{
 "cells": [

{
   "cell_type": "markdown",
   "id": "bd8d3700",
   "metadata": {},
   "source": [
    "# <font  color = \"#0093AF\">Propagators and Sequences</font>"
   ]
},
{
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<a href=\"https://githubtocolab.com/alsinmr/SLEEPY_tutorial/blob/main/ColabNotebooks/Chapter1/Ch1_Propagation.ipynb\" target=\"_blank\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\"></a>"
     
            ]
},
{
   "cell_type": "markdown",
   "id": "e16e2faf",
   "metadata": {},
   "source": [
    "Propagators are responsible for moving the density matrix, $\\hat{\\rho}(t)$ forward in time in magnetic resonance simulations. For a constant Liouvillian, the propagator, $\\hat{\\hat{U}}(t,t+\\Delta t)$ is given by\n",
    "\n",
    "$$\n",
    "\\begin{equation}\n",
    "\\hat{\\hat{U}}(t,t+\\Delta t)=\\exp(\\hat{\\hat{L}}(t)\\Delta t)\n",
    "\\end{equation}\n",
    "$$\n",
    "\n",
    "such that\n",
    "\n",
    "$$\n",
    "\\begin{equation}\n",
    "\\hat{\\rho}(t+\\Delta t)=\\hat{\\hat{U}}(t,t+\\Delta t)\\hat{\\rho}(t)\n",
    "\\end{equation}\n",
    "$$\n",
    "\n",
    "Of course, the Liouvillian is often not constant, either due to rotor spinning or a pulse sequence. In this case, the propagator is constructed as a product of piecewise constant propagators.\n",
    "\n",
    "$$\n",
    "\\begin{equation}\n",
    "U(t_0,t_n)=U(t_{n-1},t_{n})*U(t_{n-2},t_{n-1})*...*U(t_1,t_2)*U(t_0,t_1)\n",
    "\\end{equation}\n",
    "$$\n",
    "\n",
    "SLEEPY handles the construction of the propagator for a time-dependent Liouvillian from propagators generated from piecewise-constant Liouvillians internally. Here, we start without pulses, where time-dependence arises only from rotor spinning."
   ]
},
{
   "cell_type": "markdown",
   "id": "3ef4a5a0",
   "metadata": {},
   "source": [
    "## Setup"
   ]
}
,
    {
     "cell_type": "code",
     "execution_count": 0,
     "metadata": {"tags": [
        "remove-cell"
    ]},
     "outputs": [],
     "source": [
      "# SETUP SLEEPY\n",
      "import sys\n",
      "if 'google.colab' in sys.modules:\n",
      "  !pip install sleepy-nmr"
     ]
    },
{
   "cell_type": "code",
   "execution_count": 2,
   "id": "acab298f",
   "metadata": {},
   "outputs": [],
   "source": [
    "import SLEEPY as sl\n",
    "import numpy as np"
   ]
},
{
   "cell_type": "markdown",
   "id": "064b6b04",
   "metadata": {},
   "source": [
    "## Build the system"
   ]
},
{
   "cell_type": "code",
   "execution_count": 3,
   "id": "417021b6",
   "metadata": {},
   "outputs": [],
   "source": [
    "ex=sl.ExpSys(v0H=600,Nucs=['1H','13C'],vr=60000) \n",
    "ex.set_inter('dipole',i0=0,i1=1,delta=44000).set_inter('CSA',i=1,delta=100,eta=1).\\\n",
    "    set_inter('CS',i=0,ppm=10) #Add a dipole, CSA to 13C, and CS to 1H\n",
    "\n",
    "ex1=ex.copy() #Copy the above\n",
    "ex1.set_inter('dipole',i0=0,i1=1,delta=44000,euler=[0,30*np.pi/180,0])\n",
    "\n",
    "L=sl.Liouvillian(ex,ex1,kex=sl.Tools.twoSite_kex(1e-5)) #Here, we produce the exchange matrix with twoSite_kex"
   ]
},
{
   "cell_type": "markdown",
   "id": "bf5afda8",
   "metadata": {},
   "source": [
    "## Propagators without pulses"
   ]
},
{
   "cell_type": "markdown",
   "id": "10a1ec45",
   "metadata": {},
   "source": [
    "We first generate the default propagator from the Liouvillian. These propagators do not have contributions from applied RF-fields."
   ]
},
{
   "cell_type": "code",
   "execution_count": 4,
   "id": "846674a7",
   "metadata": {},
   "outputs": [],
   "source": [
    "U=L.U()"
   ]
},
{
   "cell_type": "markdown",
   "id": "fba7a182",
   "metadata": {},
   "source": [
    "`U` has a few key features: it has a starting time relative to the rotor period (`t0`), a length (`Dt`), and a final time (`tf`), calculated from `t0` and `Dt`. If unspecified when creating the propagator, the length is one rotor period (`L.taur`), since this helps make computational speed faster. Defining `Dt` will override this length. If spinning is not included, then `Dt` must be specified since the rotor period is no longer defined (note that even if spinning is specified in `ex`, if no anisotropic interactions are defined, then the rotor period becomes undefined).\n",
    "\n",
    "Unless we are not spinning, it is important that when multiplying propagators, that the end of the propagator to the right (the first to be applied in time) ends when the propagator to the left starts. That is, to calculate \n",
    "```\n",
    "U1*U0\n",
    "```\n",
    "we require that `U0.tf%U0.taur==U1.t0` (% is the modulo operator, so this is the same point in the rotor period, although it is okay if `U0.tf` is multiple rotor periods after `U1.t0`, as long as they are equal modulo `taur`. Note that `taur` is available in most SLEEPY objects, although defaults to None (has no value) if spinning is not included.\n",
    "\n",
    "`t0` may also be specified when a propagator is created, but if omitted, then `t0` is set to `ex.current_time`. `ex.current_time` is updated whenever a propagator is created, to match the time at the end of that propagator. Then, if propagators are created in the order in which they will be applied, they will always be created with the correct `t0`. Note that `ex.reset_prop_time()` or `L.reset_prop_time()` can be used to change the time at which the next propagator will be created.\n",
    "\n",
    "A density matrix also has a time associated with it, and to multiply\n",
    "```\n",
    "U*rho\n",
    "```\n",
    "we require, similarly, that `U.t0==rho.t%rho.taur`\n",
    "\n",
    "\n",
    "For example, we generate two propagators, and see how their initial times are defined."
   ]
},
{
   "cell_type": "code",
   "execution_count": 5,
   "id": "5a61a900",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "U0: t0=0.000 us, tf=5.556 us, Dt=5.556\n",
      "U1: t0=5.556 us, tf=16.667 us, Dt=11.111\n"
     ]
    }
   ],
   "source": [
    "L.reset_prop_time()\n",
    "U0=L.U(Dt=L.taur/3)\n",
    "U1=L.U(Dt=L.taur*2/3)\n",
    "print(f'U0: t0={U0.t0*1e6:.3f} us, tf={U0.tf*1e6:.3f} us, Dt={U0.Dt*1e6:.3f}')\n",
    "print(f'U1: t0={U1.t0*1e6:.3f} us, tf={U1.tf*1e6:.3f} us, Dt={U1.Dt*1e6:.3f}')"
   ]
},
{
   "cell_type": "markdown",
   "id": "7227355a",
   "metadata": {},
   "source": [
    "Note that while the first propagator has t0=0, the second starts at 5.556 μs, when the previous propagator ended. This happens because when U0 was created, ex.current_time was updated to match the end of U0. Then, we can easily take their product without having problems with mis-matched starting and ending times. When we do so, we get a propagator with length of one rotor period. Note that t0 of the new propagator is not 16.667 μs, which corresponds to the end of U1, but rather 0. This is because t0 is always return as its modulus with respect to the rotor period , where 16.667 μs is the rotor period length."
   ]
},
{
   "cell_type": "code",
   "execution_count": 6,
   "id": "0a047c9c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Propagator with length of 16.667 microseconds (t0=0.000,tf=16.667)\n",
       "Constructed from the following Liouvillian:\n",
       "\tLiouvillian under the following conditions:\n",
       "\t\t2-spin system (1H,13C)\n",
       "\t\tB0 = 14.092 T (600.000 MHz 1H frequency)\n",
       "\t\trotor angle = 54.736 degrees\n",
       "\t\trotor frequency = 60.0 kHz\n",
       "\t\tTemperature = 298 K\n",
       "\t\tPowder Average: JCP59 with 99 angles\n",
       "\t\n",
       "\tThe individual Hamiltonians have the following interactions\n",
       "\t\tHamiltonian #0\n",
       "\t\t\tdipole between spins 0,1 with arguments:\n",
       "\t\t\t\t(delta=44000.00)\n",
       "\t\t\tCSA on spin 1 with arguments: (delta=100.00,eta=1.00)\n",
       "\t\t\tCS on spin 0 with arguments: (ppm=10.00)\n",
       "\t\t\n",
       "\t\tHamiltonian #1\n",
       "\t\t\tCSA on spin 1 with arguments: (delta=100.00,eta=1.00)\n",
       "\t\t\tCS on spin 0 with arguments: (ppm=10.00)\n",
       "\t\t\tdipole between spins 0,1 with arguments:\n",
       "\t\t\t\t(delta=44000.00,euler=[0.00,30.00,0.00])\n",
       "\t\t\n",
       "\t\t\n",
       "\tHamiltonians are coupled by exchange matrix:\n",
       "\t\tarray([[-50000.,  50000.],\n",
       "\t\t       [ 50000., -50000.]])\n",
       "\t\n",
       "<SLEEPY.SLEEPY.Propagator.Propagator object at 0x7fe2e85758d0>"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "U1*U0"
   ]
},
{
   "cell_type": "markdown",
   "id": "76ca7021",
   "metadata": {},
   "source": [
    "Note that if we multiply `U0` by itself, we get a warning, since the end of `U0` is not the same time as the beginning of `U0` (relative to the rotor period)"
   ]
},
{
   "cell_type": "code",
   "execution_count": 7,
   "id": "3c075ac1",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/albertsmith/Documents/GitHub/SLEEPY/SLEEPY/Propagator.py:260: UserWarning: \n",
      "First propagator ends at 5.555555555555556e-06 but second propagator starts at 0.0\n",
      "  warnings.warn(f'\\nFirst propagator ends at {U.tf%self.taur} but second propagator starts at {self.t0%U.taur}')\n"
     ]
    }
   ],
   "source": [
    "_=U0*U0"
   ]
},
{
   "cell_type": "markdown",
   "id": "f597cf5a",
   "metadata": {},
   "source": [
    "As with the Hamiltonian and Liouvillian, propagators can also be plotted:"
   ]
},
{
   "cell_type": "code",
   "execution_count": 8,
   "id": "fe362e51",
   "metadata": {},
   "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": [
    "_=U0.plot(mode='abs')"
   ]
},
{
   "cell_type": "markdown",
   "id": "f060c5bb",
   "metadata": {},
   "source": [
    "### Special propagators"
   ]
},
{
   "cell_type": "markdown",
   "id": "7d5604e6",
   "metadata": {},
   "source": [
    "The Liouvillian can also produce propagators for $\\delta$-pulses (i.e. pulses with zero length) and an identity propagator. Both types have zero length, and they can be multiplied with propagators ending at any time (they do acquire an initial time, but this is not checked when multiplying these types of propagators). The identity propagator does not have any effect on other propagators or density matrices, but is occasionally useful as a kind of propagator pre-allocation."
   ]
},
{
   "cell_type": "code",
   "execution_count": 9,
   "id": "ddee0cee",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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f+c1S1nBQHlVR2z7S9ods35LiKlURtb3Y9nzb8+dssXnRbIIgqAgb1q1XrhBUw8g9gbyqoraP7rStWkU0CILxZ4idxYIKGLkRGFVVtIiKaCiIFidE54JxwsRwUNOUNRzUUhXdQ9LubfFzaVMVlXSppIskHZPiuqaXtJukS3oJyNm+3/YJJdU9CILGyCcZEc7j6iirEbgGOJxMFfRSSfdIuhPYE/h0StNSFT0ReEeK65o+XvJBMDvY+UJQDaXMDiqqKhoqokEw29iwfn0sFmuS0u5+naqioSAaBNNDDAc1S53aQblVRQepiKaZSOGtDIIpIIZ6mqW2RmAYVdFQEW2GEJ0LmiBmBzVLrSqiQRAE7ZjQBWqa3GPzOQTikLSXpH/sCNt15FPYPgTkgmDKcPgEmiZXTyCnQBy270rne+XTSzAul30QBFNI+AQaJe9wUB6BuDyMaj8UISAXBONPDAc1S97hoDwCcXkY1X4oQkAuCMafWCzWLAMbgXaBONtryCShFwKPkQm+Lchb2Kj2HfXqKy0RBMH409IOik1lmmPgcNCoAnFd8hvJvi2f+4ETohGoniKic4PsggAAg0MmulHyDgcVEojrpKhgXBAEU4xzhqAS8jqGrwFOBR4AvippS2AtsJLkeOUFgbjrJC0FvtElny262ScHcXzVB8HMEUM9TZOrESgqENctH0oSjOsjLREEwSQRX/mNknvFsO29ByRpCcTdTonCdH3q01VaIgiCCSJ8Ao1T5sv6KuBdkhYxQCCukyFUQZcATxeuYRAEY4hyhqAKStMO6iYQJ+njwM7AHrYP7mOb66s+BOSCYAqJ4aBGqUxATtJLgbemwzurKidoltizOBiZaAQapfBwUA5BuflkU0uPAF7XZtdTQC5HmSEgFwTThAErXwgqoVBPIKeg3JuAm9PvR5NdTwG5NPuoc4bP8bYfLVLHIAgmA69vugazTdHhoDyCcvOA/cgagAtTXM/0VSiIhoBcEEwA8ZXfKEWHg/IIytn2kbY/ZPuWFBcCckEQvAg5XwiqYehGIK+gnO2jO23LFJALgmAKyCsZEY1AZQw9HDSqoFwZAnKSdgPOAF5h+8hhryGoh9izOBhMOH2bpqhPoCUodxew3vZ9KX4ubYJywPnAs8By29+QtEe39MO+1ENBNAimiHAMF0bS5cBz6fAR238zbB5FG4GignIhIBcEwYuJoZ5R+IHtr8DzempDU6gRKCooV6aAXBAEU0BrnUBQlMMkrQeut31vkQwKLxazvbfttX2StATlRiqnG0NoDQVBMOaUNTtI0sGSfi5ppaTTu5yXpHPS+Tsl7ZPXttB1ScdJui+F49rid5V0a4pfKmmTfvkM4H3AL8h02y4ukkGVap+5BeWGFZCz/YTtk2y/JiSkg2DCKWF2kKQ5wHnA24E9gaMl7dmR7O3A7iksBBYNYduv7A2UDCRtDZxJtlZqX+BMSa3FSmcBZ9veHXgKOCFvWZ3Y/qXtZba/aPsDRfKoTDuom6Bcn7QhIBcEM0pJawD2BVamSSMth+lhwD1taQ4DLrNt4BZJW0raHtill62k15A1EK8EfgucaPtnOerzNuBG20+mPG8km0xzOXAg8N6U7lLg06QGaRQkvR94wPbyYexKawQk3QQcBJxGdoHryPz+H7R9q6TNgM8ALye7mdfZvknSGZ3pyXwJ3dIuBxbYXlVWvYP6iT2LgxeR3yewraQVbceLbS9Ov3cAHmo7t5rsK7ydbml2GGC7GDjJ9n1J5+x8spf4IHqVtQ3ZaMbajvgy+DbwhmGNSmkEcmoJnQx8B3gQuMj2qX20hDZIW0Y9gyAYM4ZbCPa47fk9znVrSTpz7pWma7ykLcgmsnxTej7JpvD8V/dfp7jXAsskPUv2JX7EsGV1iRuapLN2/bB2ZfUE8mgJ7QN8hUxV9IoU1zV9cth0pg2CYBopZzhoNbBT2/GOwMM502zSI34jsq/2uZ2F2f468HXIfAJsOEKxGti/I8/lwOPAlpI2Tr2BbvXMTRnrBMpyDOfRErqZTCV0X+AtKa5X+m5ph0bSQkkrJK1Yt+Y3RbMJgqBCtD5fGMBtwO5p5s0mwFHAtR1prgWOTbOE3gg8Y/uRXra2fwU8IOnd8PzsokHb7La4HjhI0lbJIXwQ2TROk73fWotijyNbd1WUH9h+n+33kTmch2bknkC7llA6nkf24j6ATBvodNtLbG8wfcn2mh7pC0116pL/YrIxPTbdeadYkhIE40gJ/zJtr5X0YbKX7xzga7bvlnRSOn8BsIzsg3Ulma/x/f1sU9bHAIskfQp4CXA5cEeO+jwp6XNkDQzAZ1tOYjK/6eWSPg/8BLhkhEsfeZ3AyI1AU1pCkg4HDgW2A86zfcNoVxIEQd2UqRBqexnZi7497oK23wZOyWub4h8Aem6Nm9Ls3yP+a8DXusTfTzbKUQbvA/YG3inptUWmiZblE6hdS8j21cDVqav1d2RDS0EQTBqxYrgwtn8J/FLSGbbfXCSPshqBJrWEPkU2jzeYAmLP4hkkBmrLYPuihqU0Ak1oCSmbs/Ul4Du2f1wkjyAImieH0zcAJH2V7B16F/BT278uI9/SFovZHuQ1b2kJ3U45s5I+AvwF8Io0FtZ7lVEQBONJ7Bo2DHcBf0zmrH69pF/xQqPwX4pmWplsRBeuAs6VdCg5tISAL5C0hNr0gZYATwPYPgc4p7LaBkFQD9EI5KJtdTQAknYkaxT2osAisRa1NQJlaAmFdlAQTCHRCBTC9mqyEZYNZjUNQy2NQPIXPGP7wTrKC4JgcojhoGapqycwD1hFpgU0kF5ic5XVLpgIYs/iKSUagUYp7KCVdJOkjSWdIeluZZs03J6U9lpp9kz7BBwHfEzSBZJe1ZFPN/uWgNyXgbmtBqCbbncQBBNMzg1lordQHYV6AjlVQ7F9D3CSpAXAqk6d6z4qoiEgFwSzQrzgG6XocFAe1dA89FIRbQnIQaa1PfKGC0EQjCnRCDRK0UbgELIv+LXA30q6F/gusNT2P3cm7jOr54Zu9mUJyElaSFqxPGerrQakDoKgbkQM9TTN0D6BdtVQ22vInL4LgcfIVEAX5M1rFHtJh0u6SNI1kg7qkf9i2/Ntz5+zxeZ5qxUEQV24NCnpoCBD9wRGVQ3tkl8h+xCQC4IpIXoCjVJ0OKiQamhnJqOoiLYRAnIzTojOTTjRCDRK0Smi1wCHk6mAXirpHkl3AnsCn05pWqqhJwLv6JFPV3vb99s+oV8F0i4/ZxECckEw0cQU0WYp1BMoqhraLR8KqogSAnJBMPkYiPH+Rim8YrgB1dDO8kNALgimgPjKb5bSX85tXAW8S9IiBqiGdiJpm7TS+A2SPtl2aglJRTQIginBOUNQCZVpBw2jGtrFNlREg2BGiJ5As9S5n0AQ1E6Izk0A0Qg0SmUCcpI2k/RlSRdKOlvSgSm+p+BcjjJDQC4Ipoi8M4Oit1AdVQrItZRAHwQusn1qH8G4Vr578YJmUIvjbT9apJ5BEEwA8YJvlCoF5LopgfYVnLN9F1kjEQTBjBBf+c1SdDjoEODbZFINO0m6V9L5kv6sLU1LCXRf4C0prl/60pG0UNIKSSvWrflNlUUFQVCUmB3UKEP3BNoF5NLxPLKX/AFkAnCn217STQnU9ppe6Ue4hp6kjZkXA2y6807xZxQE40j8y2yU2gXkRhWcayHpcOBQYDvgPNshIBcEk4ZDIbRpahWQK0kwDggV0WA0iojODbILihE+gWapW0CusGBcH0JFNAgmmfAJNEqtAnIjCsa9CEkCvkSoiAbBRBM9gWYpvFjM9t621/ZJ0hKQG6mcPrRURI+UFH30IJhE8vYCoqGojCplI64CzpV0KAME5CRtA3yBJBhnu3PBWIslJAG5UBENgslHhGO4aSppBNJQ0TO2cwnI9RKM65JuyYhVC4Jg3Iiv/EapSkp6HrBbRXkHQTBFyM4VgmrI3ROQdBNwEHAa8F4yZ+964IO2b01p9gT+CtgD+LWko4Azbf9HWz5n9LHvqx0kaTmwwPaqoa80CHISexbXSIz3N06uRiCnYBy27wFOkrQAWGV7eUc+fQXkQjsoCGaP8Ak0S96eQB7BuDyMah8EwZQRU0SbJa9PII9g3PMk7aDlXU6FgFwQBC8mpog2ysBGoF0wzvYaMqfvQuAxMgG4BXkLG9W+o16HS7pI0jWSDupR3mLb823Pn7PF5kWKCYKgSmJTmcYZOBw0qmBcl/xKEZAL7aAgmBLiBd8oeX0ChQTjOjMpU0CujdAOCmoj9iwul2yxWLQCTZLXJ1BUMK6T0gTklHEWoR0UBBNNDAc1S66eQFHBuG75UJKAHC9oB71C0mtt9/4UC4JgPAmnb+PkXixme+8BSVqCcbdT3Urk9vqEdlAQTAGxTqBZynxZXwW8S9IiBgjGdSJpG0kXkATk+iRdQhKQC4JgSih5iqik/SU9I+n2FP62R7pdJd0q6T5JSyVtkuIl6RxJKyXdKWmfYS9J0ieT/c8lva0t/guSHpK0Ztg8q6I0ATnbvwFyCcZ1sQ0BuSCYRVyZY/hfbA9SHzgLONv25ekj9ARgEfB2YPcU9ktx++UtOMnnHAW8DvivwHcl/WGaGXkdcC5wX58saqXyYZsgCIJ+NOEYTptSHQhcmaIuJZv8AplCwmXOuAXYUtL2ye4vJf0w9TAulDSnS/aHAZfb/r3tB4CVwL4Atm+x/Ui5VzMahXsCeQTlcuazgaAcmYP5M8DLgd8C19m+KQTkgnEl9iwegfwv+G0lrWg7Xmx7cY+0b5J0B/Aw8Anbd3ec3wZ4um1jrNXADun3DsBDbWlXAztI2hJ4D/Bm289JOh84BrisI+8dgFs67QdeXUMUagTyCsrlUAXtJSh3MvAd4EHgItunFqlnEATjjRjqK/9x2/NzpPsx8GrbayQdQqZ7tnuXojvxgHN/TqZ4cFvWkeBlwKNd0vbLe+wo2hPIJSiXQxW0q31yxHwFOAK4omAdgyAYd+wsjIikU4AT0+Ehth/OsveypFO2bcf76XGyYZ6NU29gR7JeA2Rf7ju1pW2dE3Cp7RdNXpF0BHBmOvxAH/uxpKhPYChBuT70sr+ZrAexL/CWgnUMAbkgmAC0Pl/oh+3zbM+1PRdYn8b8kbQv2XvuiY70JnvPtBQKjiNbFAtwLXBsmiX0RrJdEh8Bvke2p/l2Ke+tJb3a9rdaZdtekeyPkrSppF3JeiE/HPE2VcbQPYF2Qbl0PI/sRX0AmSDc6Xln8aTuWjf7i4etV4/8FwOLATbdeaex7Y4FwSxTwWrgI4GTJa0FfgcclV76SFoGfCD1FE4DLpf0eeAnwCXJfhnZh+5KMp/k+yHbL0XSp4AbJG0EPAecQpLOaWH7bklXAPcAa4FT0swgJP0vMh/oZpJWAxfb/nTpd2AIhm4ExkVQTtLhwKHAdsB5tkNALggmDQMlTxG1fS7ZNMxu5w5p+30/adZORxqTvdy72S8FluaowxeAL3SJ/xvgbwbZ10nR4aCWoNwektodLnNpE5STdKkyuedjumXSy17SbpIukXRlNzvIVESTTtECMo99EASTSOwn0ChFHcPXAKcCDwBfTVOn1pJ1nxamNC1BueskLQU2UBUlE5TbwD45cE7o1wi0ESqiwVgTexb3J1REm6VQIzAOgnLJ8fMlQkU0CCaaUAhtlsIrhm3v3bbQohstQbmRyulDS0X0SEmz89kUBNNE3qGgaCgqozTtoC5cBZwr6VAKCMqROVXeIOmTtlsLzpaQBORCRTQIJp9ssVi84ZukskagCkG5EJALgikkpKQbpcqeQBAEQX+qUxENclJrIyDp48DOwB62D66z7CAYR2LP4nJkI4Li1NYISHop8NZ0eGdd5QZBMN7E7KBmKW3WjqSbJG0s6QxJd6cdeW6X1NqMYT7ZIrMjyDZbaNltkF7SZpK+nPS6z5Z0YEq7XNIuZdU5CIIxoCUiNygElVBKTyCntPSbyASbIMmvhpR0EMw4jj2Gm6as4aA80tLzyLZoexS4MMVVKiUtaSFpBfOcrbYqmk0QBFUSjuFGKWs4KI+0tG0faftDacs2+qQvRUra9mLb823Pn7PF5kWzCYKgQmTnCkE1jNwTyCstbfvoTtuqpaSDIJgA4gXfKCM3AqNKSxeRkg4Z6WDamRnROROLxRqmLJ9AS1r6LmC97ftS/FzapKWB84FngeW2vyFpj27pJe0GnAG8wvaRdGD7auBqSVsBf0c2rBQEwYQhYqinacpqBIpKS48qJR0y0kEw6ayPrkCTlNIIFJWWLiolHTLSQTAlxHBQ45S2Ytj23gOStKSlb2f0WUktGelXSHqt7d5r7IMgGGtiOKhZ6tQOyi0tPUhKOmSkg2CKiEagUWprBIaRlg4p6SDozVSJztnhE2iYkJIOgqBZog1olNxj8zkE4pC0l6R/7AjbdeRT2D4E5IJg+ogVw82SqyeQUyAO23el873y6SUYl8s+CIIpJF7wjZJ3OCiPQFweRrUfihCQC4Ixx4SAXMPkHQ7KIxCXh1HthyIE5IJg3EmO4TwhqISBjUC7QJztNWSS0AuBx8gE3xbkLWxU+456HS7pIknXSDqoSB5BEIwBsalMowwcDhpVIK5LfiPZt+VzNaEfFATPU0R0bpBd5cRwUOPk9QkUEojrzKSoYNwAQj8oCCYWg2Oop0ny+gSuAQ4nE3y7VNI9ku4E9gQ+ndK0BOJOBN7RI5+u9rbvt33CMBVXxlmEflAQTDYxHNQouXoCRQXiuuVDAcG4HoR+UBBMOgbWRU+gSXKvGK5ZIC5PfUI/KAimgfjKb5QyX9ZXAe+StIgBAnGdSNpG0gUkwbg+SZcATxeuYRAEY0bOoaBoKCqjTCnpDQTiJH0c2BnYw/bBfWy7CsZ1SbdkxGoGQTBOmFgD0DCVCchJeinw1nR4Z1XlBEEwmLHeszi+8hul8HBQDkG5+WRTS48AXtdm11NALkeZISAXBFOFM8dwnhBUQqGeQE5BuTcBN6ffjya7ngJyafbRF3kxx9t+tEgdgyCYAAyOdQKNUnQ4KI+g3DxgP7IG4MIU1zN9FQqiISAXBBNArBhulKLDQXkE5Wz7SNsfsn1LigsBuSAIXkzMDmqUoRuBvIJyto/utC1TQC4IginAhnXr8oWgEoYeDhpVUK4MATlJhwOHAtsB59kO8bggKEiRPYvnfLS88h1TRBul6HBQS1BuD0m7t8XPpU1QTtKlSe75mBTXNb2k3SRdIunKPIXbvjppFC0A3lPwGoIgaJxYLNY0RR3D1wCnAg8AX5W0JbAWWElyxPKCoNx1kpYC3yATkNsgfXIQn5C3EWgjFESDYJIJKenGKdQIFBWUK0tATpKALxEKokEw+cQU0UYpvFjM9t621/ZJ0hKUG6mcHrQURI+U1OCOGEEQjIJtvG5drpAXSX8k6QeSfi/pE33S7SrpVkn3SVoqqbVmSZLOkbQyLWrdZ9TrlLRpKmNlKnOXtnPHpTrcJ+m4UcsalirVPnMLyg0rIGf7HNvzbJ8UEtJBMNl4vXOFIXgS+CuyHQf7cRZwtu3dgaeA1p4mbwd2T2EhsChvwZJ2kbS8y6kTgKdsvxY4O5WNpK2BM8nWVO0LnJl2S6yNyhoB27+x/X7bJ3fbZawj7RPphf4a252rhtvTLbH9dOmVDYKgObw+X8ibnf2o7duA53qlSUPKBwItP+SlZBtnQbYY9jJn3AJsKWn7ZPeXkn6YJG8ulDQnZ7UOS2WQyvzzVIe3ATfaftL2U8CNQE+xzSqoTEBu3Hj2odWPr/roJ/49HW4LPN4vfQ+K2E2bTZ1lxTXVa/MiuwHTQF9dIO8N+DVPXf9dX7ltzuQvlbSi7Xix7cUFi96GbFShNaS9Gtgh/d4BeKgt7WpghzSh5T3Am20/J+l84BjgshzlPZ+n7bWSnkl16FpWoSsqyMw0ArZf2fotaYXt+cPmUcRu2mzqLCuuqV6bUeyK0k9ivmLUJc4Dzv052WLX27KPeF7GC7po3wJ2JdNC21nS7cnuK7a/3ifPfvWohZlpBIIgmF4knQKcmA4Psf3wAJPHyYZ5Nk69gR2Bls1qYKe2tK1zAi61vYHf0vYRqR67AEts79+RpJXnakkbA68g812sBtrT7ki2mLY2Kt8GMgiCoGpsn2d7bgqDGgBsm0zl+MgUdRzZ+ieAa4Fj0yyhNwLP2H4E+B7ZjMTtIHPqSso7LHZtKoNU5k2pDtcDB0naKjmED0pxtTGrPYGi44hF7KbNps6y4prqtRnFbmyQ9AfACuDlwHpJHwX2tP0rScuAD6SG4jTgckmfB34CXJKyWEYmkrkS+C1px0Tb90j6FHCDpI3IHM+nkFQSBnAJ8PeSVpL1AI5KeT4p6XPAbSndZ20/OdINGBI5lmMHQRDMLDEcFARBMMNEIxAEQTDDRCMQBEEww0QjEARBMMNEIxAEQTDDzMwUUUnHkm1C022FXguTzZg6YNxtithN2/VMQll1PKM6ryeYPmZmiqiklwNb5klr+8FxtyliN23XMwll1fGM6qpbnnTB5DEzjUA7SR98je170/HmZCv1HkmqgRNlU8Ru2q5nEsqq4xnVeT3BlGB75gLw/4DXpt8iWy14TYo/Y9JsithN2/VMQll1PKO6712EyQ+NV6CRi4aftv1+K/Cz9Hvj9nOTYlPEbtquZxLKquMZ1X3vIkx+mNXZQe3bYh5IJgyFMzXBXvvYjbNNEbtpu55JKKuOZ1RX3YIpYWZmB3WwMok23U62fdxRQGurt15/8ONsU8Ru2q5nEsqq4xnVeT3BNNB0V6SJAGwNnEM25nlsW/z2wJ8MafMHBWxKLaeI3TjfgzG6d3Xdi9KeUd3PKcLkh5mZHSRpM+AI0sYOwFW2f5vOHQ8cQOYMW2T7d4Nsmi6niN2Aun2DbPHgj4AfACts/76Ce9C1nHG6d3X9PdR178ouK5guZskncC7wFJlW+FPA+W3n/ohs0cyPgc/ktGm6nCJ2PdPbPgaYA9wE7AUslfSxgnUrUs6g66nz3tX19zBU+hHuXdllBVPELDUCv7O9zPadtr8NtH/VmOyL6CGyjSjy2DyPpP/WQDm57CS9foj0jwI/sX1BquvOQ9Rt1HIG2dV57+r6e2ilr/relV1WMEXMkmP4XyX9A9kf8xyy7d5a7ADcT7ar0h05bdr5K+DkmsvJa/dF4H/kTH8ZcLGkVwE/Bf6YbDekPHUbtZxBdnXeu7r+HlpUfe/KLiuYImbGJ5AXSfvbXp4z7bHp53Fk/2B+ZPunY1bOtbbfkSftKNRVTiqrlntXV1k137vaygomg1nqCTzPgCXyy4fIalX6/6/T76ebKGcAXcXBKpAJKLWcOu/dGDynup5R3WUFk0DRaUWTHCh5iTxwfpPlDLC5tqa6lVpOnfeu6edU1zOqu6wIkxEar0AjF13yEnnglU2WM8Bm25rqVmo5dd67pp9TXc+o7rIiTEaYpdlB7ZS6RN72Yw2X08/m8ZrqVnY5dd67Rp9TXc+o7rKCyWAmfQIUWCJfcMy0rnKK2I3tPai5frWVVcczKlq3omUFU0DTXZEmAsWW1ReR562lnCJ243wPGrh3dd2Lyp9R3c8pwuSHxiswKYGaxkyLllNH/eq6B5NQvyJljfvfUITZDDPjE0jd2vbjXdt+Hy/p7yV9TNLLemSRa8y0Xzk5qWwMfdzvQV31G6WORcoaxqaEe5C7biXcg2AKmJlGAPhCx/EZbb97acW0s1LS5yS9i2zMdCl0HTPtVw7J5vWdcQXKKWI37vegrvqNUsciZQ1jM+o9GKZuo96DYAqYJcfwX0j6JtkS+U5MphXzE16sFdPOQuDTwLHA6ba/n+I3AT6Ys5wW7Uv3i5ZTxG7c70Fd9RuljkXKGsZm1HswTN1GvQfBNND0eFRdATitz7n/Q/aP43vA/66qnLY0XRfszPo9qKt+4/yc4h5EqDs0XoFxC8D+bb+37ji3a8fx8cDfAx8DXjZEGdd1HPctp08+Pe361Q3YckC+7xy1bsPeg3F7RmU9pzqe0bg+pwiTEWbJJwCApC0HJLm97fegMdO8Y7SddHa/B47NQtfx2aLjx1dLukHSRyTtnPL+Q0mfkHQz8Jcl1G0QPYcgxuQZQTnPqY5nVLRueeg3VBRMAbPkE2hxtaRngeuAa2w/KOkPgXcAh5JtrvHOlHbQmGneMdpOju84zjM2CxuOzxYaP7a9vzJ54HcA56dZIfcC3yL7wnyqhLoNovMetDMOz6hbHYvcizqeUdG65aHfcwqmgJlrBIb8x3Wx7bP6ZNdLd35QHTqX7g8q53nTIez61s32fwAXpdCPonXrn7i3fMFYPKMedSxyL+p4RkXrNjhxn+cUTAexn0BJaAjd+RHKuM720DM12usmaWvbT7ad29X2A1XUTdKWtp/uY9P3fNnU8YxSOUM/p45n9BHbX207d5jta6qq27g9p6BeZsonMMIipi0Hne98ueSxyVN2B11b7JLH0IvSrW6DxrW/1mkg6SMdx4flKbzIM8prl6f8Dja4F0M+o2Mlfant+NACdehFKc8pmB5mqicgaZHtk9uOL7b9gRx2y4G+Y9S23zmqTY56bNutez5MWZLuI3vhPP/gbf/PYeoxZN1awzqHAe3DOtd1GddG0m3A92yfno4X216Yo/zlFLjfdT2nIZ/R+cCVwCHA54Gv2z5imDoMU7cUP9RzCqaHWWsECr8Ai/wjqfMfVt6yJJ2Wc+y4EUZ5ARa933U9pyGe0ULbiyXNBU4BrrB9Y1n1CIJ2Zq0RGOsX4DRR1PcQL8B6qdL/EEwGM+UTKNoADOtLqNL3UFZZw1KgboV8D7YXp//fbvvEvA1AEV9Clf6HMsopQoHnVKX/IZgAZqoRGIFhX2hFna9FHHRVOXpHrdtfSPqmpCskXcHwc/SHpcjLrOgLcNh7UeeLdti63QbcIOnvUgPxygrrFowhM7dOoCB5F+IUTQ8UWiBUuKxhKVC3vPPWy+L5lxmZLyHPy6yITZF7UaicIhSo2+22b5L0JPBl4Pyq6haMJzPlEyjKsL6EOn0P4efIKOJLqMv/EH6OYJyJRmCMKOpMrYNxrlvdjLMzdZzrFown4RMYL+oa3y/CONetbsbZmTrOdQvGkPAJjBe1jO8XZJzrVje1jfEXYJzrFowh0QiMF3U7U4dhnOtWN+PsTB3nugVjSPgEgiAIZpjwCQRBEMww0QgEQRDMMNEIBEEQzDDRCARBEMww0QgEQRDMMP8fpa3X7/wZfaoAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "Ueye=L.Ueye()\n",
    "_=Ueye.plot(mode='re')"
   ]
},
{
   "cell_type": "markdown",
   "id": "785f3282",
   "metadata": {},
   "source": [
    "A $\\delta$-pulse propagator needs to be provided with a channel ('13C','1H', etc.) or a spin-index (0,1,2). The spin-index will apply the pulse just to one spin, even if there are other spins of the same nucleus type. While this is unphysical, it can be useful for creating selective pulses without requiring a full simulation of a complex pulse shape.\n",
    "\n",
    "The default flip angle for the $\\delta$-pulse is a $\\pi$-pulse with an 'x' phase, but by adjusting phi (flip angle) and phase, any $\\delta$-pulse may be obtained."
   ]
},
{
   "cell_type": "code",
   "execution_count": 10,
   "id": "bf0cf7dc",
   "metadata": {},
   "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": [
    "Ud=L.Udelta('13C',phi=np.pi/2,phase=np.pi/2)  #pi/2 y-pulse on 13C\n",
    "_=Ud.plot(mode='re')"
   ]
},
{
   "cell_type": "markdown",
   "id": "a6e7333c",
   "metadata": {},
   "source": [
    "## Sequences"
   ]
},
{
   "cell_type": "markdown",
   "id": "9b300033",
   "metadata": {},
   "source": [
    "Of course, we would also like to be able to apply pulses within propagators. This is achieved with the sequence object, which is also generated from the Liouvillian. \n",
    "\n",
    "Sequences can be used to create propagators, although in some cases, propagators are only calculated internally within rho (next chapter) when used with a sequence.\n",
    "\n",
    "We initialize a sequence by calling `L.Sequence()`. Usually, when we add channels to the sequence, we define a time-axis while doing so, which will then define the length of the sequence. However, it is possible to use the sequence to just define the amplitude (and optionally phase) of a continuously applied field (or even just a delay). In this case, under spinning, the sequence length then becomes one rotor period by default. If no spinning is used, then this case requires initializing the sequency with a length (`Dt=1e-3` would, for example, make a 1 ms sequence).\n",
    "\n",
    "Once a sequence is defined, we add channels to it. Channels may be added the usual way, by specifying the nucleus, but they may also be added to a specific spin by index."
   ]
},
{
   "cell_type": "code",
   "execution_count": 11,
   "id": "1358cbac",
   "metadata": {},
   "outputs": [],
   "source": [
    "seq=L.Sequence()"
   ]
},
{
   "cell_type": "markdown",
   "id": "56d7d835",
   "metadata": {},
   "source": [
    "We start with a sequence that has no time axis, just a constant field applied to $^{13}$C, applied in the *y*-direction. We use `seq.plot()` to visualize the result."
   ]
},
{
   "cell_type": "code",
   "execution_count": 12,
   "id": "13dfd4e9",
   "metadata": {},
   "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": [
    "seq.add_channel('13C',v1=25000,phase=np.pi/2)\n",
    "_=seq.plot()"
   ]
},
{
   "cell_type": "markdown",
   "id": "5f7c8c01",
   "metadata": {},
   "source": [
    "Alternatively, we may use pulses. We may overwrite the existing channels, or just create a new sequence. Here we overwrite the $^{13}$C channel by calling it without any arguments (the default for v1 is 0). Note that the `seq.add_channel(...)` function returns the sequence, so we can string together multiple calls to `seq`."
   ]
},
{
   "cell_type": "code",
   "execution_count": 13,
   "id": "16c79c59",
   "metadata": {},
   "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": [
    "v1=150000\n",
    "tpi=1/v1/2\n",
    "t=[0,L.taur/2-tpi,L.taur/2,L.taur-tpi,L.taur]\n",
    "_=seq.add_channel('13C').add_channel('1H',t=t,v1=[0,v1,0,v1]).plot()"
   ]
},
{
   "cell_type": "markdown",
   "id": "69c87563",
   "metadata": {},
   "source": [
    "This setup lets us easily create shaped pulses by just providing a time axis and a time-dependent amplitude. Note that for complex shapes, it is computationally advantageous to synchronize the time axis for the sequence with `n_gamma` for the rotor period, i.e. if `n_gamma` is 100, and we define a sequence over one rotor period, we would take 100 uniform time steps (i.e. `t=np.linspace(0,L.taur,ex.n_gamma)`). This is because a new propagator needs to be calculated for every new $\\gamma$-angle, but also for every new field strength, but these can be synchronized."
   ]
},
{
   "cell_type": "markdown",
   "id": "1b523df9",
   "metadata": {},
   "source": [
    "### Generating propagators from sequences\n",
    "A propagator is generated from a sequence by calling `seq.U()`. By default, the resulting propagator will have `t0` at `ex.current_time`, and `Dt` will be the length of the sequence (`seq.Dt`). However, both `t0` and `Dt` may be user defined when calling `seq.U`. Additionally, `t0_seq` may also be defined. This results in a propagator that starts partway through the sequence. Note that for a cyclic sequence (cyclic is the default sequence type, and can be changed by calling `L.Sequence(cyclic=False)`), `t0_seq` defaults to the end of the last call to the sequence, but defaults to 0 for `cyclic=False`. Then, we can interupt a sequence, for example, for detection, without having to actively re-synchronize it. \n",
    "\n",
    "We can observe this behavior below. We start with a 5 μs sequence, such that no $^1$H pulses are applied (the first pulse above starts at 5 μs). We obtain a propagator without off-diagonal terms in the imaginary part except due to exchange, since the system only has a heteronuclear dipole coupling."
   ]
},
{
   "cell_type": "code",
   "execution_count": 14,
   "id": "5fcbca2b",
   "metadata": {},
   "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": [
    "U0=seq.U(Dt=5e-6)\n",
    "_=U0.plot()"
   ]
},
{
   "cell_type": "markdown",
   "id": "ba8ebd86",
   "metadata": {},
   "source": [
    "Now, if we generate another 5 μs sequence, we obtain a very different propagator, because both `t0` and `t0_seq` have been set forward by 5 μs. Now the propagator contains a $\\pi$-pulse on $^1$H. We can check `t0_seq` before and after generating `U1` to see how this works."
   ]
},
{
   "cell_type": "code",
   "execution_count": 15,
   "id": "082d9e3f",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Before: 5.00 microseconds\n",
      "After: 10.00 microseconds\n"
     ]
    },
    {
     "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": [
    "print(f'Before: {seq.t0_seq*1e6:.2f} microseconds')\n",
    "U1=seq.U(Dt=5e-6)\n",
    "print(f'After: {seq.t0_seq*1e6:.2f} microseconds')\n",
    "_=U1.plot()"
   ]
},
{
   "cell_type": "markdown",
   "id": "c0025edb",
   "metadata": {},
   "source": [
    "We complete the sequence, and do one more check of `t0_seq`, which should now be set back to 0. We also check that multiplying all the propagators together works, since they were generated one after another to cover a rotor cycle."
   ]
},
{
   "cell_type": "code",
   "execution_count": 16,
   "id": "5b9cf096",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.0\n"
     ]
    }
   ],
   "source": [
    "U2=seq.U(Dt=seq.Dt-U0.Dt-U1.Dt)\n",
    "print(seq.t0_seq)"
   ]
},
{
   "cell_type": "code",
   "execution_count": 17,
   "id": "62a5423c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Propagator with length of 16.667 microseconds (t0=0.000,tf=16.667)\n",
       "Constructed from the following Liouvillian:\n",
       "\tLiouvillian under the following conditions:\n",
       "\t\t2-spin system (1H,13C)\n",
       "\t\tB0 = 14.092 T (600.000 MHz 1H frequency)\n",
       "\t\trotor angle = 54.736 degrees\n",
       "\t\trotor frequency = 60.0 kHz\n",
       "\t\tTemperature = 298 K\n",
       "\t\tPowder Average: JCP59 with 99 angles\n",
       "\t\n",
       "\tThe individual Hamiltonians have the following interactions\n",
       "\t\tHamiltonian #0\n",
       "\t\t\tdipole between spins 0,1 with arguments:\n",
       "\t\t\t\t(delta=44000.00)\n",
       "\t\t\tCSA on spin 1 with arguments: (delta=100.00,eta=1.00)\n",
       "\t\t\tCS on spin 0 with arguments: (ppm=10.00)\n",
       "\t\t\n",
       "\t\tHamiltonian #1\n",
       "\t\t\tCSA on spin 1 with arguments: (delta=100.00,eta=1.00)\n",
       "\t\t\tCS on spin 0 with arguments: (ppm=10.00)\n",
       "\t\t\tdipole between spins 0,1 with arguments:\n",
       "\t\t\t\t(delta=44000.00,euler=[0.00,30.00,0.00])\n",
       "\t\t\n",
       "\t\t\n",
       "\tHamiltonians are coupled by exchange matrix:\n",
       "\t\tarray([[-50000.,  50000.],\n",
       "\t\t       [ 50000., -50000.]])\n",
       "\t\n",
       "<SLEEPY.SLEEPY.Propagator.Propagator object at 0x7fe2f8c500b8>"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "U2*U1*U0"
   ]
},
{
   "cell_type": "markdown",
   "id": "dc7163c3",
   "metadata": {},
   "source": [
    "We will later see that sequences can be a powerful tool for performing more complicated calculations. For example, if we want to calculate spinning sidebands, we need to interrupt the rotor period to detect (detecting once a rotor period folds all sidebands back onto the main peak). However, we would have to then generate propagators for different parts of the rotor period and multiply them many times, always in the right order. While not impossible to code, it is somewhat tedious, and the calculations are somewhat slow. On the other hand, the `rho.DetProp` function, which will be introduced in the next section, may be combined with a sequence (possibly an empty one), to perform this operation in the eigenbases of the propagators to rapidly obtain a spectrum. This power comes from the flexibility of sequences, as compared to propagators, which are only valid starting at the same point in the rotor period."
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