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diff --git a/exercise_1-atomic_clocks/Ramsey_Fringes-students.ipynb b/exercise_1-atomic_clocks/Ramsey_Fringes-students.ipynb
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--- a/exercise_1-atomic_clocks/Ramsey_Fringes-students.ipynb
+++ b/exercise_1-atomic_clocks/Ramsey_Fringes-students.ipynb
@@ -1,213 +1,215 @@
 {
  "cells": [
   {
    "cell_type": "code",
    "execution_count": null,
    "metadata": {},
    "outputs": [],
    "source": [
     "import numpy as np\n",
     "import matplotlib.pyplot as plt\n",
     "from scipy.spatial.transform import Rotation as R\n",
     "import scipy.integrate as integrate"
    ]
   },
   {
    "cell_type": "code",
    "execution_count": null,
    "metadata": {},
    "outputs": [],
    "source": [
     "\"\"\"\n",
     "Parameters of the experiment.\n",
     "To write greek letters in Jupyter, type for example \\gamma + Tab\n",
     "\"\"\"\n",
     "\n",
     "Ω = 2*np.pi*14\n",
     "τ = np.pi/2./Ω\n",
     "T0 = 0.3"
    ]
   },
   {
    "cell_type": "code",
    "execution_count": null,
    "metadata": {},
    "outputs": [],
    "source": [
     "\"\"\"\n",
     "Here, δ will be expressed directly in Hz and corresponds to δ/2π in the exercise sheet.\n",
     "So whebn we use it as a hamiltonian parameter, it needs to be multiplied back by 2π\n",
     "Equivalently, Ω is still an angular frequency and has to be defined as 2π x a frequency in Hz\n",
     "\"\"\"\n",
     "def interaction(Ω,δ) :\n",
     "    \"\"\"\n",
     "    rotation matrix for the interaction region :\n",
     "    the argument of this function is the rotation vector (norm and direction)\n",
+    "    τ defines the interaction as π/2-pulse\n",
     "    \"\"\"\n",
+    "    τ = np.pi/2./Ω\n",
     "    return R.from_rotvec(τ*np.array([Ω, 0.,2*np.pi*δ]))\n",
     "\n",
     "def free_evolution(δ,T0) :\n",
     "    \"\"\"\n",
     "    rotation matrix for the free-evolution ------- FILL IN\n",
     "    \"\"\"\n",
     "    return R.from_rotvec(    )\n",
     "\n",
     "def sigma_expect(Ω,δ,T0) :\n",
     "    \"\"\"\n",
     "    expectation value for the Pauli operators (as a vector) ------- FILL IN\n",
     "    \n",
     "    the syntax for applying a rotation to a vector is for example :\n",
     "    interaction(Ω,δ).apply(initial_state)\n",
     "    \"\"\"\n",
     "    initial_state = np.array([0.,0.,1.])\n",
     "    return \n",
     "\n",
     "def proba_1(Ω,δ,T0) :\n",
     "    \"\"\"\n",
     "    Probability of detecting the atom in state |1> ------- FILL IN\n",
     "    \n",
     "    To take the z component of vector initial_state, the syntax would be initial_state[2]\n",
     "    \"\"\"\n",
     "    return "
    ]
   },
   {
    "cell_type": "code",
    "execution_count": null,
    "metadata": {},
    "outputs": [],
    "source": [
     "δ_array = np.arange(-50,50,0.1)   ### values of δ where we want to evaluate our function proba_1\n",
     "\n",
     "result = np.array([proba_1(Ω,δ,T0) for δ in δ_array])\n",
     "\n",
     "plt.figure(figsize=(8,5))\n",
     "plt.plot(δ_array,result)\n",
     "plt.title('Ideal Ramsey fringes',fontsize=15)\n",
     "plt.xlabel(r'detuning $δ$ (Hz)',fontsize=12)\n",
     "plt.ylabel(r'$\\mathcal{P}_1$',fontsize=15)\n",
     "plt.show()"
    ]
   },
   {
    "cell_type": "code",
    "execution_count": null,
    "metadata": {},
    "outputs": [],
    "source": [
     "\"\"\"\n",
     "Parameters and constants for velocity distribution\n",
     "\"\"\"\n",
     "m = 2.2e-25 ; kb= 1.38e-23 ; g=9.8\n",
     "\n",
     "mean_velocity = 1.4\n",
     "temperature = 1e-5\n",
     "\n",
     "def n(v,vbar,temp) :\n",
     "    \"\"\"\n",
     "    Probability to find an atom at velocity v, gievn the mean velocity vbar and temperature temp of the distribution ------- FILL IN\n",
     "    to take the square-root of a variable x , the syntax is np.sqrt(x)\n",
     "    \"\"\"\n",
     "    return \n",
     "\n",
     "def transit_time(v) :\n",
     "    \"\"\"\n",
     "    Time for an atom to cross the free-evolution region as a function of its velocity\n",
     "    \"\"\"\n",
     "    return 2*v/g\n",
     "\n",
     "def proba1_velocity (v,Ω,δ) :\n",
     "    \"\"\"\n",
     "    fringe signal for an atom at velocity v\n",
     "    \"\"\"\n",
     "    return proba_1(Ω,δ,transit_time(v))\n",
     "\n",
     "def proba1_integrand (v,Ω,δ,vbar,temp) :\n",
     "    \"\"\"\n",
     "    fringe signal for one velocity class v ------ FILL IN\n",
     "    will depend on temperature through the Blotzmann distribution n(v,vbar,temp)\n",
     "    \"\"\"\n",
     "    return \n",
     "\n",
     "def proba1_temperature (Ω,δ,vbar,temp) :\n",
     "    \"\"\"\n",
     "    integration of the fringe signal for all velocities - parameters are optimized\n",
     "    \"\"\"\n",
     "    return integrate.quad(proba1_integrand,-10.,10.,args=(Ω,δ,vbar,temp),points=np.linspace(-10.,10.,num=15))\n"
    ]
   },
   {
    "cell_type": "code",
    "execution_count": null,
    "metadata": {},
    "outputs": [],
    "source": [
     "\"\"\"\n",
     "Evaluation of the Ramsey fringes at a a temperature of 10 μK\n",
     "The integration takes a long time (a few minutes)\n",
     "You can test your code by changing δ_array = np.arange(-50,50,0.1) to δ_array = np.arange(-50,50,5)\n",
     "\"\"\"\n",
     "\n",
     "δ_array = np.arange(-50,50,0.1)\n",
     "temp=1e-5\n",
     "\n",
     "result_temp = np.array([proba1_temperature(Ω,δ,mean_velocity,temp) for δ in δ_array])\n",
     "\n",
     "plt.figure(figsize=(8,5))\n",
     "plt.plot(δ_array,result_temp[:,0])\n",
     "\n",
     "plt.title(f'Ramsey fringes at temperature T = {temp*1e6:.0f} μK',fontsize=15)\n",
     "plt.xlabel(r'detuning $δ$ (Hz)',fontsize=12)\n",
     "plt.ylabel(r'$\\mathcal{P}_1$',fontsize=15)\n",
     "plt.show()"
    ]
   },
   {
    "cell_type": "code",
    "execution_count": null,
    "metadata": {},
    "outputs": [],
    "source": [
     "\"\"\"\n",
     "Evaluation of the Ramsey fringes at a a temperature of 1 mK\n",
     "The integration takes a long time (a few minutes)\n",
     "You can test your code by changing δ_array = np.arange(-50,50,0.1) to δ_array = np.arange(-50,50,5)\n",
     "\"\"\"\n",
     "\n",
     "δ_array = np.arange(-50,50,0.1)\n",
     "temp=1e-3\n",
     "\n",
     "result_temp = np.array([proba1_temperature(Ω,δ,mean_velocity,temp) for δ in δ_array])\n",
     "\n",
     "plt.figure(figsize=(8,5))\n",
     "plt.plot(δ_array,result_temp[:,0])\n",
     "\n",
     "plt.title(f'Ramsey fringes at temperature T = {temp*1e3:.0f} mK',fontsize=15)\n",
     "plt.xlabel(r'detuning $δ$ (Hz)',fontsize=12)\n",
     "plt.ylabel(r'$\\mathcal{P}_1$',fontsize=15)\n",
     "plt.show()"
    ]
   }
  ],
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