Physical realization of bound states in the continuum

Abstract

We show within the framework of quantum-defect theory that bound states in the continuum (BIC) can occur in systems of at least three coupled Coulombic channels due to the interference of resonances belonging to different channels, As a physical realization of such BIC, we discuss the Schrodinger equation describing a hydrogen atom in a uniform magnetic field. Below a well-defined continuum threshold, the solutions of a Schrodinger equation are square integrable and the energy eigenvalues are discrete. Above the threshold, the eigenvalues are distributed continuously and the eigenstates are generally not normalizable. ' Narrow resonances with large but finite lifetimes can occur if a slight modification of the Hamiltonian would lead to an effectively higher threshold. Examples are long-lived resonances behind a large potential barrier, or quasibound states in closed channels of a system with weakly coupled channels. However, in special cases, local potentials have been constructed where, for discrete energies above threshold, the Schrodinger equation -has square integrable solutions which have no obvious connection to bound eigenstates of such an approximate Hamiltonian. In examples given by Stillinger and Herrick and Fonda and Newton, nonseparability of the Hamiltonian or relatively strong coupling of channels are important for the occurrence of such bound states in the continuum (BIC). The aim of this Brief Report is to show how BIC can occur naturally in systems of coupled Coulombic channels and to present, for the first time, a physically real example of BIC. Coupled Coulombic channels are comprehensively described in the, framework of multichannel quantum-defect theory (MQDT) as developed by Seaton. 5 The basic equation of MQDT is det[ tan(n ];)5;, + R;, ] = 0, where the elements R"" define Seaton's real symmetric R matrix, and describe the dynamic effect of the short-range deviations of the true potential from the case of uncoupled pure Coulomb potentials. If all important channels are included, the R"" vary only weakly with energy. In closed channels, the quantity ]; in Eq. (1) is related to the energy E and the respective channe1 threshold I; by denly by m, and the width I is' (4) where E, is the turning point of 5(E). For three coupled channels, Eq. (1) becomes R ]2T3+R 3]T2 -2R ]2R 23R 3] tan5(E) = R ]]-T2T3 -R 23 with T2 and T3 as defined in Eq. (3). The phase shift 5(E) jumps by rr near poles of the right-hand side of Eq. (5). To understand the general behavior of 5(E), it is useful to study the zeros and poles of the fraction on the righthand side of Eq. (5). This quantity stays regular when either T2 or T3 has a pole. Poles of the fraction occur when the denominator D (E) = T2T3 -R p3 has zeros, and zeros occur when the numerator N (E) = R ]2T3+ R 3]T2 -2R ]2R 23R 3]vanishes.Near resonances, i.e. , near zeros of D, the derivative of the phase shift is d 5 D'(E) dE D(E) 0 N(E) In general, D(E) has no multiple zeros, so Eq. (8) shows that an infinite derivative of 5 corresponding to a vanishing width ...

Mechanisms of Fano Resonances in Coupled Plasmonic Systems

“…Finally, thanks to the tunability of the plasmonic system, we show a plasmonic equivalent of a bound state in the continuum. 56,57 This effect, which can only be obtained with close interacting modes, is a further proof of the extremely fine control achieved with this system.…”

“…This effect, which is exclusively associated with the positions of DA and DN and not with any other parameter like the coupling, is the plasmonic equivalent of a quantum mechanical effect known in literature as bound state in the continuum (BIC). 56,57 This system is in fact equivalent to two states that are in interaction with radiation. The tunability of plasmonic systems enables positioning the diabatic states very close to each other while maintaining appreciable coupling.…”

Bound states in the continuum

“…One finds that when Examples of such BICs were first proposed in atoms and molecules 118,119 , and its effect has been observed experimentally as a suppressed autoionization in certain doubly excited Rydberg states of barium 120 . More recently, Friedrich-Wintgen BICs have been studied in continuum shell models 121 , cold-atom collisions 122 , 2D topological insulators with defect 123 , and for quantum graphs 124 , quantum billiards 125 , or impurity atoms 126,127 attached to leads.…”

Electronic transport through a parallel-coupled triple quantum dot molecule: Fano resonances and bound states in the continuum

“…and the broadenings are given by We interpret the formation of BICs in this system in the same sense of reference [23], that is, as result of the quantum interference between resonances of different channels through the multiple connected quantum dots. The levels of the quantum dots are hybridized through the common leads, forming these states of infinite lifetimes.…”