i Contents Introduction 1 1 Open Quantum Systems 5 1.1 Reduced state and its evolution ..................... 5 1.1.1 Complete positivity ....................... 7 1.2 Lindblad equation ............................. 8 2 Decoherence 9 2.1 Gallis-Flemming master equation .................... 10 2.1.1 Short wavelength limit ...................... 12 2.1.2 Long wavelength limit ...................... 12 2.2 Rotational Decoherence ......................... 13 3 Quantum Brownian Motion 18 3.1 The model ................................. 18 3.2 The Calderira-Leggett master equation ................. 20 3.2.1 Complete positivity problem .................. 23 3.3 Non-Markovian Quantum Brownian motion ............. 23 3.3.1 The adjoint master equation ................... 24 3.3.2 The Master Equation for the statistical operator ....... 27 3.3.3 Complete Positivity ....................... 29 3.3.4 Time evolution of relevant quantities ............. 31 3.3.5 Non-Gaussian initial state .................... 34 4 Gravitational time dilation 37 4.1 Model for universal decoherence .................... 38 4.2 Heat capacity for gravitational decoherence .............. 39 4.3 Competing effects ............................. 40 4.3.1 Comparison of the effects .................... 42 5 Collapse Models 45 5.1 Continuous Spontaneous Localization Model ............. 45 5.1.1 Imaginary noise trick ....................... 49 5.2 Optomechanical probing collapse models ............... 49 5.3 Gravitational wave detectors bound collapse parameters space ... 53 5.3.1 Interferometric GW detectors: LIGO .............. 56 5.3.2 Space-based experiments: LISA Pathfinder .......... 57 5.3.3 Resonant GW detectors: AURIGA ............... 58 5.4 Ultra-cold cantilever detection of non-thermal excess noise ..... 61 5.5 Hypothetical bounds from torsional motion .............. 67 5.5.1 Experimental feasibility ..................... 69 ii 6 Conclusions 71 Appendices A Quantum Brownian Motion master equation 74 A.1 Explicit form of φ(t) ............................ 74 A.2 Explicit form of the adjoint master equation .............. 74 A.3 Derivation of the master equation for the states ............ 75 A.4 Explicit expression for ⇤dif(t) and E(t) ................. 76 B Gravitational time dilation 78 C Collapse Models 80 C.1 CSL Diffusion coefficients ........................ 80 C.2 Effective frequencies and damping constants ............. 81 C.3 Cantilever ................................. 82 Bibliography 92 iii List of Publications Published works as outcome of the doctoral project 1. A. Vinante, R. Mezzena, P. Falferi, M. Carlesso and A. Bassi. Improved noninterferometric test of collapse models using ultracold cantilevers. Physical Review Letters, 119 110401 (2017). Link to paper: https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.119.110401 Link to ArXiv: https://arxiv.org/abs/1611.09776 The most important contents of this article are reported in Sec. 5.4. 2. M. Carlesso and A. Bassi. Adjoint master equation for quantum brownian motion. Physical Review A, 95 052119 (2017). Link to paper: https://journals.aps.org/pra/abstract/10.1103/PhysRevA.95.052119 Link to ArXiv: https://arxiv.org/abs/1602.05116 The most important contents of this article are reported in Sec. 3.3. 3. S. McMillen, M. Brunelli, M. Carlesso, A. Bassi, H. Ulbricht, M. G. A. Paris, and M. Paternostro. Quantum-limited estimation of continuous spontaneous localization. Physical Review A, 95 012132 (2017). Link to paper: https://journals.aps.org/pra/abstract/10.1103/PhysRevA.95.012132 Link to ArXiv: https://arxiv.org/abs/1606.00070 iv 4. M. Carlesso, A. Bassi, P. Falferi, and A. Vinante. Experimental bounds on collapse models from gravitational wave detectors. Physical Review D, 94 124036 (2016). Link to paper: https://journals.aps.org/prd/abstract/10.1103/PhysRevD.94.124036 Link to ArXiv: https://arxiv.org/abs/1606.04581 The most important contents of this article are reported in Sec. 5.3. 5. M. Carlesso and A. Bassi. Decoherence due to gravitational time dilation: Analysis of competing decoher- ence effects. Physics Letters A, 380 (31–32), pp. 2354 – 2358 (2016). Link to paper: http://www.sciencedirect.com/science/article/pii/S0375960116302407 Link to ArXiv: https://arxiv.org/abs/1602.01979 The most important contents of this article are reported in Chap. 4. Pre-prints 6. M. Carlesso, M. Paternostro, H. Ulbricht, A. Vinante and A. Bassi. Non-interferometric test of the Continuous Spontaneous Localization model based on the torsional motion of a cylinder. ArXiv, 1708.04812 (2017). Link to ArXiv: https://arxiv.org/abs/1708.04812 The most important contents of this article are reported in Sec. 5.5. 7. M. Carlesso, M. Paternostro, H. Ulbricht and A. Bassi. When Cavendish meets Feynman: A quantum torsion balance for testing the quantumness of gravity. ArXiv, 1710.08695 (2017). Link to ArXiv: https://arxiv.org/abs/1710.08695 v List of attended Schools, Workshops and Conferences 1. September, 2017 Training Workshop at Instituto Superior Tecnico of Lisbon, Portugal Title Lisbon Training Workshop on Quantum Technologies in Space http://www.qtspace.eu/?q=node/131 Organization Dr. R. Kaltenbaek, Dr. E. Murphy, Dr. J. Leitao and Dr. Y. Omar 2. June, 2017 Workshop at University of Milano, Italy Title Fundamental problems of quantum physics http://www.mi.infn.it/~vacchini/workshopBELL17.html Organization Dr. B. Vacchini 3. May, 2017 Workshop at Laboratory Nazionali in Frascati, Italy Title The physics of what happens and the measurement problem https://agenda.infn.it/conferenceDisplay.py?confId=13169 Organization Dr. A. Bassi, Dr. C.O. Curceanu, Dr. B. Hiesmayr and Dr. K. Pis- cicchia 4. May, 2017 Junior Symposium in Trieste, Italy Title Trieste Junior Quantum Days http://people.sissa.it/~alemiche/junior-tsqd-2017.html Organization Dr. A. Bassi, Dr. F. Benatti and Dr. A. Michelangeli 5. March, 2017 Conference and Working Group Meeting in Valletta, Malta Title QTSpace meets in Malta http://www.qtspace.eu/?q=node/112 Organization Dr. M. Paternostro, Dr. A. Bassi, Dr. S. Gröblacher, Dr. H. Ul- bricht, Dr. R. Kaltenbaek and Dr. C. Marquardt 6. November, 2016 Autumn School at LMU in Munich, Germany Title Mathematical Foundations of Physics https://light-and-matter.github.io/autumn-school Organization Dr. D.-A. Dercket and Dr. S. Petrat vi 7. May, 2016 Workshop in Pontremoli, Italy Title Quantum control of levitated optomechanics https://quantumlevitation.wordpress.com Organization Dr. A. Serafini, Dr. M. Genoni and Dr. J. Millen 8. September, 2015 International Workshop at Laboratory Nazionali in Frascati, Italy Title Is quantum theory exact? The endeavor of the theory beyond standard quantum mechanics - Second edition http:www.lnf.infn.it/conference/FQT2015 Organization Dr. A. Bassi, Dr. C.O. Curceanu, Dr. S. Donadi and Dr. K. Pisci- cchia 9. March, 2015 International Conference at Ettore Majorana Foundation in Erice, Italy Title Fundamental Problems in Quantum Physics http:www.agenda.infn.it/conferenceDisplay.py?confId=9095 Organization Dr. A. Bassi and Dr. C.O. Curceanu 10. February, 2015 51 Winter School of Theoretical Physics in Ladek Zdroj, Poland Title Irreversible dynamics: nonlinear, nonlocal and non-Markovian mani- festations http:www.ift.uni.wroc.pl/~karp51 Organization Institut of Theoretical Physics in Wroclaw, Poland 1 Introduction When I look back to the time, already twenty years ago, when the concept and magnitude of the physical quantum of action began, for the first time [. ] the whole development [from the mass of experimental facts to its disclosure] seems to me to provide a fresh illustration of the long-since proved saying of Goethe’s that man errs as long as he strivesa. And the whole strenuous intellectual work of an industrious research worker would appear [. ] in vain and hopeless, if he were not occasionally through some striking facts to find that he had, at the end of all his criss-cross journeys, at last accomplished at least one step which was conclusively nearer the truth. aJohann Wolfgang von Goethe, Faust, 1808. Max Karl Ernst Ludwig Planck Nobel Lecture, June 2, 1920 [1] The question: “How does a chicken move in the atmosphere?” would be typically answered by a physicist: “To start, let us approximate the problem by considering a spherical chicken in vacuum. ”. This is for sure a strong and rough approximation, however it can be a good starting point for solving the problem and in certain cases it is more than enough to properly describe the motion of the system of in- terest. Quantum mechanics is an example of a theory exhibiting a broad collection of theoretical results in complete agreement with experimental evidence: from the black body radiation [2–4] to the double slit experiment [5, 6], from the photo- electric effect [7–9] to the hydrogen atom, from interference fringes in a matter- wave interferometry experiment [10, 11] to Bose-Einstein condensates [12, 13] and many more. In some situations, the unitary dynamics of a quantum isolated sys- tem is not sufficient to well describe the system. One situation is of particular importance due to its ubiquity and unavoidability. Every realistic (quantum) sys- tem interacts with the surrounding environment and consequently is changed by it. In such a case, phenomena like dissipation, diffusion or decoherence emerge and may become important for the system dynamics. External influences on a quantum system must be considered explicitly to get a better description of Na- ture. This is the purpose of the theory of open quantum systems. In