Semi-classical gravity phenomenology under the causal-conditional quantum measurement prescription

The semi-classical gravity sourced by the quantum expectation value of the matter's energy-momentum tensor will change the evolution of the quantum state of matter. This effect can be described by the Schroedinger-Newton (SN) equation, where the semi-classical gravity contributes a gravitational potential term depending on the matter quantum state. This state-dependent potential introduces the complexity of the quantum state evolution and measurement in SN theory, which is different for different quantum measurement prescriptions. Previous theoretical investigations on the SN-theory phenomenology in the optomechanical experimental platform were carried out under the so-called post/pre-selection prescription. This work will focus on the phenomenology of SN theory under the causal-conditional prescription, which fits the standard intuition on the continuous quantum measurement process. Under the causal-conditional prescription, the quantum state of the test mass mirrors is conditionally and continuously prepared by the projection of the outgoing light field in the optomechanical system. Therefore a gravitational potential depends on the quantum trajectory is created and further affects the system evolution. In this work, we will systematically study various experimentally measurable signatures of SN theory under the causal-conditional prescription in an optomechanical system, for both the self-gravity and the mutual gravity scenarios. Comparisons between the SN phenomenology under three different prescriptions will also be carefully made. Moreover, we find that quantum measurement can induce a classical correlation between two different optical fields via classical gravity, which is difficult to be distinguished from the quantum correlation of light fields mediated by quantum gravity.