We theoretically study cooling of flexural modes of a mechanical oscillator by Bose-Einstein Condensate (BEC) atoms (Rb87) trapped in a magnetic trap. The mechanical oscillator with a tiny magnet attached on one of its free ends produces an oscillating magnetic field. When its oscillating frequency matches certain hyperfine Zeeman energy of Rb87 atoms, the trapped BEC atoms are coupled out of the magnetic trap by the mechanical oscillator, flying away from the trap with stolen energy from the mechanical oscillator. Thus the mode temperature of the mechanical oscillator is reduced. The mode temperature of the steady state of the mechanical oscillator, measured by the mean steady-state phonon number in the flexural mode of the mechanical oscillator, is analyzed. It is found that ground state (phonon number less than 1) may be accessible with optimal parameters of the hybrid system of a mechanical oscillator and trapped BEC atoms.
Recently, the authors have found a flaw in the basic modeling of the system presented in this paper. In modeling a hybrid system of a mechanical oscillator and BEC atoms, the hyperfine states |1, −1 and |1, 0 of 87 Rb are chosen as eigenstates in the Jaynes-Cummings Hamiltonian, which will not cool the flexural modes of the mechanical oscillator. The proper choice of eigenstates of 87 Rb in the Jaynes-Cummings Hamiltonian should be the hyperfine states |1, −1 and |2, −2. When using the hyperfine states |1, −1 and |2, −2 of 87 Rb as eigenstates in the Jaynes-Cummings Hamiltonian, the theoretical analyses given in Secs. II and III are still valid. However, this new choice of eigenstates will greatly affect the methods of cooling the mechanical oscillators. To match the energy of the hyperfine states |1, −1 and |2, −2 , the frequency of the mechanical oscillators is in the order of 6 GHz, instead of a tunable frequency range from a few MHz to a few hundred MHz in this paper. Also, the coupling strength between the mechanical oscillator and 87 Rb atoms is also reduced. Therefore, Sec. IV should be reconsidered. The cooling efficiency for mechanical oscillators in this study is greatly reduced. Since the cooling efficiency is an indispensable part of this paper, we wish to retract this paper to avoid misleading the readership. We regret the error.
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