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arXiv:1001.0964 · Physical Review B

Spectral singularities in a non-Hermitian Friedrichs-Fano-Anderson model

Stefano Longhi

Published Oct 23, 2009• 102 citations
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Abstract

Spectral singularities are predicted to occur in a non-Hermitian extension of the Friedrichs-Fano-Anderson model describing the decay of a discrete state $|a⟩$ coupled to a continuum of modes. A physical realization of the model, based on electronic or photonic transport in a semi-infinite tight-binding lattice with an imaginary impurity site at the lattice boundary, is proposed. The occurrence of the spectral singularities is shown to correspond either to a diverging reflection probability (for an amplifying impurity) or to a vanishing reflection probability (for an absorbing impurity) from the lattice boundary. In the former case, the spectral singularity of the resolvent is also responsible for the nondecay of state $|a⟩$ into the continuum, in spite of the absence of bound states.

DOI
10.1103/physrevb.80.165125

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Spectral singularities are predicted to occur in a non-Hermitian extension of the Friedrichs-Fano-Anderson model describing the decay of a discrete state ∣a⟩|a⟩∣a⟩ coupled to a continuum of modes. A physical realization of the model, based on electronic or photonic transport in a semi-infinite tight-binding lattice with an imaginary impurity site at the lattice boundary, is proposed. The occurrence of the spectral singularities is shown to correspond either to a diverging reflection probability (for an amplifying impurity) or to a vanishing reflection probability (for an absorbing impurity) from the lattice boundary. In the former case, the spectral singularity of the resolvent is also responsible for the nondecay of state ∣a⟩|a⟩∣a⟩ into the continuum, in spite of the absence of bound states.
The interaction of a discrete state coupled to a continuum is a longstanding problem of major interest in different areas of quantum and classical physics. In Hermitian models, several dynamical decoupling schemes have been suggested, in which the discrete-continuum interaction can be substantially reduced and even suppressed. In this work, we consider a discrete state interacting with a continuum via a time-dependent non-Hermitian coupling with finite (albeit arbitrarily long) duration, and show rather generally that for a wide class of coupling temporal shapes, in which the real and imaginary parts of the coupling are related each other by a Hilbert transform, the discrete state returns to its initial condition after the interaction with the continuum, while the continuum keeps trace of the interaction. Such a behavior, which does not have any counterpart in Hermitian dynamics, can be referred to as non-Hermitian pseudo decoupling. Non-Hermitian pseudo decoupling is illustrated by considering a non-Hermitian extension of the Fano–Anderson model in a one-dimensional tight-binding lattice. Such a non-Hermitian model can describe, for example, photonic hopping dynamics in a tight-binding chain of optical microrings or resonators, in which non-Hermitian coupling can be realized by fast modulation of the real and imaginary (gain/loss) parts of the refractive index of the edge microring.
The decay of a bound state weakly coupled to a non-Hermitian tight-binding unstable continuum, i.e., a continuum of states comprising energies with positive imaginary part, is theoretically investigated. As compared to quantum decay in a Hermitian continuum, in the non-Hermitian case a richer scenario can be found as a result of nonunitary dynamics. Different behaviors are observed depending on the kind of instability of the continuum. These include complete or fractional decay in convectively unstable continua, the absence of quantum decay for a bound state with energy embedded in the continuum loop, and unstable (secular) growth with pseudoexponential amplification in the absolutely unstable regime. Analytical results are presented for a nearest-neighboring tight-binding continuum with asymmetric hopping rates ${\ensuremath{\kappa}}_{1}$ and ${\ensuremath{\kappa}}_{2}$, which shows a transition from convective to absolute instability when the sign ${\ensuremath{\kappa}}_{1}{\ensuremath{\kappa}}_{2}$ changes from positive to negative. In the convectively unstable regime, the model describes the decay of a bound state coupled to a tight-binding lattice with an imaginary gauge field, which shows a pseudo-Hermitian dynamics. In the absolutely unstable regime, pseudo-Hermitian dynamics is broken and a pseudoexponential secular growth is observed.
This paper explores the novel concept of topological eigenvalues within the realm of non-Hermitian quantum mechanics, focusing on their role in unveiling spectral singularities. Unlike conventional Hermitian systems, non-Hermitian Hamiltonians, which describe open systems with gain and loss, exhibit complex eigenvalues and unique spectral features such as exceptional points (EPs) where eigenvalues and their corresponding eigenvectors coalesce. We introduce the idea that certain eigenvalues in non-Hermitian systems possess an inherent topological character, dictated by the geometry of the parameter space and the winding properties of the characteristic polynomial. These topological eigenvalues are shown to be intimately linked to spectral singularities, which are non-Hermitian generalizations of bound states in the continuum or points of perfect reflection/transmission, characterized by infinite scattering amplitudes. We develop a theoretical framework to identify and characterize these topological eigenvalues, demonstrating how they emerge from the interplay of non-Hermiticity and underlying symmetries. Our findings reveal that the robust nature of these topological eigenvalues provides a new lens through which to understand the exotic phenomena observed in non-Hermitian systems, including the non-Hermitian skin effect and the breakdown of traditional bulk-boundary correspondence. This work opens new avenues for designing robust non-Hermitian devices and understanding fundamental aspects of open quantum systems.
In a Hermitian system, bound states must have quantized energies, whereas free states can form a continuum. We demonstrate how this principle fails for non-Hermitian systems, by analyzing non-Hermitian continuous Hamiltonians with an imaginary momentum and Landau-type vector potential. The eigenstates, which we call "continuum Landau modes" (CLMs), have Gaussian spatial envelopes and form a continuum filling the complex energy plane. We present experimentally realizable 1D and 2D lattice models that host CLMs; the lattice eigenstates are localized and have other features matching the continuous model. One of these lattices can serve as a rainbow trap, whereby the response to an excitation is concentrated at a position proportional to the frequency. Another lattice can act a wave funnel, concentrating an input excitation onto a boundary over a wide frequency bandwidth. Unlike recent funneling schemes based on the non-Hermitian skin effect, this requires a simple lattice design with reciprocal couplings.
We study the effect of PT\mathcal{PT}PT-symmetric imaginary potentials embedded in the two arms of an Aharonov-Bohm interferometer on the transmission phase by finding an exact solution for a concrete tight-binding system. It is observed that the spectral singularity always occurs at $k=\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2$ for a wide range of fluxes and imaginary potentials. Critical behavior associated with the physics of the spectral singularity is also investigated. It is demonstrated that the quasispectral singularity corresponds to a transmission maximum and the transmission phase jumps abruptly by $\ensuremath{\pi}$ when the system is swept through this point. Moreover, we find that there exists a pulselike phase lapse when the imaginary potential approaches the boundary value of the spectral singularity.
We introduce and solve from first principles a continuous-time quantum walk with absorption generated by a Lindblad boundary sink of arbitrary strength. Projecting onto the surviving walker sector maps the problem to a non-Hermitian tight-binding Hamiltonian with a rank-one imaginary defect on the semi-infinite line. We obtain closed-form expressions for the exact propagator, from which the first-passage statistics follow. Weak coupling limits absorption through inefficient transfer into the sink, whereas for strong dissipation, occupation at the boundary is suppressed by Zeno-like reflection, accompanied by the emergence of a localized non-Hermitian mode. Despite the different physical origins of these suppression mechanisms, their respective absorption probabilities exhibit an asymptotic duality. The evolution is conveniently visualized in phase space, where the localized mode has an exponentially confined Wigner representation near the edge site.