Effective coupling for open billiards

Interference traps waves in an open system: bound states in the continuum

I review the four mechanisms of bound states in the continuum (BICs) in the application of microwave and acoustic cavities open to directional waveguides. The most simple are symmetry-protected BICs, which are localized inside the cavity because of the orthogonality of the eigenmodes to the propagating modes of waveguides. However, the most general and interesting is the Friedrich-Wintgen mechanism, when the BICs are the result of the fully destructive interference of outgoing resonant modes. The third type of BICs, Fabry-Perot BICs, occurs in a double resonator system when each resonator can serve as an ideal mirror. Finally, the accidental BICs can be realized in the open cavities with no symmetry like the open Sinai billiard in which the eigenmode of the resonator can become orthogonal to the continuum of the waveguide accidentally due to a smooth deformation of the eigenmode. We also review the one-dimensional systems in which the BICs occur owing to the fully destructive interference of two waves separated by spin or polarization or by paths in the Aharonov-Bohm rings. We make broad use of the method of effective non-Hermitian Hamiltonian equivalent to the coupled mode theory, which detects BICs by finding zero-width resonances.

Fano feature induced by a bound state in the continuum via resonant state expansion

We consider light scattering by an anisotropic defect layer embedded into anisotropic photonic crystal in the spectral vicinity of an optical bound state in the continuum (BIC). Using a resonant state expansion method we derive an analytic solution for reflection and transmission amplitudes. The analytic solution is constructed via a perturbative approach with the BIC as the zeroth order approximation. The solution is found to describe the collapsing Fano feature in the spectral vicinity of the BIC. The findings are confirmed via comparison against direct numerical simulations with the Berreman transfer matrix method.

Irreversibility and the breaking of resonance-antiresonance symmetry

We consider open quantum systems modeled as discrete lattices. Using a simple model of a single-site coupled to two leads as an example, we show that the time evolution of these systems can be analyzed in terms of an explicitly time-reversal symmetric resolution of unity. This resolution of unity includes both resonant states, which decay in the future, and anti-resonant states, which decay in the past. We show that a time-reversal invariant state contains both resonant and anti-resonant components with equal weights. However, this symmetry is automatically broken as the system evolves in time, with the resonant component becoming much larger than the anti-resonant component for t > 0 (and vice versa for t < 0). We argue that irreversibility is a manifestation of this symmetry breaking. We also compare our present approach with the subdynamics approach developed by Prof. Prigogine and collaborators. Finally, we suggest an extension of our present approach from the level of wave functions to the level of density matrices.

Spin polarized bound states in the continuum in open Aharonov-Bohm rings with the Rashba spin-orbit interaction

We consider the trapping of electrons with a definite spin polarization by bound states in the continuum (BSC) in the open Aharonov-Bohm rings in the presence of the Rashba spin-orbit interaction (RSOI). Neglecting the Zeeman term we show the existence of BSCs in the one-dimensional ring when the eigenstates of the closed ring are doubly degenerate. With account of the Zeeman term BSCs occur only at the points of threefold degeneracy. The BSCs are found in the parametric space of flux and RSOI strength in close pairs with opposite spin polarization. Thereby the spin polarization of electrons transmitted through the ring can be altered by minor variation of magnetic or electric field at the vicinity of these pairs. Numerical simulations of the two-dimensional open ring show similar results for the BSCs. Encircling the BSC points in the parametric space of the flux and the RSOI constant gives rise to a geometric phase.

Enhancement of Rayleigh scattering in a two-dimensional Fabry-Perot resonator loaded with impurities

We study wave transmission through a Fabry-Perot resonator (FPR) loaded with point-like impurities. We show both analytically in the framework of the coupled mode theory and numerically that there are two different regimes for transmission dependent on the quality of the FPR mirrors. For low quality, we obtain transmittance very similar to the clean FPR with slightly shifted Lorentz peaks. However, for good quality, the transmittance peaks are strongly reduced and substituted with Gaussian peaks because of multiple scattering of waves by each impurity. As a side effect, we observe the angular (channel) conversion in the disordered FPR. We demonstrate that the resonant peaks are dependent on the concentration of impurities to pave a way for resonant measurement of the concentration.

Gate controlled resonant widths in double-bend waveguides: bound states in the continuum

We consider quantum transmission through double-bend [Formula: see text]- and Z-shaped waveguides controlled by the finger gate potential. Using the effective non-Hermitian Hamiltonian approach we explain the resonances in transmission. We show a difference in transmission in the short waveguides that is the result of different chirality in Z and [Formula: see text] waveguides. We demonstrate that the potential selectively affects the resonant widths resulting in the occurrence of bound states in the continuum.

Harnessing quantum transport by transient chaos

Chaos has long been recognized to be generally advantageous from the perspective of control. In particular, the infinite number of unstable periodic orbits embedded in a chaotic set and the intrinsically sensitive dependence on initial conditions imply that a chaotic system can be controlled to a desirable state by using small perturbations. Investigation of chaos control, however, was largely limited to nonlinear dynamical systems in the classical realm. In this paper, we show that chaos may be used to modulate or harness quantum mechanical systems. To be concrete, we focus on quantum transport through nanostructures, a problem of considerable interest in nanoscience, where a key feature is conductance fluctuations. We articulate and demonstrate that chaos, more specifically transient chaos, can be effective in modulating the conductance-fluctuation patterns. Experimentally, this can be achieved by applying an external gate voltage in a device of suitable geometry to generate classically inaccessible potential barriers. Adjusting the gate voltage allows the characteristics of the dynamical invariant set responsible for transient chaos to be varied in a desirable manner which, in turn, can induce continuous changes in the statistical characteristics of the quantum conductance-fluctuation pattern. To understand the physical mechanism of our scheme, we develop a theory based on analyzing the spectrum of the generalized non-Hermitian Hamiltonian that includes the effect of leads, or electronic waveguides, as self-energy terms. As the escape rate of the underlying non-attracting chaotic set is increased, the imaginary part of the complex eigenenergy becomes increasingly large so that pointer states are more difficult to form, making smoother the conductance-fluctuation pattern.

Effective Hamiltonian and unitarity of the S matrix

The properties of open quantum systems are described well by an effective Hamiltonian H that consists of two parts: the Hamiltonian H of the closed system with discrete eigenstates and the coupling matrix W between discrete states and continuum. The eigenvalues of H determine the poles of the S matrix. The coupling matrix elements W(cc')(k) between the eigenstates k of H and the continuum may be very different from the coupling matrix elements W(cc')(k) between the eigenstates of H and the continuum. Due to the unitarity of the S matrix, the W(cc')(k) depend on energy in a nontrivial manner. This conflicts with the assumptions of some approaches to reactions in the overlapping regime. Explicit expressions for the wave functions of the resonance states and for their phases in the neighborhood of, respectively, avoided level crossings in the complex plane and double poles of the S matrix are given.

Is the concept of the non-Hermitian effective Hamiltonian relevant in the case of potential scattering?

We examine the notion and properties of the non-Hermitian effective Hamiltonian of an unstable system using as an example potential resonance scattering with a fixed angular momentum. We present a consistent self-adjoint formulation of the problem of scattering on a finite-range potential, which is based on the separation of the configuration space into two segments, internal and external. The scattering amplitude is expressed in terms of the resolvent of a non-Hermitian operator H. The explicit form of this operator depends on both the radius of separation and the boundary conditions at this place, which can be chosen in many different ways. We discuss this freedom and show explicitly that the physical scattering amplitude is, nevertheless, unique, although not all choices are equally adequate from the physical point of view. The energy-dependent operator H should not be confused with the non-Hermitian effective Hamiltonian H(eff) which is usually exploited to describe interference of overlapping resonances. We note that the simple Breit-Wigner approximation is as a rule valid for any individual resonance in the case of few-channel scattering on a flat billiardlike cavity, leaving no room for nontrivial H(eff) to appear. The physics is appreciably richer in the case of an open chain of L connected similar cavities whose spectrum has a band structure. For a fixed band of L overlapping resonances, the smooth energy dependence of H can be ignored so that the constant LxL submatrix H(eff) approximately describes the time evolution of the chain in the energy domain of the band and the complex eigenvalues of H(eff) define the energies and widths of the resonances. We apply the developed formalism to the problem of a chain of L delta barriers, whose solution is also found independently in a closed form. We construct H(eff) for the two commonly considered types of boundary conditions (Neumann and Dirichlet) for the internal motion. Although the final results are in perfect coincidence, somewhat different physical patterns arise of the trend of the system with growing openness. Formation in the outer well of a short-lived doorway state shifted in energy is explicitly demonstrated together with the appearance of L-1 long-lived states trapped in the inner part of the chain.

Effective Hamiltonian for a microwave billiard with attached waveguide

In a recent work the resonance widths in a microwave billiard with attached waveguide were studied in dependence on the coupling strength [E. Persson et al., Phys. Rev. Lett. 85, 2478 (2000)], and resonance trapping was experimentally found. In the present paper an effective Hamiltonian is derived that depends exclusively on billiard and waveguide geometry. Its eigenvalues give the poles of the scattering matrix provided that the system and environment are defined adequately. Further, we present the results of resonance trapping measurements where, in addition to our previous work, the position of the slit aperture within the waveguide was varied. Numerical simulations with the derived Hamiltonian qualitatively reproduce the experimental data.