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arXiv:1902.02106 · Physical Review A

Monopole-antimonopole instability in non-Hermitian coupled waveguides

Rosie Hayward, Fabio Biancalana

Published Apr 29, 2020• 1 citation
arXivHTMLPDF

Abstract

A non-Hermitian coupled waveguide system with periodically varying parameters, in which the Berry curvature is analogous to a hyperbolic magnetic monopole or antimonopole, is investigated. It is shown to have a purely imaginary Berry connection and is consequently influenced by a geometric multiplier. It is possible for this multiplier to induce net gain or loss in the system, corresponding to the existence of the antimonopole or monopole in parameter space, respectively. For the right choice of parameters, the system will display an apparent nonadiabatic change in behavior, which implies a switch between the dominant eigenstate in the waveguides, leading to a change in parameter space analogous to a charge reversal of the hyperbolic magnetic monopole.

DOI
10.1103/physreva.101.043846

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A non-Hermitian coupled waveguide system with periodically varying parameters, in which the Berry curvature is analogous to a hyperbolic magnetic monopole or antimonopole, is investigated. It is shown to have a purely imaginary Berry connection and is consequently influenced by a geometric multiplier. It is possible for this multiplier to induce net gain or loss in the system, corresponding to the existence of the antimonopole or monopole in parameter space, respectively. For the right choice of parameters, the system will display an apparent nonadiabatic change in behavior, which implies a switch between the dominant eigenstate in the waveguides, leading to a change in parameter space analogous to a charge reversal of the hyperbolic magnetic monopole.
We show that the analog of the geometric phase for non-Hermitian coupled waveguides with PT\mathcal{PT}PT symmetry and at least one periodically varying parameter can be purely imaginary, is hence no longer a true phase and instead a real multiplier, and will consequently result in the amplification of Floquet sidebands in the system. The sideband peaks seen in the spectrum of the system's eigenstates after evolution along the waveguides can be directly mapped to the spectrum of the derivative of the geometric function. The sidebands are magnified (becoming virtually unstable) as the exceptional point of the system is approached, and nonadiabatic effects begin to appear. Because the system cannot evolve adiabatically in the vicinity of the exceptional point, PT\mathcal{PT}PT symmetry will be observed breaking earlier than theoretically predicted.
Abstract This work revisit on the intrinsic connection between Dirac monopole theory and Berry geometric phases by extending Dirac's theory to the parameter space. Using the simplest two-mode Hamiltonian model, we explicitly visualize Dirac strings with endpoints in the parameter space, demonstrating that these endpoints correspond to accidental degeneracy points of energy eigenvalues in Hermitian systems. We show that non-integrable phase factors, induced by such Dirac strings, directly give rise to the well-known Berry connection and curvature, which can be re-derived by following the spirit of Dirac's monopole theory with the aid of the quantum adiabatic theorem. Our results indicate that the Berry geometric phase is essentially the non-integrable phase factor induced by Dirac strings with endpoints in the parameter space. These results further deepen our understanding on monopole and geometric phases, and would inspire more studies on them especially when Berry's framework is not applicable.
Concepts from non-Hermitian quantum mechanics have proven useful in understanding and manipulating a variety of classical systems, such as those encountered in optics, classical mechanics, and metamaterial design. Recently, the non-Hermitian analog of the Berry phase for adiabatic processes was experimentally measured. In non-Hermitian systems, the Berry phase can have an imaginary part, which contributes to the amplification or decay of the total wave intensity. When the imaginary part of the Berry curvature is zero, this geometric amplification factor is determined solely by the initial and final points of the adiabatic path in parameter space, and it does not depend on how these points are connected by the path. We list classes of non-Hermitian Hamiltonians where this path independence is guaranteed by suitable symmetries, and we find that, for some of these classes, the amplification factor can be written only in terms of the Petermann factors of the initial and final points. Our result can, in turn, be used to experimentally obtain the Petermann factor by observing how the norm of the wave function changes under adiabatic processes. We validate our theory using a couple of concrete examples of physical relevance.
We theoretically establish that non-Hermitian perturbations induce a topological transformation of point-like Dirac monopoles into extended monopole distributions, characterized by distinct charge configurations emergent from three distinct Berry connection forms. Using piecewise adiabatic evolution, we confirm the validity of these configurations through observations of complex geometric phases. Most critically, we find a quantitative relation Δϕd=ΔϕgΔϕ_d = Δϕ_gΔϕd​=Δϕg​, which quantifies how cumulative minute energy differences (ΔϕdΔϕ_dΔϕd​) manifest as geometric phase shifts (ΔϕgΔϕ_gΔϕg​) uniquely in non-Hermitian systems. We further propose a scheme leveraging soliton dynamics in dissipative two-component Bose-Einstein condensates, enabling direct measurement of these topological signatures. These results establish a milestone for understanding Dirac monopole charge distributions and measuring complex geometric phases in non-Hermitian systems, with far-reaching implications for topological quantum computing and non-Hermitian photonics.
A magnetic monopole is a hypothetical elementary particle with an isolated magnetic pole. Its existence would directly lead to the quantization of electric charges. In recent years, analogs of magnetic monopoles, represented by topological defects in parameter spaces, have been studied in a wide range of physical systems. These works mainly focused on Abelian Dirac monopoles in spin-1/2 or non-Abelian Yang monopoles in spin-3/2 systems. Here we propose to realize three types of spin-1 topological monopoles and study their geometric properties using the parameter space formed by three hyperfine states of ultracold atoms coupled by radio-frequency fields. These spin-1 monopoles, characterized by different monopole charges, possess distinct Berry curvature fields and spin textures, which are directly measurable in experiments. The topological phase transitions between different monopoles are accompanied by the emergence of a spin ``vortex'' and can be intuitively visualized using Majorana's stellar representation. We show how to determine the Berry curvature, hence the geometric phase and monopole charge from dynamical effects. Our scheme provides a simple and highly tunable platform for observing and manipulating spin-1 topological monopoles, paving the way for exploring new topological quantum matter.
The smooth topology change of Berry's phase from a Dirac monopolelike configuration to a dipole configuration, when one approaches the monopole position in the parameter space, is analyzed in an exactly solvable model. A novel aspect of Berry's connection Ak{\mathcal{A}}_{k}Ak​ is that the geometrical center of the monopolelike configuration and the origin of the Dirac string are displaced in the parameter space. Gauss's theorem ${\ensuremath{\int}}_{S}(\ensuremath{\nabla}\ifmmode\times\else\texttimes\fi{}\mathcal{A})\ifmmode\cdot\else\textperiodcentered\fi{}d\stackrel{\ensuremath{\rightarrow}}{S}={\ensuremath{\int}}_{V}\ensuremath{\nabla}\ifmmode\cdot\else\textperiodcentered\fi{}(\ensuremath{\nabla}\ifmmode\times\else\texttimes\fi{}\mathcal{A})dV=0$ for a volume VVV which is free of singularities shows that a combination of the monopolelike configuration and the Dirac string is effectively a dipole. The smooth topology change from a dipole to a monopole with a quantized magnetic charge ${e}_{M}=2\ensuremath{\pi}\ensuremath{\hbar}$ takes place when one regards the Dirac string as unobservable if it satisfies the Wu-Yang gauge invariance condition. In the transitional region from a dipole to a monopole, a half-monopole appears with an observable Dirac string, which is analogous to the Aharonov-Bohm phase of an electron for the magnetic flux generated by the Cooper pair condensation. The main topological features of an exactly solvable model are shown to be supported by a generic model of Berry's phase.