Fig. 1. (a) Diagram of the population jump rates between neighboring Fock states. (b) Eigenvalues of the dissipation rate operators κn^± versus number n. (c) Number statistics distribution of the steady state. The probability Pn increases versus n in the region dominated by the negative-temperature dissipation and decays in the rest region dominated by the positive-temperature one, so a peak appears in the intermediate region.

Fig. 2. (a) Cavity optomechanical scheme of feedback control. The optical cavity is coupled to the mechanical oscillator through dispersive and dissipative optomechanical interactions simultaneously. With the dispersive coupling, the oscillator undergoes a shift proportional to the radiation pressure force, i.e., to the photon number, and then changes the cavity dissipation rate κx^+ through the dissipative coupling. Except for the optomechanical dissipation, the cavity mode has a gain of rate κ− induced by the negative-temperature reservoir, and the oscillator is subjected to Brownian thermal noise. The high frequency of the optical mode makes our near-zero temperature assumption reasonable. (b) Dissipation control protocol. The positive-temperature dissipation rate κx^+ is smaller than the negative-temperature one in the region x<L but increases rapidly and overtakes it in the region x>L. The steep change occurs mainly in a region of width d.

Fig. 3. Steady-state photon number statistics obtained by approximate solution [Eq. (13)] and numerical simulation of the master equation [Eq. (7)]. (a) Photon number fluctuation Δn versus dissipation ratio γ/κ−. The approximate solution is plotted in a red solid line, whereas, the numerical results are marked with a “+.” (b) Numerical results for the steady-state probability distribution of photon number for increasing γ/κ−. All results for g0=7.07×102ωm, (κ0,κ−,κv)=(10−1,10−2,10−3)ωm, and (d,L)=(14,7×104)xzpf.