<p>When $N$ non-interacting atoms interact with a single frequency laser with no phase difference between the photons interacting with the atoms, their interaction can be described collectively \cite{dicke}. For instance, suppose that there is a two level atom with state $|\Psi_1\rangle=\alpha_1|a\rangle+\beta_1|b\rangle$, and another two level atom with state $|\Psi_2\rangle=\alpha_2|a\rangle+\beta_2|b\rangle$. We assume that the two internal states $|a\rangle$ and $|b\rangle$ are indistinguishable between the atoms. Since they are non-interacting atoms, the total state of the system with the two atoms is $|\Psi\rangle_C=\alpha_1\alpha_2|aa\rangle+\alpha_1\beta_2|ab\rangle+\beta_1\alpha_2|ba\rangle+\beta_1\beta_2|bb\rangle$. By rotating the states $|ab\rangle$ and $|ba\rangle$, we can redefine the system using two new basis states, $|+\rangle=(|ab\rangle+|ba\rangle)/\sqrt{2}$ and $|-\rangle=(|ab\rangle-|ba\rangle)/\sqrt{2}$. The state of the system with these states is $|\Psi\rangle_C=\alpha_1\alpha_2|aa\rangle+(\alpha_1\beta_2+\beta_1\alpha_2)/\sqrt{2}|+\rangle+(\alpha_1\beta_2-\beta_1\alpha_2)/\sqrt{2}|-\rangle+\beta_1\beta_2|bb\rangle$. If the two atoms interact with the same field, they evolve in the same way, so that $\alpha_1=\alpha_2\equiv\alpha$ and $\beta_1=\beta_2\equiv\beta$. Hence, the $|-\rangle$ state, which is the antisymmetric state, vanishes, and only the symmetric states remain in the system, so that the total state of the system can be described by $|\Psi\rangle_C=\alpha</p><p>2|aa\rangle+\sqrt{2}\alpha\beta|+\rangle+\beta</p><p>2|bb\rangle$. The remaining states are what are known as the symmetric Dicke states, symmetric collective states, or collective spin states. This two atom case can be generalized to $N$ atoms; for $N$ atoms, there are $N+1$ symmetric collective states. They have been studied since Dicke's seminal paper in the 1950s , especially with respect to superradiance \cite{bonifacio,rehler,skribanowitz,gross,kaluzny,l ambert} and squeezed states \cite{kitagawa,kuzmich01,hald,sorenson}.
We first studied the symmetric collective states for the purpose of quantum computing. Using Rydberg atoms, we showed that with the proper choice of experimental parameters, the excitation of the collective states can be confined to just the ground state and the first excited state by way of differential light shifts. We called this the Rydberg assisted light shift imbalance induced blockade. Such a two level system is important in quantum computing because it can be used as a qubit, a building block of quantum computers. Although the collective state description of Rydberg atoms is quite complicated, since it requires more than just the two traditional hyperfine ground states of an alkali atom, we were able to successfully simplify the system and find the conditions necessary for the proper light shifts to occur to our advantage. The simulations supported our results and we published the results \cite{tu}.
We then moved on to study whether the collective states could be used to make atomic clocks and interferometers. In the case of a collective state atomic clock (COSAC), we found that the Ramsey fringes narrowed by a factor of $\sqrt{N}$ compared to a conventional clock -- $N$ being the number of non-interacting atoms -- without violating the uncertainty relation. This narrowing is explained as being due to interferences among the collective states, representing an effective $\sqrt{N}$ fold increase in the clock frequency, without entanglement. We discuss the experimental inhomogeneities that affect the signal and show that experimental parameters can be adjusted to produce a near ideal signal. The detection process collects fluorescence through stimulated Raman scattering of Stokes photons, which emits photons predominantly in the direction of the probe beam for a high enough optical density. By using a null measurement scheme, in which detection of zero photons corresponds to the system being in a single collective state, we detect the population in a collective state of interest. The quantum and classical noise of the ideal COSAC is still limited by the standard quantum limit and performs only as well as the conventional clock. However, when detection efficiency and collection efficiency are taken into account, the detection scheme of the COSAC increases the quantum efficiency of detection significantly in comparison to a typical conventional clock employing fluorescence detection, yielding a net improvement in stability by as much as a factor of 10. For the off-resonant Raman excitation based COSAC, the theory and results from simulations were published together \cite{kim}; the experiment is underway, and we hope to publish the results in a few months. The COSAC can also be described in terms of the coherent population transfer (CPT) states. The theory behind it is being polished and will be published soon. The collective state atomic interferometer is also possible, with similar inhomogeneities being present in such system, as well \cite{shahriar03}.
This dissertation is organized as follows: In Chapter \ref{chap1}, the fundamental atomic interactions with the electric field and magnetic field are used to derive the interaction Hamiltonian and the density matrix formulation. Chapter \ref{chap3} comprises of the theoretical work and simulation results regarding Rydberg assisted light shift imbalance induced blockade. This chapter introduces collective states. For a more thorough investigation into these states, recommended reading includes Dicke's seminal paper \cite{dicke}, and other references \cite{sargent,mandel}. Chapter \ref{chap4} discusses the off-resonant Raman-Rabi excitation based COSAC, and Chapter \ref{chap5} follows it up with the discussion on coherent population trapping based COSAC. In Chapter \ref{chap6}, I discuss the ongoing experimental progress and the preliminary results we have obtained thus far. I conclude in Chapter \ref{chap7} with future work. Finally, some of the key programs used for simulations are included in the appendices. In Appendix \ref{a1}, some of the MATLAB programs used in the evolution of the density matrix, and the steady state solution of the master equation, in Chapter \ref{chap3} are included. In Appendix \ref{a2}, the more sophisticated Python programs used for Chapter \ref{chap4} are included. Despite the rumors I have heard about no one actually reading anyone else's dissertations, I hope that someone will find the information in here useful in the future.
| Identifer | oai:union.ndltd.org:PROQUEST/oai:pqdtoai.proquest.com:3741428 |
| Date | 24 December 2015 |
| Creators | Kim, May E. |
| Publisher | Northwestern University |
| Source Sets | ProQuest.com |
| Language | English |
| Detected Language | English |
| Type | thesis |
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