5 optimizing over the realizable coupling between the phonons and the ion’s internal levels, ˆ Xσ$_{θ}$ , and periods of free evolution. Surprisingly, even this meager palette is sufficient to implement a red sideband-like operation and drive energy from the phonons to the ions faster than the trapping frequency. Furthermore, the principles de- scribed above are applicable to a very wide range of sys- tem and robust enough to be implementable in ongoing experiments. Acknowledgments — A.R. acknowledges the support of EPSRC project number EP/E045049/1 and MBP ac- knowledges support from the Royal Society and the EU STREP project HIP and the Alexander von Humboldt Professorship. [1] T.W. Hnsch and A.L. Schawlow, Optics Communications 13 , 68, 1975. [2] D. Wineland and H. Dehmelt, Bull. Am. Phys. Soc, 20 , 637 (1975). [3] P. Lett et al. Phys. Rev. Lett. 61 169 (1988). [4] J. Dalibard et al.Atomic Physics 11 . [5] J. Dalibard and C. Cohen-Tannoudji J. Opt. Soc. Am. B 6 , 2023 (1989). [6] D. J. Wineland et al.J. Opt. Soc. Am. B 9 , 32 (1992). [7] A. Aspect et al.J. Opt. Soc. Am. B 6 , 2112 (1989). [8] M. Kasevich and S. Chu Phys. Rev. Lett. 69 , 1741 (1992). [9] G. Morigi et al. Phys. Rev. Lett. 85 , 4458 (2000). [10] C. Monroe et al. , Phys. Rev. Lett. 75 , 4011 (1995). [11] P. Rabl et al. , Phys. Rev. B 79 , 041302 (2009). [12] I. Wilson-Rae, P. Zoller, and A. Imamoglu, Phys. Rev. Lett. 92 , 075507 (2004). [13] S. Machnes, quant-ph/0708.0478, http://qlib.info [14] T. Monz et al. , Phys. Rev. Lett. 103 , 200503 (2009). [15] D. Leibfried, R. Blatt, C. Monroe and D.J. Wineland., Rev. Mod. Phys. 75 , 281 (2003) [16] J. I. Cirac, L. M. Duan, and P. Zoller, quant-ph/0405030 [17] M. Suzuki, Physica A 194 , 432 (1993) [18] A. Retzker, J. I. Cirac, and B. Reznik, Phys. Rev. Lett. 94 050504 (2005) [19] H. Baker, Proc Lond Math Soc (1) 34 347–360 (1902) ibid (1) 35 (1903) 333–374; ibid (Ser 2) 3 24–47 (1905) [20] J. Campbell, Proc Lond Math Soc 28 381–390 (1897) ibid 29 (1898) 14–32. [21] F. Hausdorff, Ber Verh Saechs Akad Wiss Leipzig 58 19–48 (1906) [22] M.W. Reinsch, J. Math. Phys. 41 (4):2434 (2000). [23] D.J. Wineland, R.E. Drullinger, F.L. Walls, Phys. Rev. Lett. 40 1639 (1978) [24] C. Monroe, et al. , Phys. Rev. Lett. 75 , 4011 - 4014 (1995) [25] V. Vuletic, C. Chin, A.J. Jerman, S. Chu, Phys. Rev. Lett. 81 , 5768 (1998) [26] The mirror can be mounted to the electrodes of the ion trap as currnetly done in the Ion trap group in Ulm. Private communication with Kilian Singer. [27] I. Wilson-Rae et al. , Phys. Rev. Lett. 99 , 093901 (2007). [28] F. Marquardt et al. , Phys. Rev. Lett. 99 , 093902 (2007).