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arXiv:1812.05499 · Physical Review A

Designing lasing and perfectly absorbing potentials

V. V. Konotop, Evgeny Lakshtanov, Boris Rufimovich Vainberg

Published Apr 26, 2019• 20 citations
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Abstract

Existence of a spectral singularity (SS) in the spectrum of a Schr\"odinger operator with a non-Hermitian potential requires exact matching of parameters of the potential. We provide a necessary and sufficient condition for a potential to have a SS at a given wavelength. It is shown that potentials with SSs at prescribed wavelengths can be obtained by a simple and effective procedure. In particular, the developed approach allows one to obtain potentials with several SSs and with SSs of the second order, as well as potentials obeying a given symmetry, say, $\mathcal{PT}$-symmetric potentials. We illustrate all these opportunities with examples. We also describe splitting of a second-order SS under change of the potential parameters, and discuss possibilities of experimental observation of SSs of different orders.

DOI
10.1103/physreva.99.043838

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Existence of a spectral singularity (SS) in the spectrum of a Schr"odinger operator with a non-Hermitian potential requires exact matching of parameters of the potential. We provide a necessary and sufficient condition for a potential to have a SS at a given wavelength. It is shown that potentials with SSs at prescribed wavelengths can be obtained by a simple and effective procedure. In particular, the developed approach allows one to obtain potentials with several SSs and with SSs of the second order, as well as potentials obeying a given symmetry, say, PT\mathcal{PT}PT-symmetric potentials. We illustrate all these opportunities with examples. We also describe splitting of a second-order SS under change of the potential parameters, and discuss possibilities of experimental observation of SSs of different orders.
If spectrum of a Schr\{o}dinger oparator with a non-Hermitian potential contains a spectral singularity (SS), the latter requires exact matching of the parameters characterizing the potential. We provide a necessary and sufficient condition for a potential to have a SS at a given wavelength. It is shown that potentials with SS at prescribed wavelengths can be obtained by a simple and effective procedure. In particular, the developed approach allows one to obtain potentials with several SSs or with SSs of the second order and potentials obeying a given symmetry, say, PT-symmetric potentials. Also, the problem can be solved when it is required to obtain a potential obeying a given symmetry, say, PT−\mathcal{PT}-PT−symmetric potential. We illustrate all opportunities with examples. We also describe splitting of the SSs of the second order, under change of the potential parameters, and discuss possibilities of experimental observation of SSs of different orders.
We investigate a two-parametric family of one-dimensional non-Hermitian complex potentials with parity-time (PT\mathcal{PT}PT) symmetry. We find that there exist two distinct types of phase transitions, from an unbroken phase (characterized by a real spectrum) to a broken phase (where the spectrum becomes complex). The first type involves the emergence of a pair of complex eigenvalues bifurcating from the continuous spectrum. The second type is associated with the collision of such pairs at the bottom of the continuous spectrum. The first transition type is closely related to spectral singularities (SSs), at which point the transmission and reflection coefficients are divergent simultaneously. The second is associated with the emergence of bound states. In particular, under specific parameter conditions, we construct an exact bound state solution. By systematically exploring the parameter space, we establish a universal relationship governing the number of SSs in these potentials. These findings provide a fundamental theoretical framework for manipulating wave scattering in non-Hermitian systems, offering promising implications for designing advanced optical and quantum devices.
We develop a perturbative method of computing spectral singularities of a Schr\"odinger operator defined by a general complex potential that vanishes outside a closed interval. These can be realized as zero-width resonances in optical gain media and correspond to a lasing effect that occurs at the threshold gain. Their time-reversed copies yield coherent perfect absorption of light that is also known as antilasing. We use our general results to establish the exactness of the nnnth-order perturbation theory for an arbitrary complex potential consisting of nnn delta functions, obtain an exact expression for the transfer matrix of these potentials, and examine spectral singularities of complex barrier potentials of arbitrary shape. In the context of optical spectral singularities, these correspond to inhomogeneous gain media.
We propose a scheme for a coherent perfect absorber (CPA) and a laser in an atomic medium with gain and loss, obeying simultaneously a spectral singularity and a time-reversed spectral singularity, both occurring at different wavevectors (or frequencies). We term this system a CPA- and -laser and investigate its features allowing to obtain asymmetric lasing and absorbing properties, switches, etc. We show that the CPA-and-laser can be obtained by modifying characteristics of a CPA-laser of an initial -symmetric configuration, provided there are at least three tunable parameters. The physical mechanism of emergence of a CPA-and-laser in this way, is based on splitting of a self-dual spectral singularity of the -symmetric CPA-laser into the spectral singularity and time-reversed spectral singularity. After the discussion of a particular example of a bilayer consisting of one active and one passive slabs, we suggest a realistic physical system for implementing a CPA-and-laser. It consists of two adjacent atomic cells filled with isotopes of Λ-type three-level rubidium atoms interacting with probe and control fields, allowing for the required number of tunable parameters.
A spectral singularity is a mathematical notion with an intriguing physical realization in terms of certain zero-width resonances. In optics it manifests as lasing at the threshold gain. We explore the recently developed nonlinear generalization of spectral singularities for a complex barrier potential and study the associated resonance effect for an infinite planar slab of gain medium. In particular, for a Kerr nonlinearity, we show that the first-order perturbative equation that determines the nonlinear spectral singularities provides an explicit expression for the intensity of the emitted waves from the slab. This is a mathematical derivation of the known linear relationship between the output intensity of the slab and its gain coefficient. We also discuss the implications of our results for the time-reversed system which acts as a coherent perfect absorber.
The spectral singularity (SS) from a non-Hermitian potential is one of the most remarkable scattering feature of non-Hermitian quantum mechanics. At the spectral singular point, the scattering amplitudes diverge to infinite. This phenomena have been extensively studied over the last two decades. The previous studies have suggested the need of extremely fine control of the various system parameters for practical applications of SS. However no study have been carried out to understand the behavior of the scattering amplitude in the vicinity of SS. Such study would be important towards the practical application of SS where it is desired to maintain outgoing scattering amplitude to a specific value. The behavior of the loci of constant scattering amplitude in the neighborhood of SS are studied for a pure imaginary barrier potential iViViV, V∈R+V \in R^{+}V∈R+. We show that these loci are elliptical and self-similar to each other in E−VE-VE−V plane where EEE is energy of the wave. The orientations of these ellipses for reflection (RRR) and transmission (TTT) amplitude around a given SS are same. This is surprising due to the different mathematical forms of RRR and TTT. In the vicinity of SS, R≈TR \approx TR≈T.