape and tone in freq domain:"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "8oTzt75IXLPC",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 283
        },
        "outputId": "a22d3c74-1f72-432a-aea1-5919ee43a8e0"
      },
      "source": [
        "\n",
        "M=1024\n",
        "\n",
        "a_m=np.zeros(M,dtype=complex)   # complex fft parameters\n",
        "\n",
        "a_m[1] = 1j*M/2\n",
        "a_m[0] = -1j*M/4\n",
        "a_m[2] = -1j*M/4\n",
        "\n",
        "STFT_block=np.fft.ifft(a_m).real  # take real part \n",
        "plt.plot(STFT_block)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "[<matplotlib.lines.Line2D at 0x7fc094557ba8>]"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 7
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "t_8wbk3dXvxw",
        "colab_type": "text"
      },
      "source": [
        "# With higher freq tone:"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "hSdE2H7RXaXu",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 283
        },
        "outputId": "3d24a06b-4bec-4c19-8fa7-4f28a1cf20fb"
      },
      "source": [
        "M=1024\n",
        "\n",
        "a_m=np.zeros(M,dtype=complex)   # complex fft parameters\n",
        "\n",
        "a_m[200] = 1j*M/2\n",
        "a_m[199] = -1j*M/4\n",
        "a_m[201] = -1j*M/4\n",
        "\n",
        "STFT_block=np.fft.ifft(a_m).real  # take real part \n",
        "plt.plot(STFT_block)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "[<matplotlib.lines.Line2D at 0x7fc0944c3470>]"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 8
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "cZHvGb37Yh5w",
        "colab_type": "text"
      },
      "source": [
        "# Remarks:\n",
        "\n",
        "* There is a correspondence between freq. and time resolution, essentially like Heisenberg's uncertainty principle.\n",
        "\n",
        "* A signal limited in time (aka pulse) has an infinite spectrum. The higher freq components just get smaller and smaller the longer the pulse is. (and windowing can change the freq. distibution)\n",
        "\n",
        "* In a digital representation (DFT/FFT) the spectrum and the time domain are essentially circles, aka periodic with one period beeing the number of samples/number of representable frequencies."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "N3ule1Twjq1O",
        "colab_type": "text"
      },
      "source": [
        "#Phase Space:\n",
        "\n",
        "\n",
        "The phase space trajectory with tone amplitudes $a_m$ representing one tone AND the (Hann) shape\n",
        "\n",
        "$\t\\alpha(t)=-i\\frac{\\Omega}{2}\\sum_m\\int_{0}^{t}a_m e^{-i\\delta_m x}dx$\n",
        "\n",
        "should be equivalent to\n",
        "\n",
        "$\t\\alpha(t)=-i\\frac{\\Omega}{2}\\int_{0}^{t}P(x) e^{-i\\delta x}dx$\n",
        "\n",
        "with $P(x)$ being the Hann window.\n",
        "\n",
        "\n",
        "\n",
        "\n",
        "\n",
        "\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "O--oKj8yrZH5",
        "colab_type": "text"
      },
      "source": [
        "# Multiloop Gates\n",
        "There should be a correspondence of the Multiloop parameters $K,\\tau\\delta$ and freq. representation of the (higher order) Hann shape:\n",
        "\n",
        "multiloop gate multiplies both by $\\sqrt{K}$\n",
        "\n",
        "K=1: gatetime $\\tau= 500\\mu s$ det $\\delta=2kHz$\n",
        "\n",
        "K=2: gatetime $\\tau= 707.1\\mu s$ det $\\delta=2.818 kHz$\n",
        "\n",
        "K=3: gatetime $\\tau= 866.02\\mu s$ det $\\delta=3.464 kHz$"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "VtTJsAQSsoBd",
        "colab_type": "text"
      },
      "source": [
        "# Question:\n",
        "\n",
        "Can we engineer the STFT Parameters so that the conditions\n",
        "\n",
        "$\t\\alpha(\\tau)=0$  and $Im(\\Phi(\\tau))=-\\frac{\\pi}{8}$\n",
        "\n",
        "are met while maintaining an overall shape that:\n",
        "\n",
        "* Doesn't \"kick\" the ion into a superposition of some unwanted motional states (can I say it like this?)\n",
        "* Is robust against detuning error [[Sussex](https://arxiv.org/pdf/1805.07351.pdf)]\n",
        "* Is robust against premature pulse termination\n",
        "* Maxwell permits the synthesis and amplification of (a hard turn-on will result in ringing and overshoot)\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "X1docm6O88XX",
        "colab_type": "text"
      },
      "source": [
        "#Conclusion:\n",
        "\n",
        "Many of the requirements are related or even equivalent. \n",
        "\n",
        "[[Farhang Haddadfarshi and Florian Mintert 2016 New J. Phys. 18 123007](https://spiral.imperial.ac.uk/bitstream/10044/1/42177/9/Haddadfarshi_2016_New_J._Phys._18_123007.pdf)] have optimized for some condition I don't understand but I think their pulse will also start and end at 0 and be \"smooth\" (essentially some weird window defined by the tone coefficients). \n",
        "\n",
        "One could also optimize for the pulse and its derivatives to be 0 at $t=0$ and $t=\\tau$."
      ]
    }
  ]
}