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Phase space methods are ubiquitous in quantum mechanics. From the Weyl-Wigner Moyal formalism to coherent states and discrete phase spaces we see the imprints of the classical world again and again. In this presentation, we address one of two major developments introduced by Aharonov and his collaborators: The concept of weak values that stems from a time-symmetric view of quantum physics. We look at the weak measurement through two distinct geometric frames: The geometry of the measuring apparatus and the geometry of the measured system. We …nalize with some brief comments on the second major conceptual due to Aharonov and collaborators: the theory of modular variables and how Schwinger´s discrete phase space structure helps to shed light on it. We conclude then with the following mantra: It´ s all about phase space.

In classical mechanics, an action is defined only modulo additive terms which do not modify the equations of motion; in certain cases, these terms are topological quantities. We construct an infinite sequence of higher order topological actions and argue that they play a role in quantum mechanics, and hence can be accessed experimentally.

Curiosity about how the world works can lead to beneficial progress in technology, and vice-versa. This kind of interplay can be found in quantum nanoscience, where foundationally motivated experiments and technologically motivated experiments often use similar materials and techniques, because both involve extending the realm of non-classical behaviour. At a higher level, curiosity about ultimate questions such as meaning and purpose can create an environment that is conducive to scientific breakthroughs, and many of the best minds in science have also been curious about deeper realities. Eugene Wigner described the miracle of the effectiveness of mathematics as a wonderful gift which we neither understand nor deserve. The same could be said of curiosity.

Superconducting circuit technology has rapidly developed over the past several years to become a leading contender for realizing a scalable quantum computer. Modern circuit designs are based on the transmon qubit, which coherently superposes macroscopic charge oscillations. Measurements of a transmon are fundamentally weak and continuous in time, with projective measurements emerging only after a finite duration. Adding gates, such measurements may then implement ancilla-based measurements of controllable strength. Recent experiments have used both types of weak measurement to great effect: for monitoring qubit evolution, and for showing violations of a hybrid Bell-Leggett-Garg inequality.

The Kochen-Specker (KS) theorem can gives rise to logical paradoxes under pre- and post-selection in which the contextual behavior is confined to specific observables of a system. Weak measurements allow direct experimental observation of the nonclassical behavior of these specific observables. This presents an experimental advantage over other tests of KS inequalities which rule out a particular class of counterfactual noncontextual hidden variable models, but can never specify where the contradiction occurs, nor make any direct observation of its consequences. The confined contextuality can always be interpreted as a logical pre- and post-selection paradox, such as the 3-box paradox, the Quantum Cheshire Cat, or the Quantum Pigeonhole Effect. This confined contextuality was recently observed using neutron inferometry for KS sets of up to 17 qubits. Details of the theory and experimental results will be presented.

Quantum tunneling is one such phenomenon that is essential for a number of devices that are now taken for granted. However, our understanding of quantum tunneling dynamics is far from complete, and there are still a number of theoretical and experimental challenges. The dynamics of the quantum tunneling process can be investigated if we can create a large tunneling region. We have achieved this using a linear Paul trap and a quantum tunneling rotor, which has resulted in the successful observation of the Aharonov–Bohm effect in tunneling particles. Also, this result shows that the spatially separated phonon can be interfered.

The question of the time reversibility of quantum mechanics with measurements is one that has been debated for some time. In this talk, I will present new work exploring our ability to distinguish the forward from the time-reverse measurement records of continuous quantum measurements. The question involves both the conditions for the time-reversibility of the quantum trajectory equations of motion, as well as statistical distinguishability of the arrow of time. I will present the case with and without postselection on the final state, and connect the issue to a similar topic in nonequilibrium statistical physics. This work generalizes and pushes the two-time reformulation of quantum mechanics developed by Yakir Aharonov and collaborators beyond arbitrarily weak measurements. I will also discuss how this proposal can be implemented with continuously monitored superconducting quantum circuits. In collaboration with Alexander Korotkov, Justin Dressel, Areeya Chantasri, and Kater Murch Funded by the John Templeton Foundation.