The present state-of-the-art in cooling mechanical resonators is a version of "sideband" cooling. Here we present a method that uses the same configuration as sideband cooling -coupling the resonator to be cooled to a second microwave (or optical) auxiliary resonator -but will cool significantly colder. This is achieved by varying the strength of the coupling between the two resonators over a time on the order of the period of the mechanical resonator. As part of our analysis, we also obtain a method for fast, high-fidelity quantum information-transfer between resonators.PACS numbers: 85.85.+j,42.50.Dv,85.25.Cp, There is presently a great deal of interest in cooling high-frequency micro-and nano-mechanical oscillators to their ground states. This interest is due to the need to prepare resonators in states with high purity to exploit their quantum behavior in future technologies [1,2]. The key measure of a cooling scheme is the cooling factor, which we will denote by f cool . The cooling factor is the ratio of the average number of phonons in the resonator at the ambient temperature, n T , to the average number of phonons achieved by the cooling method, which we will denote by n cool . The present state-of-the-art for cooling mechanical resonators is sideband cooling, which was originally developed in the context of cooling trapped ions [3][4][5]. This method is a powerful and practical technique, able to achieve large cooling factors, and these have been demonstrated in the laboratory [6][7][8][9][10][11][12][13][14][15].In the context of mechanical resonators, sideband cooling involves coupling the resonator to be cooled (from now on the "target") to a microwave or optical resonator (the "auxiliary") whose frequency is sufficiently high that it sits in its ground state at the ambient temperature. The resonators are coupled together by a linear interaction, and one that is straightforward to implement experimentally. In particular, if we denote the annihilation operators for the target and auxiliary resonator by a and b, respectively, then the full Hamiltonian of the two resonators iswhere x a = a + a † and x b = b + b † are the position operators of the respective resonators. The coupling is modulated at the difference frequency between the resonators, ν = Ω − ω. This converts the high frequency of the auxiliary resonator so that the two resonators are effectively on-resonance, and thus exchange energy at the coupling rate g. With this frequency conversion, the auxiliary constitutes a source of essentially zero entropy (and thus zero temperature) for the target resonator [16].When the rate of the coupling, g, is significantly smaller than the frequency ω of the target resonator (so that one is within the rotating-wave approximation (RWA)-see, e.g. [17]), then the linear coupling between the resonators is merely excitation (phonon/photon) exchange between the two. If the auxiliary is now damped sufficiently rapidly, then the excitation exchange, combined with the relatively fast damping of the auxiliary at effec...