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.

### Harish-Chandra Research Institute

Huelga, and M. 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.

## The quantum theory of time, the block universe, and human experience

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.

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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.

- Relativistic Structures.
- Action at a Distance in Quantum Mechanics?
- Quantum Field Theory.
- Introduction.

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.

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. 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.

Detection of small forces is at heart of many fundamental physics experiments. Here, we are primarily interested in understanding how experiments detecting weak classical forces using mechanical systems can be used to test modifications of quantum mechanics, such as spontaneous collapse models.

These techniques are often regarded as non-interferometric, as opposed to direct tests making use of quantum superposition states. I will introduce the general features of weak force detection experiment, and illustrate the main sources of noise and dissipation in relevant experiments, including and ultrasensitive force microscopy and gravitational wave detectors. Finally, we will discuss the state of the art in testing collapse models using mechanical systems. We will discuss trapping and cooling experiments of optically levitated nanoparticles [1]. We will report on the cooling of all translational motional degrees of freedom of a single trapped silica particle to 1mK simultaneously at vacuum of 10 -5 mbar using a parabolic mirror to form the optical trap.

We will further report on the squeezing of a thermal motional state of the trapped particle by rapid switch of the trap frequency [2]. We will further discuss ideas to experimentally test quantum mechanics by means of collapse models [3] by both matter-wave interferometry [4] and non-interferometric methods [5]. While first experimental bounds by non-interferometric tests have been achieved during the last year by a number of different experiments according to non-interferometric experiments [4], we shall also report on different matterwave interferometry experiments to test the quantum superposition principle directly for 1 million atomic mass unit amu particles.

We will further discuss some ideas to probe the interplay between quantum mechanics and gravitation by levitated optomechanics experiments. One idea is to seek first experimental evidence about the fundamentally quantum or classical nature of gravity by using the torsional motion of a non-spherical trapped particle, while a second idea is to test the effect of the gravity related shift of energy levels of the mechanical harmonic oscillator, which is predicted by semi-classical gravity the so-called Schr dinger-Newton equation [6].

The idea is to complement topic covered during the first week of this school from the experimental perspective. Feeback Control: taming atoms and nano-drums with electronic feedback. Over last few decades, spectacular sophistication has been achieved in experiments involving macroscopic systems that behave quantum mechanically. These include laser-cooled and trapped atoms, ions and condensates, artificially fabricated mesoscopic quantum circuits, micro and nano-mechanical as well as a myriad of hybrid systems with a mixed combination.

Experimentally, a skeletal structure that guides almost all these experiments is provided by control theory, which explains how to observe fluctuations mostly classical and compensate for it in real time. In this short lecture, I will pick up examples to explain the basics of control theory, as applicable to experiments with quantum systems. In particular, I will talk about how feedback control is used in widely different cases, for experiments pursued in our laboratory, with cold atoms, cavities and graphene resonators.

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## The quantum theory of time, the block universe, and human experience

The tutorial problems will mostly consists of understanding few simple op-amp circuits, debugging and interpreting them in the language of feedback control. We will end with a short overview of how these ideas extend to quantum feedback control. These two lectures will deal with the experimental attempts at measuring the Quantum Measure. There have been several attempts at experimentally bounding the Quantum Measure using photons, atoms, NMR among others [2—6]. Recently, it has been measured to be a definite non-zero for the first time [7].

These lectures will discuss these experiments, the usual difficulties faced in such precision experiments and future experiments which could be attempted in this genre.