Individual electrodynamically trapped and laser cooled ions are addressed in frequency space using radio-frequency radiation in the presence of a static magnetic field gradient. In addition, an interaction between motional and spin states induced by an rf field is demonstrated employing rfoptical double resonance spectroscopy. These are two essential experimental steps towards realizing a novel concept for implementing quantum simulations and quantum computing with trapped ions.PACS numbers: 37.10. Vz, 37.10.Ty, 32.60.+i Quantum simulations addressing a specific scientific problem and universal quantum computation are expected to yield new insight into as of yet unsolved physical problems that withstand efficient treatment on a classical computer (e.g., [1]). Already a small number of qubits (i.e., a few tens) used for quantum simulations could solve problems even beyond the realm of quantum information science. Creating and investigating entanglement in large physical systems is a related important experimental challenge with implications for our understanding of the transition between the elusive quantum regime and the classical world [2].Laser cooled atomic ions confined in an electrodynamic cage have successfully been used for quantum information processing (QIP) [3] and advantages and difficulties associated with this system have been and still are subject to detailed investigations. The electromagnetic radiation used to coherently drive ionic resonances that serve as qubits needs to be stable against variations in frequency, phase, and amplitude over the course of a quantum computation or simulation. Experimentally this is particularly challenging when laser light is used for realizing quantum gates. When employing laser light additional issues need to be dealt with to allow for accurate qubit manipulation such as the intensity profile of the laser beam, its pointing stability, and diffraction effects. Furthermore, the motional state of the ion chain strongly affects the gate fidelity which requires ground state cooling and low heating rates during the gate operation [4]. Also, spontaneous scattering of laser light off excited electronic states may pose a limit for the coherence time of a quantum many-body state. The probability for scattering can be reduced by increasing the detuning from excited states (when two laser light fields are used that drive a Raman transition between hyperfine or Zeeman states) which, however, leads to an increasing demand for laser power [5].For generating Raman laser beams with a desired frequency difference, first a radio-frequency (rf) or microwave signal at this difference frequency has to be generated that is then "imprinted" on the laser light and send to the ions. Using rf or microwave radiation directly for coherent driving of qubit transitions is impeded in usual ion trap schemes, since, (i) individual addressing of qubits by focusing radiation on just one ion is difficult due to the long wavelength of rf radiation, and (ii) the required coupling between qubit stat...

Coherent operations constitutive for the implementation of single and multi-qubit quantum gates with trapped ions are demonstrated that are robust against variations in experimental parameters and intrinsically indeterministic system parameters. In particular, pulses developed using optimal control theory are demonstrated for the first time with trapped ions. Their performance as a function of error parameters is systematically investigated and compared to composite pulses.In order to experimentally implement a device capable of performing fault-tolerant universal quantum computation (QC), quantum gate operations involving one or multiple qubits have to be carried out with demandingly high accuracy (see, for instance, [1,2]). According to recent theoretical investigations, the experimentally required accuracy of quantum gates for fault-tolerant universal quantum computation no longer seems daunting or even prohibitive [2]. But still, the desired error probability per gate (EPG) should be as small as possible in order to keep the experimental overhead necessary for quantum computation within a feasible limit. Thus a low error probability is prerequisite for scalable fault-tolerant QC.Any quantum algorithm can be decomposed into a sequence of unitary operations applied to individual qubits (single-qubit gate) and conditional quantum dynamics with at least two qubits [3]. Multi-qubit gates (involving two or more qubits) are synthesized by applying a sequence of elementary unitary operations on a collection of qubits. Each of these elementary operations is often similar, or identical, to what is needed for singlequbit gates, and therefore each operation has to be implemented with an error probability well below the tolerable EPG characterizing the full gate operation.If electrodynamically trapped ions are used as qubits, then a unitary operation amounts to letting ions interact with electromagnetic radiation with prescribed frequency, phase, amplitude, and duration of interaction in order to implement quantum gates. Recently, impressive experimental progress was demonstrated in entangling up to eight ions, and performing 2-qubit quantum gates [4,5,6]. Architectures allowing for scalable QC with trapped ions have been proposed (e.g., [7]), and building blocks necessary for achieving this ambitious goal are currently being investigated using various types of ions.The error budget, for instance, of the geometrical phase gate demonstrated in [6] is dominated by the frequency and amplitude uncertainty of the laser light field. These errors are also responsible for a part of the EPG of the controlled-NOT gate reported in [5]. If an "ion spin molecule", that is, trapped ions coupled via a long range spin-spin interaction, is to be used for quantum information processing, then the exact transition frequency of a particular ionic qubit depends on the internal state of other ions [8]. Therefore, here too, it is important to have quantum gates at hand that are insensitive to the detuning of the radiation driving the qubit tran...

We use the three-cornered-hat method to evaluate the absolute frequency stabilities of three different ultrastable reference cavities, one of which has a vibration-insensitive design that does not even require vibration isolation. An Nd:YAG laser and a diode laser are implemented as light sources. We observe approximately 1 Hz beat note linewidths between all three cavities. The measurement demonstrates that the vibration-insensitive cavity has a good frequency stability over the entire measurement time from 100 ms to 200 s. An absolute, correlation-removed Allan deviation of 1.4 x 10(-15) at s of this cavity is obtained, giving a frequency uncertainty of only 0.44 Hz.