Quantum feedback cooling of a trapped nanoparticle by using a low-pass filter

We propose a low-pass-filter (LPF) feedback control for cooling a trapped particle with a low-pass filter, which utilizes a shift of the potential caused by the feedback operation. By incorporating this shift in the energy cost function, we show that the LPF control can achieve the minimum phonon occupation number that is lower than cold damping with a band-pass filter, that with delayed feedback, and linear--quadratic--Gaussian (LQG) control, the last two of which are the standard methods of ground-state cooling of a levitated nanoparticle. For the detection efficiency of $90\%$, the achievable phonon occupation number with the LPF control is about one third, two fifths and one half of that of cold damping with a band-pass filter, that with delayed feedback, and LQG control, respectively. Thus our method has a decisive advantage to reach the absolute ground state.