The realization of quantum simulation and quantum computation requires an exquisite level of control over composite quantum systems. This includes their isolation from environmental noise while at the same time retaining the ability to control the system and realize high fidelity quantum operations between individual quantum particles that make up the system. Perhaps the current frontrunner in the race towards these goals are ion traps which trap chains of ions in ultrahigh vacuum and permit their control by external microwave and laser fields. Several Nobel Prizes have been awarded for their development including the 2012 award to Dave Wineland. In close collaboration with several experimental physics teams we are exploring the development of ion traps as quantum information processors, quantum simulators and as testbeds for the statistical mechanics of mesoscopic systems.

Quantum Gates:

A classical computer strings together all its operations from a set of basic logical gates. In a quantum information processor this is accomplished by quantum gates, i.e. usually basic two-particle interactions whose properties we need to control carefully. If we ever want to realize a quantum computer we will need those gates to possess error rates that are of the order of 1 part in 10000. We are taking steps towards this goal by finding tricks to make fundamental quantum gates more robust against noise or by making them faster so that the environment does not have time to introduce noise.

- A. Bermudez, P.O. Schmidt, M.B. Plenio and A. Retzker.
*Robust Trapped-Ion Quantum Logic Gates by Microwave Dynamical Decoupling**.**Physical Review A***85**, 040302(R) (2012) - A. Bermudez, T. Schaetz and M.B. Plenio,
*Dissipation-assisted quantum information processing with trapped ions**.**Physical Review Letters***110**, 110502 (2013) - A. Lemmer, A. Bermudez and M.B. Plenio.
*Driven Geometric Phase Gates with Trapped Ions*.*New Journal of Physics***15***,*083001 (2013)

The ideas for the robust trapped-ion quantum logic gates were put into practice in the labs of NIST (Boulder, USA)

- T. R. Tan, J. P. Gaebler, R. Bowler, Y. Lin, J. D. Jost, D. Leibfried, and D. J. Wineland.
*Demonstration of a Dressed-State Phase Gate for Trapped Ions*.*Physical Review Letters***110***,*263002 (2013)

Quantum Many Body Dynamics in Ion Traps:

Besides quantum information processing a goal of considerable current interest is the realization of specific quantum many body systems and the observation of their dynamics. This may allow us eventually to test models for complex quantum systems whose dynamics we cannot reproduce on a classical computer. Such devices also provide flexibility in the choice of their parameters that is much wider than for the real system that they are modeling.

- A. Bermudez and M.B. Plenio.
*Quantum Spin-Peierls Phase Transition in Trapped-Ion Crystals**.*Phys. Rev. Lett.**109**, 010501 (2012) - A. Bermudez, J. Almeida, F. Schmidt-Kaler, A. Retzker and M.B. Plenio.
*Frustrated Quantum Spin Models with Cold Coulomb Crystals**.*Phys. Rev. Lett.**107**, 207209 (2011) - A. Bermudez, J. Almeida, K. Ott, H. Kaufmann, S. Ulm, F. Schmidt-Kaler, A. Retzker and M.B. Plenio.
*Quantum Magnetism of Spin-Ladder Compounds with Trapped-Ion Crystals**.*New J. Phys.**14**, 093042 (2012)

Statistical Mechanics in Ion Traps:

Because of the exquisite control that ion traps can exert over ion crystals we are now beginning to be in a position to use them to test a wide variety of phenomena in non-equilibrium statistical mechanics. This includes for example the Kibble-Zurek mechanism, namely the defect formation in a symmetry breaking phase transition, which has originally been proposed in the study of quantum fields in the early universe and that we have predicted and observed in ion trap experiments.

With increasing complexity of such systems it is becoming a challenge to measure their properties. Hence we have started to extend the toolbox of spectroscopic techniques for trapped-ion experiments in order to better characterize the dynamics of crystals of trapped ions. To this end, we have adapted the technique of 2D spectroscopy from the field of NMR and devised a protocol for the interrogation of the dynamics of crystals of trapped ions.

- K. Pyka, J. Keller, H. Partner, R. Nigmatullin, T. Burgermeister, D.M. Meier, K. Kuhlmann, A. Retzker, M.B. Plenio, W.H. Zurek, A. del Campo, T.E. Mehlstäubler.
*Causality and Defects as Relics of Symmetry Breaking in Ion Traps**.**Nature Communications***4**, 2291 (2013) - S. Ulm, J. Roßnagel, G. Jacob, C. Degünther, S.T. Dawkins, U.G. Poschinger, R. Nigmatullin, A. Retzker, M.B. Plenio, F. Schmidt-Kaler and K. Singer.
*Observation of the Kibble-Zurek mechanism in ion crystals**. Nature Communications***4**, 2290 (2013) - A. Bermudez, M. Bruderer and M.B. Plenio.
*Controlling and measuring quantum transport of heat in trapped-ion crystals**.**Physical Review Letters***111***,*040601 (2013) - A. Lemmer, C. Cormick, C. T. Schmiegelow, F. Schmidt-Kaler, and M.B. Plenio.
*Two-dimensional Spectroscopy for the Study of Ion Coulomb crystals**. Physical Review Letters***114***,*073001 (2015)