Theoretical Study On The Quantum Behavior Of Macroscopic Objects In Optical Interferometer

As the requirement of the precision of the gravitational waves detection is getting morestringent, meanwhile, the quantum behavior of macroscopic objects is becoming an im-portant research topic in these years, and the analysis on optical interferometry from thequantum aspect is now compulsory.Michelson interferometry and Sagnac interferometry, as being the physical modelsto test quantum mechanics for macroscopic objects, are studied by theoretical analysis inSchro¨dinger picture and Heisenberg picture. Through the optomechanical coupling betweenthe photon and the movable mirror in the interferometers, the mirror's movement affects thephoton field. Photon at the output readout carries the information about the state of themacroscopic mirror, which allows us to study the quantum behaviors of the whole systemby detecting the output photon.The theoretical calculation in Schro¨dinger picture for a Michelson interferometer withdual output readout, shows that a non-localized, linear superposition of two Schro¨dingercats (states) could be generated by the opto-mechanical coupling of a single photon witha movable macroscopic quantum mirror in one arm of the interferometer. In a balancedhomodyne detection scheme, a CHSH inequality may be established, the violation of whichserves to verify the non-localization of the two spacelike-separated Schro¨dinger cats. Therelevance of this result to our understanding of possible wave function collapse dynamicsis brie?y discussed. As well as the macroscopic entangled state of two mirrors generatedthrough opto-mechanical coupling and the associated CHSH inequality are also discussed.Similar analysis is done for Sagnac interferometry. It is shown that, due to the struc-tural symmetry of the ring cavity of a Sagnac interferometer in the clockwise and counter-clockwise direction, the radiation pressure exerted by a photon on a single macroscopicmirror within the cavity generates a linear superposition of temporally separated quantumstates of the mirror, in contrast to the standard Schro¨dinger cat in a linear superposition of spatially separated quantum states at the same instant of time. For the case of optomechan-ical coupling of a single photon with two mirrors within the cavity, the radiation pressureforce mediates a transfer (swapping) of the which-path entanglement of the photon to thetemporally (but not spatially) separated quantum states of the two mirrors. Some fundamen-tal issues related to the enigmatic nature of time in quantum theory and possible quantumgravity theory raised by the present work are brie?y touched upon.Meanwhile, a detail analysis of the input-output relation is performed in Heisenbergpicture. It reveals that by modulating some parameters in the interferometer, the Sagnacinterferometry can be a position meter or speed meter, detecting the movable mirror'sposition or momentum, such that the final output signal of Michelson interferometrycontains both the position and momentum information. Moreover the photon numberspectrum readout of Sagnac interferometry at the bright and dark ports contains informationconcerning the non-commutativity of position and momentum of the macroscopic mirror. Afeasible experimental scheme to probe the commutation relation of a macroscopic quantummirror is outlined for further explore of the possible frontier boundary between the classicaland quantum regimes.