Properties of the Pellian Sequence

The purpose of this thesis is to show that the Pellian sequence possesses a great deal of symmetry and regularity. The English method shall be introduced to show that there is a least positive integer x that makes Ax^+i a square if and only if A is a non-square. This method will be shown to produce the entire infinite solution set for any positive integer A. The method of continued fractions will be described and will be used to construct a simple algorithm to produce the Pellian sequence given in both the table and the diskette of Pellian numbers. It will also be shown that an intimate relationship holds between this method and the English method. Finally, properties of the Pellian sequence will be proved to demonstrate regularity in the sequence. It will be shown that every non-negative integer occurs infinitely often in the sequence and, for some classes of integers, the complete set of occurrences in the sequence will be determined prior to solving the Pell equation. A strong connection will be shown to hold between a given number\u27s continued fraction expansion and its set of occurrences. Lastly, it will be conjectured that a given number is a prime power according to its first occurrence in the sequence

Rydberg excitation of a single trapped ion

We demonstrate excitation of a single trapped cold $^{40}$Ca$^+$ ion to Rydberg levels by laser radiation in the vacuum-ultraviolet at 122 nm wavelength. Observed resonances are identified as 3d$^2$D$_{3/2}$ to 51 F, 52 F and 3d$^2$D$_{5/2}$ to 64F. We model the lineshape and our results imply a large state-dependent coupling to the trapping potential. Rydberg ions are of great interest for future applications in quantum computing and simulation, in which large dipolar interactions are combined with the superb experimental control offered by Paul traps.Comment: 4 pages, 3 figure

Focusing a deterministic single-ion beam

We focus down an ion beam consisting of single 40Ca+ ions to a spot size of a few mum using an einzel-lens. Starting from a segmented linear Paul trap, we have implemented a procedure which allows us to deterministically load a predetermined number of ions by using the potential shaping capabilities of our segmented ion trap. For single-ion loading, an efficiency of 96.7(7)% has been achieved. These ions are then deterministically extracted out of the trap and focused down to a 1sigma-spot radius of (4.6 \pm 1.3)mum at a distance of 257mm from the trap center. Compared to former measurements without ion optics, the einzel-lens is focusing down the single-ion beam by a factor of 12. Due to the small beam divergence and narrow velocity distribution of our ion source, chromatic and spherical aberration at the einzel-lens is vastly reduced, presenting a promising starting point for focusing single ions on their way to a substrate.Comment: 16 pages, 7 figure

Entangled states of trapped ions allow measuring the magnetic field gradient of a single atomic spin

Using trapped ions in an entangled state we propose detecting a magnetic dipole of a single atom at distance of a few $\mu$m. This requires a measurement of the magnetic field gradient at a level of about 10$^{-13}$ Tesla/$\mu$m. We discuss applications e.g. in determining a wide variation of ionic magnetic moments, for investigating the magnetic substructure of ions with a level structure not accessible for optical cooling and detection,and for studying exotic or rare ions, and molecular ions. The scheme may also be used for measureing spin imbalances of neutral atoms or atomic ensembles trapped by optical dipole forces. As the proposed method relies on techniques well established in ion trap quantum information processing it is within reach of current technology.Comment: 4 pages, 2 fi

New analysis in the field of open cluster Collinder 223

The present study of the open cluster Collinder 223 (Cr 223) has been mainly depended on the photoelectric data of Claria & Lapasset (1991; hereafter CL91). This data of CL91 has been used with the cluster's image of AAO-DSS in order to re-investigate and improve the main parameters of Cr 223. Stellar count has been achieved to determine the stellar density, the cluster's center and the cluster's diameter. In addition, the luminosity function, mass function, and the total mass of the cluster have been estimated.Comment: 12 pages, 8 figure

Quantum gate in the decoherence-free subspace of trapped ion qubits

We propose a geometric phase gate in a decoherence-free subspace with trapped ions. The quantum information is encoded in the Zeeman sublevels of the ground-state and two physical qubits to make up one logical qubit with ultra long coherence time. Single- and two-qubit operations together with the transport and splitting of linear ion crystals allow for a robust and decoherence-free scalable quantum processor. For the ease of the phase gate realization we employ one Raman laser field on four ions simultaneously, i.e. no tight focus for addressing. The decoherence-free subspace is left neither during gate operations nor during the transport of quantum information.Comment: 6 pages, 6 figure

Controlling the transport of an ion: Classical and quantum mechanical solutions

We investigate the performance of different control techniques for ion transport in state-of-the-art segmented miniaturized ion traps. We employ numerical optimization of classical trajectories and quantum wavepacket propagation as well as analytical solutions derived from invariant based inverse engineering and geometric optimal control. We find that accurate shuttling can be performed with operation times below the trap oscillation period. The maximum speed is limited by the maximum acceleration that can be exerted on the ion. When using controls obtained from classical dynamics for wavepacket propagation, wavepacket squeezing is the only quantum effect that comes into play for a large range of trapping parameters. We show that this can be corrected by a compensating force derived from invariant based inverse engineering, without a significant increase in the operation time

Experiments towards quantum information with trapped Calcium ions

Ground state cooling and coherent manipulation of ions in an rf-(Paul) trap is the prerequisite for quantum information experiments with trapped ions. With resolved sideband cooling on the optical S1/2 - D5/2 quadrupole transition we have cooled one and two 40Ca+ ions to the ground state of vibration with up to 99.9% probability. With a novel cooling scheme utilizing electromagnetically induced transparency on the S1/2 - P1/2 manifold we have achieved simultaneous ground state cooling of two motional sidebands 1.7 MHz apart. Starting from the motional ground state we have demonstrated coherent quantum state manipulation on the S1/2 - D5/2 quadrupole transition at 729 nm. Up to 30 Rabi oscillations within 1.4 ms have been observed in the motional ground state and in the n=1 Fock state. In the linear quadrupole rf-trap with 700 kHz trap frequency along the symmetry axis (2 MHz in radial direction) the minimum ion spacing is more than 5 micron for up to 4 ions. We are able to cool two ions to the ground state in the trap and individually address the ions with laser pulses through a special optical addressing channel.Comment: Proceedings of the ICAP 2000, Firenz