Testing alternative theories of Quantum Mechanics with Optomechanics, and effective modes for Gaussian linear Optomechanics Thesis by Bassam Helou In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy CALIFORNIA INSTITUTE OF TECHNOLOGY Pasadena, California 2019 Defended 12/11/2018 ii © 2019 Bassam Helou ORCID: 0000-0003-2760-7622 All rights reserved iii ACKNOWLEDGEMENTS My time at Caltech wasn’t just an academic journey. I would like to first thank the people who made a lasting impact on my self. Haixing Miao was a great mentor, and his outlook on life continues to inspire me. William H. Hurt welcomed me to his family, re-ignited a love of reading, and shared his worldview with me. Robert Wills cultivated a love of the outdoors. Karthik Seetharam cultivated a love of working out. Zhewei Chen cultivated a love of climbing and other outdoor adventures. Locatelli Rao gave me some tools and understanding of physical therapy. Lee Coleman introduced me to mindfulness meditation. I am also thankful for friends who have me kept me sane over the years: Anandh Swaminathan, Stephan Zheng, Rose Yu, Li Zhiquin, Tejas Deshpande, Zachary Mark, Linhan Shen, Eric Wolff, Min-feng Tu, Nicole Yunger-Halpern, Pavan Bilgi, Evan Miyazano, Ivan Papusha, Jenia Mozgunov, Howard Hui, Daniel Naftalovich, Stephen Perry, Krzysztof Chalupka, Xiaoyu Shi, Becky Schwantes, Jacklyn Phu, Corwin Shiu, Tung Wongwaitayakornkul, Mark Harfouche, Rebecca Herrera, Luis Goncalves, Vipul Singhal, Tobias Bischoff, Chris Rollins, Julius Su, Reed Scharff, Elena Murchikova and Tal Einav. I’d also like to thank the Alpine club for organizing many amazing trips. I’d also like to thank ’the drunkest table’ for staying in touch and organizing many events. Specifically, I’d like to thank Tony Liao, Utkarth Gaur, Mike Wang, Andrew Ng, Velora Chan, the two Jon Ngs, Jimmy Lu, Hecheng Wang, Roshny Francis, Adrienne Wang and Patricia Ko. I am thankful for many stimulating discussions with other researchers that made me feel like I was part of a community. I would like to thank Maaneli Derakhshani, Antoine Tilloy, Huan Yang, Yiqiu Ma, P. C. E. Stamp, Dan Carney, Belinda Pang and Xiang Li. I’d like to thank my family for their unwavering support: Hikmat Helou, Mohamad Helou, Akram Helou, Randa Helou, Rym Helou, Randa Kabbara, Malek Kabbara, Khaled Helou, my grandfather Akram Helou and Houda Helou. I’d like to also thank extended family for their support: Amer Mikkewi, Mo Mikkewi, Kareem Mikkewi, Jihad Mikkewi, Vivian Cunanan, Wilfredo Cunanan, Bernadette Glen, Douglas Murray, Alex Murray, Molly Purnell, Kelley Purnell and Mark Purnell. I’d like to thank the ISP office, JoAnn Boyd, Jennifer Blankenship, Alfrida King and Christy Jenstad for their kindness and amazing administrative support. I’d like to thank CTLO and the Caltech library staff for being supportive of my ideas. iv I’d also like to thank Kerry Vahala for allowing me to experiment with TA ideas. Finally, I’d like to thank Yanbei Chen for his patience, support and the freedom he gave me. His boundless creativity and curiosity is inspirational. I am a much better researcher and problem solver because of him! v ABSTRACT Optomechanics has made great strides in theory and experiments over the past decade, which culminated in the first direct detection of gravitational waves in 2015 by LIGO. This thesis explores how optomechanics can be used to test fundamental physics other than the theory of general relativity. Our emphasis will be on falsifiable theories (ultimately, only experiments can decide whether a theory is correct) that address two outstanding issues in quantum mechanics: the measurement problem, and reconciling quantum mechanics with the theory of general relativity. In partic- ular, we show that the space experiment LISA pathfinder places aggressive bounds on two objective collapse models, which are non-linear stochastic modifications of the Schroedinger equation that can resolve the measurement problem. Moreover, we show that state-of-the-art torsion pendulum experiments can test the Schroedinger- Newton theory, which is the non-relativistic limit of a non-linear theory combining quantum mechanics with a fundamentally classical spacetime. Along the way, we propose how to resolve two major difficulties with determining the predictions of non-linear quantum mechanics in an actual experiment. First, we cannot use the density matrix formalism in non-linear quantum mechanics and so we have to suggest and justify a particular ensemble for the thermal bath. Separating out quantum and classical fluctuations helped us propose a reasonable ensemble. Second, most researchers believe that deterministic non-linear quantum mechanics must violate the no-signaling condition. We show this isn’t necessarily the case because different interpretations of quantum mechanics make different predictions in non-linear quantum mechanics. We propose an interpretation, the causal-conditional prescription, that doesn’t violate causality by noticing that once we fix an initial state, the evolution of a system under many non-linear theories is equivalent to evolution under a linear Hamiltonian with feedback. The mapping allows us to leverage the tools of quantum control, and it tells us that if the non-linear parameters of a non- linear Hamiltonian respond causally (i.e. with an appropriate delay) to measurement results, then the theory can be made causal. We also contribute to the theory of quantum optomechanics. We introduce two new bases that one can view environment modes with. In linear optomechanics a system interacts with an infinite number of bath modes. We show that the interaction can be reduced to one with finite degrees of freedom. Moreover, at any particular time, the system is correlated with only a finite number of bath modes. We show that if we make the assumption that we can measure any commuting environment modes, then this basis allows us to understand the one-shot quantum Cramer-Rao bound in a simple way, and allows us to sweep large parameter regimes and so find promising optomechanics topologies for quantum state preparation tasks that we can then analyze without the assumption of being able to measure any observable of the environment. We also use this basis to show that when we are interested in the conditional dynamics of a test mass, we can only adiabatically eliminate vi a lossy cavity when we measure the optomechanical system at a slow enough rate. Finally, we develop an analytic filter for obtaining the state of a generic optomechanical system that interacts linearly with its environment and is driven by Gaussian states, and where the outgoing light is measured with a non-linear photon-counting measurement. We hope that our work will help researchers explore optomechanics topologies that make use of photon counters. vii PUBLISHED CONTENT AND CONTRIBUTIONS B. Helou, J. Luo, H. Yeh, C. Shao, B. J. J. Slagmolen, D. E. McClelland, and Y. Chen (2017). “Measurable signatures of quantum mechanics in a classical spacetime”. In: PRD 96, 044008. doi: 10.1103/PhysRevD.96.044008. BH participated in the conception of the project, co-wrote the manuscript, co-carried the calculations, carried out the simulations and co-analyzed the results. B. Helou, B J. J. Slagmolen, D. E. McClelland, and Y. Chen (2017). “LISA pathfinder appreciably constrains collapse models”. In: PRD 95, 084054. doi: 10.1103/Phys- RevD.95.084054. BH participated in the conception of the project, co-wrote the manuscript, co-carried the calculations, carried out the simulations and co-analyzed the results. B. Helou, and Y. Chen (2017). “Different interpretations of quantum mechanics make different predictions in non-linear quantum mechanics, and some do not violate the no-signaling condition”. In: arXiv:1709.06639. BH conceived of the project, co-developed the results, and wrote the manuscript viii TABLE OF CONTENTS Acknowledgements............................... iii Abstract..................................... v Published Content and Contributions...................... vii Table of Contents................................viii List of Illustrations ............................... xi List of Tables . xviii Chapter I: Introduction ............................. 1 1.1 Overview................................ 1 1.2 Introduction to collapse models.................... 3 1.3 Alternatives to quantum gravity.................... 5 1.4 Overview of contributions to the theory of optomechanics . 11 Bibliography................................. 17 Chapter II: LISA pathfinder appreciably constrains collapse models . 21 2.1 Introduction .............................. 21 2.2 Constraining the collapse models ................... 23 2.3 Discussion............................... 24 2.4 Acknowledgments........................... 25 Bibliography................................. 27 Chapter III: Measurable signatures of quantum mechanics in a classical space- time ..................................... 29 3.1 Introduction .............................. 29 3.2 Free dynamics of an optomechanical setup under the Schroedinger- Newton theory............................. 31 3.3 Nonlinear quantum optomechanics with classical noise . 39 3.4 Measurements in nonlinear quantum optomechanics......... 50 3.5 Signatures of classical gravity..................... 54 3.6 Feasibility analysis........................... 60 3.7 Conclusions.............................. 69 3.8 Appendix: Conservation of energy in the SN theory......... 72 3.9 Appendix: Derivation of p0 ! ξ and p0 ξ . 74 ˆ¹ º 3.10 Appendix: