{ "cells": [ { "cell_type": "markdown", "id": "5b54da29", "metadata": { "id": "5b54da29", "tags": [], "user_expressions": [] }, "source": [ "(chapter:NV-Centers)=\n", "# TEXT: Introduction to NV centers in diamond and cavity-based magnetometry\n", "1. NV centers in diamond\n", "2. Microwave cavity\n", "3. Coupling between NV centers and microwave cavity" ] }, { "cell_type": "markdown", "id": "2J6vs8Fk8iXa", "metadata": { "id": "2J6vs8Fk8iXa" }, "source": [ "### Pre-labs\n", "This module contains three pre-labs, due on the first, second, and third days, respectively. The pre-lab questions can be found in this text." ] }, { "cell_type": "markdown", "id": "5e028db6", "metadata": { "id": "5e028db6" }, "source": [ "## 1. NV centers in diamond" ] }, { "cell_type": "markdown", "id": "d84b0733", "metadata": { "id": "d84b0733" }, "source": [ "Watch the Youtube video below to learn how diamonds are valuable to quantum information science!" ] }, { "cell_type": "code", "execution_count": null, "id": "ab2db03a", "metadata": { "id": "ab2db03a", "outputId": "444b7f62-219a-43af-a47c-722300acc475" }, "outputs": [ { "data": { "text/html": [ "\n", " \n", " " ], "text/plain": [ "" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" } ], "source": [ "from IPython.display import IFrame\n", "IFrame('https://www.youtube.com/embed/VCT0wDLyvSs', width=700, height=350)" ] }, { "cell_type": "markdown", "id": "93caddc7", "metadata": { "id": "93caddc7" }, "source": [ "**For a comprehensive review of NV centers in diamond, you may also read over the slides in the download link below:**\n", "\n", "**Click [here](https://drive.google.com/file/d/1UuIW939mPKZ2AaD8i0lnlmR72GFbgirs/view?usp=share_link) to download.**" ] }, { "cell_type": "markdown", "id": "dafc1502", "metadata": { "id": "dafc1502" }, "source": [ "### 1.1 Crystallographic structure and level diagram\n", "\n", "Diamond is a particularly interesting material in the realm of quantum information science. It is an excellent host substrate for optically active quantum emitters due to its wide bandgap (5.5 eV) and large transparency window. Among the many _color centers_ in diamond, the negatively charged NV center is most famous for applications in quantum sensing and communication. It is a defect in the carbon lattice consisting of a substitional nitrogen atom accompanied by a vacancy site (see the left figure below).\n", "\n", "\n", ":::{figure-md} NV-center-fig\n", "\n", "\n", "Figures adapted from quTools, quNV brochure.\n", ":::\n", "\n", "\n", "Effectively, NV center is an artificial atom in a solid state environment with well-defined optical transitions. More importantly, NV center's spin states have energy splitting sensitive to the environment's magnetic field strength. The greater the magnetic field strength, the greater the splitting. In a way, NV center itself is a _magnetic field sensor_. One can then readout the spin states (at microwave frequencies) to measure ambient magnetic field.\n", "\n", "Now we briefly introduce level structure for NV center. If you are not familiar with atomic level structures, fear not! It is simply a diagram that locates the relative locations of different energy states.\n", "\n", "On the left-hand-side of the level structure, there are a ground state $^3$A and an excited state $^3$E. On the right, we can see how this states split into multiple nergy level. We will look into why that is further down in this text. The diagram is also labeled with arrows, representing transitions that can occur between the energy states.\n" ] }, { "cell_type": "markdown", "id": "ric-PYgcp-Nb", "metadata": { "id": "ric-PYgcp-Nb" }, "source": [ "### 1.2 Triplet and singlet states:\n", "\n", "An important property of the NV centers here is that they have two unpaired electrons. Remember that electrons are spin $\\pm\\frac{1}{2}$ particles with spin $+\\frac{1}{2}$ or $-\\frac{1}{2}$, which we will denote as $⇑$ and $⇓$, respectively. The spins of these elecrons can be summed using [Clebesh-Gordan coefficients](https://en.wikipedia.org/wiki/Clebsch%E2%80%93Gordan_coefficients), resulting in sum states of the form $|S,m_s⟩$, where $S$ is the magnitude of the spin, and $m_s$ is the projection onto the axis for which we are measuring spin. From two spin $\\frac{1}{2}$ particles, we get the resulting states:\n", "\n", "$$|1,1⟩ = |⇑⇑⟩ \\\\\n", "|1,0⟩ = \\frac{1}{\\sqrt2}(|⇑\\Downarrow⟩ + |\\Downarrow⇑⟩) \\\\\n", "|1,-1⟩ = |\\Downarrow\\Downarrow⟩ \\\\\n", "|0,0⟩ = \\frac{1}{\\sqrt2}(|⇑\\Downarrow⟩ - |\\Downarrow⇑⟩)$$\n", "\n", "In the first three cases, where $S=1$, we consider the electron spins to be \"aligned\", in that their individual spin vectors add up to a magntidue of 1. These are called _triplet states_. Note that in the $|1,0⟩$ state, the projection of the two spins onto our measurement axis is opposite, but the other components do not cancel out, resulting the magnitude of this sum still being 1.\n", "\n", "In the fourth case, with $|0,0⟩$, the spin vector of the electrons point in opposite directions and cancel out leading to zero momentum in all axes. This is a _singlet state_.\n", "\n", "A visual representation of these can be seen in the figure below:\n", "\n", ":::{figure-md} triplet-singlet_spins_fig\n", "\n", "\n", "Figure from paper by [Matthias Eschrig](https://iopscience.iop.org/article/10.1088/0034-4885/78/10/104501?utm_source=researchgate.net&utm_medium=article).\n", ":::\n" ] }, { "cell_type": "markdown", "id": "le1Gfw7f3GSF", "metadata": { "id": "le1Gfw7f3GSF" }, "source": [ "### 1.3 Breaking degeneracy in triplet states:\n", "\n", "In an isolated atom (or artificial atom, like an NV center), all individual states of a triplet state would be degenerate, or having the same energy. We can see in the NV center energy level diagram, however, that this is not the case here.\n", "\n", "The first thing we can look at is the energy difference between the $m_s=0$ states and the $m_s=\\pm 1$ states, which occurs without the presence of a magnetic field. This energy difference is therefore called the _zero-field splitting_. It is caused by [crystal field splitting](https://en.wikipedia.org/wiki/Crystal_field_theory). The NV center is located in a crystal lattice, with surrounding atoms. Electrostatic influence from nearby atoms breaks the symmetry of the orbitals, leading to distinct energy levels. For this lab we are concerned with the zero-field splitting of the triplet ground state, which corresponds to 2.87GHz.\n", "\n", "Second, we also note a further energy splitting between the $m_s=+1$ and $m_s=-1$ states in the presence of the external magnetic field. This is caused by the [Zeeman effect](https://en.wikipedia.org/wiki/Zeeman_effect). Due to its spin, orbit, and charge properties, an electron has a magnetic moment:\n", "\n", "$$\\vec{\\mu}=\\mu_B g \\vec{J} /ħ =\\mu_B(g_l \\vec{L} + g_s \\vec{S})/ħ$$\n", "\n", "where $\\mu_B$ is the [Bohr magneton](https://en.wikipedia.org/wiki/Bohr_magneton) and $g$ is the [Lande g-factor](https://en.wikipedia.org/wiki/Land%C3%A9_g-factor). The Lande g-factor depends on $\\vec{L}$ and $\\vec{S}$, and on constants $g_l=1$ and $g_s≈2$, where $g_s$ is known as the \"anomalous gyromagnetic ratio\". In the NV center, however, the ground triplet state has zero angular momentum, and $S=1$ so this reduces to:\n", "\n", "$$\\vec{\\mu}=\\mu_B g_s \\hat{s} / ħ$$\n", "\n", "\n", "\n", "A magnetic dipole in an a magnetic field will have a potential energy. We can treat the presence of a magnetic field here as a pertubation, and shift energy levels in accordance.\n", "\n", "$$ΔE=\\vec{\\mu}⋅\\vec{B}$$\n", "\n", "Quantization forces electronspins to align parallel or anti-parallel to the magnetic field. Thus, the $m_s=\\pm 1$ ground states get energy shifts of $ΔE=\\pm\\mu_B g_s B / \\hbar $. It is this energy shift that we will use to measure magnetic field.\n", "\n" ] }, { "cell_type": "markdown", "id": "KSl74r4ZFQGd", "metadata": { "id": "KSl74r4ZFQGd" }, "source": [ "### 1.4 Energy transitions in NV centers\n", "\n", "On the left-hand-side of the level structure, there is the ground triplet state $^3A$ and an excited triplet state $^3E$. By shining the NV center with 532 nm light, it can be pumped from $^3A$ to $^3E$. The NV center will them proceed to decay from $^3E$ to $^3A$ along some pathway.\n", "\n", "One important determining factor for the decay pathway is the spin state the NV center started in. Generally speaking, transitions between states tend to be spin-preserving. For example, an NV center in the $m_s=0$ ground state will be pumped to the $m_s=0$ excited state and will generally decay directly back to the $m_s=0$ ground state, emitting a 637 nm photon. This describes most transitions, becaues the $m_s=0$ ground state is the lowest energy state, and thus will have the highest population.\n", "\n", "NV centers in the $m_s = \\pm 1$ ground states will generally likewise be pumped to the $m_s = \\pm 1$ excited states, and can radiatively decay to the $m_s = \\pm 1$ ground states.\n", "\n", "However, NV centers contain a meta-stable singlet state $^1A$, which provides an alternate (and more likely), non-radiative decay pathway for NV centers in the $m_s = \\pm 1$, $^3E$ states. By definition, singlet states have $m_s=0$. However, NV centers can still decay from $^3E$, $m_s=\\pm 1$ states to $^1A$ due to \"spin-orbit coupling\". This is a relativistic effect causing electron spins to interact with their orbits, allowing for transitions where spin angular momentum can exchange with orbital angular momentum. Therefore, some transitions are allowed where spin is not conserved (but total angular momentum will still be conserved). Such a transition is known as an \"intersystem crossing\". NV centers that end up in this $^1A$ state will then preferentially decay to the $m_s = 0$, $^3A$ state.\n", "\n", "To summarize, an NV center pumped from the $^3A$ state to the $^3E$ state will usually return to the $m_s = 0$, $^3A$ state, even if it started at $m_s = \\pm 1$. In steady state, an NV center will then most likely be in $m_s=0$. This mechanism is called _spin-polarization_. A decay from $^3E$, $m_s = 0$ will emit a photon, a decay from $^3E$, $m_s = \\pm 1$ will tyically not emit a photon." ] }, { "cell_type": "markdown", "id": "Cm-pnJQXPtaR", "metadata": { "id": "Cm-pnJQXPtaR" }, "source": [ "### 1.5 Rabi Cycle\n", "\n", "The [Rabi cycle](https://en.wikipedia.org/wiki/Rabi_cycle) a cyclic behavior in the population probability of a two-level system. By driving a two-level system at a frequency that corresponds to the energy difference between two states, we cause a cyclic population transfer between the states.\n", "\n", "The states we will consider here are the $^3A$, $m_s = 0$ state, and the $^3A$, $m_s= \\pm 1$ states. First, we can consider the case where $B=0$, so the $^3A$, $m_s = \\pm 1$ states are degenerate. We then get Rabi oscillations by driving the system at 2.87 GHz. With the NV center pumped by 532 nm light, this results in a dip in the fluoresence, as more NV centers start in the $m_s = \\pm 1$ ground states, which are more likely to decay non-radiatively. This phenomenon is called [optically detected magnetic resonance (ODMR)](https://en.wikipedia.org/wiki/Optically_detected_magnetic_resonance).\n", "\n", ":::{figure-md} ODMR-fig\n", "\n", "\n", "Optical detected magnetic resonance. Figures adapted from quTools, quNV brochure. \n", ":::\n", "\n", "When we apply a magnetic field, we see two fluorescence dips instead of 1. Can you see why? Effectively, by detecting the optical signal coming from the NV center and noting where the changes in the intensity occur, we can extract information about the applied magnetic field. This means the NV center itself acts as a magnetometer!\n", "\n", "HOWEVER, realistically, the change in fluorescence is quite minimal. To make a magnetometer that is practical, we will pursue an alternative approach in this lab. Instead of inferring information about the magnetic field through _optical_ detection, we will attempt _microwave_ detection.\n" ] }, { "cell_type": "markdown", "id": "ee52650d", "metadata": { "id": "ee52650d" }, "source": [ "## 2. Microwave cavity\n", "\n", "### 2.1 Cavity mode of a cylindrical resonator\n", "\n", "To do so, we need to first understand what a microwave cavity is and how it works. The cavity that we will be using in the experiment is a [cylindrical resonator](https://ieeexplore.ieee.org/document/1126654) (you may also recall from your E&M class Chapter 8 in Classical Electrodynamics by J. D. Jackson). The resonator mode of interest is the so-called TE$_{01\\delta}$ mode, which has $H_z,H_{\\rho},$ and $E_{\\phi}$ components in [cylindrical coordinates](https://en.wikipedia.org/wiki/Cylindrical_coordinate_system). Explicitly,\n", "\n", "$$H_z = H_1 J_0(k_c\\rho) f(\\tilde{z})$$\n", "$$H_{\\rho} = -\\frac{j\\beta}{k_c} H_1 J_1(k_c\\rho) f'(\\tilde{z})$$\n", "$$E_{\\phi} = -\\frac{j\\omega\\mu_0}{k_c} H_1 J_1(k_c\\rho)f(\\tilde{z})$$\n", "\n", "where $k_c=x_{01}/r_d$, where $x_{01}$ is the first root of the Bessel function $J_0(x)=0$ and $r_d$ is the outer radius of the resonator. $J_1$ is the Bessel function of the first kind of order 1. $f(\\tilde{z})$ is a factor that gives longitudinal dependence. If you are interested in its exact form, you may consult the Supplement of [this paper](https://www.nature.com/articles/s41467-021-21256-7).\n", "\n", "Crucially, the $H$ components imply that it is _magnetic_ cavity, with the field profile shown in the figure below:\n", "\n", "\n", ":::{figure-md} magnetic-cavity-fig\n", "\n", "\n", "Simulation results from [Eisenach _et al._](https://www.nature.com/articles/s41467-021-21256-7)\n", ":::\n", "\n", "where the black dashed lines mark the outline of the resonator.\n", "\n", "### 2.2 Cavity resonance frequency\n", "\n", "The resonance frequency $\\omega_c$ can be calculated by solving the coupled equations:\n", "\n", "$$\\zeta \\tan\\left(\\frac{\\zeta L}{2}\\right) = \\zeta_0$$\n", "$$\\zeta = \\sqrt{\\epsilon_r \\frac{\\omega_c^2}{c^2}-\\frac{x_{01}^2}{r_d^2}}$$\n", "$$\\zeta_0 = \\sqrt{\\frac{x_{01}^2}{r_d^2}-\\frac{\\omega_c^2}{c^2}}$$\n", "\n", "where $\\epsilon_r$ is the relative dielectric constant and $x_{01}$ is again the first root of the Bessel function $J_0(x)=0$. $L$ is the length of the resonator.\n", "\n", "For more details on the derivation, you may consult [this paper](https://aip.scitation.org/doi/full/10.1063/1.2976033) or the previous [cylindrical resonator](https://ieeexplore.ieee.org/document/1126654) reference.\n" ] }, { "cell_type": "markdown", "id": "3e628f0b", "metadata": { "id": "3e628f0b" }, "source": [ "### 2.3 Inductive coupling between a microwave cavity and a loop coupler\n", "\n", "From our beloved right-hand rule, you may know that the magnetic field should follow the black arrows in the figure below:\n", "\n", ":::{figure-md} magnetic-cavity-RHR-fig\n", "\n", "\n", "Field components of a cylindrical microwave cavity.\n", ":::\n", "\n", "\n", "How do we experimentally _measure_ the cavity resonance? It's simple! We can simply use a metallic loop that _inductively_ couples to the magnetic cavity. As such, we call it a loop coupler. The magnetic field exiting out of the cavity in the $z$ direction (pointing up shown in the figure above) threads through the loop coupler, thereby inducing a current that is measurable. We will leverage this fact and read out the cavity resonance via a [vector network analyzer (VNA)](https://en.wikipedia.org/wiki/Network_analyzer_(electrical)), which you should have already gotten familiar with during the training session." ] }, { "cell_type": "markdown", "id": "cc0e403a", "metadata": { "id": "cc0e403a" }, "source": [ "### 2.4 Quality factor\n", "\n", "One important metric to the quality of a microwave cavity is its [quality factor](https://en.wikipedia.org/wiki/Q_factor), which is defined as:\n", "\n", "$$Q = \\omega_d/\\kappa_c$$\n", "\n", "where $\\kappa_c$ is the total cavity decay rate (in units of radian-frequency). The higher the $Q$, the longer the electric field is \"trapped\" inside the cavity. Since $\\kappa_c$ is inversely proportional to $Q$, higher $Q$ also corresponds to _narrower_ cavity resonance feature (a dip in reflection for our case) in frequency.\n", "\n", "A cavity's $Q$ is also the geometric mean of its intrinsic quality factor $Q_0$ and quality factors due to _leakage_. For an instance, if we bring a loop coupler near the microwave cavity, we are artificially introducing a channel into which the energy of the cavity mode can escape. In this case, the loop coupler is a leakage source and introduces its own \"quality factor\", $Q_1$. Hence,\n", "\n", "$$\\frac{1}{Q} = \\frac{1}{Q_0}+\\frac{1}{Q_1}$$\n", "\n", "Conventionally, when we purposefully introduce a leakage channel, we call the cavity $Q$ the loaded quality factor.\n", "\n", "It is also often easier to consider in terms of decay rates. We can rewrite the above expression as\n", "\n", "$$\\kappa_c = \\kappa_{c0}+\\kappa_{c1}$$\n", "\n", "where $\\kappa_{c0}$ is the intrinsic cavity loss rate and $\\kappa_{c1}$ is the additional decay rate into the loop coupler.\n" ] }, { "cell_type": "markdown", "id": "zhWeaXucxPgN", "metadata": { "id": "zhWeaXucxPgN" }, "source": [ "### Pre-Lab 1 (Day 1)\n", "\n", "In the first session of this experiment, you will first be characterizing the microwave cavity by measuring its resonance frequency and quality factor. Recall that a cavity $Q$ is defined to be the ratio of its resonance frequency to its decay rate (also known as the cavity _linewidth_).\n", "\n", "1. Calculate the $Q$ of a cavity with resonance frequency at 3 GHz and linewidth of 100 kHz.\n", "2. Typically the cavity linewidth is defined at the [full width at half maximum](https://en.wikipedia.org/wiki/Full_width_at_half_maximum). Insert a cell below and write a Python script plotting a resonance peak centered at 3 GHz with linewidth of 100 kHz. Make the x-axis frequency with a span of 1 MHz and the y-axis normalized cavity transmission. Note that the cavity lineshape is usually [Lorentzian](https://mathworld.wolfram.com/LorentzianFunction.html).\n", "\n", "The second session involves characterizing the [Helmholtz coil](https://en.wikipedia.org/wiki/Helmholtz_coil) by measuring its generated magnetic field. Recall that due to the Zeeman effect, magnetic field is used to shift the energy (frequency) difference between the $m_s=0$ and $m_s= \\pm 1$ states of NV centers. Importantly, the correct amount of magnetic field must be applied to ensure one of the energy differences matches the microwave cavity's resonance frequency.\n", "\n", "3. The amount of frequency shift due to magnetic field strength $B$ is $\\delta f = \\mu B$, where $\\mu = \\mu_B g/h$. $\\mu_B$ is the [Bohr magneton](https://en.wikipedia.org/wiki/Bohr_magneton), $g\\approx 2$ is the Lande factor, and $h$ is the Planck constant. Calculate $\\mu$ in units of MHz/Gauss. Note the SI unit for magnetic field is Tesla, where 1 Tesla = $10^4$ Gauss.\n", "4. Assume you have a cavity resonant at 3GHz. Which energy difference would you align to the cavity resonance: $m_s=0$ and $m_s=+1$, or $m_s=0$ and $m_s=-1$? Why? Recall the starting point (due to crystal field splitting) of the energy difference between $m_s=0$ and $m_s=\\pm 1$ is 2.87 GHz.\n", "5. Calculate the needed magnetic field strength (in Gauss) if the cavity resonance frequency is at 3 GHz." ] }, { "cell_type": "markdown", "id": "9b11d395", "metadata": { "id": "9b11d395" }, "source": [ "## 3. Coupling between NV centers and microwave cavity" ] }, { "cell_type": "markdown", "id": "ee8c35f1", "metadata": { "id": "ee8c35f1" }, "source": [ "So far, we have discussed two distinct entities, NV centers in diamond and microwave cavity. In our experiment, we will combine the two systems by simply putting a diamond sample full of NV centers inside the microwave resonator. There are two conditions we must first satisfy in order to observe quantum sensing with solid state spins. One, we need to first excite the entire diamond with 532 nm light, hence spin polarizing an ensemble of NV centers. Second, we need to apply the appropriate amount of magnetic field to shift the energy level of $m_s=+1$, such that the frequency difference between $m_s=+1$ and $m_s=0$ matches the resonance frequency of the microwave resonator. When the aforementioned conditions are met, we can observe the quantum mechanical response of the coupled system (consisted of NV centers and microwave cavity) by sending a microwave signal and measuring its reflection coefficient. Experimentally, we will have the VNA send microwave pulses across different frequencies down the loop coupler, which returns reflected pulses from the microwave cavity back to the receiving port of the VNA.\n", "\n", "The reflection coefficient $\\Gamma$ is given by:\n", "\n", "$$\\Gamma = -1+\\frac{\\kappa_{c1}}{\\frac{\\kappa_c}{2}+j(\\omega-\\omega_c)+\\frac{g^2}{\\frac{\\kappa_s}{2}+j(\\omega-\\omega_s)+\\alpha}}$$\n", "\n", "where $\\kappa_c=\\kappa_{c0}+\\kappa_{c1}$ is the sum of the unloaded (intrinsic) and input port (loop coupler) loss rates, respectively. $\\kappa_s=2/T_2$ is the homogeneous width of the spin resonance where $T_2$ is the [coherence time](https://en.wikipedia.org/wiki/Relaxation_(NMR)). $\\alpha$ captures the effect of saturation and is beyond the scope of this introductory discussion. $g$ is the effective coupling strength between the NV centers and the cavity mode. $\\omega$ and $\\omega_s$ are the probe (sent out of the VNA) and the spin resonance frequencies, respectively. Since there are parameters regarding the NV centers themselves, the _quality_ of the diamond indeed affects the reflection contrast. You will explore different types of diamond in this lab.\n", "\n", "For those who are interested in deriving the reflection coefficient formulae (via input-output formalism similar to the case in quantum optics), you may consult [this paper](https://www.nature.com/articles/s41467-021-21256-7)." ] }, { "cell_type": "markdown", "id": "1f51fa4c", "metadata": { "id": "1f51fa4c" }, "source": [ "Here we compare the conventional ODMR result (optical-based) and the introduced microwave cavity-based approach by showing you experimental results by [Eisenach _et al._](https://www.nature.com/articles/s41467-021-21256-7):\n", "\n", ":::{figure-md} ODMR-vs-microwave-fig\n", "\n", "\n", "ODMR and microwave readout results from [Eisenach _et al._](https://www.nature.com/articles/s41467-021-21256-7)\n", ":::\n", "\n", "Due to coupling with the cavity, the NV centers' microwave response (blue) exhibits much greater [signal-to-noise ratio (SNR)](https://en.wikipedia.org/wiki/Signal-to-noise_ratio) than conventional ODMR (red)!\n", "\n", "In the first portion of the lab, you will use the given formulae for the reflection coefficient and assess how each parameter affects the NV centers' microwave response." ] }, { "cell_type": "markdown", "id": "6CL4OxAw5iia", "metadata": { "id": "6CL4OxAw5iia" }, "source": [ "### Pre-lab 2 (Day 2)\n", "Here are some questions to better prepare you for the lab exercise.\n", "\n", "1. Consult this [product link](https://e6cvd.com/us/application/quantum-radiation/dnv-b14-%203-0mmx3-0mm-0-5mm.html) for one of the diamond types. Obtain the nitrogen concentration under \"Materials Properties\" and assume that all nitrogen defects in the crystal lattice are converted to NV centers. Estimate the density of NV centers in diamond in units of NV/m$^3$.\n", "\n", " Hint: you may need to look up the atomic density of diamond and convert to its units to number of atoms/m$^3$.\n", "\n", "2. Given our diamond dimensions, 3mm x 3mm x 0.5mm, what is the effective coupling strength $g$? Assume a single spin coupling strength is $g_0=0.01$ Hz. You may find the paper by [Eisenach _et al._](https://www.nature.com/articles/s41467-021-21256-7) helpful.\n", "\n", "Now you should be equiped to do the lab exercise!" ] }, { "cell_type": "markdown", "id": "VkBIqoJuwHmv", "metadata": { "id": "VkBIqoJuwHmv" }, "source": [ "### Pre-lab 3 (Day 3)\n", "#### Simulation with hyperfine transitions\n", "\n", "In the previous session, you were able to simulate avoided crossing for a single hyperfine transition. Since NV centers in the diamond(s) that you will be using in the Lab Exercise parimarily constitute N-14 isotope, which is spin-1 (there are 3 nuclear spin states), you should expect to see _3_ hyperfine transitions. The transitions differ slightly between each nuclear spin state,\n", "\n", ":::{figure-md} hyperfine-transitions-fig\n", "\n", "\n", "Ground-state energy level structure of a single NV center. Left: electron spin levels. The solid black arrow depicts optical electron spin polarization to the $m_S$ = 0 state. Right: coupled electron-nuclear spin levels in the $|m_S\\rangle$ = +1  electronic spin state. Solid red arrows show\n", "nuclear spin preserving transitions ($1_+, 2_+$, and $3_+$). Dashed green arrows show nuclear spin exchanging transitions ($a_+, b_+, c_+,$ and $d_+$). Figure adapted from [Huillery et al.](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.103.L140102).\n", ":::\n", "\n", "Please write a Python script that shows the 2D reflection spectra for 3 hyperfine transitions, spaced 2 MHz apart." ] }, { "cell_type": "code", "execution_count": null, "id": "GhHQSZpnwFfv", "metadata": { "id": "GhHQSZpnwFfv" }, "outputs": [], "source": [] }, { "cell_type": "markdown", "id": "hrjcPcOqwmpH", "metadata": { "id": "hrjcPcOqwmpH" }, "source": [ "#### Laser safety\n", "\n", "In this week, you will be running experiments with a free-space 532nm laser that goes up to **3W**. Given that this is a [class 4](https://www.lasersafetyfacts.com/laserclasses.html) laser, you must first undergo proper laser safety training.\n", "\n", "In addition to the training session conducted by MIT EHS, we strongly urge you to read through the Laser Safety page [here](https://ehs.mit.edu/radiological-program/laser-safety/).\n", "\n", "The teaching staff will also supply laser safety glasses ([LG12 from Thorlabs](https://www.thorlabs.com/thorproduct.cfm?partnumber=LG12) with OD7 at 532$~$nm). You must be wearing these during the alignment step (step ii in Lab Exercise below) and during the experimental sequence (step vii).\n", "\n", "**If you have any questions or concerns regarding laser safety, do not hesitate to reach out to the teaching staff and MIT EHS (contact Melissa Spencer, me22023@mit.edu).**" ] }, { "cell_type": "code", "execution_count": null, "id": "TTWu8hS7wnFg", "metadata": { "id": "TTWu8hS7wnFg" }, "outputs": [], "source": [] } ], "metadata": { "colab": { "provenance": [] }, "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": "3.13.11" }, "widgets": { "application/vnd.jupyter.widget-state+json": { "state": {}, "version_major": 2, "version_minor": 0 } } }, "nbformat": 4, "nbformat_minor": 5 }