Gravity of Light, Light in Gravitational Fields, and Metrological Implications

This thesis deals with the interplay of gravitation and light. It is split into four parts, each of them giving an overview of one of our projects: In the first and second part, we study the gravitational properties of laser light and use other light rays to illustrate these properties. In the third part, light rays are used as a tool to determine the frequency spectrum of an optical resonator in a background gravitational field. Finally, in the fourth part, light plays both the role of the source of the gravitational field and the means to perform a measurement. As the gravitational field of light is weak, its effects are too small to be experimentally measured. However, with the progress of technology, they might be detected in the future. They are of conceptual interest, revealing fundamental properties of the nature of light. In the first part, we determine the gravitational field of a laser beam: The laser beam is described as a solution of Maxwell’s equations and has a finite wavelength and circular polarization. This description is beyond the short-wavelength approximation, and allows to find novel gravitational properties of light. Among these are frame-dragging due to the laser beam’s spin angular momentum and the deflection of parallel co-propagating test light-rays that overlap with the source laser-beam. Further, the polarization of a test light-ray in the gravitational field of the laser beam is rotated. This is analyzed in the second part. The rotation consists of a reciprocal con- tribution associated to the gravitational analogue of optical activity, and a non-reciprocal part identified as the gravitational analogue of the electromagnetic Faraday effect. There- fore, letting light propagate back and forth between two mirrors, the gravitational Faraday effect accumulates, while the effect due to the gravitational optical activity cancels. Inter- estingly, using only classical general relativity, our analysis shows gravitational spin-spin coupling, which is a known effect in perturbative quantum gravity. In the third part, we study the effect of a gravitational field and proper acceleration on the frequency spectrum of an optical resonator. The resonator is modelled in two different ways: As a rod of matter with two attached mirrors at its ends, and as a dielectric rod whose ends function as mirrors. The resonator can be deformed in the gravitational field depending on the material properties of the rod. The frequency spectrum turns out to depend on the radar length, which is the length an observer measures by sending a light signals back and forth between the mirrors and measuring the time difference. The results for the frequency spectrum may be used for measuring gravitational fields or acceleration based on frequency shifts of the light. Also in the fourth part we look at an optical resonator, this time a cubic cavity. While in the third part we considered a background gravitational field, now the light inside the cubic cavity is the source of the gravitational field. With this setup, we consider an observer making a specific measurement of the speed of light and analyze the precision of the measurement. Using quantum parameter estimation theory and analyzing the effect of the gravitational field, we determine the number of photons inside the cavity which leads to the best precision of the measurement

Rotation of polarization in the gravitational field of a laser beam - Faraday effect and optical activity

We investigate the rotation of the polarization of a light ray propagating in the gravitational field of a circularly polarized laser beam. The rotation consists of a reciprocal part due to gravitational optical activity, and a non-reciprocal part due to the gravitational Faraday effect. We discuss how to distinguish the two effects: Letting light propagate back and forth between two mirrors, the rotation due to gravitational optical activity cancels while the rotation due to the gravitational Faraday effect accumulates. In contrast, the rotation due to both effects accumulates in a ring cavity and a situation can be created in which gravitational optical activity dominates. Such setups amplify the effects by up to five orders of magnitude, which however is not enough to make them measurable with state of the art technology. The effects are of conceptual interest as they reveal gravitational spin-spin coupling in the realm of classical general relativity, a phenomenon which occurs in perturbative quantum gravity

Frequency spectrum of an optical resonator in a curved spacetime

The effect of gravity and proper acceleration on the frequency spectrum of an optical resonator—both rigid or deformable—is considered in the framework of general relativity. The optical resonator is modeled either as a rod of matter connecting two mirrors or as a dielectric rod whose ends function as mirrors. Explicit expressions for the frequency spectrum are derived for the case that it is only perturbed slightly and variations are slow enough to avoid any elastic resonances of the rod. For a deformable resonator, the perturbation of the frequency spectrum depends on the speed of sound in the rod supporting the mirrors. A connection is found to a relativistic concept of rigidity when the speed of sound approaches the speed of light. In contrast, the corresponding result for the assumption of Born rigidity is recovered when the speed of sound becomes infinite. The results presented in this article can be used as the basis for the description of optical and opto-mechanical systems in a curved spacetime. We apply our results to the examples of a uniformly accelerating resonator and an optical resonator in the gravitational field of a small moving sphere. To exemplify the applicability of our approach beyond the framework of linearized gravity, we consider the fictitious situation of an optical resonator falling into a black hole

Optimal estimation with quantum optomechanical systems in the nonlinear regime

We study the fundamental bounds on precision measurements of parameters contained in a time-dependent nonlinear optomechanical Hamiltonian, which includes the nonlinear light-matter coupling, a mechanical displacement term, and a single-mode mechanical squeezing term. By using a recently developed method to solve the dynamics of this system, we derive a general expression for the quantum Fisher information and demonstrate its applicability through three concrete examples: estimation of the strength of a nonlinear light-matter coupling, the strength of a time-modulated mechanical displacement, and a single-mode mechanical squeezing parameter, all of which are modulated at resonance. Our results can be used to compute the sensitivity of a nonlinear optomechanical system to a number of external and internal effects, such as forces acting on the system or modulations of the light--matter coupling.Comment: 24 pages, 3 figure

Optimal estimation of time-dependent gravitational fields with quantum optomechanical systems

We study the fundamental sensitivity that can be achieved with an ideal optomechanical system in the nonlinear regime for measurements of time-dependent gravitational fields. Using recently developed methods to solve the dynamics of a nonlinear optomechanical system with a time-dependent Hamiltonian, we compute the quantum Fisher information for linear displacements of the mechanical element due to gravity. We demonstrate that the sensitivity can not only be further enhanced by injecting squeezed states of the cavity field, but also by modulating the light--matter coupling of the optomechanical system. We specifically apply our results to the measurement of gravitational fields from small oscillating masses, where we show that, in principle, the gravitational field of an oscillating nano-gram mass can be detected based on experimental parameters that will likely be accessible in the near-term future. Finally, we identify the experimental parameter regime necessary for gravitational wave detection with a quantum optomechanical sensor.Comment: Main text: 19 pages and 4 figures, Appendix: 16 pages and 2 figure

The gravitational field of a laser beam beyond the short wavelength approximation

Light carries energy, and therefore, it is the source of a gravitational field. The gravitational field of a beam of light in the short wavelength approximation has been studied by several authors. In this article, we consider light of finite wavelengths by describing a laser beam as a solution of Maxwell's equations and taking diffraction into account. Then, novel features of the gravitational field of a laser beam become apparent, such as frame-dragging due to its spin angular momentum and the deflection of parallel co-propagating test beams that overlap with the source beam. Even though the effects are too small to be detected with current technology, they are of conceptual interest, revealing the gravitational properties of light.© 2018 IOP Publishing Lt