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arXiv:1512.04183 · Physical review. D/Physical review. D.

Optomechanical tests of a Schrödinger-Newton equation for gravitational quantum mechanics

Ching Chuan Gan, Craig M. Savage, Sabina Scully

Published Jun 20, 2016• 27 citations
arXivHTMLPDF

Abstract

We show that optomechanical systems can test the Schr\"odinger-Newton equation of gravitational quantum mechanics due to Yang et al. Phys. Rev. Lett. 110, 170401 (2013). This equation is motivated by semiclassical gravity, a widely used theory of interacting gravitational and quantum fields. From the many-body Schr\"odinger-Newton equation follows an approximate equation for the center-of-mass dynamics of macroscopic objects. This predicts a distinctive double-peaked signature in the output optical quadrature power spectral density of certain optomechanical systems. Since the Schr\"odinger-Newton equation lacks free parameters, these will allow its experimental confirmation or refutation.

DOI
10.1103/physrevd.93.124049

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We show that optomechanical systems can test the Schr"odinger-Newton equation of gravitational quantum mechanics due to Yang et al. Phys. Rev. Lett. 110, 170401 (2013). This equation is motivated by semiclassical gravity, a widely used theory of interacting gravitational and quantum fields. From the many-body Schr"odinger-Newton equation follows an approximate equation for the center-of-mass dynamics of macroscopic objects. This predicts a distinctive double-peaked signature in the output optical quadrature power spectral density of certain optomechanical systems. Since the Schr"odinger-Newton equation lacks free parameters, these will allow its experimental confirmation or refutation.
We propose an optomechanics experiment that can search for signatures of a fundamentally classical theory of gravity and in particular of the many-body Schr\"odinger-Newton (SN) equation, which governs the evolution of a crystal under a self-gravitational field. The SN equation predicts that the dynamics of a macroscopic mechanical oscillator's center-of-mass wave function differ from the predictions of standard quantum mechanics [H. Yang, H. Miao, D.-S. Lee, B. Helou, and Y. Chen, Phys. Rev. Lett. 110, 170401 (2013)]. This difference is largest for low-frequency oscillators, and for materials, such as tungsten or osmium, with small quantum fluctuations of the constituent atoms around their lattice equilibrium sites. Light probes the motion of these oscillators and is eventually measured in order to extract valuable information on the pendulum's dynamics. Due to the nonlinearity contained in the SN equation, we analyze the fluctuations of measurement results differently than standard quantum mechanics. We revisit how to model a thermal bath, and the wave-function collapse postulate, resulting in two prescriptions for analyzing the quantum measurement of the light. We demonstrate that both predict features, in the outgoing light's phase fluctuations' spectrum, which are separate from classical thermal fluctuations and quantum shot noise, and which can be clearly resolved with state of the art technology.
The semiclassical gravity sourced by the quantum expectation value of the matter's energy-momentum tensor will change the evolution of the quantum state of matter, which can be described by the Schr\"odinger-Newton (SN) equation. Understanding the phenomenology of the SN equation is important for experimentally testing the quantumness of gravity. In the SN theory, semiclassical gravity contributes a gravitational potential term depending on the matter's 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/preselection prescription. This work will focus on the phenomenology of SN theory under the causal-conditional prescription, which fits the standard intuition of the continuous quantum measurement process. We found that under the causal-conditional prescription, the quantum state of the test mass mirrors is conditionally prepared by the continuous projection of the outgoing light field in an optomechanical system. Hence a quantum-trajectory-dependent gravitational potential is created, which significantly changes the system evolution. This work provides an extensive analysis of this new picture of system evolution, and shows that various experimentally measurable signatures predicted by SN theory under causal-conditional prescription cannot be distinguished from that predicted by quantum gravity unless a very extreme experimental parameter region is assumed. Therefore, our new understanding of SN phenomenology provides an important caution toward the experimental verification of quantum gravity.
The Schrödinger-Newton equation, a theoretical framework connecting quantum mechanics with classical gravity, predicts that gravity may induce measurable deviations in low-frequency mechanical systems—an intriguing hypothesis at the frontier of fundamental physics. In this study, we developed and operated an advanced optomechanical platform to investigate these effects. The system integrates an optical cavity with finesse over 3.5 × 10 5 and a torsion pendulum with an ultralow eigenfrequency of 0.6 mHz, achieving a high mechanical quality factor exceeding 5 × 10 4 . We collected data for three months and reached a sensitivity of 0.3 μ rad / Hz at the Schrödinger-Newton frequency of 2.5 mHz where deviations from the standard quantum mechanics may occur. While no evidence supporting semiclassical gravity was found, we identify key challenges in such tests and propose new experimental approaches to advance this line of inquiry. This work demonstrates the potential of precision optomechanics to probe the interplay between quantum mechanics and gravity.
The evolution of quantum states influenced by semiclassical gravity is distinct from that in quantum gravity theory due to the presence of a state-dependent gravitational potential. This state-dependent potential introduces nonlinearity into the state evolution, of which the theory is named the Schr\"odinger-Newton (SN) theory. The formalism for understanding the continuous quantum measurement process on the quantum state in the context of semiclassical gravity theory has been previously discussed using the Schr\"odinger picture in Paper I [Semiclassical gravity phenomenology under the causal-conditional quantum measurement prescription, Phys. Rev. D 107, 024004 (2023).]. In this work, an equivalent formalism using the Heisenberg picture is developed and applied to the analysis of two optomechanical experiment protocols that targeted testing the quantum nature of gravity. This Heisenberg picture formalism of the SN theory has the advantage of helping the investigation of the covariance matrices of the outgoing light fields in these protocols and further the entanglement features. We found that the classical gravity between the quantum trajectories of two mirrors under continuous quantum measurement in the SN theory can induce an apparent entanglement of the outgoing light field (though there is no quantum entanglement of the mirrors), which could serve as a false alarm for those experiments designed for probing the quantum gravity induced entanglement.
We apply the many-particle Schrödinger-Newton equation, which describes the coevolution of a many-particle quantum wave function and a classical space-time geometry, to macroscopic mechanical objects. By averaging over motions of the objects' internal degrees of freedom, we obtain an effective Schrödinger-Newton equation for their centers of mass, which can be monitored and manipulated at quantum levels by state-of-the-art optomechanics experiments. For a single macroscopic object moving quantum mechanically within a harmonic potential well, its quantum uncertainty is found to evolve at a frequency different from its classical eigenfrequency-with a difference that depends on the internal structure of the object-and can be observable using current technology. For several objects, the Schrödinger-Newton equation predicts semiclassical motions just like Newtonian physics, yet quantum uncertainty cannot be transferred from one object to another.