Normal Mode Splitting, Nonlinear Multistability And Quantum Entanglement In Optomechanical Cavities

With the rapid development of semiconductor materials and experimentalprocess, the optomechanical system has been microminiaturized up to micrometeror nanometer scales, which makes the quantum effect of the nanomechanicalresonator more obvious. To observe the quantum effect of the nanomechanicalresonator, some people try to cool the nanomechanical resonator in various ways,and so far the lower temperature105Khas been reached. After cooling of thenanomechanical resonator, since the quantum state plays an important role in thequantum computation and quantum information and the quantum non-demolitionmeasurement, so that some people want to control the quantum state of thenanomechanical resonator in order to show the clear picture from the quantumworld to the classical world. In the context of the above expression, we study thenormal mode splitting, nonlinear multistability, and quantum entanglement basedon three different optomechanical systems in this paper, respectively.First, we study the effect from the quadratic term of the reactive coupling onthe normal splitting when the microdisk-waveguide optomechanical system issteady state. With the help of the numerical simulation, we find that the normalmode splitting is not sensitive to the quadratic term for a weak pump field, so thequadratic term can be safely neglected. When the pump field is strong enough, wefind that the generated lower-frequency Stokes mode moves to the more lowerfrequency and the generated higher-frequency Stokes mode dose not almostchange with the coefficient of the quadratic term increasing. In addition, we alsofind that the four-wave-mixing process is more efficient among the pump field,the generated lower-frequency Stokes mode, and the waveguide-resonator. When the pump field is strong enough, we find the amplitudes of the generatedlower-frequency Stokes mode and the anti-Stokes field originating from thefour-wave-mixing will be amplified for the large coefficient of the quadratic termof the reactive coupling due to the efficient energy transfer from the pump field.Second, beyond the weak cavity field approximation we study the steady-state behaviors for the nanomechanical resonator, the quantified cavity field, andthe cold atoms confined in the cavity, respectively. With the help of the numericalsimulation, under the relevant parameters determined we find that three branchesof steady-state solutions exist for the system when the original cavity frequency islarger than the transition frequency of the atom. On the contrary, only one branchof steady-state solutions emerges. In addition, when there are a lot of photonsinside the cavity the weak cavity field approximation will results in the largeerrors for the first and second branches of steady-state solutions in the regionswhere three branches of steady-state solutions simultaneously exist. In particular,the first branch of steady-state solutions will show the bistable behaviordepending critically on the coupling intensity between the pump field and theatom and the driving power. The above results should be attributed to an effectivefeedback mechanism originating from the strong coupling betweenatom-light-resonator interactions.Finally, when the quantum-well-assisted optomechanical system is steadystate, we study the bipartite entanglement among the exciton in the quantum well,the cavity field, and the nanomechanical resonator, especially the bipartiteentanglement between the exciton and the nanomechanical resonator. And theinteraction between the exciton and the nanomechanical resonator is indirect.With the help of the numerical simulation, under the relevant parametersdetermined we find that three bipartite possible entanglements will together existand the bipartite entanglement between the exciton and the nanomechanical resonator reaches its maximal value when the cavity field resonates with theanti-Stokes sideband of the driving field and the exciton resonates with the Stokessideband of the driving field. And we also find that the maximal value for thebipartite entanglement between the exciton and the nanomechanical resonatorgradually decreases with the temperature increasing and finally vanishes whenT≈20K. Further, we also find that the maximal value for the bipartiteentanglement between the exciton and the nanomechanical resonator firstincreases then decreases with the nanomechanical resonator mass increasing andfinally vanishes when m≈100ng.The maximal value for the bipartiteentanglement between the exciton and the nanomechanical resonator depends onthe input power of the driving field in a similar way as the mass. In addition,under the relevant parameters determined, we find that the bipartite entanglementbetween the exciton and the nanomechanical resonator still remains the identicalmaximal value in the elliptic regions formed by the detunings, when the cavityfield approximately resonates with the anti-Stokes sideband of the driving fieldand the exciton approximately resonates with the Stokes sideband of the drivingfield. Under the relevant parameters determined, we also find that thenanomechanical resonator mass around the optimal mass and the driving poweraround the optimal driving power, make the maximal value of the bipartiteentanglement between the exciton and the nanomechanical resonator be equal toidentical maximal value, when the cavity field resonates with the anti-Stokessideband of the driving field and the exciton resonates with the Stokes sideband ofthe driving field.We expect that the relevant results for normal mode splitting is significant todesign the proposal used to cool the nanomechanical resonator and to understandthe mechanism of nanomechanical cooling deeply. And, we also think that therelevant results for nonlinear multistability is potentially valuable to detect thesteady state of nanomechanical resonator by optical method and to study the nonlinear phenomenon based on the optomechanical system. Further, weanticipate that the relevant results for quantum entanglement is instructive toreveal the process from the quantum world to classical world and to provide thecarrier of quantum state for the quantum computation and quantum informationand quantum non-demolition measurement.