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We propose a possibility of a phonon laser by coupling a Bose-Einsteincondensate to a nanomechanical cantilever with a magnetic tip. Due to themagnetic coupling, atomic spin flips induce cantilever motion which can be usedto produce a phonon laser. The system is described by the equivalent of theJaynes-Cummings Hamiltonian. By controlling the number of atoms and thepopulation inversion, one can obtain either a continuous wave (cw) or transientlasing. The two-body atom-atom interaction is also shown to coherentlymanipulate the lasing process. We also show that in the strong coupling limit,the same system can undergo a Dicke-Hepp-Lieb superradiant phase transition.Exotic phase diagrams can be obtained by tuning the two body atom-atominteraction.

Phonon Laser Effect and Dicke-Hepp-Lieb Superradiant Phase Transition in Magnetic Cantilever Coupled to a Bose Einstein Condensate