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In this article, we are interested in situations where the existence of a contiguous cascade of quantum resonant transitions is predicated on the validity of a particular statement in number theory. The setting is a tailored one-atom one-dimensional potential with a prescribed spectrum under a weak periodic perturbation. The former is, by now, an experimental reality [Cassettari et al., PNAS Nexus 2, pgac279 (2022)]. As a case study, we look at the following trivial statement: “Any power of 3 is an integer.” Consequently, we “test” this statement in a numerical experiment where we demonstrate an unimpeded upward mobility along an equidistant ln (3)-spaced subsequence of the energy levels of a potential with a log-natural spectrum under a frequency ln (3) time-periodic perturbation. We further show that when we “remove” 9 from the set of integers—by excluding the corresponding energy level from the spectrum—the cascade halts abruptly. 

This work is motivated by an article by Wang, Casati, and Prosen[Phys. Rev. E vol. 89, 042918 (2014)] devoted to a study of ergodicityin two-dimensional irrational right-triangular billiards. Numericalresults presented there suggest that these billiards are generally notergodic. However, they become ergodic when the billiard angle is equalto \pi/2 π / 2 times a Liouvillian irrational, morally a class of irrational numberswhich are well approximated by rationals. In particular, Wang etal. study a special integer counter that reflects the irrationalcontribution to the velocity orientation; they conjecture that thiscounter is localized in the generic case, but grows in the Liouvilliancase. We propose a generalization of the Wang-Casati-Prosen counter:this generalization allows to include rational billiards intoconsideration. We show that in the case of a 45°\!\!:\!45°\!\!:\!90° 45 ° : 45 ° : 90 ° billiard, the counter grows indefinitely, consistent with theLiouvillian scenario suggested by Wang et al. 

We discuss an interferometric scheme employing interference of bright solitons formed as specific bound states of attracting bosons on a lattice. We revisit the proposal of Castin and Weiss [Phys. Rev. Lett. vol. 102, 010403 (2009)] for using the scattering of a quantum matter-wave soliton on a barrier in order to create a coherent superposition state of the soliton being entirely to the left of the barrier and being entirely to the right of the barrier. In that proposal, it was assumed that the scattering is perfectly elastic, i.e. that the center-of-mass kinetic energy of the soliton is lower than the chemical potential of the soliton. Here we relax this assumption: By employing a combination of Bethe ansatz and DMRG-based analysis of the dynamics of the appropriate many-body system, we find that the interferometric fringes persist even when the center-of-mass kinetic energy of the soliton is above the energy needed for its complete dissociation into constituent atoms. 

QBism regards quantum mechanics as an addition to probability theory. The addition provides an extra normative rule for decision-making agents concerned with gambling across experimental contexts, somewhat in analogy to the double-slit experiment. This establishes the meaning of the Born Rule from a QBist perspective. Moreover it suggests that the best way to formulate the Born Rule for foundational discussions is with respect to an informationally complete reference device. Recent work [DeBrota, Fuchs, and Stacey, Phys. Rev. Res. 2, 013074 (2020)] has demonstrated that reference devices employing symmetric informationally complete POVMs (or SICs) achieve a minimal quantumness: They witness the irreducible difference between classical and quantum. In this paper, we attempt to answer the analogous question for real-vector-space quantum theory. While standard quantum mechanics seems to allow SICs to exist in all finite dimensions, in the case of quantum theory over the real numbers it is known that SICs do not exist in most dimensions. We therefore attempt to identify the optimal reference device in the first real dimension without a SIC (i.e., d=4) in hopes of better understanding the essential role of complex numbers in quantum mechanics. In contrast to their complex counterparts, the expressions that result in a QBist understanding of real-vector-space quantum theory are surprisingly complex. 

We consider a toy model for emergence of chaos in a quantum many-body short-range-interacting system: two one-dimensional hard-core particles in a box, with a small mass defect as a perturbation over an integrable system, the latter represented by two equal mass particles.To that system, we apply a quantum generalization of Chirikov's criterion for the onset of chaos, i.e. the criterion of overlapping resonances.There, classical nonlinear resonances translate almost automatically to the quantum language. Quantum mechanics intervenes at a later stage: the resonances occupying less than one Hamiltonian eigenstate are excluded from the chaos criterion. Resonances appear as contiguous patches of low purity unperturbed eigenstates, separated by the groups of undestroyed states-the quantum analogues of the classical KAM tori. 

We study a gas of attracting bosons confined in a ring shape potential pierced by an artificial magnetic field. Because of attractive interactions, quantum analogs of bright solitons are formed. As a genuine quantum-many-body feature, we demonstrate that angular momentum fractionalization occurs and that such an effect manifests on time of flight measurements.As a consequence, the matter-wave current in our system can react to very small changes of rotation or other artificial gauge fields. We worked out a protocol to entangle such quantum solitonic currents, allowing us to operate rotation sensors and gyroscopes to Heisenberg-limited sensitivity.Therefore, we demonstrate that the specific coherence and entanglement properties of the system can induce an enhancement of sensitivity to an external rotation.