Abstract:
In this thesis, a system including one optical fibre and one microtoroid system is first inspected. This system provides us a so-called resolved sideband (RSB) cooling mechanism which can cool the phonon number of a resonator to be lower than unit. We deduce the master equation via adiabatic elimination and evaluating the spectrum of this system by using master equation and Langevin equation. Then based on this system and its cooling mechanism, a system consisting of one optical fibre and two microtoroids is explored. We consider two cases. The first is called one-way driving system because there is one input field travelling in the fibre. The second is called two-way driving system in which an additional input field comes from the opposite direction in the fibre. In one-way driving system, two microtoroids are correlated by laser light. Small fractions of laser light are sent from the fibre into toroids to form clockwise propagating optical cavity modes. The optical cavity mode is coupled to the mechanical toroid mode via radiation pressure force. For the optical cavity mode in the first (second) toroid, a blue (red) resonant frequency detuning is turned by amount of the frequency of the mechanical mode which invokes a significantly effective heating (cooling) process. We conduct our investigation under two regimes. First, the frequency of the mechanical mode is much larger than the decay rate of the cavity mode and they are both much greater than the damping rate of the mechanical mode. Second, the heating (cooling) rates induced by the laser detuning is much larger than the damping rate of the mechanical mode. In the interesting regimes, the reduced master equation describing the two correlated microtoroids is applied to investigate how an EPR type entanglement between them can be created. It is concluded small initial phonon numbers of the toroids, low dissipation rates of mechanical modes and high laser induced rates can lead the system to be close to perfect correlation. We desire the frequency of the mechanical mode to be greater than the decay rate of optical cavity mode as much as possible. In two-way driving system, because of the extra input field, there is an additional counterclockwise propagating optical cavity field in each toroid. The field is also detuned as for the clockwise optical cavity modes and similarly give rise to heating and cooling effects respectively in the two toroids. An additional condition that the rates induced by the extra detuned laser should be as small as possible is required. By Langevin equation, we can compute the equations of motion of optical cavity modes and furthermore obtain a expression for the output field of a system. We find the output fields of the two systems are both in terms of mechanical modes of toroids and input fields. This suggests we can combine master equation and Langevin equation to evaluate the spectra of them. To attain a state close to EPR entanglement, extremely low initial thermal phonon numbers of toroids are necessary. Before performing entangling operation, by applying resolved sideband cooling scheme for the one-way driving system, theoretically it is possible to cool both phonon numbers of toroids to be less than unit. But it needs a longer period to cool the second one. And the dissipation rate of the mechanical mode may be required to be even lower.