Twin Bound States in the Continuum in a Waveguide Fabry–Pérot Resonator

We study bound states in the continuum (BICs) in a Fabry–Pérot (FP) resonator within a quantum-mechanical waveguide. We show that besides typical FP BICs corresponding to a discrete set of cavity lengths, there is a pair of symmetric and antisymmetric twin BICs (TBICs) if isolated mirrors possess BIC themselves. In contrast to the FP BIC, the TBIC does not require the formation of standing waves of the FP resonance between the mirrors. Therefore, the energy and parameters of TBICs are almost independent of the length of the FP resonator, and their wavefunctions have an exponentially small amplitude between the mirrors. Results of the numerical simulation of the 2D quantum-mechanical waveguide with attractive potentials (“impurities”) playing the role of mirrors of an FP resonator are supported by the illustrative analytical model.


INTRODUCTION
Bound states in the continuum (BICs) are localized states with energies embedded in the continuum of propagating states.BIC is basically a fundamental wave interference phenomenon, which can be observed in physical systems of different nature: quantum-mechanical, electromagnetic, acoustic (see, e.g., the reviews [1][2][3]).BIC can be considered as a resonance with zero width and formally infinite Q-factor (without taking into account material losses).BICs are orthogonal to and are decoupled from the propagating states.The decoupling condition corresponds to a certain point in the parameter space of a system that can hardly be achieved exactly in practice.However, near this point, BIC turns into narrow resonance (quasi-BIC), which can possess an extremely high Q-factor.Hence, studying BICs provides a regular way to construct structures with high Q-factors that is advantageous for a large variety of practical applications including small cavity lasers [4][5][6][7], higher harmonics generation [8], sensing [9], etc. Q-factor can be further increased near the point of BIC merging [10][11][12].In [13][14][15] BIC merging has been interpreted as an annihilation of particle-like objects with topological charges of opposite sign.In this letter, we further extend the analogy of BICs with particle-like objects, which in our case distantly interact with each other.
Destructive interference that provides BIC decoupling from the continuum can be of symmetry origin, as in the case of symmetry protected BICs, or it can result from scattering on two or more resonances-Friedrich-Wintgen BICs [16].Fabry-Pérot (FP) BIC [11,[17][18][19][20][21][22] is sometimes considered as a separate type of BIC that is formed when the round-trip phase matching condition in an FP cavity takes place at the energy (frequency) of zero transparency of the mirrors (Fano antiresonance), with inter-mirror interaction through evanescent modes being taken into account [19].Therefore, FP BICs arise at a discrete set of cavity lengths corresponding to round-trip phase difference being integer multiple of .In the present letter, we focus on a quantum-mechanical model of the waveguide structure, which admits transparent analytical consideration and where multimode interference provides a variety of BIC formation features [23].Nevertheless, previous studies of the physics of BICs in quantum-mechanical and optical waveguides [11,23,24] give reason to believe that the main results of our work will be qualitatively valid for optical systems as well.We show that in FP geometry new BICs could exist, which appear always in pairs and almost at any cavity length (except for very small lengths).Such twin BICs (TBICs) emerge in FP resonators with mirrors possessing Friedrich-Wintgen BICs.In some sense physical mechanism of TBICs origin can be considered as an extension of an analogy of BICs with particle-like objects to the case of tunneling coupling between two Friedrich-Wintgen BICs.π 2

OPTICS AND LASER PHYSICS
NUMERICAL CALCULATIONS FP BICs require an FP cavity and perfect mirrors to arise.We consider a 2D quantum-mechanical uniform waveguide along the x axis (propagation direction) of width H surrounded by a potential barrier in the y direction with two identical scattering regions ("impurities") of length L and width h providing attractive potential , which play the role of mirrors (see Fig. 1a).It is known that such impurities demonstrate Fano antiresonances in the transmission spectrum and hence become perfectly opaque at certain energies [19,25,26].Thus, one may expect FP BIC formation at these (or close) energies for an appropriate choice of the distance between the impurities (FP cavity length).Moreover, multimode interference in such waveguide systems provides quite a rich variety of interference phenomena even for an individual impurity [23,25].Thus, under variation of a single parameter (length of the impurity L), one can manipulate its antiresonances up to the formation of BIC [19,25,26].The interaction of BICs in individual mirrors and FP BICs in the cavity is the main object of our study in the present letter.
To be specific, we focus on the motion of electron with effective mass and set eV and eV.These parameters have typical values for GaAs-based materials.For the convenience of numerical calculations, the transverse modes of the continuous spectrum were simulated by a dense set of discrete states, for which artificial infinitely high barriers were added to the structure at a distance from the waveguide.The solution to the 2D Schrödinger equation describing scattering or eigenvalue problem in this structure is derived through the transverse modes decomposition method [25,[27][28][29][30]. Specific values of the number N of the transverse modes taken into account and the parameter Δ are chosen consistently based on the convergence of numerical simulation results to the exact solution ( , ).The values and nm chosen in this letter, provide a relative error in the numerical values of the energy and parameters of the BIC, determined by the error in the probability flow conservation law, not more than 10 −4 .
Besides BICs in individual impurities [25], coalescence of antiresonances may be observed there [23], which define the presence or absence of destructive interference and hence can ruin the possibility for FP resonance to build up.These phenomena are beyond the scope of the present letter and deserve special attention to be discussed elsewhere.Therefore, parameters of the structure for the numerical simulation are chosen to reduce the influence of higher modes and prevent mentioned above coalescence phenomena.We set nm and nm and vary the remaining geometrical parameters D and L to study BICs.The energy range of interest lies between

h
the thresholds of the first and the second modes in the waveguide, which for the considered values of the parameters are meV and meV, respectively.
Figures 1b and 1c show the numerically calculated dependence of BIC energy and the required length of impurities (mirrors) on the given distance D between them.Near the energy E BIC, 1 and the value of the length L BIC, 1 corresponding to BIC in an individual impurity, there is a pair of symmetric and antisymmetric BICs, which weakly depend on the FP cavity length D and follow BIC in an individual impurity.We call these pair of BICs as twin BICs (TBICs).We note that 7.528 L the observed TBIC tunneling splitting is not related to the splitting of antiresonances of isolated mirrors, as, for example, in the [31,32], but arises due to the tunneling splitting of the BICs in isolated mirrors, in some sense similar to conventional bound states, for example, in quantum wells (although this analogy is, of course, not literal).In general, one can consider FP BICs and TBICs as complementary to each other.Indeed, FP BICs exist in a wide range of mirror lengths L, but in a narrow range of the distances between them D (a discrete set of values that satisfies the FP resonance condition).On the other hand, TBICs exist in a narrow range of L values corresponding to the formation of BIC in an isolated impurity, but at the same time, in a wide range of D.
Approaching the energy and parameter of BIC in an individual mirror, TBICs and FP BICs become hybridized and there is a continuous transformation between them under parameters variation (see insets in Figs.1b, 1c).Hybridization is provided by the coupling between the impurities through the evanescent modes and hence it vanishes as the distance between them increases.It is important to note that for a fixed value of the parameters D and , there is only one TBIC in the system.The second TBIC with similar energy exists at some close but still different parameters.In particular, the coincidence of the BIC energies, which takes place in Fig. 1b, is seeming, since these BICs correspond to different values of the parameter L (Fig. 1c); i.e., there is no quantummechanical degeneracy in the usual sense.
Away from the BIC of individual impurity, there are FP BICs that follow simple single mode FP resonance conditions with the reflection coefficient phase of the mirrors taken into account (shown by thin black lines in Fig. 1c).FP BICs naturally demand a corresponding FP resonance to build up between the mirrors.Hence, their formation condition is periodic in D and the BIC wavefunction has non-zero amplitude in the first (propagating) mode inside the FP cavity comparable to the amplitude inside the mirrors.It should be noted that the reflection of the considered mirrors (impurities) is based on the Fano antiresonance, which requires relatively high wavefunction amplitude to build up.On the other hand, TBICs do not need an FP resonance and therefore they are almost independent on the distance between the mirrors and their wavefunction is exponentially small inside the FP cavity due to the absence of the wavefunction component in the propagating mode.Figure 2 illustrates this key difference between wavefunctions of FP BIC and TBIC.
Symmetric FP BICs can be observed beginning from which corresponds to the threshold of the second mode in the waveguide .For antisymmetric states, there is always a gap for small L, where no FP BIC exists.As the number n of the FP resonance increases, this gap decreases.Following Fig. 1c, BIC existence curves for symmetric BICs are higher than the simple prediction of the FP phase matching condition, whereas, for antisymmetric states, they are lower.This is due to the contribution of higher (evanescent) modes in the cavity.Eventually, this leads to the fact that when passing through a single impurity BIC (E BIC, 1 , L BIC, 1 ) by increasing L, there is sharp round-trip phase change of FP BIC by ( for symmetric states and for antisymmetric states) with corresponding change of D. In other words, FP BICs with n and are continuously transformed into each other while changing the parameters through the region of a single impurity BIC.

ANALYTICAL MODEL
In order to provide a straightforward analytical description of the formation of TBICs, we consider a model of quantum-mechanical waveguide withfunctional scattering potentials mixing different modes, which play the role of mirrors.The scattering problem (as well as the eigenvalue problem) is effectively one-dimensional with extra dimensions taken into account by introducing appropriate transverse size-quantized modes [19,30].The Hamiltonian of the described multimode one-dimensional problem takes the form (1) where is the identity matrix, is the diagonal matrix containing the thresholds of the transverse modes, and matrix represents the intraand inter-mode δ-function potentials forming the left ( ) and right ( ) mirrors.In the following, we for simplicity, we have chosen units such that .
We focus on the energy range between the thresholds of the first and second modes of the waveguide ( ), so there is only one propagating mode, whereas all the others are evanescent.Inside and outside the cavity the waveguide is uniform with a certain set of such modes.In these regions, the solution to the Schrödinger equation with Hamiltonian (1) is straightforward and is expressed as a superposition of propagating and evanescent waves.At each δ-mirror, we impose the continuity of the wavefunction and the discontinuity of its derivatives: Here, is the column vector of the wavefunctions in all the transverse modes in the th region ( and is outside the FP cavity and is inside the cavity) and are the positions of the -mirrors.
BICs can be described either from the scattering point of view as (anti-) resonances with zero width [25,33,34] or by the direct solution of the eigenvalue problem with zero outgoing waves.In spatially symmetric systems, one can distinguish eigenstates by parity, which enables one to halve the number of unknown variables (amplitudes of the wavefunction).Therefore, in symmetric multimode structures, it is more convenient to study BICs through the eigenvalue problem.Thus, we look for the symmetric ( ) and antisymmetric ( ) BICs in the following form: (3) Here, , k is the wavenumber in the first propagating mode and is the decay constant in the nth mode, which is evanescent for .Applying the continuity condition (2a) to wavefunctions (3) in the first mode implies that or for symmetric and antisymmetric states, respectively.Together these conditions can be combined as Equation ( 4) is fulfilled if either that corresponds to the FP resonance condition in the case of point ( -functional) mirrors, or .Substituting into the conditions (2b) shows that equations for evanescent modes can be solved independently from the propagating mode thus giving the bound states in the system of evanescent modes (eigenstates of the effective BIC molecule attached to the propagating channel).In the case of FP BIC, wavefunction amplitude in the propagation mode inside the cavity is non-zero: (5) where with are corresponding elements of the matrix.Similarly, condition applied to Eqs. (2b) provides decoupling of evanescent modes from the propagating one that results in the eigenvalue problem for evanescent modes only (BIC molecule).However, in this case, the eigenvalue problem becomes provided with an additional condition , which can be formulated as the vanishing of the RHS in Eq. ( 5).This requirement simply means the decoupling from the propagating mode.Thus, for TBICs, wavefunction amplitude is zero along the whole propagating mode, and for both symmetric and antisymmetric TBICs tend to the BICs in individual mirrors.
The simplest toy model possessing FP BICs is a two-mode waveguide with mirrors modeled by two identical δ-wells in the second mode coupled locally to the first propagating mode that was thoroughly discussed in [19].TBICs in this system do not exist, because the RHS of Eq. ( 5) will have a single term in the two-mode case, and turning it to zero requires identically zero wavefunction or trivially absence of any coupling between the propagating and evanescent modes.
A much richer picture takes place if there are BICs in individual mirrors, for which at least two resonances in each mirror are required [16].We consider two evanescent modes with thresholds each having two δ-wells at (see Fig. 3a).Intermode coupling matrix has the form: (6) Using the formalism [2,34], one can easily show that BIC exists in the individual mirror if the following condition is satisfied: (7) where .Applying boundary conditions ( 2) with (6) to ansatz (3), one can derive required parameters for FP BICs and TBICs formation.Following the general consideration proposed above, symmetric and antisymmetric FP BICs arise if and Equation ( 8) defines eigenstates of the BIC molecule, i.e., bound states in two coupled evanescent modes with four δ-wells.In the limit condition (8) reduces to the = 0 corre- sponding to bound states in evanescent modes for an isolated mirror, and hence, as can be easily shown, defining its perfect opaqueness.Due to the requirement of FP resonance build-up, FP BIC wavefunction has non-zero amplitude in the propagating mode inside the FP cavity according to Eq. ( 5) with and being taken from Eq. ( 6).Conditions for the formation of a pair of symmetric and antisymmetric TBICs are derived similarly and give the following two equations: (9) As expected, for large distance between the mirrors, BIC energy E BIC and required parameter value ΔU BIC calculated from Eqs. (9) tend to the values E BIC, 1 and ΔU BIC, 1 corresponding to individual mirror that satisfy conditions ( 7): (10) where .Such TBICs do not require an FP resonance to build up and hence they are almost independent of the distance between the mirrors and their wavefunction is exponentially small inside the FP cavity due to the absence of the wavefunction component in the propagating mode ( ).That is what was observed in the numerical simulation (see Figs. 1 and 2).
Equation (8) together with and Eqs. ( 9) can be solved numerically to get the energy E BIC and the required parameter (split between the thresholds of the evanescent modes ΔU BIC ) for FP BICs and TBICs, respectively.Figures 3b and 3c illustrates the dependence of E BIC and ΔU BIC on the cavity length D. Similar to the results of numerical simulations (compare with Figs.1b, 1c), one can see that FP BICs are periodic in D, whereas TBICs demonstrate quite a weak dependence on D, especially in the limit.However, in contrast to the results of the numerical calculations, where FP BICs and TBICs were hybridized, here they are fully independent, which is the result of zero width of the mirrors in the simplified analytical model.

DISCUSSION AND CONCLUSIONS
In the present letter, we have studied both numerically and analytically BICs in quantum mechanical waveguide structures with two coupled identical scattering potentials.Due to destructive multimode interference, these potentials act as mirrors constituting a Fabry-Pérot (FP) resonator together with the cavity between them.We have shown that besides the well- known FP BICs arising due to resonant confinement between the mirrors, there are twin BICs (TBICs) related to BICs in isolated mirrors.TBICs can be considered as a result of the tunneling interaction of two particle-like objects (meta-atoms) each corresponding to a BIC in an isolated mirror.This result is nontrivial because BIC existence in a single resonator requires spatial symmetry in the propagation direction, which decreases the number of effective channels necessary for BIC formation [12,35].In FP structure spatial symmetry for each individual mirror is absent locally due to the presence of another mirror, although it exists for the whole structure.Hence, it was not obvious a priori that BICs of the single mirror will survive in a paired structure.
Our consideration of TBICs in the present letter is restricted to a quantum-mechanical model for illustrative purposes.However, the main BICs' practical applications at the present moment are based on classical wave systems: electromagnetic or acoustic [2,3].In waveguides, the Schrödinger and Helmholtz equations for normal incidence ( ) of waves in TE polarization are similar, but, in particular, due to the frequency dependence of the parameters, the latter becomes a nonlinear eigenvalue problem.Hence, the analytic consideration of electromagnetic waveguides revealing basic physical mechanisms is much less transparent than the quantum-mechanical one.Nevertheless, due to the fundamental analogy of wave phenomena, the qualitative picture of TBICs formation should be valid for waveguides of another physical nature (e.g., electromagnetic) as well.It should also be noted that in the case of TM polarization, there exists a zero mode (with a constant value of the field across the waveguide), which can have a qualitative effect on all interference phenomena, including BICs.Therefore, the formation and behavior of BICs in the case of such a polarization requires a separate study.
TBICs do not require FP resonance accompanied by standing wave build-up.Hence, they exist almost at all distances between the mirrors except for very small inter-mirror separation, where intra-mirror structures of different mirrors start to interact with each other.In this region, TBIC energy becomes dependent on cavity length and can be continuously tuned by its variation, which is important for practical applications.
Wavefunctions of TBICs have vanishing amplitude in the cavity.However, the wavefunction of TBIC is coherent all over the structure.An intriguing question arises whether TBIC can serve as an interaction mediator of remote quantum objects (e.g., qubits), which deserves a special study.Another interesting topic is the properties of larger molecules with three or more BIC meta-atoms and arrays of interacting BICs and their relation to BIC metasurfaces, which will be studied elsewhere.

Fig. 1 .
Fig. 1. (Color online) (a) Scheme of the considered 2D quantum-mechanical waveguide with two attractive impurities.The numerically calculated (b) BIC energy E BIC and (c) corresponding impurities length L BIC versus a given distance .The red thick and the blue thin lines describe symmetric and antisymmetric BICs, respectively.Horizontal thin dashed black lines indicate the energy meV and the corresponding length nm of the BIC in the isolated impurity.Insets depict BIC behavior in the vicinity of BIC in individual impurity.Away from the BIC of individual impurity, FP BICs follow the FP phase matching condition (see thin black solid lines in panel (c)).Black stars 1 and 2 indicate symmetric FP BIC and TBIC, respectively, which probability density distributions are shown in Fig. 2.

Fig. 2 .
Fig. 2. (Color online) Probability density distribution in the (a) symmetric FP BIC and (b) TBIC for the same distance nm between the impurities.Energy and corresponding impurity length of these BICs are shown in Fig. 1: nm, meV for FP BIC and nm, meV for TBIC.= 10 D ≈ BIC 2 L ≈ BIC 472.324E ≈ BIC 7.528 L ≈ BIC 396.583E

Fig. 3 .
Fig. 3. (Color online) (a) Schematic view of the toy model describing the formation of FP BICs and TBICs.The (b) BIC energy and (c) required energy split between the evanescent modes versus the distance D between the δ-mirrors in the toy model.Solid and dashed lines correspond to TBICs and FP BICs, respectively (thick red and thin blue lines indicate symmetric and antisymmetric states, respectively).Horizontal thin dashed black lines show the BIC energyand the corresponding parameter ΔU BIC, 1 for an isolated mirror given by Eqs.(7).The inset in panel (c) illustrates the Q-factor dependence of symmetric (thick red lines) and antisymmetric (thin blue lines) quasi-TBICs on for D = (solid lines) 3 and (dash-dotted lines) 10.The rest parameters are , , , , and .