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Generation of noon states in trapped ions and cavities

Diana Rodriguez Mendez

Published Oct 1, 2011• 0 citations
OpenAlex

Abstract

In the first part of this thesis, by using the two-photon Jaynes - Cummings Model (JCM),\nwe show how NOON states may be generated entangling two cavities, by passing atoms\nthrough them. The atoms interact with each cavity via two-photon resonant transitions.\nWe take advantage of the fact that depending on the state the atom enters (excited or\nground), it leaves or takes two photons per interaction and leaves the cavities in a pure\nstate.\nIn the second part of this thesis, we consider the Hamiltonian of a single ion vibrating in\ntwo dimensions interacting with a laser, we show how NOON states may be generated in\nion traps. We use the individual interaction of light with each of two vibrational modes of\nthe ion to entangle them. This allows us to generate NOON states with N = 8.

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•Aug 24, 2006•2 citations
In the first part of this thesis, by using the two-photon Jaynes - Cummings Model (JCM),\nwe show how NOON states may be generated entangling two cavities, by passing atoms\nthrough them. The atoms interact with each cavity via two-photon resonant transitions.\nWe take advantage of the fact that depending on the state the atom enters (excited or\nground), it leaves or takes two photons per interaction and leaves the cavities in a pure\nstate.\nIn the second part of this thesis, we consider the Hamiltonian of a single ion vibrating in\ntwo dimensions interacting with a laser, we show how NOON states may be generated in\nion traps. We use the individual interaction of light with each of two vibrational modes of\nthe ion to entangle them. This allows us to generate NOON states with N = 8.
We propose a method to generate path-entangled NOON-state photons from quantum dots and coupled nanocavities. In the systems we considered, cavity mode frequencies are tuned close to the biexciton two-photon resonance. Under appropriate conditions, the system can have the target NOON state in the energy eigenstate, as a consequence of destructive quantum interference. The NOON state can be generated by a resonant laser excitation. This method, first introduced for a two-photon NOON state (N=2N=2N=2), can be extended toward a higher NOON state (N>2N>2N>2) based on our recipe, which is applied to the case of N=4N=4N=4 as an example.
We describe a mathematical solution for the generation of entangled N-photon states in two field modes. A simple and compact solution is presented for a two-mode Jaynes-Cummings model by combining the two field modes in a way that only one of the two resulting quasimodes enters in the interaction term. The formalism developed is then applied to calculate various generation probabilities analytically. We show that entanglement, starting from an initial field and an atom in one defined state may be obtained in a single step. We also show that entanglement may be built up in the case of an empty cavity and excited atoms whose final states are detected, as well as in the case when the final states of the initially excited atoms are not detected.
In this thesis we investigate the possibilities to create non-classical states by using light-matter interactions. Our basic approach to describe these interactions is the Jaynes-Cummings model. It describes interactions between a two-level atom and a quantized field in a lossless cavity (ultra high Q quality factor). The model can be easily generalized to describe multi-level atoms, Stark shifts, Kerr interactions, micromasers, etc. We use some of these generalizations in our study. Our main interest is in the production of Schrodinger cat states, that is, the coherent superposition of two or more classical macroscopic states; and the production of sub-Poissonian photostatistics. First of all, we use the simple Jaynes-Cummings model to investigate the interaction of coherent states with two-level atoms, and compare it to the case of interaction with a statistical mixture to show that we can discern between the two. We investigate as well the interaction of two level atoms and squeezed light, where we show that these states have a strong coherent character. However, the model itself is very simple, and as we have said before, some generalizations are necessary if we are to describe more realistically the atom field interactions. In particular, we use a phenomenological Hamiltonian, which incorporates Stark shifts in the atomic levels due to the electric field in the cavity, and show that some experimental results concerning the lineshapes in micromasers may be explained by such effects. Another step is to allow the atoms to enter and leave the cavity, which leads us to investigate a micromaser system. Within this structure, we allow for one- and two-photon transitions. In the latter case, it is shown that sub-Poissonian fields (by using special atom-field interaction times) and multi Schrodinger cat states (by measuring the state of the atoms as they leave the cavity) can be generated.
In this chapter, we discuss two more experimental realizations of quantum optical phenomena, namely the interaction of an effective two-level atom with a quantized electromagnetic field in a high Q microwave cavity, the subject usually referred to as cavity QED, or sometimes CQED, and in the quantized motion of a trapped ion. Strictly speaking, these experiments are not optical, but they do realize interactions of exactly the type that are of interest in quantum optics, namely the Jaynes–Cummings interaction between a two-level system (an atom) and a bosonic degree of freedom, a single-mode cavity field in the case of a microwave cavity, and a vibrational mode of the center-of-mass motion of a trapped ion, the quanta being phonons in this case. We shall begin with a description of the useful properties of the so-called Rydberg atoms that are used in the microwave CQED experiments, proceed to discuss some general considerations of the radiative behavior of atoms in cavities, the CQED realization of the Jaynes–Cummings model, and then discuss the use of the dispersive, highly off-resonant, version of the model to generate superpositions of coherent states, i.e. the Schrödinger cat states of the type discussed in Chapters 7 and 9 for traveling wave optical fields but this time for a microwave cavity field. Finally, we discuss the realization of the Jaynes–Cummings interaction in the vibrational motion of a trapped ion.
In this paper, we generalize the ordinary two-photon Jaynes-Cummings model (TPJCM) by considering the atom (or ion) to be trapped in a simple harmonic well. A typical setup would be an optical cavity containing a single ion in a Paul trap. Due to the inclusion of atomic vibrational motion, the atom-field coupling becomes highly nonlinear what brings out quite different behaviors for the system dynamics when compared to the ordinary TPJCM. In particular, we derive an effective two-photon Hamiltonian with dependence on the number operator of the ion's center-of-mass motion. This dependence occurs both in the cavity induced Stark-shifs and in the ion-field coupling, and its role in the dynamics is illustrated by showing the time evolution of the probability of occupation of the electronic levels for simple initial preparations of the state of the system.