Collective modes in an open microwave billiard

Microwave experiments in the realm of fidelity

In this review article, we will demonstrate the power of microwave experiments in the realm of fidelity also known as Loschmidt echoes. As the determination of the fidelity itself is experimentally tedious and error prone, we will introduce the scattering fidelity which under the conditions of chaotic systems and weak coupling approaches the fidelity itself. The main ingredient in fidelity investigations is the type and strength of a perturbation. The perturbations presented here will be both global and local boundary perturbations, as well as local perturber movements but also the change of coupling to the environment. All these perturbations will produce their own fidelity decay as a function of the perturbation strength, which will be discussed in this article.

A review of progress in the physics of open quantum systems: theory and experiment

This report on progress explores recent advances in our theoretical and experimental understanding of the physics of open quantum systems (OQSs). The study of such systems represents a core problem in modern physics that has evolved to assume an unprecedented interdisciplinary character. OQSs consist of some localized, microscopic, region that is coupled to an external environment by means of an appropriate interaction. Examples of such systems may be found in numerous areas of physics, including atomic and nuclear physics, photonics, biophysics, and mesoscopic physics. It is the latter area that provides the main focus of this review, an emphasis that is driven by the capacity that exists to subject mesoscopic devices to unprecedented control. We thus provide a detailed discussion of the behavior of mesoscopic devices (and other OQSs) in terms of the projection-operator formalism, according to which the system under study is considered to be comprised of a localized region (Q), embedded into a well-defined environment (P) of scattering wavefunctions (with Q   +   P   =   1). The Q subspace must be treated using the concepts of non-Hermitian physics, and of particular interest here is: the capacity of the environment to mediate a coupling between the different states of Q; the role played by the presence of exceptional points (EPs) in the spectra of OQSs; the influence of EPs on the rigidity of the wavefunction phases, and; the ability of EPs to initiate a dynamical phase transition (DPT). EPs are singular points in the continuum, at which two resonance states coalesce, that is where they exhibit a non-avoided crossing. DPTs occur when the quantum dynamics of the open system causes transitions between non-analytically connected states, as a function of some external control parameter. Much like conventional phase transitions, the behavior of the system on one side of the DPT does not serve as a reliable indicator of that on the other. In addition to discussing experiments on mesoscopic quantum point contacts that provide evidence of the environmentally-mediated coupling of quantum states, we also review manifestations of DPTs in mesoscopic devices and other systems. These experiments include observations of resonance-trapping behavior in microwave cavities and open quantum dots, phase lapses in tunneling through single-electron transistors, and spin swapping in atomic ensembles. Other possible manifestations of this phenomenon are presented, including various superradiant phenomena in low-dimensional semiconductors. From these discussions a generic picture of OQSs emerges in which the environmentally-mediated coupling between different quantum states plays a critical role in governing the system behavior. The ability to control or manipulate this interaction may even lead to new applications in photonics and electronics.

Phase rigidity and avoided level crossings in the complex energy plane

We consider the effective Hamiltonian of an open quantum system, its biorthogonal eigenfunctions phi(lambda), and define the value r(lambda)=(phi(lambda)|phi(lambda))/<phi(lambda)|phi(lambda)> that characterizes the phase rigidity of the eigenfunctions phi(lambda). In the scenario with avoided level crossings, r(lambda) varies between 1 and 0 due to the mutual influence of neighboring resonances. The variation of r(lambda) is an internal property of an open quantum system. In the literature, the phase rigidity rho of the scattering wave function Psi(C)(E) is considered. Since Psi(C)(E) can be represented in the interior of the system by the phi(lambda), the phase rigidity rho of the Psi(C)(E) is related to the r(lambda) and therefore also to the mutual influence of neighboring resonances. As a consequence, the reduction of the phase rigidity rho to values smaller than 1 should be considered, at least partly, as an internal property of an open quantum system in the overlapping regime. The relation to measurable values such as the transmission through a quantum dot, follows from the fact that the transmission is, in any case, resonant at energies that are determined by the real part of the eigenvalues of the effective Hamiltonian. We illustrate the relation between phase rigidity rho and transmission numerically for small open cavities.

Spectroscopic properties of large open quantum-chaotic cavities with and without separated time scales

The spectroscopic properties of an open large Bunimovich cavity are studied numerically in the framework of the effective Hamiltonian formalism. The cavity is opened by attaching two leads to it in four different ways. In some cases, the transmission takes place via standing waves with an intensity that closely follows the profile of the resonances. In other cases, short-lived and long-lived resonance states coexist. The short-lived states cause traveling waves in the transmission while the long-lived ones generate superposed fluctuations. The traveling waves oscillate as a function of energy. They are not localized in the interior of the large chaotic cavity. In all considered cases, the phase rigidity fluctuates with energy. It is mostly near to its maximum value and agrees well with the theoretical value for the two-channel case.

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.

Whispering gallery modes in open quantum billiards

The poles of the S matrix and the wave functions of open two-dimensional quantum billiards with convex boundary of different shape are calculated by using the method of complex scaling. Two leads are attached to the cavities. The conductance of the cavities is calculated at energies with one, two, and three open channels in each lead. Bands of overlapping resonance states appear that are localized along the convex boundary of the cavities and contribute coherently to the conductance. These bands correspond to the whispering gallery modes known from classical calculations.

Effective coupling for open billiards

We derive an explicit expression for the coupling constants of individual eigenstates of a closed billiard that is opened by attaching a waveguide. The Wigner time delay and the resonance positions resulting from the coupling constants are compared to an exact numerical calculation. Deviations can be attributed to evanescent modes in the waveguide and to the finite number of eigenstates taken into account. The influence of the shape of the billiard and of the boundary conditions at the mouth of the waveguide are also discussed. Finally we show that the mean value of the dimensionless coupling constants tends to the critical value when the eigenstates of the billiard follow random-matrix theory.

Dynamics of quantum systems

A relation between the eigenvalues of an effective Hamilton operator and the poles of the S matrix is derived that holds for isolated as well as for overlapping resonance states. The system may be a many-particle quantum system with two-body forces between the constituents or it may be a quantum billiard without any two-body forces. Avoided crossings of discrete states as well as of resonance states are traced back to the existence of branch points in the complex plane. Under certain conditions, these branch points appear as double poles of the S matrix. They influence the dynamics of open as well as of closed quantum systems. The dynamics of the two-level system is studied in detail analytically as well as numerically.

Observation of resonance trapping in an open microwave cavity

The coupling of a quantum mechanical system to open decay channels has been theoretically studied in numerous works, mainly in the context of nuclear physics but also in atomic, molecular, and mesoscopic physics. Theory predicts that with increasing coupling strength to the channels the resonance widths of all states should first increase but finally decrease again for most of the states. In this Letter, the first direct experimental verification of this effect, known as resonance trapping, is presented. In the experiment a microwave Sinai cavity with an attached waveguide with variable slit width was used.

Spectroscopic studies in open quantum systems

The Hamiltonian H of an open quantum system is non-Hermitian. Its complex eigenvalues E(R) are the poles of the S matrix and provide both the energies and widths of the states. We illustrate the interplay between Re(H) and Im(H) by means of the different interference phenomena between two neighboring resonance states. Level repulsion may occur along the real or imaginary axis (the latter is called resonance trapping). In any case, the eigenvalues of the two states avoid crossing in the complex plane. We then calculate the poles of the S matrix and the corresponding wave functions for a rectangular microwave resonator with a scatter as a function of the area of the resonator as well as of the degree of opening to a waveguide. The calculations are performed by using the method of exterior complex scaling. Re(H) and Im(H) cause changes in the structure of the wave functions which are permanent, as a rule. The resonance picture obtained from the microwave resonator shows all the characteristic features known from the study of many-body systems in spite of the absence of two-body forces. The effects arising from the interplay between resonance trapping and level repulsion along the real axis are not involved in the statistical theory (random matrix theory).