Solutions of the basic equations of a simplified model of the cavity quantum electrodynamics are presented under the condition that the single-photon Rabi frequency is much larger than the cavity decay rate. Such solutions are used to calculate the quantum-statistical properties of the field and atomic observables under a variety of initial conditions involving the states of the field and atom. Effects of increasing cavity damping and of the addition of thermal photons on collapse and revival phenomena are discussed. Phase-sensitive aspects of the cavity field are also treated. Quantum effects are shown to be most easily isolated by sending an atom initially prepared in a dressed state for such a state does not evolve further under the influence of a classical field. The appearance of squeezing in the cavity field is demonstrated. The squeezing is most prominent for a coherently prepared atom passing through an empty cavity. The quantum features of the complex dipole moment and their detectability are also discussed in detail.
Solutions of the basic equations of a simplified model of the cavity quantum electrodynamics are presented under the condition that the single-photon Rabi frequency is much larger than the cavity decay rate. Such solutions are used to calculate the quantum-statistical properties of the field and atomic observables under a variety of initial conditions involving the states of the field and atom. Effects of increasing cavity damping and of the addition of thermal photons on collapse and revival phenomena are discussed. Phase-sensitive aspects of the cavity field are also treated. Quantum effects are shown to be most easily isolated by sending an atom initially prepared in a dressed state for such a state does not evolve further under the influence of a classical field. The appearance of squeezing in the cavity field is demonstrated. The squeezing is most prominent for a coherently prepared atom passing through an empty cavity. The quantum features of the complex dipole moment and their detectability are also discussed in detail.
The Jaynes-Cummings model provides a useful means of studying several quantum electrodynamic effects in matter-radiation interaction. One such effect is the phenomenon of collapses and revivals of Rabi oscillations. Various aspects of this phenomenon are studied in terms of the dynamical evolution of the atom as well as the statistical properties of the field. In particular, the sensitivity of collapses and revivals on the initial photon statistics, the cavity field damping mechanism is highlighted. For this purpose the basic model of Jaynes and Cummings has been extended and modified. Some multiatom effects are also discussed.
We have in this thesis looked at two very different problems related to quantum optics. In Part I we have studied the Jaynes-Cummings model with cavity damping and a thermal background. To give an introduction to the field we started Chapter 2 by repeating how one in general derives the master equations for a system coupled to a reservoir. We then repeat the Jaynes-Cummings model and show extensions of the Jaynes-Cummings model where cavity-damping and a thermal background are taken into account. We proceed, in Chapter 3, by finding an analytical solution to the Jaynes-Cummings model with cavity damping in a thermal background when there is no detuning in the system, i.e. we have resonance between the energy of the photons in the cavity and the energy difference in the excited and ground state of the atom. We derive analytical expressions for P+(t), the probability of finding the atom in an excited state after some transition time t given that the atom was prepared in an excited state, and P++(t), the probability of finding the atom in an excited state after some transition time t when a previous atom was measured to be in the excited state. Both atoms where initially prepared in the excited state. We also, in Section 3.7 make use of the Poisson summation formula to rewrite the analytical solution P+(t) in terms of a Fourier series, from which we can predict the location of the revivals in the system. In Section 3.8 we derive stationary results, i.e. when the transition time t→∞, for the probabilities P+(t) and P++(t) and the mean number of photons ‹ n(t) › in the cavity. From our analytical results one is, for instance, led to believe that the probability P+(t) approaches 0 as the transitiontime increases, when it in reality approaches the value nb/(1 + 2nb). The results of our study of the extended Jaynes-Cummings model is presented in the article “Macroscopic Interference Effects in Resonant Cavities” and is published in Physica Scripta [5]. The article is reprinted as Article 1 at the end of this thesis. In Chapter 4 we present the result of a numerical solution of the Jaynes-Cummings model with cavity damping in a thermal background. Here we consider both the resonant case and also the case where detuning is present in the system. When we compare our numerical simulation with our analytical solutions we find that the correlation between the two are close to 1, at least when experimental values of Walther et al. [22] and Brune et al. [15, 16] are used. We also present some figures that show how the revival times and probabilities previously discussed is affected by detuning. In Section 4.2 we find an analytical formula for the revival time where detuning is embedded. In Chapter 4 we also, shortly, discuss how the detuning changes the effective number of photons in the cavity and what criteria should be present in order to separate the revival times.In Part II we study some aspects of quantum mechanical phase. We start, in Section 5.1, by repeating the definition and some results of the Susskind-Glogower phase operator. We then turn our attention to the Pegg-Barnett phase operator [26, 27, 28] in Section 5.4. We describe how to evaluate expectation-values and variances of the Pegg-Barnett phase. In Section 5.5 we establish an upper and lower bound on the phase based on the Pegg-Barnett phase description.In Section 5.9 we repeat some details of a new formalism of the phase which we have conveniently called the NFM-formalism after its inventors Noh, Foug`eres and Mandel [29, 30, 31]. We then describe how to find expectation-values using the Pegg-Barnett phase operator definition applied to the NFM-experiment described in Section 5.9. Our work on the quantum mechanical phase operator is presented in the paper “On the Quantum Phase Operator for Coherent States” and is published in Physica Scripta [6]. This paper is reprinted as Article II at the end of this thesis.
In cavity-quantum-electrodynamics experiments, two-level Rydberg atoms and single-photon microwave fields can be seen as qubits. Quantum gates based on resonant and dispersive atom-field effects have been realized, which implement various kinds of conditional dynamics between these qubits. We have also studied the interaction between a single atom and coherent fields stored in the cavity. By progressively increasing the number of photons in these fields, we have explored various aspects of the quantum-classical boundary. We have realized a complementarity experiment demonstrating the continuous evolution of an apparatus from a quantum to a classical behaviour. We have also prepared 'Schrödinger-cat'-like states of the field made of a few photons, and observed their decoherence. We present a brief review of these experiments along with a proposal to study larger systems, i.e. coherent fields with more photons. Fundamental limits to the size of mesoscopic superpositions of field states in a cavity will be briefly discussed.
It is shown that a single atom making two-photon transitions in an ideal cavity (Q = ∞) in the presence of an initial coherent state field will, as expected, continue to show the collapses and revivals of atomic inversion and the intensity–intensity correlation of the field; however, these collapses and revivals are both compact and regular, in contrast to the one-photon case. This is because, although the several Rabi frequencies involved are incommensurate, for the two-photon case, unlike for the one-photon problem, they become commensurate in the limit of intense field. We derive analytic expressions for the atomic inversion and the intensity–intensity correlation in the case of direct, as well as homodyned, detection of the intracavity field. The effects of the Stark shift are discussed.
Quantum Rabi oscillation in cavity quantum electrodynamics is at the very heart of the interaction of an atomic system with a single electromagnetic field in a resonator. It has proved a powerful tool to entangle the atom with the field in the case of an empty cavity (vacuum Rabi oscillation). On the other side, as soon as the field is macroscopic, its state seems unaffected and one observes the well-known phenomenon of classical Rabi oscillation which only affects the atomic populations. We propose to experimentally investigate the frontier between those two behaviours, when one has to take into account the fluctations of the photon number. In order to do so, we let interact circular Rydberg atoms with a microwave resonator in which a mesocopic field is injected (a few tens of photons). We observe a strong correlation between specific atomic states and the phase of the injected field, which is measured by a homodyne technique. Those correlations explain in an easy way all the standard features of quantum Rabi oscillations (especially the collapses and revivals). Moreover they allow for the preparation of atom-field entangled state as well as non classical Schroedinger cat like states for the field.