Since the original proposal from Cirac and Zoller in 1995 to use trapped ions as a quantum computer and couple a chain of ions through their collective quantize motion, most of the experimental efforts to couple trapped ions have been performed with the use of laser light. To achieve a high fidelity of the implemented quantum gates, the lasers' frequency, intensity, phase, beam quality, pointing stability and diffraction must be controlled. Additionally, spontaneous emission affects the gate operation. In 2001 a method was proposed to use long-wavelength radiation, such as microwaves or radio-waves, to implement the quantum gates. In practice, an additional inhomogeneous magnetic field is applied to the cooled chain of trapped ions and in this way the ions can be individually addressed in frequency space. Furthermore, the gradient induces a coupling between the ions' internal and motional states and the Ising-type spin-spin coupling (J-coupling) between the ions internal states. We call this method Magnetic Gradient Induced Coupling, MAGIC. By employing MAGIC, it is possible to use microwave radiation instead of optical light, avoiding the fundamental technical limitations mentioned above.
In this thesis a new experimental setup to implement MAGIC is described. Initially, the setup was characterized and then different experiments have been performed that evidenced the spin-spin coupling. The coupling constants have been measured for a two and three ion chain. The measured values are in good agreement with the calculated dependence J propto b^2/vz^2, where b is the magnetic field gradient and vz is the axial trap frequency of the common mode. The measurements demonstrated that the coupling constants can be varied by adjusting the axial trapping potential.
Spin-spin coupling can be used to implement controlled-NOT (CNOT) gates. First, a CNOT gate between two neighboring ions has been performed to demonstrate that the MAGIC method can be used for conditional dynamics. Furthermore, the CNOT gate has been implemented between non-neighboring ions in a three-ion chain as a proof-of principle of a quantum bus employing the ion chain. This has been done here for the first time using the MAGIC method. The quantum nature of a conditional gate is verified via creation of a bipartite entangled Bell state with a fidelity that exceeds the Bell limit and thus proves the existence of entanglement.