Quantum Dynamics Of Cold Atoms And Quantum Information Processing Of Superconducting Qubits In Cavities

There are two works in the article, one is studying of dynamic behaviors of cold atoms in an optical cavity, another relates to quantum information processing of superconducting qubits in a transmission line cavity.Dynamical behaviors of cold atoms trapped in optical lattices have attracted enormous attention both theoretically and experimentally. A lot of research shows the nonlinearity from the atom-atom interactions plays a very important role for dynamic properties of Bose-Einstein condensate (BEC) in optical lattices. The quantum phase transition, tunneling dynamics, the stabilities, self-trapping of BEC are affected dramatically by atoms interactions.The recent years have witnessed great advances in the overlap of both the fields of cold atom physics and cavity quantum electrodynamics (QED), which allows one to investigate dynamical behaviors of cold atoms in an optical cavity. Deterministic loading of individual atoms in a microcavity has been successfully achieved, which offers an idea flatform to study BEC. Other than free-space optical lattices, the quantized field in a cavity provides a periodic potential to the atom ensemble, and meanwhile the atomic back-action on the light field can serve as moving refractive media, which modifies the field intensity determining the optical potential for trapping the atoms. This could entail an additional nonlinearity induced due to the presence of a cavity, which is substantially different from that due to the atom-atom interaction, thus it substantially modifies the dynamics of the atoms in the lattice.In the article, we investigate modulational instability (MI) and self-trapping phenomena of BEC in a cavity. The effects of the cavity parameters on these dynamical behaviors are demonstrated, and our investigations show a lot of interesting physical behaviors.Quantum computing studies the principle of coherent superposition and entanglement, thus it is an intercross subject about the generation, transmission, control and readout of quantum information. As quantum computation solves many problems much faster than that on a conventional classical computer, it has attracted much interest in the past years for a lot of studies on the theoretical and practical aspects of quantum computing. Since cavity QED system offers an almost ideal one for the generation of entangled states and the implementation of quantum information processing, it is regarded as one of the most promising system.Among several necessary requirements, high-fidelity readout of the qubit state is an important aspect of all experimental efforts in quantum information science. In the circuit QED system, where a superconducting qubit is strongly coupled to a transmission line resonator, the qubit can be readout via the cavity using microwave signals. The readout can be accomplished by detecting the qubit state-dependent shift of the resonator frequency. Based on the master equation, we propose a high-efficiency scheme to detect a quantum state by using a series of quantum nondemolition (QND) measurements. Our results show that the detected transmission spectra should reveal multiple peaks:each of them marks one of the basis states, and the relative height of such a peak is related to the probability of the corresponding basis state superposed on the detected state. We use this method to tomographically reconstruct an unknown quantum state, and find that only one kind of QND measurement is sufficient to determine all the diagonal elements of the density matrix of the detected quantum state. Compared with the tomographic reconstructions by the usual destructive projective measurements, where one such measurement can determine only one diagonal element of the density matrix, the present reconstructive approach exhibits significantly high efficiency.As an example, we investigate a three-qubit readout via spectral joint measurements of the qubits, by which we propose a scheme to test the tripartite Mermin inequality in the quantum theory. First, we show how to encode various local variables into the prepared Greenberger-Horne-Zeilinger (GHZ) state. Then spectral joint measurements are utilized to obtain the values of the correlation functions, and to further test the Mermin inequality.The major results are listed as follows:Introduction gives the main content and the framework of the article. In Chapter1, the background of BEC, some dynamical behaviors of BEC in optical lattices, and a lot of research about the BEC in a cavity are reviewed briefly.In Chapter2, we investigate the MI of two-component BECs in an optical lattice formed inside a cavity. By adjusting the pump-cavity detuning and the pump amplitude, the optical potentials are modified, and then the hopping parameters are altered. The relevant MI scenarios due to these changes are analyzed and numerically confirmed. Our results show that the driven cavity can be utilized as a possible tool to control the MI of BECs. The numerical results confirm the analytical predictions and present some new content beyond the analytical prediction, during the period of the dynamical evolution.In Chapter3, we investigate the self-trapping of a Bose Josephson junction, which is coupled to an optical cavity. The cavity-induced nonlinearity is presented analytically, and its effect results in the appearance of the self-trapping for the Bose-Einstein condensates in the Josephson oscillation regime. In addition, there exists competition between the nonlinearities induced by the interatomic interaction and by the driven cavity for the emergences of self-trapping. Our results show that the driven cavity can be utilized as a possible tool to produce the self-trapping for the condensates with weak interatomic interaction. In order to clearly demonstrate the mechanism of the occurrence of the cavity-induced self-trapping, we use the algebra about an effective potential of a classical particle motion, and get the idea results.In Chapter4, we briefly review a lot of research about cavity QED, and Jaynes-Cummings (JC) model is introduced to show the dynamical evolution of the cavity QED system. We also present the theoretical and experimental prospect of circuit QED, which is formed by a transmission line cavity and superconducting qubits.In Chapter5, we propose a high-efficiency scheme to tomographically reconstruct an unknown quantum state by using a series of QND measurements. The proposed QND measurements of the superconducting qubits are implemented by probing the stationary transmissions through a transmission line cavity. It is shown that the detected quantum state can be determined by transmission spectra. Our generic proposal presents the tomographic reconstructions of a single qubit and a two-qubit. Quantum logic operations and experimental feasibility are demonstrated by the circuit QED systems with a few Josephson charge qubits. Compared with the tomographic reconstructions based on the usual destructive projective measurements, the present reconstructive approach exhibits significantly high efficiency.In Chapter6, we propose a scheme to test the tripartite Mermin inequality with three qubits by using spectral joint measurements of the qubits. It is the expansion of the work in Chapter5. First, we show how to generate a three-qubit GHZ state by only one-step quantum operation. Then spectral joint measurements are introduced to directly confirm such tripartite entanglement. Assisted by a series of single-qubit operations, these measurements are further utilized to test the Mermin inequality. It shows that the Mermin inequality support quantum mechanics and rule out the local hidden-variable theories.