Author: Steve Mintz

Physicists develop a linear response theory for open systems having exceptional points

Linear analysis plays a central role in science and engineering. Even when dealing with nonlinear systems, understanding the linear response is often crucial for gaining insight into the underlying complex dynamics. In recent years, there has been a great interest in studying open systems that exchange energy with a surrounding reservoir. In particular, it has been demonstrated that open systems whose spectra exhibit non-Hermitian singularities called exceptional points can demonstrate a host of intriguing effects with potential applications in building new lasers and sensors.

At an exceptional point, two or modes become exactly identical. To better understand this, let us consider how drums produce sound. The membrane of the drum is fixed along its perimeter but free to vibrate in the middle. As a result, the membrane can move in different ways, each of which is called a mode and exhibits a different sound frequency. When two different modes oscillate at the same frequency, they are called degenerate. Exceptional points are very peculiar degeneracies in the sense that not only the frequencies of the modes are identical but also the oscillations themselves. These points can exist only in open, non-Hermitian systems with no analog in closed, Hermitian systems.

Over the past years, ad-hoc analysis of the scattering coefficients of non-Hermitian systems having exceptional points has revealed a puzzling result, namely that sometimes their frequency response (the relation between an output and input signals after interacting with the system as a function of the input signal’s frequency) can be Lorentzian or super Lorentzian (i.e. a Lorentzian raised to an integer power). In contrast, the response of a standard linear, isolated oscillator (excluding situations where Fano lineshapes can arise) is always Lorentzian.

Now, an international team of physicists led by Prof. Ramy El-Ganainy from Michigan Technological University, along with several collaborators from Penn State, the Humboldt University in Berlin, and the University of Central Florida, has tackled this problem in their recent Nature Communications article titled “Linear response theory of open systems with exceptional points”. In that work, the team presents a systematic analysis of the linear response of non-Hermitian systems having exceptional points. Importantly, they derive a closed-form expression for the resolvent operator quantifying the system’s response in terms of the right and left eigenvectors and Jordan canonical vectors associated with the underlying Hamiltonian.

A schematic representation of a complex non-Hermitian open system with many degrees of freedom made of coupled optical microdisk cavities. The linear response theory developed in this work provides a full characterization of the relation between output and input signals (indicated by green and yellow arrows, respectively) in terms of the eigenmodes and the canonical states of the underlying non-Hermitian Hamiltonian.

“In contrast to previous expansions of the resolvent operator in terms of the Hamiltonian itself, the formalism developed here provides direct access to the linear response of the system and demonstrates exactly when and how Lorentzian and super-Lorentzian responses arise” says Prof. El-Ganainy. “As it turned out, the nature of the response is determined by the excitation (input) and collection (output) channels” says Amin Hashemi, the first author of the manuscript. The presented theory describes this behavior in detail and is generic enough to apply to any non-Hermitian systems having any number of exceptional points of any order, which makes it instrumental for studying non-Hermitian systems with large degrees of freedom.

Lucas Simonson, physics PhD candidate awarded scholarship to study in Germany

Lucas Simonson is off to study in Germany

Lucas Simonson has been awarded a scholarship by the German Academic Exchange Service (DAAD). He will study at the Max Planck Institute for the Physics of Complex Systems in Dresden.

The German DAAD is a joint organization of the universities and other institutions of higher education in the Federal Republic of Germany, and the world’s largest funding organization of its kind. Supported by public funds, the DAAD promotes international academic cooperation, especially through the exchange of students and academics. DAAD scholarships are awarded by selection committees comprising a panel of independent academics.

He looks forward to studying under Professor Kurt Busch starting October 2022 to the end of April 2023. “The rationale for this trip is that joining my advisor in Germany will allow me to proceed with my research activities at a fast pace without any delay due to his absence. It will also allow me to interact with world-class optics research groups at the Humboldt-Universität Berlin,” he says. “It’s a significant milestone in my academic career and will allow me to experience other cultures outside of those in the US to broaden my worldview,” says Lucas.

Studying in Germany adds another frame of reference in his study of physics. “Lucas is bringing a unique perspective to our group by combining an interdisciplinary education in both electrical engineering and physics,” says Ramy El-Ganainy, associate professor of physics.

Lucas obtained an MS in Applied Physics (back in the spring of 2021). He entered the PhD candidacy at the end of this past spring semester. Upon getting his PhD, Lucas plans to pursue R&D-related work at Ft. Belvoir in Virginia for The Command, Control, Communications, Computers, Cyber, Intelligence, Surveillance and Reconnaissance (C5ISR) Center, the U.S. Army’s information technologies and integrated systems center.

Physics Major Anthony Palmer Wins Best Poster at Computing [MTU] Showcase

Michigan Tech physics and applied and computational mathematics double major Anthony Palmer, along with computer science PhD candidate Elijah Cobb, won the best poster recently in the Computing [MTU] Showcase for “Universal Sensor Description Schema: An extensible metalanguage to support heterogenous, evolving sensor data.”

Image of Anthony Palmer and Elijah Cobb in front of their poster at Michigan Tech’s Computing [MTU] Showcase
Anthony Palmer (left) and Elijah Cobb present their poster at Michigan Tech’s Computing [MTU] Showcase

Collecting and processing underwater sensor data is a critical need for U.S. Navy operations. Differences in sensor data types and forms presents a challenge for complete and accurate use of these data. The Universal Sensor Description Schema (USDS) project seeks to design, evaluate, and deploy a unified, extensible metalanguage for supporting legacy and future sensor data across multiple programming languages and environments. Michigan Tech is collaborating with Applied Research in Acoustics LLC to develop a robust programming environment for development of data-intensive applications.

Anthony came up with the idea for the project while interning at ARiA (a small research-and-development firm serving the Navy, government and industry). It’s been the basis for his senior thesis in physics. Anthony says “This project in particular has helped me learn alot about how programming languages work and are made. It also helped me learn a new functional programming language called “Racket”. Finally, it introduced me to some awesome people in the MTU computer science department including my partner Elijah Cobb and my advisor, Dr. Charles Wallace.”

Eye-opening describes the experience for Anthony.  He says, “I would say that I was surprised by the intricacy of how programming languages are built and function. I would also say that it was unexpected how useful recursion can be for solving problems in computing.” Recursion reduces time complexity, adds clarity and reduces the time needed to write and debug code.

Anthony graduates in a few short weeks. HIs attention will turn to the Navy, where he will be a submarine officer. Eventually he hopes to go into graduate school.