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arXiv:2007.04591 · Annalen der Physik

Non‐Adiabatic Transitions in Parabolic and Super‐Parabolic PT‐Symmetric Non‐Hermitian Systems in 1D Optical Waveguides

Chon‐Fai Kam, Yang Chen

Published Jan 18, 2021• 2 citations
arXivHTMLPDF

Abstract

Abstract Exceptional points, which are spectral degeneracy points in the complex parameter space, are fundamental to non‐Hermitian quantum systems. The dynamics of non‐Hermitian systems in the presence of exceptional points differ significantly from those of Hermitian ones. Here, non‐adiabatic transitions in non‐Hermitian ‐symmetric systems are investigated, in which the exceptional points are driven through at finite speeds which are quadratic or cubic functions of time. Different transmission dynamics separated by exceptional points are identified, and analytical approximate formulas for the non‐adiabatic transmission probabilities are derived. Possible experimental realizations with a ‐symmetric non‐Hermitian 1D tight‐binding optical waveguide lattice are discussed through non‐Hermitian Bloch oscillations between different bands.

DOI
10.1002/andp.202000349

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Abstract Exceptional points, which are spectral degeneracy points in the complex parameter space, are fundamental to non‐Hermitian quantum systems. The dynamics of non‐Hermitian systems in the presence of exceptional points differ significantly from those of Hermitian ones. Here, non‐adiabatic transitions in non‐Hermitian ‐symmetric systems are investigated, in which the exceptional points are driven through at finite speeds which are quadratic or cubic functions of time. Different transmission dynamics separated by exceptional points are identified, and analytical approximate formulas for the non‐adiabatic transmission probabilities are derived. Possible experimental realizations with a ‐symmetric non‐Hermitian 1D tight‐binding optical waveguide lattice are discussed through non‐Hermitian Bloch oscillations between different bands.
Non-Hermitian quantum systems with explicit time dependence are of ever-increasing importance. There are only a handful of models that have been analytically studied in this context. Here, a PT-symmetric non-Hermitian N-level Landau-Zener type problem with two exceptional points of N-th order is introduced. The system is Hermitian for asymptotically large times, far away from the exceptional points, and has purely imaginary eigenvalues between the exceptional points. The full Landau-Zener transition probabilities are derived, and found to show a characteristic binomial behaviour. In the adiabatic limit the final populations are given by the ratios of binomial coefficients. It is demonstrated how this behaviour can be understood on the basis of adiabatic analysis, despite the breakdown of adiabaticity that is often associated with non-Hermitian systems.
We investigate the theory of exceptional points (EPs); degeneracies of non-hermitian systems famous for the swaps of eigenstates that arise when encircling the EP according to the adiabatic approximation. We review various characterizations of EPs that are used in the literature. We observe that they are not equivalent and consequently a unanimous definition is lacking. The theory of PT symmetric systems is studied as an aid in the study of EPs. In particular, we deal explicitly with a three-channel waveguide system originating from PT theory and supporting various EPs. We show how one can properly compose the eigenvalue permutations arising from EPs and how this generates a non-abelian group which we call the Λ-group. We finish with the needed geometry as induced by the adiabatic approximation. The adiabatic change as measured in experiment can be modelled naturally on a principal C× o Sn-bundle consisting of the eigenframes of the operator family. To the best of our knowledge this bundle is not found in the literature to date. The adiabatic approximation induces a canonical connection on this bundle. The special shape of the holonomy group induces three groups, one of which is the Λ-group, that fit together in a short exact sequence (SES). This SES can be used to describe and distinguish degeneracy structures. In particular, the Λ-group allows one to phrase a new definition of exceptional point that specializes existing definitions. Explicitly, one can define an EP by demanding that swaps of eigenvalues occur arbitrarily close to the EP. This definition is widely applicable, yet guarantees that the eigenvalue and eigenstates sheets are connected in a non-trivial way.
Exceptional points (EPs) have attracted extensive research interest due to their intriguing properties. One of the hallmarks of EP physics is that dynamically encircling the EPs induces chiral mode switching, arising from the breakdown of adiabaticity due to the presence of a complex spectrum in the system's Hamiltonian. While such chiral mode behavior has been widely observed experimentally, achieving truly adiabatic, and thus symmetric, state transfer, regardless of the winding direction, in time-modulated non-Hermitian systems has remained elusive. In this work, we demonstrate that this long-sought adiabatic state dynamics can indeed be restored. By steering a two-mode photonic setup along specifically designed trajectories in parameter space, we realize conditions where the associated non-Hermitian evolution operator acquires a purely real spectrum. Moreover, our experimental platform enables controlled switching between symmetric (adiabatic) and chiral (non-adiabatic) state-transfer regimes for the same set of initial modes, thus effectively implementing a universal symmetric-asymmetric two-mode switch. Our results therefore open new avenues for harnessing unique topological spectral properties of non-Hermitian systems, paving the way for the practical design of versatile optical wave-manipulation devices and for advancing both classical and quantum information technologies.
Non-Hermitian systems have attracted much interest in recent decades, driven partly by the existence of exotic spectral singularities, known as exceptional points (EPs), where the dimensionality of the system evolution operator is reduced. Among various intriguing applications, the discovery of EPs has suggested the potential for implementing a symmetric mode switch, when encircling them in a system parameter space. However, subsequent theoretical and experimental works have revealed that dynamical encirclement of EPs invariably results in asymmetric mode conversion; namely, the mode switching depends only on the winding direction but not on the initial state. This chirality arises from the failure of adiabaticity due to the complex spectrum of non-Hermitian systems. Although the chirality revealed has undoubtedly made a significant impact in the field, a realization of the originally sought symmetric adiabatic passage in non-Hermitian systems with EPs has since been elusive. In this work, we bridge this gap and theoretically demonstrate that adiabaticity, and therefore a symmetric state transfer, is achievable when dynamically winding around an EP. This becomes feasible by specifically choosing a trajectory in the system parameter space along which the corresponding evolution operator attains a real spectrum. Our findings, thus, offer a promise for advancing various wave manipulation protocols in both quantum and classical domains.
The spectral and transport properties of a non-Hermitian tight-binding lattice with unidirectional hopping are theoretically investigated in three different geometrical settings. It is shown that, while for the infinitely-extended (open) and for the ring lattice geometries the spectrum is complex, lattice truncation makes the spectrum real. However, an exceptional point of order equal to the number of lattice sites emerges. When a homogeneous dc force is applied to the lattice, in all cases an equally-spaced real Wannier-Stark ladder spectrum is obtained, corresponding to periodic oscillatory dynamics in real space. Possible physical realizations of non-Hermitian lattices with unidirectional hopping are briefly discussed.
Non-Hermitian Hamiltonians with complex eigenenergies are useful tools for describing the dynamics of open quantum systems. In particular, parity and time (PT) symmetric Hamiltonians have generated interest due to the emergence of exceptional-point degeneracies, where both eigenenergies and eigenvectors coalesce as the energy spectrum transitions from real to complex valued. Because of the abrupt spectral response near exceptional points, such systems have been proposed as candidates for precision quantum sensing. In this Letter, we emulate a passive PT dimer using a two-mode, non-Hermitian system of superconducting qubits comprising one high-coherence qubit coupled to an intentionally lossy qubit via a tunable coupler. The loss is introduced by strongly coupling the qubit to a continuum of photonic modes in an open waveguide environment. Using both pulsed and continuous-wave measurements, we characterize the system dynamics near the exceptional point. We observe a behavior broadly consistent with an ideal passive PT dimer with some corrections due to the tunable coupler element. We extract the complex eigenenergies associated with the two modes and calculate the sensitivity as a function of the coupling strength. Confirming theoretical predictions, we observe no sensitivity enhancement near the quantum exceptional point. This Letter elucidates the limitations of exceptional-point systems as candidates for quantum-enhanced sensing. We establish waveguide quantum electrodynamics as a versatile platform for exploring non-Hermitian quantum dynamics in superconducting circuits.