Background
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
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
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 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
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
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 Er=(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 Er and the quantized motional energy of the trapped atom, 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
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/2N.
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 Jij 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.