The self-orthogonality phenomenon

We study the mixing and degeneracy of two unbound energy eigenstates (resonances) in a two coupled channel model of scattering and reactions. We derive the necessary and sufficient conditions for existence of an exceptional point in the extended spectrum of bound and resonance energy eigenvalues in this model and show that these are not the same as in the single channel case. When these conditions are satisfied, in the complex energy plane, the two simple resonance poles of the scattering matrix merge into one double pole at the exceptional point. In parameter space, the surface of the eigenenergies has a branch point of square root type and branch cuts in its real and imaginary parts that start at the exceptional point and extend in opposite directions. The rich phenomenology of crossings and anticrossings of energies and widths of the doublet of unbound states, as well as the changes of identity of the poles of the scattering matrix observed when one control parameter is varied while the other is kept constant, is fully explained in terms of sections of the eigenenergy surfaces.

The known methods, due for instance to G.W. Mackey and T.F. Jordan, which exploit the transformation properties with respect to the Euclidean and Galileian group to determine the formalism of the Quantum Theory of a localizable particle, fail in the case that the considered transformations are not symmetries of the physical system. In the present work we show that the formalism of standard Quantum Mechanics for a particle without spin can be completely recovered by exploiting the covariance properties with respect to the group of Euclidean transformations, without requiring that these transformations are symmetries of the physical system.

At the exceptional point where two eigenstates coalesce in open quantum systems, the usual diagonalization scheme breaks down and the Hamiltonian can only be reduced to the Jordan block form. Most of the studies on the exceptional point appearing in the literature introduce a phenomenological effective Hamiltonian that essentially reduces the problem to that of a finite non-Hermitian matrix for which it is straightforward to obtain the Jordan form. In this paper, we demonstrate how the microscopic total Hamiltonian of an open quantum system reduces to the Jordan block form at an exceptional point in an exact manner that treats the continuum without any approximation by extending the problem to include eigenstates with complex eigenvalues that reside outside the Hilbert space. Our method relies on the Brillouin-Wigner-Feshbach projection method according to which we can obtain a finite-dimensional effective Hamiltonian that shares the discrete sector of the spectrum with the total Hamiltonian. Because of the eigenvalue dependence of the effective Hamiltonian due to the dynamical nature of the coupling between the discrete states via the continuum states, a coalescence of eigenvalues results in the coalescence of the corresponding eigenvectors of the total Hamiltonian, which means that the system is at an exceptional point. We also introduce an extended Jordan form basis away from the exceptional point, which provides an alternative way to obtain the Jordan block at an exceptional point. The extended Jordan block connects continuously to the Jordan block exactly at the exceptional point implying that the observable quantities are continuous at the exceptional point.

Exceptional points (EPs) in photonics are associated with non-Hermitian Hamiltonians with energy gain or loss, a notable example is EPs in parity-time symmetric Hamiltonians. We show that, counterintuitively, actual energy gain or loss is not required to generate this type of non-Hermitian degeneracy, since the eigenvalue of the optical Hamiltonian is the mode's propagation constant and not the energy. This is demonstrated by a simple three-layer insulator-metal-insulator (I-M-I) plasmonic waveguide, the eigenmodes of which are known to experience degeneracy at a certain point in the parametric space suggested to be used for rainbow trapping. We identify this point to be, in fact, an EP with the coalescence of the eigenmodes, despite the system having neither parity nor time symmetry. Furthermore, we demonstrate the manifestation of a third-order EP, which is generated by merging two separate EPs in the parametric space of the I-M-I waveguide. The presented results reveal unconventional properties of the Hamiltonian of a simple plasmonic waveguide and provide an insight into the nature of EPs in non-Hermitian plasmonic systems in general, suggesting the possibility of accessing even higher-order EP regimes in simple photonic structures without the need for optical gain or loss.

The adiabatic theorem, a corollary of the Schrödinger equation, manifests itself in a profoundly different way in non-Hermitian arrangements, resulting in counterintuitive state transfer schemes that have no counterpart in closed quantum systems. In particular, the dynamical encirclement of exceptional points (EPs) in parameter space has been shown to lead to a chiral phase accumulation, non-adiabatic jumps and topological mode conversion1-8. Recent theoretical studies, however, have shown that contrary to previously established demonstrations, this behaviour is not strictly a result of winding around a non-Hermitian degeneracy9. Instead, it seems to be mostly attributed to the non-trivial landscape of the Riemann surfaces, sometimes because of the presence of an EP in the vicinity9-11. Here, in an effort to bring this counterintuitive aspect of non-Hermitian systems to light and confirm this hypothesis, we provide a set of experiments to directly observe the field evolution and chiral state conversion in an EP-excluding cycle in a slowly varying non-Hermitian system. To do so, a versatile yet unique fibre-based photonic emulator is realized that utilizes the polarization degrees of freedom in a quasi-common-path single-ring arrangement. Our observations may open up new avenues for light manipulation and state conversion, as well as providing a foundation for understanding the intricacies of the adiabatic theorem in non-Hermitian systems.

Since the very early days of quantum theory there have been numerous attempts to interpret quantum mechanics as a statistical theory. This is equivalent to describing quantum states and ensembles together with their dynamics entirely in terms of phase-space distributions. Finite dimensional systems have historically been an issue. In recent works [Phys. Rev. Lett. 117, 180401 and Phys. Rev. A 96, 022117] we presented a framework for representing any quantum state as a complete continuous Wigner function. Here we extend this work to its partner function -- the Weyl function. In doing so we complete the phase-space formulation of quantum mechanics -- extending work by Wigner, Weyl, Moyal, and others to any quantum system. This work is structured in three parts. Firstly we provide a brief modernized discussion of the general framework of phase-space quantum mechanics. We extend previous work and show how this leads to a framework that can describe any system in phase space -- putting it for the first time on a truly equal footing to Schr\"odinger's and Heisenberg's formulation of quantum mechanics. Importantly, we do this in a way that respects the unifying principles of "parity" and "displacement" in a natural broadening of previously developed phase space concepts and methods. Secondly we consider how this framework is realized for different quantum systems; in particular we consider the proper construction of Weyl functions for some example finite dimensional systems. Finally we relate the Wigner and Weyl distributions to statistical properties of any quantum system or set of systems.

Geometric phase, effective conductivity enhancement, and invisibility cloak in thermal convection-conduction

Quantum mechanical irrebersibility

With a view to increasing the scope of applications of supersymmetric quantum mechanics (SUSY QM), we formulate the same in a complex phase space. Within this framework, the concept of shape invariance is reinvestigated and an insight into the eigenvalue spectra of non-Hermitian Hamiltonians is sought. The results are applied to a variety of potentials. We claim that the shape invariance for these potentials in the complex phase space can be retrieved in terms of the prescriptions already proposed in the conventional SUSY QM, in that the transformation of potential parameters takes the form of a reflection in the parameter space. Interestingly, some of these features turn out to be the generalization of the concept of quasi-parity used recently in the context of SUSY QM of PT-symmetric potentials. Further, a correspondence of the present approach with other complex formulations of SUSY QM and also in two real dimensions is demonstrated.

Eigenmode coalescence imparts remarkable properties to non-hermitian time evolution, culminating in a purely non-hermitian spectral degeneracy known as an exceptional point (EP). Here, we revisit time evolution at the EP and classify two-level non-hermitian Hamiltonians in terms of the M\"obius group. We then leverage that classification to study dynamical EP encircling, by applying it to periodically-modulated (Floquet) Hamiltonians. This reveals that Floquet non-hermitian systems exhibit rich physics whose complexity is not captured by an EP-encircling rule. For example, Floquet EPs can occur without encircling and vice-versa. Instead, we show that the elaborate interplay between non-hermitian and modulation instabilities is better understood through the lens of parametric resonance.

When a molecule is exposed to a laser field which leads to dissociation, all field‐free vibrational bound states become resonances, with complex eigenenergies described by non Hermitian quantum mechanics. It is possible to produce the coalescence of pairs of such eigenenergies, with appropriate choices of laser frequency and intensity, these particular values defining an exceptional point. We review some of the aspects of this very special situation. There are two principal characteristics of the adiabatic dynamics of a pair of vibrational states when the laser pulse encloses their exceptional point: the complete population inversion and the different type of transient resonance states. According to clock or anticlockwise loops in the frequency‐intensity plane, the resonances can either be of Shape or Feshbach‐type. This leads to a variety of different responses according to the initial state of the molecule that can be used to build a finite state logic machine based on the exceptional points of molecular photodissociation. The molecule chosen for this study is Na2.

In this chapter, we cover the fundamentals of linear algebra and provide a mathematical formulation of quantum mechanics for use in later chapters. It is necessary to understand these topics since they form the foundation of quantum information processing discussed later. In the first section, we cover the fundamentals of linear algebra and introduce some notation. The next section describes the formulation of quantum mechanics. Further, we examine a quantum two-level system, which is the simplest example of a quantum-mechanical system. Finally, we discuss the tensor product and matrix inequalities. More advanced discussions on linear algebra are available in Appendix.

2 - The Formalism of Quantum Mechanics

This paper attempts an interpretation of Everett's relative state formulation of quantum mechanics that avoids the commitment to new metaphysical entities like ‘worlds’ or ‘minds’. Starting from Everett's quantum mechanical model of an observer, it is argued that an observer's belief to be in an eigenstate of the measurement (corresponding to the observation of a well-defined measurement outcome) is consistent with the fact that she objectively is in a superposition of such states. “Subjective states” corresponding to such beliefs are constructed. From an analysis of these subjective states and their dynamics it is argued that Everett's pure wave mechanics is subjectively consistent with von Neumann's classical formulation of quantum mechanics. It follows from the argument that the objective state of a system is in principle unobservable. Nevertheless, an adequate concept of empirical reality can be constructed.

The real number line has served classical physics well as a model continuum for both space and time. Quantum mechanics preserves this continuum picture of space and time while extending the range of functions on spacetime to complex numbers. The resulting theory is precise, elegant, experimentally verified and universally accepted. However, since the existence of a ‘realistic’ basis for quantum mechanics remains an open question, there is as yet no consensus on a satisfactory interpretation of the theory. The step in the transition from classical to quantum physics that characteristically impedes attempts to construct realistic models is the special extension of functions from R to C. In quantum mechanics this step is formal and algebraic in nature. This talk discusses an alternative in which the geometry of a spacetime path automatically builds the extension from Z to C, with all spacetime functions simply counting contributions of a single path. The model produces the correct continuum propagator in the continuum limit, but the realistic origin of the propagator is only apparent when the smallest scale in the system (Planck scale) remains finite. This type of model is consequently missed by current formulations of quantum mechanics because the continuum limit is assumed at the outset. This suggests that it may be fruitful to look to discrete formulations of quantum mechanics in the quest for an ontological version of the theory.