Control of Ultracold Atoms Trapped in an Optical Lattice by Jieqiu Shao B.S., University of Iowa, 2020 A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirement for the degree of Master of Science Department of Mechanical Engineering 2020 Committee Members: Marco Nicotra Shalom Ruben Kaushik Jayaram ii Shao, Jieqiu (M.S., Mechanical Engineering) Control of Ultracold Atoms Trapped in an Optical Lattice Thesis directed by Assistant Professor Marco M. Nicotra This thesis is a first step towards the integration of quantum physics and control engineering. The goal of the work is to model and control ultracold atoms trapped in an optical lattice. First, after introducing the fundamental concept of quantum mechanics, we model the quantum system and use Bloch basis decomposition to obtain an ordinary differential equation. We then discuss the geometric properties of the system and show that the system is fully controllable. After that, we design and apply two different control strategies to the system, the Lyapunov control method and optimal control method. Finally, we discuss the simulation results from different control methods and detail the next step of this work. iii Contents 1 Introduction 1 1.1 Introduction to Quantum Mechanics . 2 1.2 Modeling of Trapped Ultracold Atoms . 3 1.2.1 Potential Energy Operator . 3 1.2.2 Bloch Basis Decomposition . 4 1.3 Stationary Solutions . 5 2 Geometric Properties 9 2.1 General Concepts . 9 2.2 Geometric Properties of Quantum Systems . 12 2.3 Controllability . 12 3 Control Design 16 3.1 Lyapunov Function . 17 3.1.1 Bilinear Case . 18 3.1.2 Nonlinear Case . 22 3.2 Optimal Control . 24 3.3 Shooting Method . 26 3.4 Newton Step Method . 26 3.4.1 Finite Element Method . 28 3.4.2 Continuation Method . 29 3.4.3 Simulation Results . 29 iv 4 Conclusions 33 Bibliography 34 v List of Figures 1.1 Optical lattice, the blue ball represents the ultracold atom and the red wave represents the laser beam; the lattice wave number is kL = 2π/λL for a lattice wavelength λL; The lattice depth is V0; φ(t) is a time varying phase. 3 1.2 Eigenstates of the Bloch Hamiltonian; this figure shows the eigenstates asso- ciated to the four lowest eigenvalues λ1, λ2, λ3, λ4: ............... 7 2 3 2 2 2 2.1 Top Left: the unit sphere S = f(x1; x2; x3) 2 R jx1 + x2 + x3 = 1g is a 2 manifold; Top Right: the tangent space TxS is a plane that contains x and @F is normal to the vector @x = [2x1 2x2 2x3]; Bottom Left: the two vector T T fields f(x) = [−x2 x1 0] and g(x) = [−x3 0 x1] belong to the tangent space; Bottom Right: the systemx _ = f(x) + ug(x) is controllable. Given u = 0, 2 x_ = f(x) spans the entirety of the level-set fx 2 S j x3 = cg. Given u 6= 0, g(x) allows the system to move from one level-set to another. 15 3.1 Left: the control input u vs time using Lyapunov control law on the bilinear model, given K = 0:2; Right: the population vs time using Lyapunov control law on the bilinear model, given K = 0:2. 20 3.2 Left: the control input u vs time using Lyapunov control law on the bilinear model, given K = 0:5; Right: the population vs time using Lyapunov control law on the bilinear model, given K = 0:5. 20 vi 3.3 Left: the control input u vs time using Lyapunov control law on the bilinear model, given K = 1; Right: the population vs time using Lyapunov control law on the bilinear model, given K = 1. 21 3.4 Left: the control input u vs time using Lyapunov control law on the bilinear model, given K = 2; Right: the population vs time using Lyapunov control law on the bilinear model, given K = 2. 21 _ _ ∗ dV 3.5 V vs u, given α = −1 and β = −0:1; u is the solution to du = 0 that minimizes V_ ; the green region is the control law u = sin−1(K sin u?), where K 2 (0; 1]...................................... 22 3.6 Left: the control input u vs time using Lyapunov control law on the nonlinear model, given K = 1; Right: the population vs time using Lyapunov control law on the nonlinear model, given K = 1. 23 3.7 Left: the control input u vs time using Lyapunov control law on the nonlinear model, given K = 0:1; Right: the population vs time using Lyapunov control law on the nonlinear model, given K = 0:1. 23 3.8 Left: the control input u vs time using optimal control law (Newton method) on the bilinear model; Right: the population vs time using using optimal control law (Newton method) on the bilinear model. 30 3.9 Left: the control input u vs time using optimal control law (Newton method) on the nonlinear model; Right: the population vs time using using optimal control law (Newton method) on the nonlinear model. 30 3.10 The optimal control input u vs time obtained from bilinear and nonlinear model. 31 3.11 Left: the population vs time using optimal control input from bilinear model applied to nonlinear model; Right: the population vs time using using opti- mal control law (Newton method) on nonlinear model. 32 1 Chapter 1 Introduction This thesis investigates the control of ultracold atoms trapped in optical lattice. Ultra- cold atoms are atoms that are maintained at temperatures close to 0 Kelvin (absolute zero). Ultracold atoms offer remarkable opportunities for investigating quantum problems that are relevant to fields as diverse as condensed matter physics, statistical physics, quantum chem- istry, and high-energy physics [1]. The high degree of controllability and new observation tools, which enable the detection of each individual atom, make ultracold atoms ideal as analog quantum simulators [2, 3, 4, 5] high-accuracy measurements such as quantum clocks [6] and interferometers [7]. The optical lattices used in the experiments are typically formed by interfering several laser beams in order to realize a fully controllable periodic light struc- ture meant to mimic the crystal lattice of a solid. Quantum technology is currently at the tipping point and is aiming to transition from experimental physics laboratories to real-world engineering applications. This transition requires a strong degree of collaboration between quantum physicists and specialists from other disciplines, such as control engineering. This thesis brings together quantum mechanics and control engineering. I am a mechanical engi- neering student with little previous knowledge in quantum theory, and my role is to adapt classic tools from systems and control theory to the new and exciting setting of quantum systems. Before we delve deeper into the specifics of our system, we provide a brief overview of quantum mechanics. The objective is to provide just enough background information to 2 enable someone with a traditional engineering background to gain insight into the intricacies of quantum theory. 1.1 Introduction to Quantum Mechanics Quantum mechanics describes the physical properties of small objects (atomic scale) that are well isolated from other objects. The purpose of this section is to introduce the fundamental concept of quantum mechanics. Quantum mechanics follows from four primary postulates: Wave Function: The state of a quantum system is fully described by a wave function Ψ(x; t), which is a complex valued function of two real variables, where x is position and t is time. The modulus square of the wave function, jΨ(x; t)j2, is the probability density, which represents the probability of finding the atom in a given position at a given time. If there are many atoms, then the wave function represents the density of atoms at given position and time. Observables: An observable is a physical quantity that can be measured and is captured by a Hermitian operator applied to the wave function Ψ(x; t). For example, the kinetic energy of a particle is K^ (x)Ψ(x; t), where 2 K^ (x) = − ~ r2 (1.1) 2m x is the kinetic energy operator, m is the mass of particle and ~ is the reduced Planck constant. Measurements: For a given observable A, the only possible result of the measurement is one of the eigenvalues ai of the operator, where ai is associated to an eigenvector νi satis- fying A(x)νi(x) = aiνi(x). This means that the more closely the wave function matches an eigenvector of the observable νi, the more likely the measurement is to reveal the associated eigenvalue. This property is fundamentally different from classical mechanics, where a mea- surement can yield a continuum of results. Indeed, the whole premise of quantum mechanics is that measurements are \quantized" and any measurement can only yield a discrete set of values. 3 Schr¨odingerequation: The dynamics of the wave function is described by the Schr¨odinger equation i~Ψ(_ x; t) = H^ (x)Ψ(x; t); (1.2) where H^ (x) = K^ (x) + U^(x) is the Hamiltonian, K^ (x) is the kinetic energy operator and U^(x) is the potential energy operator. This can be paralleled with classical mechanics where the total energy of the system is equal to the sum of kinetic and potential energy. 1.2 Modeling of Trapped Ultracold Atoms 1.2.1 Potential Energy Operator For our optical lattice, the laser beam defines the potential energy of the wave function referring to Figure 1.1, it is possible to define the potential energy operator as V U^(x) = − 0 cos(2k x + φ(t)); (1.3) 2 L where V0 is the depth of the lattice, λL is the wavelength, kL = 2π/λL is the wave number, and φ(t) is the phase.