# Overview

## Background [next]

*Quantum mechanics* is a tremendously successful theory playing a central role in natural sciences even beyond physics, and has been verified in countless experiments. Despite its great success and its history reaching back more than hundred years, still today the laws and interpretation of quantum mechanics challenge our intuition that has been formed by an environment governed by classical physical laws.

In addition to puzzling us with fundamental questions, quantum mechanics holds the opportunity to put its laws to practical use. In recent years, researchers have started to learn to generate at demand and manipulate entangled quantum systems, thus verifying experimentally a most counter-intuitive prediction of quantum theory: the non-local correlation of quantum systems that A. Einstein considered as "spooky action at a distance".

The creation and manipulation of entangled states is essential in the field of quantum information processing (QIP) and communication, where basic elements of computers are explored relying on quantum mechanical laws. A quantum computer would be able to solve problems that, for all practical purposes, cannot be handled by classical computers and communication devices.

## Quantum Computing [previous next top]

The basic element of a *quantum computer* is a qubit: a
quantum mechanical system that can exist in two distinct states
0 and 1, similar to the "bit" forming the elementary switching
unit of a classical computer. The crucial difference being that
a qubit may also exist simultaneously in state 0 and 1. Such
quantum mechanical superposition states are very fragile, and
soon cease to exist when the qubit comes in contact with the
rest of the world (which is the reason why we do not observe
them in our macroscopic environment.) The basis for the
practical realisation of quantum computation is the virtuous
manipulation of these delicate quantum superposition states.
Experimental schemes that have been successfully applied to
tame qubits and even demonstrate complete quantum algorithms
include nuclear magnetic resonance (NMR) using molecules in a
liquid state, and electrodynamically trapped ions manipulated
by laser light.

## Quantum Simulation [previous next top]

One way to overcome difficulties associated with universal quantum computing is to use one quantum system to simulate the dynamics of another quantum system. This general idea was originally conceived by R. Feynman. Recently, concrete proposals for quantum simulations have been made using a chain of pairwise coupled spins described by a Heisenberg model or a variant thereof. Such models serve as a starting point for numerous theoretical investigations in various branches of Physics, and in particular in Condensed-Matter Physics where collective effects, like (anti-)ferromagnetism and superconductivity, are investigated. To date, experimental quantum simulations have only just started to emerge and have been realized using only few qubits.

We develop a spin chain made up of individual electrodynamically trapped ions is being developed with the ability to coherently manipulate and read-out individual members of this many body system. Furthermore, the spin-spin coupling strength and range of interaction is controllable by the experimenter.

## Quantum Control [previous next top]

In order to physically implement universal quantum computation and apply it to interesting problems, quantum gates have to be executed with demandingly high precision and need to be applied to a large numbers of qubits. A way to cope with the daunting requirements of universal quantum computing is the development of practical schemes for better control of the quantum evolution of individual building blocks for quantum information processing, that is, error correction, characterization and subsequent control of decoherence, and of robust and optimized quantum gates [Experiments section].

## RF Radiation for Coherent Manipulation of Trapped Ions [previous next top]

The potential that trapped ions have as a physical system
for quantum information processing (QIP) was first recognized
in [PRL
**74** (1995)], and since then many important
experimental steps have been undertaken towards the realization
of an elementary quantum computer with this system. At the same
time, the advanced state of experiments with trapped ions
reveals the difficulties that still have to be overcome.

In addition to the ability to perform arbitrary single-qubit
operations, a second fundamental type of operation is required
for QIP: conditional quantum dynamics with, at least, two
qubits. Any quantum algorithm can then be synthesized using
these elementary building blocks [PRA
**51** (1995), PRA
**52** (1995) ]. While two internal states of
each trapped ion serve as a qubit, communication between these
qubits, necessary for conditional dynamics, is achieved via the
vibrational motion of the ion string in a linear trap (the
''bus-qubit'') [PRL
**74** (1995)]. Thus, it is necessary to
couple external (motional) and internal degrees of freedom.
Common to most experiments performed to date - related either
to QIP or other research fields - that require some kind of
coupling between internal and external degrees of freedom of
atoms is the use of **optical** radiation for this
purpose. The recoil energy
*E _{r}=(ħk)^{2}/2m* taken up by an atom
upon absorption or emission of a photon may change the atom's
motional state (

*k=2πλ*,

*λ*is the wavelength of the applied electromagnetic radiation, and

*m*is the mass of the ion.) In order for this to happen with appreciable probability with a harmonically trapped atom, the ratio between

*E*and the quantized motional energy of the trapped atom,

_{r}*ħν*should not be too small (

*ν*is the angular frequency of the vibrational mode to be excited.) Therefore, in usual traps, driving radiation in the optical regime is necessary to couple internal and external dynamics of trapped atoms.

The distance between neighboring ions *δz* in a
linear electrodynamic ion trap is determined by the mutual
Coulomb repulsion of the ions and the time averaged force
exerted on the ions by the electrodynamic trapping field.
Manipulation of individual ions is usually achieved by focusing
electromagnetic radiation to a spot size much smaller than
*δz*. Again, only optical radiation is useful for this
purpose.

In references [QIS3,
QIS8,
QIS10,
QIS13,
QIS14]
a concept for ion traps is introduced that allows for
experiments requiring individual addressing of ions and
conditional dynamics with several ions even with radiation in
the **radio frequency** (rf) regime (i.e., with
frequencies ranging from kHz to GHz). It is shown how an
additional magnetic field gradient applied to an electrodynamic
trap individually shifts ionic qubit resonances making them
distinguishable in frequency space. Thus, individual addressing
for the purpose of single qubit operations becomes possible
using long-wavelength radiation [QIS22,
QIS33].
At the same time, magnetic gradient-induced coupling (MAGIC)
between internal and motional states arises [QIS3,
QIS22].
Thus, conditional quantum dynamics can be carried out in this
modified electrodynamic trap, and in such a new type of trap
all schemes originally devised for **optical** QIP
in ion traps can be applied in the rf regime, too.

## Trapped Ions as a Designer Spin-Molecule [previous top]

Many phenomena that were only recently studied in the
optical domain form the basis for techniques belonging to the
standard repertoire of coherent manipulation of nuclear and
electronic magnetic moments associated with their spins.
Nuclear magnetic resonance (NMR) experiments have been
tremendously successful in the field of QIP taking advantage of
highly sophisticated experimental techniques. However, NMR
experiments usually work with macroscopic ensembles of spins
and considerable effort has to be devoted to the preparation of
pseudo-pure states of spins with initial thermal population
distribution. This preparation leads to an exponentially
growing cost (with the number *N* of qubits) either in
signal strength or the number of experiments involved [arXiv:quant-ph/0012108
(2000)], since the fraction of spins in their ground state
is proportional to *N/2 ^{N}*.

Trapped ions, on the other hand, provide individual qubits -
for example, hyperfine states as described in this work - well
isolated from their environment with read-out efficiency near
unity. It would be desirable to combine the advantages of
trapped ions and NMR techniques in future experiments using
either ''conventiona'' ion trap methods, but now with rf
radiation as outlined above [QIS3],
or treating the ion string as a *N*-qubit molecule with
adjustable spin-spin coupling constants [QIS8,
QIS14,
QIS33]:
In a suitably modified ion trap, ionic qubit states are
pairwise coupled by MAGIC. This spin-spin coupling can be
formally described in the same way as J-coupling in molecules
used for NMR, even though the physical origin of the
interaction is very different. Thus, successful techniques and
technology developed in spin resonance experiments, like NMR or
ESR, can immediately be applied to trapped ions. An advantage
of an artificial ''molecule'' in a trap is that the coupling
constants *J _{ij}* between qubits

*i*and

*j*can be chosen by the experimenter by setting the magnetic field gradient, the secular trap frequency, and the type of ions used. In addition,

**individual**spins can be detected state selectively with an efficiency close to 100% by collecting scattered resonance fluorescence.