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ABSTRACT HYBRID OPTOMECHANICS AND THE DYNAMICAL CASIMIR EFFECT by Robert A. McCutcheon We explore various aspects of hybrid optomechanics. A hybrid optomechanical model in- volving a cavity, atom, and moving mirror is introduced as the intersection between cavity quantum electrodynamics and cavity optomechanics. We describe how numerical simulations of such systems are performed and demonstrate a method that can reduce the computational scope of certain simulations. A method for calculating time-dependent spectra is also ex- plained. Finally, we explain how the dynamical Casimir effect can be achieved using an optical cavity with and without an atom and look at notable behavior of this system. HYBRID OPTOMECHANICS AND THE DYNAMICAL CASIMIR EFFECT Thesis Submitted to the Faculty of Miami University in partial fulfillment of the requirements for the degree of Master of Science by Robert A. McCutcheon Miami University Oxford, Ohio 2017 Advisor: Perry Rice Reader: Samir Bali Reader: Karthik Vishwanath c 2017 Robert A. McCutcheon This thesis titled HYBRID OPTOMECHANICS AND THE DYNAMICAL CASIMIR EFFECT by Robert A. McCutcheon has been approved for publication by the College of Arts and Science and Department of Physics (Perry Rice) (Samir Bali) (Karthik Vishwanath) Contents 1 Introduction 1 1.1 Cavity . .2 1.2 Cavity and Atom . .5 1.3 Cavity and Mirror . .9 1.4 Cavity, Atom, and Mirror . 11 2 Theoretical Frameworks 13 2.1 Wavefunctions and Density Matrices . 13 2.2 Correlation Functions and Emission Spectrum . 16 2.3 Spectrum of Squeezing . 18 2.4 QuTiP Implementation . 19 3 Steady-State Value Computational Method 21 3.1 Procedure . 21 3.2 Results . 23 4 Time-Dependent Spectrum 28 5 Dynamical Casimir Effect 31 5.1 Photon Generation in a Cavity . 31 5.2 Photon Generation in a Cavity with an Atom . 35 6 Conclusion 40 6.1 Summary . 40 6.2 Future Work . 41 A Steady-State Value Computational Method Code 43 B Time-Dependent Spectrum Code 48 iii C Dynamical Casimir Effect - Cavity Code 51 D Dynamical Casimir Effect - Cavity with Atom Code 54 iv List of Figures 1.1 System configuration . .2 1.2 Driven cavity . .5 1.3 Vacuum Rabi oscillations . .8 1.4 Vacuum Rabi doublet . .9 1.5 Cavity-mirror dynamics . 11 2.1 First-order correlation function . 17 2.2 Second-order correlation functions . 18 3.1 Method results for optomechanical system . 25 3.2 Results comparison . 25 3.3 Method results for hybrid optomechanical system . 27 4.1 Time-dependent spectrum . 30 5.1 Photon generation . 33 5.2 First-order correlations . 34 5.3 Second-order correlation . 34 5.4 Spectrum of squeezing . 35 5.5 Photon and atomic expectation values . 37 5.6 First-order correlations, emission spectra, and spectra of squeezing . 38 5.7 Impurity . 39 v List of Tables 1.1 List of common parameters. 12 vi Acknowledgements I would like to thank Dr. Perry Rice for the interesting and engaging classes over the last four years, for introducing me to the dynamical Casimir effect, and for advice on writing this thesis. I would also like to thank Dr. James P. Clemens for first getting me started on quantum optics and working with me for two and a half years, and for suggesting the topic of hybrid optomechanics. vii Chapter 1 Introduction The field of quantum optics deals with light and its interaction with matter in a funda- mentally "quantum" way. Light is treated as photons, and here matter consists of atoms whose electrons absorb and emit photons to transition between discrete energy levels. These relatively simple interactions lead to rich and diverse phenomena that can give insight into a wider variety of physical systems and have applications on small and large scales. For example, the principles of quantum optics can be used to create extremely precise force and position sensors [1][2] and study the crossover into quantum behavior of macroscopic objects [3][4], and they have applications in quantum computing [5]. We will look at several system configurations that are all built around a cavity created between two reflective mirrors. Photons reflect off the mirrors and can be contained in the cavity, although they occasionally "leak" out. From here, the system can be modified in many ways to elicit new behavior. For example, a two-level atom can be added to the cav- ity. The interaction between an atom and photons in a cavity is known as cavity quantum electrodynamics (cQED). Alternatively, one of the mirrors could be allowed to oscillate as if on a spring. The study of photons in a cavity interacting with this mirror, a macroscopic mechanical object, is called cavity optomechanics. Of course, we can include an atom to- gether with the moving mirror, which has become known as hybrid optomechanics [6]. A schematic of this hybrid system is shown in Fig. 1.1 [4][7]. These relatively simple hybrid systems can be used to demonstrate seemingly endless interesting behavior including many non-classical effects, such as photon anti-bunching and squeezed light [8][9][10]. Over the last three years, we have taken a somewhat exploratory approach to various cavity systems. In this thesis, we look at some of the outcomes of this approach. In the rest of this chapter, we describe the cavity, cavity-atom, cavity-mirror, and cavity-atom-mirror systems in more detail while also explaining the important characteristics and parameters that are used throughout the paper. In Chapter 2, we explain some of the methods in quantum theory that are used to calculate the quantities of interest, and describe how these are implemented in the Quantum Toolbox in Python (QuTiP), which we used for numerical 1 simulations. In Chapter 3, we explain a method for reducing the computational scope of these simulations for systems with large numbers of photons and phonons. In Chapter 4, we explain a filter-cavity method which can be used to calculate time-dependent spectra of cavity systems. Finally, in Chapter 5, we use a cavity system to model the dynamical Casimir effect, which results in the creation of photon pairs out of a quantum vacuum state. Figure 1.1: A hybrid optomechanical system with a cavity, two-level atom, and movable mirror. The labeled parameters are explained in subsequent sections. 1.1 Cavity A cavity is used to keep photons localized to a region in space. Since photons reflecting off the mirrors typically make many round trips in the cavity, this strengthens the interactions with the atom and movable mirror when they are added later. The use of a cavity also limits the frequencies of photons found inside. Classically, a cavity of length L between two mirrors can contain an electric field subject to the standing wave condition for allowed frequencies, cm f = ; (1.1) m 2L 2 or πcm ! = ; (1.2) m L where c is the speed of light and m is an integer index starting at 1. We assume that only one mode of the cavity electric field is filled, so that the photons present are only of frequency !1 ≡ !c, each with energy E = ~!c: (1.3) The cavity is modeled as a quantum harmonic oscillator with discrete energy levels given by 1 E = ! n + : (1.4) n ~ c 2 The ladder operators a and ay, which obey the commutation relation a; ay = 1, are used on number states of the oscillator to add and remove quanta (photons); the annihilation operator a removes one photon from the state jni and the creation operator ay adds one, and together aya is a number operator which gives the number n of photons in state jni as follows: p ajni = njn − 1i; (1.5a) p ayjni = n + 1jn + 1i; (1.5b) aya = hnjayajni = hnjnjni = n: (1.5c) These are also related to the electric field; an electric field can be quantized and written in terms of a and ay. A single mode of the field in the cavity of volume V can be written as follows if the field does not vary much over the length of the cavity [11]: r ! E(t) = i ~ ae−i!t − ayei!t : (1.6) 20V The Hamiltonian which represents the energy of the cavity mode is y H = ~!ca a: (1.7) The cavity can be driven by an external laser to add photons to it. This setup results in 3 adding a term to (1.7): y i!Lt y −i!Lt H = ~!ca a + ~Y ae + a e ; (1.8) where !L is the laser photon frequency and Y is the amplitude of the laser driving. (1.8) is in the Schr¨odingerpicture, in which states evolve with time due to the Hamilto- nian and operators are constant. We usually transform to the interaction picture, in which states evolve due to interaction terms of the Hamiltonian and operators evolve due to non- interaction terms. This results in a time-independent Hamiltonian which is easier to use in simulations: y y H = ~∆a a + ~Y a + a ; (1.9) where ∆ is the detuning between the cavity resonance frequency and the laser frequency, !c − !L. We must also include some form of decay, since realistically the cavity is \leaky," and photons occasionally pass through a mirror and leave the system. We explain more about y the system's interaction with the environment in Section 2.1. For now, we just add −iκca a to the Hamiltonian, where κc is the cavity damping rate, and finally we have y y y H = ~∆a a + ~Y a + a − i~κa a: (1.10) In Fig. 1.2, we show the expected photon number aya for a driven cavity as a function y of detuning ∆.
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