Quantum-State Manipulation With Trapped Ions And Electrons On Liquid Helium

Manipulating quantum states can test the fundamental quantum principles (such as EPR and Shrodinger cat "paradox") and implement the quantum information processes (e.g., implementing quantum computer, quantum telegraph, and quantum cryptography). Therefore, manipulation of quantum states has attracted considerable attention in recent years. Up to now, several kinds of physical systems, e.g., trapped ions, cavity-QEDs, superconducting Josephson junctions, nuclear magnetic resonances (NMRs), coupled quantum dots, nano-mechanical oscillators, and electrons on the liquid Helium, etc., have been proposed to physically implement the quamtum computation. In this paper, we study the quantum-state manipulation with the trapped ions and electrons on the liquid Helium.Currently, manipulating quantum states with trapped ions are usually based on the Lamb-Dicke (LD) approximation. Such an approximation requires that the coupling between the qubits and data bus must being sufficiently weak. As a consequence, the times of the qubit-operations are relatively long. However, the implementation of quantum computation should be completed within the decoherence times of the relevant quantum systems. Thus, it is necessary to develop the relevant theories of implementing quantum computation in trapped ions with strong couplings, i.e., beyond the LD approximations. In this paper, we propose a series of methods to generate certain typical quantum states and implement the fundamental quantum logic gates beyond the LD approximation.A single electron trapped on the liquid Helium has two freedoms:the hydrogen-liked levels along the direction perpendicular to the liquid Helium surface (due to its dielectric image potential), and the vibrational quanta parallel to the liquid Helium surface (due to the confinments of a micro-electrode set below the surface of liquid Helium). Its hydrogen-liked levels could be encoded as a qubit for implementing desirable quantum computation. In the paper, we first show that these two freedoms of the trapped electrons could be utilized to realize the famous Jaynes-Cummings (JC) model. Furthermore, the we show that trapping the electrons in a high-finesse cavity could realize the strong coupling between the electrons and cavity fields, and thus the desirable displaced Fock states, coherence states, and Shrodinger cat states of the cavity could be easily prepared. In chapter 1, we introduce the background, principles, and developments of the systems of trapped ions and electrons on the liquid Helium.In chapter 2, we derivate the dynamical evolution of a single trapped ion (driven by a laser beam) with and beyond the LD approximation. The numerical results show that the LD approximation could work well with the deceasing of LD parameters.Based on the quantum dynamics of laser-ion interaction beyond the LD limit, in chapter 3, we discuss how to superpose a series of vibrational number states of a single trapped ion from the motional ground state. It is shown that, by controlling the durations and phases of the sequentially applied laser pulses, these superposed quantum states could be well approached to the various target quantum states, e.g., coherence states, squeezed coherent states, squeezed odd/even coherent states and squeezed vacuum states etc.It has been shown that a quantum computer can be built by a series of one-qubit rotating operations and two-qubit controlled-NOT gates, because any computation can be decomposed into a sequence of these basic logic operations. Therefore, a precondition work is to effectively implement these fundamental logic gates. However, most of the experiments for realizing these fundamental quantum gates are operated within the LD limit, i.e., the so-called LD parameters should be sufficiently small. In the chapter 4, we propose some methods to realize CNOT gates (of the internal and external freedoms of the trapped ion) beyond the LD approximation. If the durations of each laser pulses are properly set, then the desirable CNOT gates with arbitrary LD parameters could be realized. Second, we propose a simplified method to generate CNOT gate without performingLD approximation, where only two sequential laser beams are needed. Finally, we present an interesting method to realize the CNOT gates with arbitrary LD parameters by one step laser operation. All the above methods are based on the quantum dynamics of laser-ion interaction beyond the LD limit. By properly setting the relevantly experimental parameters, e.g., the frequencies, phases, durations, and amplitudes of the applied laser pulses, the CNOT gate could be effectively implemented.In the chapter 5, we apply a classical laser field to the electron trapped on the liquid Helium. Two freedoms (i.e., the vertical direction's qubit and the in-plane vibrational bosonic modes) of the trapped electrons could be coupled together by the applied classical laser field. This is similar to the laser-assisted coupling in the system of trapped ions. If the frequencies of the applied laser fields are properly set, e.g., the so-called first red and blue-sideband excitations, then the famous JC and ant-JC models could be realized with a electron on the liquid Helium. The numerical results show that, for the typical experimental parameters, the desirable JC and ant-JC models could be well realized.In the chapter 6, we show that an electron trapped in a high-finesse cavity could realize the electron-cavity coupling. In principle, the electron can be always trapped in the cavity by the applied micro-electrode. This is different from the proposal in chapter 5. Here, a cavity, instead the quantized in-plane vibration of the electron is introduced to couple the trapped electron. It is shown that under the excitation of the cavity and an additionally classical laser field, the desirable JC and driven JC models of electron-cavity coupling could be realized. These models could be used to generate the displaced Fock states, coherence states, and Shrodinger cat states of the cavity. The numerical results show that the strong coupling between the electron and the THz cavity could be obtained. Compared to the usual system of a natural atom interacting with the cavity, an advantage of the present system is that it possesses a sufficiently long electron-cavity interaction time.