Abstract Simulation of Electromagnetically

ABSTRACT SIMULATION OF ELECTROMAGNETICALLY INDUCED TRANSPARENCY AND ABSORPTION by Thomas Jenkins We investigated Electromagnetically Induced Transparency and Absorption in Lambda, Vee, and degenerate two level system systems. By using the Quantum Toolbox in Python (QuTiP) we were able to simulate the coupling and probe absorption spectra by calculating steady state density matrix and finding the expectation values of the dipole transitions. We also used quantum trajectories to model the evolution of each of the quantum states which proved to be helpful in analyzing the absorption spectra seen in the degenerate two level system. SIMULATION OF ELECTROMAGNETICALLY INDUCED TRANSPARENCY AND ABSORPTION A Thesis Submitted to the Faculty of Miami University in partial fulfillment of the requirements for the degree of Master of Science Department of Physics by Thomas Jenkins Miami University Oxford, Ohio 2013 Advisor: (Dr. Perry Rice) Reader: (Dr. Samir Bali) Reader: (Dr. James Clemens) Contents 1 Introduction 1 1.1 Motivation . 1 1.2 Three Level Systems . 2 1.3 Classical Analog of EIT . 3 2 Dark States 7 3 Theoretical Background 10 3.1 Open Systems and the Density Operator . 10 3.2 The Master Equation with Dissipation . 12 3.3 Quantum Trajectories . 15 3.3.1 Two Level Atom in a Field . 18 4 EIT and EIA in Three Level Systems 20 4.1 Λ Structures . 20 4.1.1 Non-Degenerate Systems . 20 4.1.2 Degenerate Systems . 23 4.2 V Structures . 26 5 Degenerate Two Level Systems 30 5.1 Circularly Polarized Probe Beam . 31 5.2 Linearly Polarized Probe Beam . 48 6 Conclusion 63 6.1 Summary . 63 6.2 Future Work . 64 A Classical EIT Code 65 B Code for EIT in Non Degenerate Λ System 67 C Code for EIT in Degenerate Λ System 71 D Code for EIT and EIA in Degenerate V systems 76 ii E Code for Six Level System 82 References 92 iii List of Figures 1.1 Transition Diagram for a Non Degenerate Three Level Atom . 2 1.2 Diagram of Coupled RLC Circuits . 4 1.3 Power Absorption of Coupled Oscillator . 6 3.1 Schematic for Constructing a Quantum Trajectory . 17 3.2 Quantum Trajectories for Two Level Atom . 19 4.1 EIT in a Non Degenerate Λ Structure . 21 4.2 Index of Refraction and Absorption in Λ Structure . 22 4.3 Transition Diagram for Degenerate Λ Structure . 23 4.4 EIT in a Degenerate Λ system . 24 4.5 Quantum Trajectories for Degenerate Λ Structure . 24 4.7 3D plot of EIT in Degenerate Λ Structure . 26 4.8 3D plot of EIT in a Degenerate V Structure . 27 4.9 EIA in Degenerate V Structure . 27 4.10 Quantum trajectories for EIA in Degenerate V Structure . 28 5.1 Transition Diagram for a Six Level Atom . 30 5.3 3D Plots of Absorption Spectra in Six Level Atom with σ− Probe Beam 33 5.11 Quantum Trajectories for Six Level Atom with σ− Probe Beam . 39 5.19 3D Plots of Absorption Spectra in Six Level Atom with π Probe Beam 49 5.31 Quantum Trajectories for Six Level Atom with π Probe Beam . 55 iv ACKNOWLEDGEMENTS First and foremost, I cannot thank my family enough for giving me the support that has allowed me to reach this point. Without them I would not be where I am today. Chase will always be the king. I would also like to thank my advisor, Dr. Perry Rice. There is no doubt that Dr. Rice has helped me to grow as a physics student during my time at Miami University, and it is because of his guidance and patience that I was able to complete this project. I always enjoyed our meetings because of his personality and sense of humor. I knew that Perry was a good guy when I discovered that he was a Browns fan. Thank you for everything. Finally I would like to thank Drs. James Clemens and Samir Bali for taking the time to go through this thesis and for giving me helpful feedback. I feel it is also necessary to thank Samir for setting up weekly volleyball matches, without them I would have never gotten revenge on him for his EM course. v Chapter 1 Introduction 1.1 Motivation Electromagnetically Induced Transparency (EIT) and Electromagnetically Induced Absorption (EIA) are phenomena that are well known in the field of quantum optics. EIT is the product of quantum interference and is characterized by an atomic system becoming effectively transparent to a laser, meaning that there is no absorption at certain frequencies. Along with this we EIT results in a rapid change of the refractive index which produces a low group velocity, meaning that the speed of light can be slowed down. Because of this effect EIT is useful for quantum information processing [12,13]. EIA, the counterpart to EIT, is also the effect of quantum interference and it is characterized by increased absorption. There are several different atomic systems in which we can produce EIT, ranging from systems with three quantum states to more complicated spectra seen in degenerate two level atomic structures. Typically the master equation must be solved in order to model EIT and EIA, however the density matrix theory that is used to solve the the master equation averages away information that we may be interested in such as spontaneous emission events. By doing this we effectively hide what we call quantum jumps. We will do this by finding the steady state solution to the master equation so that we can model EIT and EIA, but we are also going to use what is known as quantum trajectory theory so that we model the evolution of each of the quantum states in the atomic structures. A single quantum trajectory allows us to see how the atoms oscillate through the various states between each quantum jump. Many trajectories averaged together give us an accurate portrayal of how the population evolves with time. This will prove especially useful in degenerate two level systems because knowing which states the population is in will help us understand what type of interference may occur, and subsequently why we see the complex absorption spectra that are exhibited. 1 1.2 Three Level Systems There are various 3 level atom structures that can exhibit EIT and EIA effects under the right circumstances. In this section we will be investigating Λ and V structures in particular. The Λ structure, which resembles the greek letter Λ, is characterized by having a single excited state and two ground states, as seen in Fig. 1.1a. There are electro- magnetic fields (in our case, lasers) that interact with the three states of the material. One beam probes the jg1i ! jei transition while the second couples the jg2i ! jei transition. The jg1i ! jg2i transition is forbidden. It is important to understand that EIT and EIA arise from quantum interference. In the Λ structure, both of the ground states are being driven to a single excited state. When an atom becomes excited, one cannot determine which path the atom took to get into that state which results in interference similar to that found in Young's double slit experiment. The probability amplitude for jg1i ! jei destructively interferes with the probability amplitude for jg1i ! jei and effectively renders the medium transparent to the probe beam. What this means is that the atoms can no longer absorb photons and that eventually all of the atoms will become trapped in the ground states. Even if atoms were in the excited state to begin with, they would decay via spontaneous emission and would have no means of becoming excited again. When the atoms become trapped in the ground states, they are in what is known as a dark state [1]. a.) b.) Figure 1.1: Transition diagram for a a.) Λ and b)V atomic structure. The spontaneous emission frequencies are denoted by γ and Γ while Ωp and Ωc represent the rabi frequencies of the probe and coupling laser beams. ∆ and φ are the detunings in the system. EIT is not constrained to Λ structures, for instance it can also be produced in an atoms with V structures. A V structure (shown in Fig 1.1b) is similar to a Λ structure, it has three quantum states with two lasers driving two transitions, the difference is that it has two excited states with a single ground state. By having two lasers driving the atoms from the ground state to the two ground states V structures can also exhibit EIT. In this case however, the destructive interference comes from 2 the atoms decaying from each of the excited states to the single ground state. Along with EIT, V systems can also exhibit Electromagnetically Induced absorption (EIA) if the lasers are used drive the atoms from the excited states to the ground state. While EIT results in no absorption of photons, EIA is characterized by constructive quantum interference which leads to an increase of absorption. In EIA, the atoms become trapped in the excited states in what is known as a bright state. The atoms cannot go into the ground state via stimulated emission. However, it should be noted that the atoms can still decay into the ground state through spontaneous emission. In V structures we produce either both EIT and EIA based on whether we drive the atoms into the ground or excited states. This is not true for Λ systems. In Λ systems we can produce EIT by driving the atoms into the excited state, however if we were to drive the atoms into the ground states we would not get EIA, we would just see all of the atoms ending up in the ground states because there would not be any interference.

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