= ASSESSMENT OF CURRENT RESULTS AND OUTLOOK ON FUTURE EFFORTS =

QUANTUM COMPUTING

NEUTRAL ATOMS, MOLECULES AND CAVITY QED

A. Physical approach and perspective

Neutral atoms and molecules provide a promising test bed for the development of scalable general purpose quantum processors, and for quantum simulators as special purpose quantum computers involving a very large number of qubits.

As in the case of ions, qubits can be represented by long-lived internal atomic and molecular states in electronic ground states (hyperfine levels, rotational states), or in metastable excited electronic states, which can be manipulated by optical and microwave fields. The unique promises of neutral atom quantum computing rest in particular on the well developed cooling and trapping techniques, as exemplified by laser cooling, realization of Bose Einstein condensates and quantum degenerate Fermi gases, in combination with optical, magnetic and electric traps, realized in free space or in cavities or on atom chips. Such techniques provide an ideal starting point to build and prepare large scale quantum registers with high fidelity. At present these trapping and cooling techniques are being extended to molecules, including, for example, electric on-chip traps for polar molecules. The scenarios of quantum computing with neutral atoms are directly linked to the development of specific trapping techniques. First, traps can be developed allowing the independent manipulation of the centre-of-mass degrees of freedom of individual atoms and molecules, including the addressing of single qubits, which is a necessary requirement for general purpose quantum computing; and massively parallel, identical manipulations of large number of qubits, as realized, for example, in the context of optical lattices with only global addressing, as is relevant in the context of quantum simulators of translational invariant condensed matter systems.

Entanglement of neutral atom or molecule qubits is based on the following physical mechanisms

• controlled qubit-dependent two-particle interactions, as for example in cold coherent collisions, or the dipole-dipole interactions between (highly) excited atomic states;
• cavity QED setups, where distant qubits are entangled via photon exchange, which plays the role of a quantum data bus.

In the cavity QED scenario, the atomic or molecular qubit is strongly coupled to a high-Q cavity, either in the optical domain by coupling to an electronic excitation, or in the microwave regime for a transition between Rydberg states or rotational states of a polar molecule. Two-qubit gates between distant qubits can be achieved via photon exchange as quantum data bus, in close formal analogy to the phonon data bus of collective oscillation modes in trapped ions. These cavity QED setups also provide a natural interface to quantum communication with photons.

Atoms and molecules can be stored in optical lattices, corresponding to an array of microtraps generated by counterpropagating laser fields. The dynamics of cold atoms loaded into optical lattices can be described by a Hubbard model, with atoms hopping between lattice sites, and interacting via collisions. Thus cold atoms in optical lattices provide a direct way to simulating condensed matter systems with a large number of bosons or fermions. In addition, loading an optical lattice from an atomic Bose Einstein condensate provides via the superfluid-Mott insulator transition the preparation of a Mott phase with exactly one atom per lattice site, and thus the preparation of a very large number of atomic qubits. These can be entangled in parallel operations with qubit-dependent controllable 2-particle interactions, provided, for example, by coherent collisional interactions in combination with movable qubit (spin) dependent optical lattices. This provides the basis for a digital quantum simulator, for example of a spin lattice system, where the time evolution generated by the Hamiltonian is decomposed into a series of single and two-qubit gates performed in parallel on all qubits (spins).

For single atoms strongly coupled to an optical cavity, single photons for the purpose of exchanging quantum information between remote locations can be generated on demand and with high quantum efficiency. Protocols for generating a stream of photons with entanglement mediated and controlled by a single intracavity atom have been proposed. In addition to these deterministic mechanisms for entanglement, probabilistic protocols can be developed which are based on free space atoms emitting photons where entanglement is achieved by appropriate photon detection.

Currently, quantum computing with neutral atoms is investigated experimentally in several dozen laboratories worldwide, with half of them located in Europe. The European groups working with a controllable number of atoms include I. Bloch (Mainz, DE), T. Esslinger (Zurich, CH), P. Grangier (Orsay, FR), S. Haroche (Paris, FR), D. Meschede (Bonn, DE), G. Rempe (Garching, DE), and H. Weinfurter (Munich, DE). Several other groups are presently setting up new experiments, including W. Ertmer (Hannover, DE), E. Hinds (London, UK), J. Reichel (Paris, FR), and J. Schmiedmayer (Vienna, AT). The experimental program is strongly supported by implementation-oriented theory groups like H. Briegel (Innsbruck, AT), K. Burnett (Oxford, UK), J. I. Cirac (Garching, DE), A. Ekert (Cambridge, UK), P. L. Knight (London, UK), M. Lewenstein (Barcelona, ES), K. Mølmer (Aarhus, DK), M. B. Plenio (London, UK), W. Schleich (Ulm, DE), P. Tombesi (Camerino, IT), R. Werner (Braunschweig, DE), M. Wilkens (Potsdam, DE), and P. Zoller (Innsbruck, AT). In fact, European theory groups have played a crucial role in the development of QIPC science from the very beginning. The close collaboration between experiment and theory in Europe is unique, partly because of the support provided by the European Union.

B. State of the art

I. Quantum memories: The strength of using neutral atoms for QIPC is their relative insensitivity against environmental perturbations. Their weakness comes from the fact that only shallow trapping potentials are available. This disadvantage is compensated by cooling the atoms to very low temperatures. So far, several different experimental techniques for trapping and manipulating neutral atoms have been developed:

Optical tweezers and arrays of optical traps allow for the preparation of a well-defined quantum state of atomic motion, as can be achieved by either cooling single atoms into the ground state of the trapping potential, or by loading a Bose-Einstein condensate into an optical lattice. While the first approach allows for individual atom manipulation, both methods offer the possibility of a massive parallelism, with many pairs of atoms colliding at once. The landmark results attained are:

::* Single atoms were trapped with a large aperture lens, thus providing a three-dimensional sub-wavelength confinement.

::* Single atoms were also loaded into the antinodes of a one-dimensional standing wave, and excited into a quantum superposition of internal states.

::* This superposition was preserved under transportation of the atoms; coherent write and read operations on individual qubits were performed.

::* A small number of atoms were loaded into a two-dimensional array of dipole traps made with a microlens array, and the atoms were moved by moving the trap array.

::* Single atoms were loaded into the antinodes of a three-dimensional optical lattice, by starting from a Bose-Einstein condensate and using a Mott transition.

Atom chips: The ability to magnetically trap and cool atoms close to a surface of a micro-fabricated substrate (for example using micro-magnetic potential wells produced by micron-sized current carrying wires or microscopic permanent magnets) has led to an explosive development of atom chips in the past few years. Such devices are very promising building blocks for quantum logic gates due to their small size, intrinsic robustness, strong confinement, and potential scalability. The main accomplishments they have attained include:

::* Cooling of atoms to quantum degeneracy (Bose-Einstein condensation).

::* Transport of an ensemble of atoms using a magnetic conveyor belt.

::* Very long coherence times by using appropriate qubit states.

::* Multilayer atom chips with sub-µm resolution and smooth magnetic potentials.

::* On-chip single-qubit rotation via two-photon transitions on hyperfine qubits.

::* Single-atom detection using a fibre Fabry-Perot cavity.

Traps for polar molecules at the individual level have recently been proposed, based on microwave or electric fields, and are the subject of growing experimental investigation. On the experimental side,

::* cold polar molecules at millikelvin temperatures have been produced by several different techniques, including deceleration of supersonic molecules, filtering of slow molecules from a thermal ensemble, and Helium buffer gas cooling in a cryogenic environment,

::* ensembles of cold polar molecules have been stored in magnetic or electric bottles.

II. Entangling gates: a variety of schemes have been proposed theoretically, based on interactomic interactions either direct (for instance collisional – possibly enhanced by Feshbach resonances – or between dipoles of Rydbrg excited atoms) or mediated by a quantum data bus, i.e. a different degree of freedom (for instance photons in a high-finesse cavity mode).

Optical tweezers and arrays of optical traps are ideal to perform collisional gates, which require the preparation of a well-defined quantum state of atomic motion. In this field, a highly parallelized quantum gate was implemented by state-selectively moving the atoms, and making them interact using cold collisions. This landmark experiment has pioneered a new route towards large-scale massive entanglement and quantum simulators with neutral atoms.

Cavity QED, possibly in combination with optical dipole traps, is the most promising technique for realizing an interface between different carriers of quantum information, implemented either with free-space atoms emitting photons in a random direction (probabilistic approach), or with atoms in high-finesse cavities where the strong atom-photon coupling guarantees full control over photon emission and absorption (deterministic approach). The latter approach can be realized both with Rydberg atoms in microwave cavities as well as with ground-state atoms in optical cavities. If each atom resides in its own cavity, the scheme guarantees addressability and scalability in a unique way. As quantum information is exchanged via flying photons, the individual qubits of the quantum register can easily be separated by a large distance. The photon-based scheme is therefore ideal to build a distributed quantum network. The main achievements in this sector include:

1. Probabilistic approach in free space:

::* A single trapped atom has been entangled with a single photon.

2. Deterministic approach using microwave cavities: Circular Rydberg atoms and superconducting cavities are proven tools for fundamental tests of quantum mechanics and quantum logic:

::* Complex entanglement manipulations on individually addressed qubits with long coherence times have been realized.

::* Gates have been demonstrated.

::* New tools for monitoring decoherence of mesoscopic quantum superpositions have been developed.

3. Deterministic approach with optical cavities:

::* The strong atom-photon coupling has been employed to realize a deterministic source of flying single photons, a first step towards a true quantum-classical interface.

::* With single photons, two-photon interference effects of the Hong-Ou-Mandel type have been observed. These experiments demonstrate that photons emitted from an atom-cavity system show coherence properties well suited for quantum networking.

::* Moreover, single atoms were optically trapped inside a cavity for such a long time that experiments can be performed with just one single atom.

::* A novel cooling technique avoiding spontaneous emission was successfully implemented.

::* Single or a small number of individually addressable atoms was deterministically transported in and out of a cavity by means of an optical conveyor belt.

All of the achievements reported in this section have been realized within European labs, and in many cases they are purely European achievements, in the sense that they are not to be found in labs outside Europe.

C. Present challenges

Most neutral-atom systems have not yet demonstrated two-qubit operations, mainly because the technology to perform single-atom experiments is relatively new (less than 10 years).

Optical tweezers and arrays of optical traps are most advanced in manipulating neutral-atom qubits.

1. In optical tweezers and small-scale dipole trap arrays, the main challenges are first to implement a two-qubit quantum gate, e.g., using a controlled collision of two atoms, and then to increase the size of the quantum register to more than 2 atoms.
2. In optical lattices, full addressability of each individual qubit of the closely spaced register is one of the main challenges.
3. In both scenarios, the speed of a gate must eventually be increased by implementing a collision which exhibits a large cross section, for example by involving Rydberg atoms or molecular (e.g., Feshbach) interactions.

Atom chips: Despite their recent achievements, experiments with atom chips are still facing a large number of challenges for implementing QIPC.

1. A quantum memory, that is the reading and writing of quantum information into single atoms or atomic ensembles must be realized.
2. Next, a two-qubit quantum gate, for example by employing a controlled collision, must be implemented.
3. The full demonstration of the potential provided by atom chips requires the realization of large-scale integration, e.g., with several 10 qubits.
4. Potential roughness very close (µm) to micro-fabricated structures is of concern for qubit storage and transport. Even though for current-carrying structures the problem can be solved and compensated for by the design and fabrication methods as developed recently, micro-structures with fewer defects might be needed for permanent magnets.
5. Merging atom-chip technology and cavity QED is promising. High-finesse miniature optical or microwave cavities can be coupled to ground state or Rydberg atoms trapped on a chip. Coherence preserving trap architectures are an important first step towards a fully scalable architecture combining the best of both worlds.

Polar molecules: Research with polar molecules has just started and, hence, is still facing a large number of experimental challenges. Some of these are:

1. As laser cooling methods developed for atoms fail for molecules, new cooling techniques need to be developed to reach the ultracold regime.
2. The number of molecules and their density needs to be increased before collisions can be observed in electric trapping experiments.
3. Efficient molecule detection techniques must be developed in particular for experiments involving only single or a few molecules.

Cavity QED: The main difficulty in implementing QIPC protocols in present demonstration experiments is the enormous technological complexity required to obtain full control over both atoms and photons at the single-particle level.

1. The probabilistic approach suffers from the low efficiency of photon generation and detection, and the large solid angle of photon emission for a free-space atom.
2. The deterministic approach employing microwave cavities has intracavity-photon generation and absorption efficiencies close to 100%, and the implementation of simple algorithms is in view.

::* One of the main challenges is the demonstration of scalability. The preparation of a non-local entangled and possibly mesoscopic quantum state shared between two remote cavities is a major task.

::* Another challenge is the realization of quantum feedback or error correction schemes to preserve the quantum coherence of the field stored in a cavity with a finite quality factor. 3. The deterministic approach utilizing optical cavities has led to photon-emission efficiencies of up to about 30 %. Challenges are

::* To entangle in a deterministic manner a single atom with a single photon, and to teleport the quantum states between distant photon-emitting and photon-receiving atoms.

::* In order to integrate individual quantum-network nodes into a scalable quantum-computing network, a set of individually addressable atoms located in different cavities must be implemented.

::* Moreover, single-photon quantum repeaters which are necessary to communicate quantum information over large distances need to be developed.

::* Ultimately, the gate speed should be increased by installing a few-wavelength long cavity. The combination of such a micro-cavity with presently available trapping and cooling techniques is a challenge.

In the microwave domain, a method of deterministically transporting single atoms in and out of a cavity, for example by means of an optical conveyor belt, is needed to address the individual atoms of a stationary quantum register.

A major challenge for theory is to characterize and optimize the suitability of each of the available and proposed experimental systems as platforms for general-purpose quantum computing or rather for quantum simulation.

All of the strategic challenges in this section represent current or planned activity at European labs.

D. Key references

A tutorial review on QIPC with atoms, ions and photons can be found in, e.g.:

[1] C. Monroe, Quantum Information Processing with Atoms and Photons, Nature 416, 238-246 (2002).

[2] J.I. Cirac and P. Zoller, New Frontiers in Quantum Information with Atoms and Ions, Physics Today 38-44 (March 2004).

Category:ERA Quantiki Project