Asymmetric perfect absorption and lasing of nonlinear waves by a complex δ -potential
AAsymmetric Perfect Absorption and Lasing of Nonlinear Waves by a Complex δ -potential Dmitry A. Zezyulin and Vladimir V. Konotop , ITMO University, St. Petersburg 197101, Russia Centro de F´ısica Te´orica e Computacional, Faculdade de Ciˆencias da Universidade de Lisboa,Universidade de Lisboa, Campo Grande, Edif´ıcio C8, Lisboa 1749-016, Portugal Departamento de F´ısica, Faculdade de Ciˆencias da Universidade de Lisboa,Campo Grande, Edif´ıcio C8, Lisboa 1749-016, Portugal
Spectral singularities and the coherent perfect absorption are two interrelated concepts that haveoriginally been introduced and studied for linear waves interacting with complex potentials. Inthe meantime, the distinctive asymptotic behavior of perfectly absorbed waves suggests consideringpossible generalizations of these phenomena for nonlinear waves. Here we address perfect absorptionof nonlinear waves by an idealized infinitely narrow dissipative potential modeled by a Dirac δ -function with an imaginary amplitude. Our main result is the existence of perfectly absorbedflows whose spatial amplitude distributions are asymmetric with respect to the position of theabsorber. These asymmetric states do not have a linear counterpart. Their linear stability is verifiednumerically. The nonlinear waveguide also supports symmetric and constant-amplitude perfectlyabsorbed flows. The stability of solutions of the latter type can be confirmed analytically. PACS numbers:
I. INTRODUCTION
The concept of spectral singularities (SSs) is known inmathematics already for a long time [1, 2]. The relatedphysical phenomenon, known today as coherent perfectabsorption (CPA) [3], was discovered independently [4–6] (see [7] for a chronological review on the topic) andis characterized by the asymptotic behavior of the fieldcorresponding to only incoming wave. The link betweenmathematical singularities and asymptotic behaviour ofthe respective solutions is established by the theorem dueto Vainberg [8]. In the last decade, the interest in physi-cal effects related to the SSs was revitalized due do to aseries of works [3, 9, 10] establishing direct links betweenmathematical properties of SSs and their relevance forphysical applications, as well as due to the first experi-mental implementation of a CPA [11].The early proposals of CPA addressed the absorp-tion of the electromagnetic radiation by a layer withthe complex-valued dielectric permittivity. This, in par-ticular, can be implemented using a confined plasmalayer [5]. More recently, CPA was reported for a varietyof photonic systems, including plasmonic metasurfaces,graphene-based systems electromagnetic waves interact-ing with graphene and plasmonic metasurfaces, micro-cavities, etc., see a recent review [12] on physical applica-tions of photonic coherent perfect absorbers. Moreover,the paradigm of CPA was enriched by addressing theabsorption of waves of various nature, such as acousticwaves interacting with a fluid absorber [13] and quantumsuperfluids depleted by a focused electron beam appliedto an atomic Bose-Einstein condensate (BEC) [14].By its definition, SS is an essentially linear object.However, considering it from the physical point of view,i.e., focusing on the distinctive asymptotic behavior ofthe solutions associated with SSs, one can extend the paradigm to nonlinear setups. One of the ways to dothis is by using nonlinear properties of a confined ab-sorbing layer embedded in a linear medium [15, 16]. An-other possible way is to consider waves propagating in anonlinear medium and interacting with a linear absorb-ing potential. This generalization of the concept of CPAwas suggested in [17] and validated in experiments withatomic BECs [14]. Furthermore, the two types of nonlin-earities (that of the potential and that of the medium)can be combined [18]. Strictly speaking, in the nonlinearcase there is no interference of states, and thus the co-herence loses the meaning it has in the linear theory, butone still can consider perfect absorption.These recent developments raise new questions in thetheory of perfect absorption, such as the effects of even-tual instabilities and nonlinearity-induced symmetry-breaking. The aim of this paper is to show that waveg-uides with a spatially uniform nonlinearity and a local-ized dissipation in the form of the idealized δ -functionsupport perfectly absorbed modes with asymmetric am-plitude distribution. In sharp contrast to constant-amplitude CPA solutions, nonlinear asymmetric currentscannot be reduced to the linear limit by decreasing thebackground solution intensity.In a more general context of the nonlinear waves the-ory, the found asymmetric states are remarkable, be-cause they are supported by only a single absorbinglayer. This situation contrasts with the well-studied con-ventional spontaneous symmetry-breaking of nonlinearwaves which typically requires a double-well spatial po-tential (see e.g. [19–22]) or results from the competi-tion between spatially inhomogeneous linear and nonlin-ear potentials [23] .The rest of our paper is organized as follows. In Sec-tion II we introduce the main model and provide somepreliminary discussion, and in Section III we present and a r X i v : . [ n li n . PS ] O c t discuss the main results of the study. Section IV con-cludes the paper. II. THE MODEL
We consider the spatially one-dimensional defocusingnonlinear Schr¨odinger equation (NLSE) i Ψ t = − Ψ xx − iγδ ( x )Ψ + | Ψ | Ψ , (1)where γ (cid:54) = 0 is a real parameter which governs thestrength of the dissipation (for γ >
0) or energy gain(for γ < δ ( x ) is the Dirac delta function. Themodel (1) was introduced in Ref. [24] as a limitingcase, modeling scanning electron microscopy of ultra-cold atomic gasses [25]. In such an experimental set-ting, a Bose-Einstein condensate is affected by a nar-row electronic beam, which in the meanfield approxi-mation is described by a localized dissipative potentialin the Gross-Pitaevskii equation [26]. Similar scenar-ios with the spatially confined absorption of nonlinearwaves can be implemented in other experimental setups,like nonlinear optical and magnon waveguides, plasmonicnanostructures, exciton-polariton condensates, etc. (seee.g. [17, 27] for schematics of possible systems).In what follows, we present our main results mainly forthe perfectly absorbed flows supported by a δ -function-shaped dissipation situated at x = 0, and therefore weassume γ >
0. In the meantime, most of our results canbe generalized straightforwardly on the case of a lasingpotential by inverting the sign of γ .Looking for stationary states Ψ = e − iµt ψ ( x ), wherereal µ has the meaning of the chemical potential ofthe condensate, we use the hydrodynamic represen-tation of the time-independent wavefunction ψ ( x ) = ρ ( x ) exp (cid:8) i ´ x v ( s ) dx (cid:9) , where | ρ | = | ψ | is the amplitudeof the wavefunction, and v ( x ) is the hydrodynamics ve-locity. The respective current density, j ( x ), is defined as j ( x ) = 2 v ( x ) | ψ | . Now Eq. (1) reduces to the system ρ xx + (cid:18) R + J R (cid:19) ρ − ρ − j ρ = 0 , (2) j x + 2 γδ ( x ) ρ = 0 . (3)We are looking for perfectly absorbed flows directedfrom the left and the right infinities towards the cen-ter. The corresponding solutions are determined by theboundary conditionslim x →±∞ ρ ( x ) = R, lim x →±∞ j ( x ) = ∓ J, (4)where constants R ≥ J ≥ µ = R + J R . (5) Since the perfectly absorbed flows are directed fromthe infinity to the center, the limiting (below also calledbackground) current density is negative ( − J ) for largepositive x and positive (+ J ) for large negative x . Thecase of lasing solutions emitted by a δ -function amplify-ing potential with γ < J <
0. In this situation it follows from (4) thatthe current j ( x ) is positive for x > x < j ( x ) = − J sign x , and the background cur-rent density is related to the amplitude at x = 0 as J = γρ (0) . (6)Using that j ( x ) = J is constant for x (cid:54) = 0, and integrat-ing Eq. (2) we obtain a first-order nonlinear differentialequation, in which J plays the role of a parameter: ρ x + (cid:18) R + J R (cid:19) ρ − ρ J ρ = R J R . (7)Using relation (6), from Eq. (7) one can express thederivative of the amplitude at the origin through the pa-rameters of the problem:[ ρ x (0)] = 14 γ R ( γR − J ) (2 R − γJ ) . (8) III. THE MAIN RESULTSA. Symmetric and asymmetric perfectly absorbedflows
Let us now discuss possible types of perfectly absorbedsolutions that can be found in the introduced model.First, we notice that there exists an immune to the dis-sipation solution in the form of a dark soliton pinned to x = 0 [24]: ψ = R tanh( Rx/ √ j ( x ) ≡
0. So-lutions of the second type correspond to the constant-amplitude nonlinear CPA modes and have uniform am-plitudes ρ ( x ) = R . Such modes are characterized by thebackground current densities J = γR and represent thedirect nonlinear generalization, parametrized by R , of thelinear CPA corresponding to the SS of the absorbing δ -potential [28]. These nonlinear modes exist for arbitrarilystrong dissipation γ .Solutions of the third type are characterized by a spa-tially nonuniform amplitude with a dip. These solutionsdo not exist in the linear limit. They feature nonzero cur-rent density at the dissipative spot and have nontrivialand asymmetric amplitude landscapes. The possibility ofexistence of such currents becomes evident from the in-spection of the phase space ( ρ, ρ x ) corresponding to theidentity (7) which for any J in the interval (0 , √ R )features a homoclinic orbit connecting the saddle point( ρ, ρ x ) = ( R,
0) to itself (according to the introducedboundary conditions, the latter saddle point correspondsto x = ±∞ ). Further analysis of the differential equation(7) shows that the homoclinic orbits exemplified in Fig. 1are associated with exact solutions of the form ρ ( x ) = R − D sech (cid:18) D ( x − x ) √ (cid:19) , (9)where the new parameters D and x are defined from therelations D = R − J R , (10) x = ± R √ R − J acosh (cid:32) R (cid:115) R − J R − J/γ ) (cid:33) . (11)Notice that the quantity D which characterizes thedepth of the dip in the squared amplitude is positiveprovided that J < √ R . Additionally, for the solutionto be meaningful, two more constraints must be imposed.The first condition reads J < γR . (12)This condition is necessary to guarantee that the argu-ment of acosh in Eq. (11) is real. In the asymptotic limitwhere J approaches γR from below, the argument ofacosh diverges, and the position of the intensity dip x tends to ±∞ . Another condition that needs to be im-posed for the solutions to be meaningful reads J ≤ R /γ. (13)It implies that the right-hand side of (8) is nonnegative,and the argument of acosh in Eq. (11) is greater than orequal to unity.Notice that found solutions are generically asymmet-ric , i.e., the minimum of amplitude is achieved at x which is generically different from zero. Notice also thatfound states are essentially nonlinear since they cannotbe reduced to the linear limit by sending the nonlinear-ity coefficient to zero, i.e., they have no counterpart inthe asymptotically linear limit of small background am-plitudes R →
0. Such asymmetric perfectly absorbedstates exist in pairs, with positive and negative x , whichis reflected by the ± sign in (11). To summarize the sit-uation, in Fig. 2 we show a representative existence dia-gram for perfectly absorbed solutions of different types.Since the main characteristic of a perfectly absorbed so-lution is the associated flux J , the existence diagramis presented in the plane J vs. γ for fixed backgroundamplitude R . Emerging of asymmetric nonlinear statesupon the increase of the dissipation strength is illustratedin Fig. 3(a) for two fixed values of the background cur-rent J . Asymmetric states emerge when the dissipationstrength exceeds J/R (the latter value corresponds tothe dissipation that is necessary to support the constant-amplitude CPA state with the given current J ). Emerg-ing nonlinear states are characterized by the infinitely FIG. 1: Examples of orbits generated by the differential equa-tion (7) in the phase space ( ρ, ρ x ) for several different valuesof J . The heteroclinic orbits shown with red lines correspondto the dark soliton solution that is immune to the dissipationand bears zero background current J = 0. Homoclinic orbitswith J = 0 . , , . ρ, ρ x ) = ( R, R is the background amplitude. Depending on the position ofthe point that corresponds to the dissipative spot, i.e., x = 0in our case, each orbit can represent either a symmetric oran asymmetric perfectly absorbed flow. In this figure, thebackground amplitude of nonlinear flows is fixed as R = 1. large position of the amplitude dip x (mathematicallydiverging x is explained by the fact that the argumentof acosh in (11) is infinitely large). The further increaseof the dissipation strength γ decreases x and eventually,at γ = 2 R /J , the amplitude distribution becomes sym-metric with x = 0. Thus symmetric perfectly absorbedstates correspond to the equality sign in (13). The exis-tence of symmetric dip solutions can be explained by thefact that in a realistic system that cannot support in-finitely large background currents J , one should expectthat the increase of the dissipation strength destroys theconstant amplitude of the steadily absorbed state andeventually results in the decrease of the background cur-rent rather than to its increase (compare red and greencurves in Fig. 2). This behavior can be attributed to themacroscopic Zeno effect studied previously in nonlinearwaveguides with localized dissipation [27]. B. Stability of perfectly absorbed nonlinearcurrents
In order to study spectral stability of the perfectly ab-sorbed flows, we consider a perturbed stationary mode inthe form Ψ( x, t ) = e − iµt [ ψ ( x ) + u ( x ) e iωt + v ∗ ( x ) e − iω ∗ t ],where u and v are small-amplitude perturbations. A lin-earization of the main equation (1) with respect of u and FIG. 2: The existence diagram for solutions of three differenttypes discussed in Sec. III A. The diagram is shown in theplane J vs. γ for fixed background density R = 1. Thegray domain corresponds to asymmetric dips, whereas red andgreen lines correspond to the constant-amplitude solutionsand symmetric dips, respectively.FIG. 3: (a) Dependencies of the position of the amplitude dip x defined by (11) on the absorption strength γ for the fixedvalues of the background current: J = 1 / J = 1 (blue curve). (b) Squared amplitude of asymmetricstates that exist at γ = 3 / J = 1 / J = 1 (blue curve). In both panels R = 1. Only solutions with x ≥ x ≤ v leads to the system( ∂ x + µ + iγδ ( x ) − | ψ | ) u − ψ v = ωu, (14)( ∂ x + µ − iγδ ( x ) − | ψ | ) v − ( ψ ) ∗ u = − ωv. (15)The spectrum of eigenvalues ω associated with boundedeigenfunctions u ( x ) and v ( x ) characterizes stability of the stationary solution: the perfectly absorbed flow is unsta-ble if there is an eigenvalue ω with negative imaginarypart: Im ω <
0. Since for the defocusing nonlinearitythe modulational instability of the uniform background isabsent, and it is intuitively clear that the thin absorbinglayer cannot excite spatially unbounded unstable modes,we expect that the eventual instability can be causedonly by spatially localized eigenmodes. We are thereforeinterested in eigenfunctions u , v that decay as x → ±∞ .In general, eigenvalue problem (14)–(15) can only besolved numerically. However, for the constant-amplitudesolutions, the analytical treatment is possible. Let us firstconsider this problem only in the right half-axis x > ψ ( x ) = Re − iγx/ , and chemical potential reads µ = R + γ /
4. Using substitutions u ( x ) = U ( x ) e − iγx/ , v ( x ) = V ( x ) e iγx/ , the stability equations take the formof a constant-coefficient ODE problem ∂ x U + A U = 0 , (16)where U = U∂ x UV∂ x V , (17)and A = − − ω − R − iγ − R
00 0 0 − − R ω − R iγ . (18)Searching for solutions proportional to e − kx , the expo-nent k is determined from the characteristic equation formatrix A : k + ( γ − R ) k + 2 iγωk − ω = 0 . (19)For solutions that decay at x → ∞ , roots k must be-long to the right complex half-plane. Using the Routh-Hurwitz theorem [29], we conclude that for any ω withnegative imaginary part the characteristic equation (19)has exactly two roots k , with positive real parts. Gener-ically, those two roots are different, and the most generalsolution that decays at x → ∞ is a linear combination oftwo independent exponents. In terms of functions u ( x )and v ( x ) this solution has the form u + ( x ) = ( U , + e − k x + U , + e − k x ) e − iγx/ , (20) v + ( x ) = ( U , + Λ( k ) e − k x + U , + Λ( k ) e − k x ) e iγx/ , (21)where U , , + are complex-valued constant coefficients,and Λ( k ) = ( k + iγk − ω ) /R − x <
0. The analysis can be performed ina similar by replacing γ to − γ and k with − k (becausenow we consider growing solutions U, V ∝ e kx ). As aresult, the general solution reads u − ( x ) = ( U , − e k x + U , − e k x ) e iγx/ , (22) v − ( x ) = ( U , − Λ( k ) e λ x + U , − Λ( k ) e k x ) e − iγx/ . (23)Applying the continuity conditions u + (0) = u − (0), v + (0) = v − (0) and using that Λ( k ) (cid:54) = Λ( k ), we con-clude that U , + = U , − = U and U , + = U , − = U .Integrating equations (14)–(15) across x = 0, we obtainconditions for jumps of the derivatives: u + ,x (0) − u − ,x (0) = − iγu ± (0) , (24) v + ,x (0) − v − ,x (0) = iγv ± (0) . (25)It is easy to check that these conditions imply U = U =0; therefore no unstable localized mode is possible.For the peculiar case when the characteristic equation(19) has a double root k = k in the right half-plane,inspecting the structure of this polynomial one can es-tablish that the latter root is positive: k >
0, and,respectively, if the corresponding eigenvalue ω exists,then it is purely imaginary: ω = iλ , λ <
0. Inthis case without loss of generality one can consider so-lutions of the form U ( x ) = V ∗ ( x ). The solution has theform U ( x ) = ( a ± a x ) e ∓ k x , where upper and lowersigns correspond to x > x <
0, respectively,and a , are some coefficients. From the condition ofthe derivative jump it follows that a = ka . On theother hand, from the ODE system (16)-(17) one can de-rive ( k + iλ + R ) a + R a ∗ = 0, which is impossiblefor real k and λ . Therefore, in the case of the doubleroot the instability of constant-amplitude currents can-not take place.Thus we have demonstrated that no localized eigen-modes of the stability problem exists for any ω withnonzero real part. This implies stability of constant-amplitude nonlinear CPA states.For dark solitons and dip solutions linear stabilityequations (14)–(15) does not admit simple analyticaltreatment, but the spectrum of eigenvalues ω can be com-puted numerically. A systematic stability study demon-strates that these perfectly absorbed solutions are alsostable for all parameters in their existence domain. IV. CONCLUSION
In this paper, we have studied nonlinear stationaryflows perfectly absorbed by an idealized infinitely narrowdissipative spot of infinite strength, which can be mod-eled by a Dirac δ -function potential with a purely imagi-nary amplitude. The found solutions have been classifiedinto three types. Solutions of the first type have the formof conventional dark solitons. Their amplitude is identi-cally zero at the dissipative spot, and these solutions aretherefore immune to dissipation and do not generate su-perfluid flows. Solutions of the second class have constantamplitude and represent a direct nonlinear generalizationof linear coherently absorbed modes corresponding to thespectral singularities of the underlying δ -function poten-tial. Solutions of the third type represent intensity dipsin the uniform amplitude. Remarkably, these solutionsdo not have linear counterparts, and their amplitude dis-tribution can be asymmetric. Constant-amplitude CPAflows and (a)symmetric dips are supported by superfluidflows directed towards the dissipation. Using the linearstability approach, we have demonstrated analyticallythat constant-amplitude flows are stable. The stabilityof dark solitons and dip solutions has been confirmed nu-merically. Acknowledgments
Work of D.A.Z. is funded by Russian Foundation forBasic Research (RFBR) according to the research projectNo. 19-02-00193. V.V.K. acknowledges supported fromthe Portuguese Foundation for Science and Technology(FCT) under Contract no. UIDB/00618/2020. [1] Naimark, M. A. Investigation ofthe spectrum and theexpansion in eigenfunctions of a nonselfadjoint operatorof the second order on a semi-axis.
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