Due to the improvement within cavity quantum electrodynamics experiments during the last decades, what was former seen as 'toy models' are today realized in laboratories. A controlled isolated coherent evolution of one or a few atoms coupled to a single mode inside a cavity is achievable. Such systems are well suited for studying purely quantum mechanical effects, and also for performing quantum gates, necessary for quantum computing. The Jaynes-Cummings model has served as a theoretical description of the interaction. However, as the experimental techniques are improved, for example, atom cooling, the use of multi-level atoms or multi-modes and driving of atoms or elds by external lasers, extensions of the original Jaynes-Cummings model are needed. In this thesis we study some of these extended models, and in particular multi-level models, time-dependent models and quantized motion models. Both analytical and numerical analysis are considered. The two-level structure of the Jaynes-Cummings model leads to applications of known solvable time-dependent two-level Schrödinger equations. In other cases, di erent forms of adiabatic approximate solutions are used, and with the analytically solvable models, the amplitudes of non-adiabatic contributions may be estimated. For higher dimensional systems, STIRAP and multi-STIRAP methods are applied. It is shown how the time-dependent models may be used for preparation of various kinds of non-classical states, and also to generate universal sets of quantum gates, both on atomic and eld qubits. When the atoms are cooled to very low temperatures, their velocities must be treated quantum mechanically, and we have studied the dynamics of such cases for di erent coupling shapes. Again numerical and analytical approaches have been used and compared, wave-packet propagations of the atom, approaching and traversing the cavity, have been performed. For periodic couplings, standing wave cavity modes, the dynamics has been described by e ective parameters; group velocity or atomic index of refraction and effective mass. Tunneling resonances for ultra cold atoms have been exhibited in the STIRAP models for certain initial conditions.
Due to the improvement within cavity quantum electrodynamics experiments during the last decades, what was former seen as 'toy models' are today realized in laboratories. A controlled isolated coherent evolution of one or a few atoms coupled to a single mode inside a cavity is achievable. Such systems are well suited for studying purely quantum mechanical effects, and also for performing quantum gates, necessary for quantum computing. The Jaynes-Cummings model has served as a theoretical description of the interaction. However, as the experimental techniques are improved, for example, atom cooling, the use of multi-level atoms or multi-modes and driving of atoms or elds by external lasers, extensions of the original Jaynes-Cummings model are needed. In this thesis we study some of these extended models, and in particular multi-level models, time-dependent models and quantized motion models. Both analytical and numerical analysis are considered. The two-level structure of the Jaynes-Cummings model leads to applications of known solvable time-dependent two-level Schrödinger equations. In other cases, di erent forms of adiabatic approximate solutions are used, and with the analytically solvable models, the amplitudes of non-adiabatic contributions may be estimated. For higher dimensional systems, STIRAP and multi-STIRAP methods are applied. It is shown how the time-dependent models may be used for preparation of various kinds of non-classical states, and also to generate universal sets of quantum gates, both on atomic and eld qubits. When the atoms are cooled to very low temperatures, their velocities must be treated quantum mechanically, and we have studied the dynamics of such cases for di erent coupling shapes. Again numerical and analytical approaches have been used and compared, wave-packet propagations of the atom, approaching and traversing the cavity, have been performed. For periodic couplings, standing wave cavity modes, the dynamics has been described by e ective parameters; group velocity or atomic index of refraction and effective mass. Tunneling resonances for ultra cold atoms have been exhibited in the STIRAP models for certain initial conditions.
Based on a multi-mode multi-level Jaynes-Cummings model and multi-photon resonance theory, a set of universal two-qubit and three-qubit gates has been realized where dual-rail qubits are encoded in cavities. In this way, the information has been stored in cavities and the off-resonant levels have been eliminated by the theory of an effective two-level Hamiltonian. A further model, namely the spin-J model, has been introduced so that a complete population inversion for levels of interest has been achieved and periodic multilevel multi-photon models have been performed. The combination of the two models has been employed to address two-level, three-level, four-level, and even five-level configurations. Considering the present cavity-QED experiments, several numerical simulations have been designed in order to check the robustness of the logic gates to variations in experimentally important parameters including the coupling constants and the detunings. Finally, based on Liouville's equation, and the wave-function treatments, the impact of decoherence processes on the fidelity of the qubit states in the iSWAP and the Fredkin gates has been studied. This thesis may have applications to quantum information processing, involving logic with simple quantum bits, with the possible application to the building of a quantum computer.
The object of this PhD Thesis is light and matter quantum interaction including studies on cavity and circuit quantum electrodynamics (CQED and cQED), quantum information theory and entanglement dynamics in open systems. In recent years both theoretical and experimental efforts have been devoted to study problems regarding the transfer of quantum correlations, quantum memories, entanglement protection against decoherence and new regimes accessible in cQED. One of the fundamental model underlying these topics is the well known Jaynes-Cummings (JC) one for light-matter interaction, introduced in quantum optics already in 1963. This paradigmatic model describes the coherent exchange of a single excitation between a qubit (a two-level system) and a mode of a quantized harmonic oscillator. The JC model describes a lot of purely quantum effects such as Rabi oscillations and generation of entangled quantum states, which were confirmed by several sophisticated experiments in CQED. This model has been then applied in many other frameworks. Analytical results can be derived only under the so-called rotating wave approximation (RWA), which allows to neglect the energy contributions coming from the counter-rotating terms, which describe the simultaneous excitation (or de-excitation) of the qubit and the field mode. In this Thesis a first interesting investigation, based on the JC model, is related to the realization of quantum memories. The main aspect is the transfer of generic quantum entangled states of a propagating radiation to some qubits, such as two-level atoms placed in separated optical cavities. Generalizing previous results for bipartite entangled systems, I investigated the more complex case of tripartite systems, where the very quantification of entanglement is still an open problem. This issue is broadened to open systems in contact with a common environment which induces decoherence effects unavoidable in realistic implementations. The main result of this analysis is that it is possible to map in an optimal way a quantum state from radiation to qubits in the unitary dynamics, and even in the presence of a dissipative environment a significative amount of entanglement can be transferred. This study has been carried on with either theoretical calculations or numerical simulations performed adopting the powerful Monte Carlo wave function method. A second important problem for quantum memories faced in this PhD Thesis, regards the protection of qubits entanglement for long enough times in order to implement quantum information tasks. A possible way to obtain this goal is to add an external coherent field driving two or more two-level atoms interacting with a common mode of a cavity electromagnetic field. In fact it is possible to demonstrate that, under certain conditions, an effective Hamiltonian of the whole system involves, together with the familiar JC terms, also the counter-rotating ones. The main feature that comes out from this analysis is the possibility to freeze, during system dynamics, peculiar quantum entangled states of the atomic qubits in so called decoherence-free subspaces. This important property allows to store the entanglement in matter qubits, isolating the quantum state from environment induced decoherence. Though the system includes dissipation, analytical results have been obtained for multipartite entanglement and decoherence by solving the master equation through a projection on a suitable basis and using the method of characteristics. In particular it is possible to monitor the rate of decoherence via measurements of joint atomic probabilities, shading light on the fascinating border between the quantum and the classical world. A last topic investigated in this Thesis concerns the recently developed and quite promising implementation of the JC model well outside CQED, that is cQED. Here superconductive circuits, which play the role of artificial atoms, interact on-chip with transmission line resonators, instead of the usual electromagnetic cavities. These systems have the advantage of being quite easily engineered exploiting the most advanced available technologies. Moreover they allow a great flexibility in handling the relevant parameters, since the inter-system coupling is here of capacitive or inductive nature, instead of the usual electric dipole interaction in CQED. In this way a strong and ultrastrong coupling regime have become available, giving a direct access to the physics beyond the JC model and its possible applications. I first performed a detailed characterization of system dynamics in order to test the main approximations usually introduced to describe superconductive circuits as qubits. Then I described, both analytically and numerically, the dissipative dynamics of a superconductive qubit coupled to a resonator mode in the deep strong regime.
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 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.
Cavity quantum electrodynamics (QED) has become one of themost successful tools for quantum information processing, butalso for the study of other pure quantum effects. In thisthesis we investigate one such system, the micromaser; an atomtraverse a cavity and interacts with the quantized modes of thecavity field. Today, the relaxation times of the atom and thecavity field can be made orders of magnitude longer than singleinteraction times, which makes it possible to carry out severaloperations on the system before decoherence becomessignificant. One can often, to a good approximation, treat themicromaser problem by considering the atom as a two-levelsystem interacting with just one of the modes, and losses ofthe combined system can be neglected during the interaction.Any rotations of the atomic Bloch vector during the interactionare, in principle, achievable with modern laser techniques. Forexample, the atom can be Stark shifted in a controlled fashionwith external lasers while it passes through the cavity.The micromaser is theoretically well described by the famousJaynes-Cummings model. Here we look into the problem arisingwhen the parameters of the Jaynes-Cummings model, which are theatom-field coupling and detuning, are allowed to vary in time.What possibilities may occur by letting the parameters dependon time? This question is fairly unanalyzed in the literatureand we are especially interested in preparing variousnon-classical states of the field. The states of the atoms aremeasured after the interactions with thefield and we considerboth the cases when the results of the detections are collectedor not. It is shown that, both in the non-selective andselective measurement procedure, any Fock state can begenerated. We also discuss how one can heat up or cool down thefield and how to prepare entangled states including more thanone mode. The photon field statistics are investigated when theparameters becomes time dependent and field losses areconsidered.
We study interaction between a pair of indistinguishable two-level atoms and the single-mode cavity field. It is supposed that the pair of two-level atoms is laser cooled and trapped into the ground vibrational state, in which vibrational quantum number ⟨nv⟩=0 . Two Jaynes–Cummings models are investigated. One is the Jaynes–Cummings model with intensity-dependent coupling and the another is the two-photon Jaynes–Cummings model of a pair of indistinguishable two-level atoms. It should be noted that in the present model, at initial moment t=0 , in intensity-dependent Jaynes–Cummings model the cavity field is prepared in Holstein–Primakoff SU(1,1) CS, while in two-photon Jaynes–Cummings model it is prepared in the squeezed vacuum state. Moreover, at initial moment t=0 , pair of atoms is supposed to be in the first excited state ∣e1⟩ in both models. By using exact analytical solutions for state-vectors of the coupled atom-field systems amplitude-squared squeezing of the quantized cavity field is examined as a function of the ∣ξ∣ parameter. In this situation, in both models higher-order squeezing has the tendency towards oscillations, but exact periodicity of these oscillations is violated by the analogy with the second-order squeezing.