Daniel F. V. James

In this paper we consider two problems in diagnostics of trapped ion crystals in which an analysis of the ions' collective oscillatory motion yield potentially useful results. When one of the ions in a linear crystal undergoes a collision, observation of the subsequent motion allows one to deduce the identity of which ion sustained the collision. When a linear ion crystal is formed with a dark impurity ion, analysis of the ions' motion can identify the mass (and thus give an important clue to the species) of the impurity.

We propose a method to manipulate the normal modes in a chain of trapped ions using only two lasers. Linear chains of trapped ions have proven experimentally to be highly controllable quantum systems with a variety of refined techniques for preparation, evolution, and readout, however, typically for quantum information processing applications people have been interested in using the internal levels of the ions as the computational basis. We analyse the case where the motional degrees of freedom of the ions is the quantum system of interest, and where the internal levels are leveraged to facilitate interactions. In particular, we focus on an analysis of mode-mixing of phonons in different normal modes to mimic the quantum optical equivalent of a beam splitter.

We describe a possible architecture to implement a universal bosonic simulator (UBS) using trapped ions. Single ions are confined in individual traps, and their motional states represent the bosonic modes. Single-mode linear operators, nonlinear phase-shifts, and linear beam splitters can be realized by precisely controlling the trapping potentials. All the processes in a bosonic simulation, except the initialization and the readout, can be conducted beyond the Lamb-Dicke regime. Aspects of our proposal can also be applied to split adiabatically a pair of ions in a single trap.

We develop a master equation formalism to describe the evolution of the average density matrix of a closed quantum system driven by a stochastic Hamiltonian. The average over random processes generally results in decoherence effects in closed system dynamics, in addition to the usual unitary evolution. We then show that, for an important class of problems in which the Hamiltonian is proportional to a Gaussian random process, the 2nd-order master equation yields exact dynamics. The general formalism is applied to study the examples of a two-level system, two atoms in a stochastic magnetic field and the heating of a trapped ion.

We investigate the error due to D.C. Stark effect for quantum information processing for trapped ion quantum computers using the scalable architecture proposed in J. Res. Natl. Inst. Stan. 103, 259 (1998) and Nature 417, 709 (2002). As the operation speed increases, dephasing and decoherence due to the D.C. Stark effect becomes prominent as a large electric field is applied for transporting ions rapidly. We estimate the relative significance of the decoherence and dephasing effects and find that the latter is dominant. We find that the minimum possible of dephasing is quadratic in the time of flight, and an inverse cubic in the operational time scale. From these relations, we obtain the operational speed-range at which the shifts caused by D.C. Stark effect, no matter follow which trajectory the ion is transported, are no longer negligible. Without phase correction, the maximum speed a qubit can be transferred across a 100 micron-long trap, without excessive error, in about 10 ns for Calcium ion and 50 ps for Beryllium ion. In practice, the accumulated error is difficult to be tracked and calculated, our work gives an estimation to the range of speed limit imposed by D.C. Stark effect.

We analyze nonlinear coupling between individual vibrational quanta for trapped ions. The nonlinear Coulomb interaction causes a Kerr-type Hamiltonian, for which we derive an analytical expression for the coupling constant. In contrast to a previously published formula [1], our result is in close agreement with experimental data.

We examine in detail the theory of the intrinsic non-linearities in the dynamics of trapped ions due to the Coulomb interaction. In particular the possibility of mode-mode coupling, which can be a source of decoherence in trapped ion quantum computation, or, alternatively, can be exploited for parametric down-conversion of phonons, is discussed and conditions under which such coupling is possible are derived.

We consider the feasibility of performing quantum logic operations based on stimulated Raman transitions in trapped Calcium ions. This technique avoids many of the technical difficulties involved with laser stabilisation, and only three laser wavelengths are required, none of which need have particularly stringent requirements on their bandwidths. The possible problems with experimental realisations are discussed in detail.

Classical and quantum dynamics of a harmonic oscillator in a monochromatic wave is studied in the exact resonance and near resonance cases. This model describes, in particular, a dynamics of a cold ion trapped in a linear ion trap and interacting with two lasers fields with close frequencies. Analytically and numerically a stability of the ``classical ground state'' (CGS) -- the vicinity of the point ($x=0, p=0$) -- is analyzed. In the quantum case, the method for studying a stability of the quantum ground state (QGS) is suggested, based on the quasienergy representation. The dynamics depends on four parameters: the detuning from the resonance, $δ=\ell-Ω/ω$, where $Ω$ and $ω$ are, respectively, the wave and the oscillator's frequencies; the positive integer (resonance) number, $\ell$; the dimensionless Planck constant, $h$, and the dimensionless wave amplitude, $ε$. For $δ=0$, the CGS and the QGS are unstable for resonance numbers $\ell=1, 2$. For small $ε$, the QGS becomes more stable with increasing $δ$ and decreasing $h$. When $ε$ increases, the influence of chaos on the stability of the QGS is analyzed for different parameters of the model, $\ell$, $δ$ and $h$.

We present a novel method of performing quantum logic gates in trapped ion quantum computers which does not require the ions to be cooled down to their vibrational center of mass (CM) mode ground state. Our scheme employs adiabatic passages and the conditional phase shift first investigated by D'Helon and Milburn (C.~D'Helon and G.J.~Milburn, Phys. Rev. A {\bf 54}, 5141 (1996)).

Using a displacement operator formalism, I analyse the depopulation of the vibrational ground state of trapped ions. Two heating times, one characterizing short time behaviour, the other long time behaviour are found. The short time behaviour is analyzed both for single and multiple ions, and a formula for the relative heating rates of different modes is derived. The possibility of correction of heating via the quantum Zeno effect, and the exploitation of the suppression of heating of higher modes to reduce errors in quantum computation is considered.

The theory of interactions between lasers and cold trapped ions as it pertains to the design of Cirac-Zoller quantum computers is discussed. The mean positions of the trapped ions, the eigenvalues and eigenmodes of the ions' oscillations, the magnitude of the Rabi frequencies for both allowed and forbidden internal transitions of the ions and the validity criterion for the required Hamiltonian are calculated. Energy level data for a variety of ion species is also presented.

Using simple physical arguments we investigate the capabilities of a quantum computer based on cold trapped ions. From the limitations imposed on such a device by spontaneous decay, laser phase coherence, ion heating and other sources of error, we derive a bound between the number of laser interactions and the number of ions that may be used. The largest number which may be factored using a variety of species of ion is determined.