Bound state in the continuum

A bound state in the continuum (BIC, BSC) is an eigenstate of some particular quantum system with the following properties:

Energy lies in the continuous spectrum of propagating modes of the surrounding space; The state does not interact with any of the states of the continuum (it cannot emit and cannot be excited by any wave that came from the infinity); Energy is real and Q factor is infinite, if there is no absorption in the system.

BICs are observed in electronic, photonic, acoustic systems, and are a general phenomenon exhibited by systems in which wave physics applies.

Bound states in the forbidden zone, where there are no finite solutions at infinity, are widely known (atoms, quantum dots, defects in semiconductors). For solutions in a continuum that are associated with this continuum, resonant states are known, which decay (lose energy) over time. They can be excited, for example, by an incident wave with the same energy. The bound states in the continuum have real energy eigenvalues and therefore do not interact with the states of the continuous spectrum and cannot decay.

Classification of BICs by mechanism of occurrence

Source:

BICs arising when solving the inverse problem

Wigner-von Neumann's BIC (Potential engineering) The wave function of one of the continuum states is modified to be normalizable and the corresponding potential is selected for it.

ExamplesIn the work of von Neumann and Wigner two types of spherically symmetric potentials are considered, with the potential of

localization occurs due to the "reflection" from infinity (since the material point goes to infinity for a finite time, rolling down from this potential, and the probability of being there tends to zero). A similar example in 1D considers the exponential dependence of the potential.Zafar Ahmed et al 2019 Phys. Scr. 94 105214 The second type of potentialSimon, B. On positive eigenvalues of one-body Schrödinger operators. Commun. Pure Appl. Math. 22, 531-538 (1969)Stillinger, F. H. & Herrick, D. R. Bound states in the continuum. Phys. Rev. A 11, 446-454 (1975) has a complex shape, but is periodically modulated with period . A wave function with period is reflected from such a potential to the center in analogy to the reflection from a Bragg mirror. In the one-dimensional case, such potentials can be realized using superlattices.D. R. Herrick, "Construction of bound states in the continuum for epitaxial heterostructure superlattices," Physica B 85, 44-50 (1977).Molina, M. I., Miroshnichenko, A. E. & Kivshar, Y. S. Surface bound states in the continuum. Phys. Rev. Lett. 108, 070401 (2012)

Hopping rate engineering In the tight binding approximation, the jump rates are modified so that the state becomes localized

ExamplesBICs of this type have been realized in an array of coupled optical waveguides,Corrielli, G., Della Valle, G., Crespi, A., Osellame, R. & Longhi, S. Observation of surface states with algebraic localization. Phys. Rev. Lett. 111, 220403 (2013) with the hopping rate modulated by the spacing between the waveguides. Experimentally, a state with following rates

was realized in the system which consist of 40 waveguides. There are also examples of BICs in PT-symmetric systems.Stefano Longhi. Non-Hermitian tight-binding network engineering. Phys. Rev. A 93, 022102Stefano Longhi, "Bound states in the continuum in PT-symmetric optical lattices, " Opt. Lett. 39, 1697—1700 (2014)

Boundary shape engineering Sources for BICs of different types, e.g. Fabry-Perot type are replaced by scatterers so as to create BIC of the same type.

ExamplesBICs of this type are observed for water waves.McIver, M. An example of non-uniqueness in the two-dimensional linear water wave problem. J. Fluid Mech. 315, 257—266 (1996)Kuznetsov, N. & McIver, P. On uniqueness and trapped modes in the water-wave problem for a surface-piercing axisymmetric body. Q. J. Mech. Appl. Math. 50, 565—580 (1997)Porter, R. & Evans, D. V. Water-wave trapping by floating circular cylinders. J. Fluid Mech. 633, 311—325 (2009).

BICs arising due to parameter tuning Fabry-Perot BICs For resonant structures, the reflection coefficient near resonance can reach unity. Two such structures can be arranged in such a way that they radiate in antiphase and compensate each other. ExamplesIn addition to the waves on water mentioned above, states of this type are found in many systems, such as those of pairs of quantum dots coupled to a waveguide,Cattapan, G. & Lotti, P. Bound states in the continuum in two-dimensional serial structures. Eur. Phys. J. B 66, 517–523 (2008)Sadreev, A. F., Bulgakov, E. N. & Rotter, I. Trapping of an electron in the transmission through two quantum dots coupled by a wire. JETP Lett. 82, 498–503 (2005) double chains of copper atoms,Díaz-Tendero, S., Borisov, A. G. & Gauyacq, J.-P. Extraordinary electron propagation length in a metallic double chain supported on a metal surface. Phys. Rev. Lett. 102, 166807 (2009) waveguide pairs of Z- and П- shapes Sadreev, A. F., Maksimov, D. N. & Pilipchuk, A. S. Gate controlled resonant widths in double-bend waveguides: bound states in the continuum. J. Phys. Condens. Matter 27, 295303 (2015). In photonics, states are realized, for example, in pairs of two-dimensional photonic crystal slabs,Suh, W., Yanik, M. F., Solgaard, O. & Fan, S. Displacement-sensitive photonic crystal structures based on guided resonance in photonic crystal slabs. Appl. Phys. Lett. 82, 1999—2001 (2003) double arrays of dielectric cylindersNdangali, R. F. & Shabanov, S. V. Electromagnetic bound states in the radiation continuum for periodic double arrays of subwavelength dielectric cylinders. J. Math. Phys. 51, 102901 (2010) etc.

Friedrich-Wintgen BICs

Two modes of the same symmetry of one and the same structure approach each other when the parameters of the structure are changed, and at some point an anti-crossing occurs. In this case, BIC is formed on one of the branches, since the modes as if compensate each other, being in antiphase and radiating into the same radiation channel.Friedrich, H. & Wintgen, D. Interfering resonances and bound states in the continuum. Phys. Rev. A 32, 3231-3242 (1985)

ExamplesFriedrich-Wintgen BICs (FW BICs) were considered for a hydrogen atom in a magnetic fieldFriedrich, H. & Wintgen, D. Physical realization of bound states in the continuum. Phys. Rev. A 31, 3964-3966 (1985) and experimentally manifested in inhibition of autoionization of barium atom, topological insulators with a defect Sablikov, V. A. & Sukhanov, A. A. Helical bound states in the continuum of the edge states in two dimensional topological insulators. Phys. Lett. A 379, 1775—1779 (2015) and many other quantum systems.Sadreev, A. F., Bulgakov, E. N. & Rotter, I. Bound states in the continuum in open quantum billiards with a variable shape. Phys. Rev. B 73, 235342 (2006)Texier, C. Scattering theory on graphs: II. The Friedel sum rule. J. Phys. A 35, 3389 (2002). Also, this type of BIC is found in two-dimensional and three-dimensional cylindrical open acoustic resonators.Hein, S., Koch, W. & Nannen, L. Trapped modes and Fano resonances in two-dimensional acoustical duct-cavity systems. J. Fluid Mech. 692, 257—287 (2012)Lyapina, A. A., Maksimov, D. N., Pilipchuk, A. S. & Sadreev, A. F. Bound states in the continuum in open acoustic resonators. J. Fluid Mech. 780, 370—387 (2015) In photonics, accidental BICs in periodic structures can sometimes appear as FW BICs, also modes of this type appeared in Bragg waveguide at interaction of TE and TM waves through anisotropic medium. A notable example is the theoretical realization of BIC on a layered spherical nanoparticle of specifically selected materials,Embedded Photonic Eigenvalues in 3D Nanostructures. Francesco Monticone and Andrea Alù. Phys. Rev. Lett. 112, 213903 (2014)M. G. Silveirinha, Phys. Rev. A 89, 023813 (2014). in this case BICs appear due to the interaction of different dipole modes of the same particle. By the same mechanism high-Q states can be realized in nonspherical dielectric resonators as well,K. Koshelev et al., Science 367, 288—292 (2020). but in this case each mode consist of an infinite number of multipoles,S. Gladyshev, K. Frizyuk, A. Bogdanov Phys. Rev. B 102, 075103 — Published 3 August 2020 and the Q-factor doesn't go to infinity.

Single-resonance parametric BICs Occur when a single mode can be represented as a sum of contributions, each of which varies with the structure parameters. At some point, destructive interference of all contributions occurs.

ExamplesBICs of this type are usually considered in photonic crystalsHsu, C. W. et al. Observation of trapped light within the radiation continuum. Nature 499, 188—191 (2013) and are also accidental BICs. They can appear in planar structures in -plane, which possess and symmetries,B. Zhen, C. W. Hsu, L. Lu, A. D. Stone, and M. Soljaˇci ́c, "Topological Nature of Optical Bound States in the Continuum, " Phys. Rev. Lett. 113, 257401 (2014) but their existence is not guaranteed by symmetry, they appear for particular values of structure parameters. They can also arise in cylindrically symmetric one-dimensional periodic chainsBulgakov, E. N. & Sadreev, A. F. Light trapping above the light cone in a one-dimensional array of dielectric spheres. Phys. Rev. A 92, 023816 (2015). In addition to photonic periodic structures,Porter, R. & Evans, D. V. Embedded Rayleigh-Bloch surface waves along periodic rectangular arrays. Wave Motion 43, 29-50 (2005). BICs of this type can appear for water waves,McIver, M., Linton, C. M., McIver, P., Zhang, J. & Porter, R. Embedded trapped modes for obstacles in two-dimensional waveguides. Q. J. Mech. Appl. Math. 54, 273—293 (2001). quantum waveguides,Linton, C. M. & Ratcliffe, K. Bound states in coupled guides. I. Two dimensions. J. Math. Phys. 45, 1359—1379 (2004). mechanical resonators,Chen, Y. et al. Mechanical bound state in the continuum for optomechanical microresonators. New J. Phys. 18, 063031 (2016) as a surface wavesYamanouchi, K. & Shibayama, K. Propagation and amplification of rayleigh waves and piezoelectric leaky surface waves in LiNbO3 . J. Appl. Phys. 43, 856—862 (1972). etc.

Symmetry-protected BICs Arise when the symmetry of the eigenstate differs from any of the possible symmetries of propagating modes in the continuum. ExamplesIn the simple case, states of this type are observed in the Г-point of planar photonic crystals possessing symmetry B. Zhen, C. W. Hsu, L. Lu, A. D. Stone, and M. Soljaˇci ́c, "Topological Nature of Optical Bound States in the Continuum, " Phys. Rev. Lett. 113, 257401 (2014).Z. Sadrieva, K. Frizyuk, M. Petrov, Yu. Kivshar, and A.Bogdanov «Multipolar origin of bound states in the continuum» Phys. Rev. B 100, 115303Lee, J. et al. Observation and differentiation of unique high-Q optical resonances near zero wave vector in macroscopic photonic crystal slabs. Phys. Rev. Lett. 109, 067401 (2012). They also arise in cylindrically symmetric 1D periodic chains Bulgakov, E. N. & Sadreev, A. F. Light trapping above the light cone in a one-dimensional array of dielectric spheres. Phys. Rev. A 92, 023816 (2015). Often, other types of BICs exist in the same structures. Also symmetry-protected BICs have been observed in coupled waveguide systems.Dreisow, F. et al. Adiabatic transfer of light via a continuum in optical waveguides. Opt. Lett. 34, 2405—2407 (2009)Plotnik, Y. et al. Experimental observation of optical bound states in the continuum. Phys. Rev. Lett. 107, 183901 (2011).

Separable BICs Arise when the eigenvalue problem is solved by the Separation of Variables Method, and the wave function is represented, for example, as , where both multipliers correspond to localized states, with the total energy lying in the continuum.