  An optical cavity enhances the interaction between atoms and light, and the
rate of coherent atom-photon coupling can be made larger than all decoherence
rates of the system. For single atoms, this strong coupling regime of cavity
quantum electrodynamics (cQED) has been the subject of spectacular experimental
advances, and great efforts have been made to control the coupling rate by
trapping and cooling the atom towards the motional ground state, which has been
achieved in one dimension so far. For N atoms, the three-dimensional ground
state of motion is routinely achieved in atomic Bose-Einstein condensates
(BECs), but although first experiments combining BECs and optical cavities have
been reported recently, coupling BECs to strong-coupling cavities has remained
an elusive goal. Here we report such an experiment, which is made possible by
combining a new type of fibre-based cavity with atom chip technology. This
allows single-atom cQED experiments with a simplified setup and realizes the
new situation of N atoms in a cavity each of which is identically and strongly
coupled to the cavity mode. Moreover, the BEC can be positioned
deterministically anywhere within the cavity and localized entirely within a
single antinode of the standing-wave cavity field. This gives rise to a
controlled, tunable coupling rate, as we confirm experimentally. We study the
heating rate caused by a cavity transmission measurement as a function of the
coupling rate and find no measurable heating for strongly coupled BECs. The
spectrum of the coupled atoms-cavity system, which we map out over a wide range
of atom numbers and cavity-atom detunings, shows vacuum Rabi splittings
exceeding 20 gigahertz, as well as an unpredicted additional splitting which we
attribute to the atomic hyperfine structure.
