Linear analysis plays a central role in science and engineering. Even when dealing with nonlinear systems, an understanding of the linear response is often necessary to gain insight into the underlying complex dynamics. In recent years, there has been great interest in the study of open systems that exchange energy with the surrounding reservoir. In particular, it has been demonstrated that open systems whose spectra display non-hermetic singularities called exceptional points can exhibit a range of interesting effects with potential applications in building new lasers and sensors.
At an exceptional point, two modes or modes become completely identical. To better understand this, let’s think about how drums produce sound. The cylinder diaphragm is fixed along its circumference but the freedom to vibrate in the middle.
As a result, the membrane can move in different ways, each of which is called mode and displays a different sound frequency. When two different modes oscillate at the same frequency, they are called degenerate. The exceptional points are very peculiar decays in the sense that not only are the frequencies of the patterns identical but also the oscillations themselves. These points can only exist in open and non-hierarchical systems without symmetry in closed hierarchical systems.
Over the past years, the dedicated analysis of scattering coefficients for non-hierarchical systems with exceptional points has revealed a puzzling result. Sometimes , Frequency response (The relationship between the output and Input signals After interacting with the system as a function of the input signal repeat) can be Lorentzian or Super Lorentzian (for example, Lorentzian raised to the integer power). In contrast, the response of a standard isolated linear oscillator (except for situations where Fano line shapes can appear) is always Lorentzian.
An international team of physicists led by Rami Al-Janaini, associate professor at Michigan Technological University, has recently addressed this problem. Nature Communications An article entitled “Linear Response Theory of Open Systems with Exceptional Points”. The team provides a systematic analysis of linear response for non-hierarchical systems that have exceptional points. Importantly, they derived a closed expression for the solvent operator that defines the response of the system in terms of the left and right eigenvectors and the Jordanian canonical vectors associated with the basic Hamiltonian.
In contrast to previous expansions of the solvent operator in terms of the same Hamiltonian, the formalism developed here provides direct access of the linear response of the system and shows exactly when and how the Lorentzian and Super Lorentz responses appear,” says Professor Al-Janayni.
“As it turns out, the nature of the response is determined by the excitation (input) and synthesis (output) channels,” says Amin Hashemi, first author of the manuscript. The theory presented describes this behavior in detail and is general enough to apply to any non-hierarchical systems having any number of exceptional points from any system, making it useful for studying non-hierarchical systems with great degrees of freedom.
Hashemi et al., Linear response theory for open systems with exceptional points, Nature Communications (2022). DOI: 10.1038 / s41467-022-30715-8
Michigan Tech University
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