For a given operator D ( t ) of an observable in theoretical parity-time symmetric quantum physics (or for its evolution-generator analogues in the experimental gain-loss classical optics, etc.) the instant t c r i t i c a l of a spontaneous breakdown of the parity-time alias gain-loss symmetry should be given, in the rigorous language of mathematics, the Kato’s name of an “exceptional point”, t c r i t i c a l = t ( E P ) . In the majority of conventional applications the exceptional point (EP) values are not real. In our paper, we pay attention to several exactly tractable toy-model evolutions for which at least some of the values of t ( E P ) become real. These values are interpreted as “instants of a catastrophe”, be it classical or quantum. In the classical optical setting the discrete nature of our toy models might make them amenable to simulations. In the latter context the instant of Big Bang is mentioned as an illustrative sample of possible physical meaning of such an EP catastrophe in quantum cosmology.
For a given operator D ( t ) of an observable in theoretical parity-time symmetric quantum physics (or for its evolution-generator analogues in the experimental gain-loss classical optics, etc.) the instant t c r i t i c a l of a spontaneous breakdown of the parity-time alias gain-loss symmetry should be given, in the rigorous language of mathematics, the Kato’s name of an “exceptional point”, t c r i t i c a l = t ( E P ) . In the majority of conventional applications the exceptional point (EP) values are not real. In our paper, we pay attention to several exactly tractable toy-model evolutions for which at least some of the values of t ( E P ) become real. These values are interpreted as “instants of a catastrophe”, be it classical or quantum. In the classical optical setting the discrete nature of our toy models might make them amenable to simulations. In the latter context the instant of Big Bang is mentioned as an illustrative sample of possible physical meaning of such an EP catastrophe in quantum cosmology.
In this work, taking the most general non-Hermitian Hamiltonian without parity-time symmetry and with non-Hermitian off-diagonal elements into account, we study the exceptional points of the model system. We find that there exists an exceptional ellipse in theory, compared with the exceptional ring studied in non-Hermitian symmetry Hamiltonian, in which the parameters in the model Hamiltonian take particular values. We also report on the dynamics of the model Hamiltonian and symmetry Hamiltonian. We find that they display different dynamical behaviors under different parameters. Moreover, there exists an unstable behavior of the system. The unstable behavior has a surprising property by which the gain always dominates at long times no matter how small the gain is. This phenomenon enables the compensation for losses in dissipative systems and opens a wide range of applications in quantum optics, plasmonics, and optoelectronics, where the loss is inevitable but can be compensated by the gain.
Analysis of fundamentally open systems which exhibit both spatial reflection (parity) and time-reversal symmetry, called PT-symmetric [1] systems, has played an important role in understanding the dynamics of a broad class of non-Hermitian systems. Such systems support distinct regimes of stability and instability respectively, and the transition between these plays an important role in the system dynamics. We investigate a variety of non-Hermitian systems which exhibit a PT-symmetry breaking transition in both classical and quantum contexts using an array of theoretical and numerical approaches which are directly amenable to physical experiments. Organizationally, we divide these studies into those done on classical and quantum systems. Specifically for classical systems we analyze several interesting phenomena in the context of optical waveguide arrays. These provide an important testing ground for much of the discrete, PT-symmetric theory. In our case, we analyze PT-symmetric extensions of several interesting lattice models. Firstly, we explore single gain/loss pair, PT-symmetric extensions of the Su-Schreiffer-Heeger (SSH) model [2] and the Aubry-Andre-Harper (AAH) model [3,4]. In both cases, we find a veiled symmetry which is responsible for the PT-symmetry. Secondly we expand upon the Rudner-Levitov (RL) model [5,6], a non-Hermitian extension of the SSH model with multiple gain/loss pairs, and we explore the dependence of topological phenomena on changes in the initial state as well as their robustness in the presence of gain-induced nonlinearity. For quantum systems, we explore the rich PT-symmetric phase diagram resulting from time-periodic, Floquet PT-symmetry [7,8]. This is experimentally implementable in ultracold atomic systems [9], and features a reentrant PT-symmetric phase. Secondly, we examine quantum multi-photon statistics for a simple non-Hermitian model which is transformed and mapped to a system with unitary evolution via a unitary dilation, and is fully realizable on a small-scale quantum simulator [10].
Parity-time (PT) symmetric systems are classical, gain-loss systems whose dynamics are governed by non-Hermitian Hamiltonians with exceptional-point (EP) degeneracies. The eigenvalues of a PT-symmetric Hamiltonian change from real to complex conjugates at a critical value of gain-loss strength that is called the PT breaking threshold. Here, we obtain the PT threshold for a one-dimensional, finite Kitaev chain---a prototype for a p-wave superconductor---in the presence of a single pair of gain and loss potentials as a function of the superconducting order parameter, on-site potential, and the distance between the gain and loss sites. In addition to a robust, nonlocal threshold, we find a rich phase diagram for the threshold that can be qualitatively understood in terms of the band structure of the Hermitian Kitaev model. In particular, for an even chain with zero on-site potential, we find a re-entrant PT-symmetric phase bounded by second-order EP contours. Our numerical results are supplemented by analytical calculations for small system sizes.
Abstract With an innovative idea of acceptability and usefulness of the non-Hermitian representations of Hamiltonians for the description of unitary quantum systems (dating back to the Dyson’s papers), the community of quantum physicists was offered a new and powerful tool for the building of models of quantum phase transitions. In this paper the mechanism of such transitions is discussed from the point of view of mathematics. The emergence of the direct access to the instant of transition (i.e., to the Kato’s exceptional point) is attributed to the underlying split of several roles played by the traditional single Hilbert space of states ℒ into a triplet (viz., in our notation, spaces K and ℋ besides the conventional ℒ ). Although this explains the abrupt, quantum-catastrophic nature of the change of phase (i.e., the loss of observability) caused by an infinitesimal change of parameters, the explicit description of the unitarity-preserving corridors of access to the phenomenologically relevant exceptional points remained unclear. In the paper some of the recent results in this direction are summarized and critically reviewed.
Gain and loss to the wave function of quantum mechanics can in a convenient way be modelled by effective non-Hermitian Hamiltonians. Imaginary contributions to the potential introduce source and drain terms for the probability amplitude. A special class of non-Hermitian Hamiltonians are those which possess a parity-time symmetry. In spite of their non-Hermiticity these Hamiltonians allow for real energy eigenvalues, i.e. the existence of stationary states in the presence of balanced gain and loss. This effect has been identified theoretically in a large number of quantum systems. Its existence has also been proved experimentally in coupled optical wave guides. The wave guides are, however, only optical analogues of quantum systems. In the first part of this thesis it is shown from the theoretical side that Bose-Einstein condensates in a double-well setup are an ideal candidate for a first experimental realisation of a genuine quantum system with parity-time symmetry. When particles are removed from one well and coherently injected into the other the external potential is parity-time symmetric. To investigate the system the underlying time-independent and time-dependent Gross-Pitaevskii equations are solved numerically. It turns out that a subtle interplay between the nonlinearity of the Gross-Pitaevskii equation and the gain-loss effect leads to a complicated dynamics of the condensate wave function. However, the most important result is the existence of stationary states that are sufficiently stable to be observable in an experiment. Two suggestions for experimental realisations are presented. They are based on the idea of embedding the non-Hermitian parity-time-symmetric system into a larger structure described by a Hermitian Hamiltonian. A further effect of non-Hermitian Hamiltonians are so-called exceptional points, at which two resonances coalesce such that both their eigenvalues and wave functions become identical. It is shown that an exceptional point can unambiguously be identified by a characteristic non-exponential decay of the resonances. With numerically exact calculations for the hydrogen atom in crossed electric and magnetic fields this effect is verified in an experimentally accessible quantum system. The second part of the thesis is devoted to semiclassical Gaussian approximations to the Boltzmann operator, which have become an important tool for the investigation of thermodynamic properties of clusters of atoms at low temperatures. A numerically cheap frozen Gaussian approximation to the imaginary time propagator with a width matrix especially suited for the dynamics of clusters is developed. It is applied to the cases of Ar3 and Ar6. For these clusters classical-like transitions in one step from a bounded moiety to free particles are found for increasing temperatures. Additionally, the structure of the Ar6 cluster is studied in the bound configuration and during the dissociation. Quantum effects, i.e. differences with the purely classical case, manifest themselves in the low-temperature behaviour of the mean energy and specific heat as well as in a slight shift of the transition temperature. A first-order correction to the semiclassical propagator is used to improve the results of the calculation for Ar3, and it is shown how the correction can be used to objectively assess the validity of the frozen Gaussian approximation.
Stable bound states living on a discrete one-dimensional N-point lattice are assumed generated by a certain two-parametric PT symmetric potential V=V(A,B). At any N, the coordinates A=A(EP) and B=B(EP) of the Kato's exceptional point (marking the phase-transition instant of the loss of stability) are defined, exactly, via roots of a certain elementary quartic polynomial.