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   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import warnings\n",
    "warnings.filterwarnings(\"ignore\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Lecture 3: Thermomechanical Noise: Brownian Motion of an Oscillator\n",
    "\n",
    "\n",
    "```{admonition} Expected prior knowledge\n",
    ":class: tip\n",
    "Before the start of this lecture, you should be able to:\n",
    "\n",
    "- Peform calculations with power spectral densities\n",
    "- Explain different types of fluctuating (stochastic) variables \n",
    "```\n",
    "\n",
    "```{admonition} Learning goals\n",
    ":class: important\n",
    "After this lecture you will be able to:\n",
    "\n",
    "- Calculate the power spectral density of the thermal noise spectrum of a harmonic oscillator\n",
    "- Predict how the power spectrum changes for lower or higher damping\n",
    "- Explain and draw a representation of thermal noise in an I-Q representation of signals\n",
    "```\n",
    "\n",
    "In principle, even when things are at rest (not moving), at finite temperature everything is actually moving a little tiny bit.  For example, if I knock on this desk, I can make it vibrate, and you can hear a sound.  But even if I didn't hit this desk, it is vibrating already by some small amount, due to its thermal energy producing Brownian motion.  If we look at it, of couse it doesn't look like its moving, but in principle, if I looked very carefully, I would see it bouncing up and down.\n",
    "\n",
    "You might ask yourself: what does any of this have to do with quantum mechanics?  Well, in quantum mechanics, it turns out that things are \"sort of\" still moving even at zero temperature, and if you look hard enough, you can (try) to see this quantum motion... but more on that later.\n",
    "\n",
    "### The damped harmonic oscillator \n",
    "\n",
    "But before we start, it will be useful to review the behavior of the damped harmonic oscillator (DHO).  A DHO consists of a mass on a spring under the influence of friction.  We will consider the DHO under the influence of an external driving force $F_e(t)$, giving \n",
    "\n",
    "$$\n",
    "F = ma = -kx - cv + F_e(t)\n",
    "$$\n",
    "\n",
    "[<div align=\"center\"><img src=\"figures/3/dho.PNG\" width=\"300\"/></div>](dho.png)\n",
    "\n",
    "Defining $\\omega_0 = \\sqrt{\\frac{k}{m}}$ as the undamped natural frequency and $\\gamma = \\frac{c}{m}$ as the \"damping rate\", we can write the above equation as:\n",
    "\n",
    "$$\n",
    "m \\ddot{x} + m \\gamma \\dot{x} + m \\omega_0^2 x = F_e(t)\\\\\n",
    "\\ddot{x} + \\gamma \\dot{x} + \\omega_0^2 x = \\frac{F_e(t)}{m}\n",
    "$$\n",
    "\n",
    "For an oscillating external force\n",
    "\n",
    "$$\n",
    "F_e(t) = F_0 \\cos{\\omega t}\n",
    "$$\n",
    "\n",
    "we will have solutions of the form\n",
    "\n",
    "$$\n",
    "x(t) = A(\\omega) \\cos(\\omega t + \\phi(\\omega))\n",
    "$$\n",
    "\n",
    "with\n",
    "\n",
    "$$\n",
    "A(\\omega) = \\frac{F_0}{m \\sqrt{\\omega^2 \\gamma^2 + (\\omega_0^2 - \\omega^2)^2}}\n",
    "$$\n",
    "\n",
    "and\n",
    "\n",
    "$$\n",
    "\\tan{\\phi(\\omega)} = \\frac{\\omega \\gamma}{\\omega^2 - \\omega_0^2}\n",
    "$$\n",
    "\n",
    "The phase undergoes a $\\pi$ phase shift as we pass through resonance.  For weak damping, the response is sharply peaked and can be approximated as:\n",
    "\n",
    "$$\n",
    "A(\\omega) \\approx \\frac{F_0}{2 m \\omega_0} \\frac{1}{\\sqrt{(\\omega_0 - \\omega)^2 + (\\gamma / 2)^2}}\n",
    "$$\n",
    "\n",
    "and\n",
    "\n",
    "$$\n",
    "\\tan{\\phi(\\omega)} = \\frac{\\gamma/2}{\\omega - \\omega_0}\n",
    "$$\n",
    "\n",
    "The squared amplitude $A(\\omega)^2$ follows a Lorentzian lineshape\n",
    "\n",
    "$$\n",
    "A(\\omega)^2 = \\bigg( \\frac{F_0}{2m \\omega_0} \\bigg)^2  \\frac{1}{(\\omega_0 - \\omega)^2 + (\\gamma / 2)^2}\n",
    "$$\n",
    "\n",
    "whose full width at half maximum (FWHM) is $\\Delta_{FWHM} = \\gamma$.  "
   ]
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",
      "text/plain": [
       "<Figure size 1200x400 with 2 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.subplots(figsize=(12,4))\n",
    "w = np.linspace(0,3,1000)\n",
    "def HO_response(w,gamma):\n",
    "    return 1/np.sqrt(w**2*gamma**2+(1-w**2)**2)\n",
    "\n",
    "def lorentzian(w,gamma):\n",
    "    return 1/(gamma**2/4+(1-w)**2)/4\n",
    "\n",
    "plt.subplot(121)\n",
    "for gamma in (1,0.5,0.2):\n",
    "    plt.plot(w, HO_response(w,gamma), label=\"$\\gamma = %.1f$, $Q = %.0f$\" % (gamma,1/gamma))\n",
    "plt.title(\"Harmonic oscillator response, $\\omega_0=1$, $F_0=m=1$\")\n",
    "plt.xlabel(\"Frequency ($\\omega$)\")\n",
    "plt.ylabel(\"Amplitude $A(\\omega)$\")\n",
    "plt.legend()\n",
    "\n",
    "plt.subplot(122)\n",
    "gamma = 0.2\n",
    "plt.plot(w, np.sqrt(lorentzian(w,gamma)), label=\"Lorentzian\",c='g')\n",
    "plt.plot(w, HO_response(w,gamma), c='g', ls=':', label = \"HO Response\")\n",
    "plt.title(\"Lorentzian with $Q = 5$\")\n",
    "plt.xlabel(\"Frequency ($\\omega$)\")\n",
    "plt.ylabel(\"Amplitude $A(\\omega)$\")\n",
    "plt.tight_layout()\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "```{admonition} The quality factor\n",
    ":class: note\n",
    "- The quality factor of the peak is a dimensionless parameter defined as:\n",
    "\n",
    "  $$\n",
    "  Q = \\frac{\\omega_0}{\\Delta {\\omega}} = \\frac{\\omega_0}{\\gamma}\n",
    "  $$\n",
    "\n",
    "  The smaller the damping rate $\\gamma$, the higher the quality factor.\n",
    "\n",
    "- If we shake the resonator on resonance, we find a significant enhancement of the amplitude, given by:\n",
    "\n",
    "  $$\n",
    "  A(\\omega_0) = \\frac{F_0}{m \\omega_0 \\gamma} = \\frac{F_0}{m \\omega_0^2}\\frac{\\omega_0}{\\gamma} = \\frac{Q F_0}{k}\n",
    "  $$\n",
    "\n",
    "  The amplitude is enhanced by a factor Q for the same force compared to that expected for the static limit ($x = F_0 / k$).\n",
    "\n",
    "- A useful interpretation of this is that at resonance, there is an effectively reduced restoring force, \"$k_{eff} = \\frac{k}{Q}$\", resulting in more displacement for the same force. \n",
    "```\n",
    "\n",
    "### The mechanical suscpetibility \n",
    "\n",
    "The functions $A(\\omega)$ and $\\phi(\\omega)$ are (aside from the scaling factor $F_0$) actually the amplitude and phase of something called the \"mechanical susceptibility\" $\\chi(\\omega)$. Suscetibility is actually a more general concept that  applies to  the linear response a system (in our case, the mass spring) to a stimulus (in our case, the applied force). The [wikipedia page](https://en.wikipedia.org/wiki/Linear_response_function) even has a concrete example of this for a Harmonic oscillator.\n",
    "\n",
    "1. They work a simplified equation with $m=1$ and $k=1$:  \n",
    "\n",
    "$$\n",
    "\\ddot{x}(t)+\\gamma \\dot{x}(t)+\\omega_0^2 x(t)=h(t)\n",
    "$$\n",
    "\n",
    "2. Taking the Fourier transform and writing in terms of $\\tilde x(\\omega)$, this becomes simplified since the derivatives fall out:\n",
    "\n",
    "$$\n",
    "(- \\omega^2  +i\\gamma \\omega + \\omega_0^2)\\tilde{x}(\\omega) = \\tilde{h}(\\omega)\n",
    "$$\n",
    "\n",
    "3. We then define the suscetibility in the following way:\n",
    "\n",
    "$$\n",
    "\\tilde{x}(\\omega) = \\chi(\\omega) \\tilde{h}(\\omega)\n",
    "$$\n",
    "\n",
    "This is handy since it means that if we know the Fourier transform of the force, we can find the Fourier transform of the position.\n",
    "\n",
    "4. Using the two formulas above, and we can find the following form for $\\chi$:\n",
    "\n",
    "$$\n",
    "\\chi(\\omega) = \\frac{1}{- \\omega^2  +i\\gamma \\omega + \\omega_0^2}\n",
    "$$\n",
    "\n",
    "5. In terms of the above, and filling back in our missing constants, we can see that the phase of $\\chi$ is given by the phase we derived $\\phi(\\omega)$, and the amplitude is given by $|\\chi(\\omega)| = A(\\omega)/F_0$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Thermomechanical motion\n",
    "\n",
    "Ok, so what does this have to do with Brownian motion?  Well, we can understand the Brownian motion of our harmonic oscillator as its response to a random force as a function of time $F_{th}(t)$.  What do we know about $F$?  Well, first of all, if it is really random, it has a white power spectral density:\n",
    "\n",
    "$$\n",
    "S_{FF}(\\omega) = S_{FF} \\quad (\\text{constant})\n",
    "$$\n",
    "\n",
    "This is quite handy to know! \n",
    "\n",
    "We could also ask: how large is this force?  Thermodynamics and statistical mechanics tells us that the average thermal energy for each \"degree of freedom\" is $k_B T/2$.  Furthermore, for harmonic motion, we have the convenient feature that, on average, energy is equally divided between kinetic ($\\frac{1}{2} mv^2$) and potential ($\\frac{1}{2}kx^2$) energy (this is because $\\int_{0}^{2\\pi} \\sin^2{x} dx = \\frac{1}{2}$).  This means that we can calculate the average potential energy $\\langle \\frac{1}{2} kx^2 \\rangle$ and equate this to $k_B T/2$. \n",
    "\n",
    "How do we calculate $\\langle \\frac{1}{2} kx^2 \\rangle$?  We can work easily in the \"Fourier domain\" using the formulas above!  Since we have chosen the rest position to be zero, and since the potential of the harmonic oscillator is symmetric, we we will have $\\langle x \\rangle = 0$. In this case we can directly use our formulas for $\\sigma_x^2$ from the first lecture. Why?\n",
    "\n",
    "$$\n",
    "\\sigma_x^2 = \\langle x^2 \\rangle - \\langle x \\rangle^2  = \\langle x^2 \\rangle\n",
    "$$\n",
    "\n",
    "For calculating $\\sigma_x^2$, we will need $S_{xx} = \\langle \\tilde x^2(\\omega)  \\rangle$. Now, the important point is that we can determine $S_{xx}$ from $S_{FF}$ using the susceptibility:\n",
    "\n",
    "$$\n",
    "S_{xx}(\\omega) = |\\chi(\\omega)|^2 S_{FF}\n",
    "$$\n",
    "\n",
    "where, as discussed, $S_{FF}$ is constant and independent of $\\omega$. Putting this together, we then have:\n",
    "\n",
    "$$\n",
    "\\frac{1}{2}k_B T = \\frac{k}{2} \\int S_{xx} d\\omega = \\frac{k}{2} \\int \\frac{S_{FF}}{4m^2\\omega_0^2} \\frac{d\\omega}{(\\omega - \\omega_0)^2 + (\\gamma/2)^2}\n",
    "$$\n",
    "\n",
    "Performing the integral, we get\n",
    "\n",
    "$$\n",
    "S_{FF} = 4 k_B T m \\gamma\n",
    "$$\n",
    "\n",
    "We could ask, why does the force (power spectral density) depend on the damping?  Shouldn't the force depend only on the number of gas molecules hitting my mass per second (assuming the force noise is from air?)\n",
    "\n",
    "The first answer to this is simple: if this were not the case, then our average thermal energy would not be $k_B T$, and we need it to be in order for thermodynamics to work.\n",
    "\n",
    "A second answer is a bit more subtle: the same random gas moleucles that are causing the random force are *also* the ones carrying away the energy.  This is an example of the fluctuation-dissipation theorem, which states that whenever there is dissipation, there must also be fluctuations, and vice versa.\n",
    "\n",
    "An interesting consequence of this is that if there is no dissipation, then there can be no fluctuating force.\n",
    "\n",
    "Before moving on, we note that the Johnson noise voltage of a resistor is another example of the fluctuation-dissipation theorem, which is calculated in a similar way:\n",
    "\n",
    "$$\n",
    "S_{VV} = 4 k_B T R\n",
    "$$\n",
    "\n",
    "```{admonition}  Thermomechanical noise spectrum\n",
    ":class: note\n",
    "In any case, now that we know $S_{ff}$, it is actually very easy to calculate the thermomechanical noise spectrum:\n",
    "\n",
    "$$\n",
    "S_{xx}(\\omega) = \\frac{k_B T \\gamma}{m \\omega_0^2} \\frac{1}{(\\omega - \\omega_0)^2 + (\\gamma/2)^2}\n",
    "$$\n",
    "\n",
    "Like $A(\\omega)$, this is sharply peaked at $\\omega_0$.  We can also calculate the peak value:\n",
    "\n",
    "$$\n",
    "S_{xx}(\\omega_0) = \\frac{4 k_B T}{m \\omega_0^2} \\frac{Q}{\\omega_0}\n",
    "$$\n",
    "\n",
    "We can see that although the force spectral noise density $S_{FF}$ decreases for increasing Q, the peak of $S_{xx}$ increases because $A(\\omega)^2$ increaes $\\propto Q^2$.  For this reason, it is easier to observe thermal noise in high Q oscillators.\n",
    "```\n",
    "\n",
    "### Thermal Noise in the Quadrature Plane\n",
    "\n",
    "We've now derived the thermomechanical spectrum of the Brownian motion of an oscillator.  Before we move on though, we should discuss this a bit further and also introduce some concepts that will come back later in the course.\n",
    "\n",
    "Above, when we considered the harmonic oscillator, we wrote its motion as \n",
    "\n",
    "$$\n",
    "x(t) = A \\cos(\\omega t + \\phi)\n",
    "$$\n",
    "\n",
    "describing the oscillations as an amplitude and a phase.  There is another way of expressing this expression:\n",
    "\n",
    "$$\n",
    "x(t) = I \\cos(\\omega t) + Q \\sin(\\omega t)\n",
    "$$\n",
    "\n",
    "This is known as the quadrature representation of the signal:\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "I\\cos(\\omega t) \\quad &\\rightarrow \\quad \\text{\"in phase\" quadrature}\\\\\n",
    "Q\\sin(\\omega t) \\quad &\\rightarrow \\quad \\text{\"out of phase\" quadrature}\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "Using trigonometric identities, we can show that\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "A &= \\sqrt{I^2 + Q^2}\\\\\n",
    "\\tan(\\phi) &= -Q/I\\\\\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "These definitions are suggestive of the name \"quadrature\", since you need to add $I$ and $Q$ \"quadratically\" to get $A$.  A nice way to visualize this is using a polar plot, where the x-axis is $I$ and the y-axis is $Q$.\n",
    "\n",
    "[<div align=\"center\"><img src=\"figures/3/phasor.PNG\" width=\"300\"/></div>](phasor.png)\n",
    "\n",
    "```{admonition}  Phasor Notation\n",
    ":class: note\n",
    "\n",
    "Often in physics and electrical engineering, we use the \"phasor notation\" to represent an oscillating signal.  In this case, we work with a complex valued variable $\\tilde{x}$ such that \n",
    "\n",
    "$$\n",
    "x(t) = \\mathcal{Re}(\\tilde{x}e^{i\\omega t})\n",
    "$$\n",
    "\n",
    "For a signal $x(t) = A\\cos(\\omega t + \\phi)$, we have \n",
    "\n",
    "$$\n",
    "\\tilde{x} = A e^{i\\phi}\n",
    "$$\n",
    "\n",
    "We can also see that the quadratures $I$ and $Q$ are related to the real and imaginary parts of $\\tilde{x}$, and are given by\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "\\tilde{x} &= I - iQ\\\\\n",
    "\\tilde{x}e^{i \\omega t} &= (I-iQ)(\\cos(\\omega t) + i \\sin(\\omega t)) \\\\\n",
    "&= I\\cos(\\omega t) + Q\\sin(\\omega t) + i(I\\sin(\\omega t)-Q\\cos(\\omega t)\\\\\n",
    "\\mathcal{Re}(\\tilde{x}e^{i\\omega t}) &= I\\cos(\\omega t) + Q\\sin(\\omega t)\n",
    "\\end{align}\n",
    "$$\n",
    "```\n",
    "\n",
    "### Thermomechanical Noise in Quadrates\n",
    "\n",
    "[<div align=\"center\"><img src=\"figures/3/noisy_peak.PNG\" width=\"400\"/></div>](noisy_peak.png)\n",
    "[<div align=\"center\"><img src=\"figures/3/lockin.PNG\" width=\"400\"/></div>](lockin.png)\n",
    "\n",
    "Imagine we measure a signal with a lock-in amplifier set to frequency $\\omega$ with a bandwidth $\\Delta \\omega$.  What does the amplifier actually measure?  If my signal input is \n",
    "\n",
    "$$\n",
    "V_{in}(t) = I\\cos(\\omega t) + Q\\sin(\\omega t)\n",
    "$$\n",
    "\n",
    "then it produces an output that is a DC voltage proportional to $I$ and $Q$ (without the $\\cos\\omega t, \\sin\\omega t$). \n",
    "\n",
    "If $I$ and $Q$ are slowly varying in time (more slowly than the \"time constant\" filter $\\tau = 1/\\Delta \\omega$), then the output of my lock-in amplifier will directly tell me $I(t)$ and $Q(t)$.\n",
    "\n",
    "What do we see, then, if we measure thermal noise with our lock-in amplifier?\n",
    "\n",
    "[<div align=\"center\"><img src=\"figures/3/projections.PNG\" width=\"450\"/></div>](projections.png)\n",
    "\n",
    "And if there is some coherent driving in addition to the thermal noise, we will see:\n",
    "\n",
    "[<div align=\"center\"><img src=\"figures/3/coherent_thermal.PNG\" width=\"400\"/></div>](coherent_thermal.png)"
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