Misch Donald C. Plumb Anne M. Rowling Sarah J. This idea is not only important to the foundations of physics but also leads to practical advantages: a quantum circuit with such indefinite causality performs computationally better than one with definite causality. This thesis provides one of the first comprehensive introductions to quantum causality, and presents a number of advances. We will then describe one concrete attempt to develop such an underlying theory, namely the theory of Trace Dynamics [TD], proposed by Stephen Adler and collaborators.

TD is a classical dynamical theory of matrices, possessing an important global unitary invariance. It is shown how, as a result of coarse-graining, quantum theory emerges as the equilibrium statistical thermodynamics of the underlying microscopic TD. Statistical fluctuations about equilibrium can in principle result in a stochastic nonlinear modification of quantum theory, thus providing a theoretical underpinning for the phenomenological collapse model known as Continuous Spontaneous Localisation [CSL].

We will end by mentioning the outstanding unsolved problems of this program, and attempts to include gravitation. The tutorial will be used to present some ongoing research work in this context, and to have an open-ended discussion with the audience on the subject matter of these talks. When utilized appropriately, the path-integral offers an alternative to the ordinary quantum formalism of state-vectors, self-adjoint operators, and external observers - an alternative that seems closer to the underlying reality and more in tune with quantum gravity.

- Meely LaBauve: A Novel.
- Exploring Macroscopic Quantum Mechanics in Optomechanical Devices.
- Reward Yourself!
- QND measurements for future gravitational-wave detectors - Semantic Scholar.

Nor, in general, can it be interpreted as the expectation value of a projection operator or POVM. I ntroduction to non-Markovian open quantum systems dynamics. The lectures will be devoted to introduce the basic ideas and formalism for the description of open quantum systems, namely quantum systems whose interaction with an external environment cannot be neglected. As a consequence the reduced system dynamics is irreversible and features typical phenomena such as dissipation and decoherence.

We will introduce the notion of quantum dynamical map and the basic dynamical equations governing the time evolution of an open quantum system, considering both the fundamental Gorini-Kossakowski-Sudarshan-Lindblad master equation and more general time evolutions including time-convolutionless and memory kernel master equations.

We will further consider recent developments pointing to possible definitions of memory effects in a quantum reduced dynamics. In particular we will explore a recently introduced notion of quantum non-Markovianity based on the distinguishability of quantum states as quantified by the trace distance and connecting memory effects with an information flow between system and environment.

We will consider the connection of this approach with the divisibility of quantum dynamical maps. Finally we will show how this strategy leads to the actual measurement of non-Markovianity and also allows for the detection of initial system-environment correlations. Breuer and F. Rivas and S. Gorini, A.

Kossakowski, and E. Sudarshan, J. Lindblad, Rep. Vacchini and K. Hornberger, Phys. Breuer and B. Vacchini, Phys. Breuer, E. Laine, J. Piilo, and B. Vacchini, Rev. Rivas, S. Huelga, and M.

## Applications of cavity optomechanics: Applied Physics Reviews: Vol 1, No 3

Plenio, Rep. Alonso, Rev. Laine, and J. Piilo, Phys. Plenio, Phys. Piilo, and H. Vacchini, A.

Smirne, E. Breuer, New J. Vacchini and G. Amato, Sci.

Quantum Trajectory formalism for Weak Measurements. Projective measurement is used as a fundamental axiom in quantum mechanics, even though it is discontinuous and cannot predict which measured operator eigenstate will be observed in which experimental run. The probabilistic Born rule gives it an ensemble interpretation, predicting proportions of various outcomes over many experimental runs.

Understanding gradual weak measurements requires replacing this scenario with a dynamical evolution equation for the collapse of the quantum state in individual experimental runs.

We revisit the quantum trajectory framework that models quantum measurement as a continuous nonlinear stochastic process. We describe the ensemble of quantum trajectories as noise fluctuations on top of geodesics that attract the quantum state towards the measured operator eigenstates. Investigation of the restrictions needed on the ensemble of quantum trajectories, so as to reproduce projective measurement in the appropriate limit, shows that the Born rule follows when the magnitudes of the noise and the attraction are precisely related, in a manner reminiscent of the fluctuation-dissipation relation.

That implies that the noise and the attraction have a common origin in the measurement interaction between the system and the apparatus. We analyse the quantum trajectory ensemble for the dynamics of quantum diffusion and quantum jump, and show that the ensemble distribution is completely determined in terms of a single evolution parameter, which can be tested in weak measurement experiments.

We comment on how the specific noise may arise in the measuring apparatus. In Unruh published a paper that initiated the field of Analogue Gravity. Originally just a theoretical endeavour, this is now involves a remarkably broad range of research areas such as Bose-Einstein condensates, optics, nonlinear optics, superfluids and hydrodynamics. More than that, and possibly the main legacy of the field, these different field often come together and find an overlap or common inspiration through the ideas of analogue gravity.

In a nutshell, Analogue gravity refers to the attempt to reproduce certain aspects of the quantum field theory in specific curved spacetime geometries. The second part are the Einsteina equations that describe how mass or the energy stress tensor modifies the surrounding spacetime metric and how this in turn modifies the mass distribution. A full quantum gravity theory must necessarily account for the full nonlinear dynamics of the Einstein equations - this level of quantisation is the main challenge of contemporary physics and is even celebrated in Hollywood movies.

However, a semi-classical approach is also possible whereby one quantizes the fields, e. Such classical spacetime metrics, whilst usually considered to be the result of mass or exotic oject such as balck holes, may actually be encountered in very trivial situations and therefore easily controllable on Earth-based laboratories.

## Exploring corrections to the Optomechanical Hamiltonian

The first model investigated by Unruh was simply a flowing body of water. He showed that it is possibly, by controlling the flow, to create a horizon for acoustic waves propagating inside the flow. Even more interestingly, he discovrered that the same mathematical procedures applied by hawking to predict the quantum-vacuum seeded blackbody emission from black holes also applies to these lab systems.

The implications of this is that whilst it is not possible to directly verify if a black hole emits Hawking radiation, we may certainly verify in the lab if the mathematical model that predicts this emission is indeed correct. Recent results have provided possible evidence of Hawking emission from analogue black holes or horizons created in a BEC, a flowing body of water and an optical light pulse propagating in a dielectric medium. We will take a closer look at these effects, starting from a basic overview of the main curved spacetimes that are being studies, or could be studies using analogue gravity.

We will look at the basic mathematical tools used to predict particle emission from a curved or time dependence spacetime metric and then apply this to a few simple cases.

### Bibliographic Information

We will then use this to look in detail at how optical models for gravity work with some working examples. Equations will be infrequent and incomplete with more weight given to intuitive understanding of the physics at play. Contents: General overview of analogue gravity, why analogue gravity? At first glance, the minute momentum of photons seems an unlikely candidate to provide control over the motion of massive objects. Yet, today on-chip mechanical oscillators can be manipulated with only a few photons and momentum transfer from lasers can manipulate the mechanical properties of gram-scale mirrors.

The relevance of this reasearch covers new approaches to answer fundamental questions about the quantum behaviour of massive objects, applications in quantum information science, and novel methods for sensing of forces and acceleration. All of this is part of cavity quantum optomechanics [1, 2, 3], where methods from quantum optics are employed to taylor the interaction between light and mechanical resonators. Theory on the subject has been investigated already over a rather long time and soon after the millenium first ground-breaking experiments have been conducted.

However, only the last seven years have witnessed experimens that unambiguously demonstrate optomechanics at the quantum level. A few examples are the demonstration of entanglement between a microwave field and a mechanical resonator [4], the generation of squeezed mechanical states [5], and non-classical correlations between single photons and phonons [6]. In this lecture, we will understand the underlying principles of quantum cavity-optomechanical interaction, some of the common methods to measure its effects and examples of broadly used cavity-optomechanical systems.

We will furthermore look exemplarily at a selection of the recent ground-breaking experiments to illustrate solutions and challenges in the field. These lectures will cover methods to perform matterwave interferometry experiments. The students will gain an insight into the complexity of such experiments, such as how to coherently split matterwaves.

The second lecture will give a overview of experiments done on atom and molecule matterwave interferometry, while lecture three is focusing on applications of atom and molecule interferometry for metrology and sensing. Lecture 2: Concepts and realization of matter-wave interferometers with atoms and molecules.