Researchers experimentally realize non-Abelian permutations in an acoustic system- Times Of Nation
A study led by Dr. Guancong Ma (Department of Physics, Hong Kong Baptist University, Kowloon Tong, Hong Kong, China) and Dr. Kun Ding (Department of Physics, Fudan University, Shanghai 200438, China) provides a new perspective for the study of non-Abelian phenomena and states permutations.
Permutation is of great significance for a diversity of physical problems. For instance, one effective way to distinguish bosons from fermions is to exchange two identical particles. As a new quasi-particle different from fermion and boson, the permutation of anyons is more complicated and needs to be described by a matrix. Due to the fact that the commutative law cannot be applied to matrix multiplication, non-Abelian permutations are expected in an anyon system. On top of that, how to realize non-Abelian permutations in other systems has sparked great interest.
“The geometric phase of multiple states under adiabatic evolution is essentially a unitary matrix, which can be mapped to non-Abelian groups. This makes it possible to realize non-Abelian permutation through the parallel transport of three or more degenerate states.” Ma explains.
Hermiticity is an important mathematical property of matrices or operators, which guarantees the existence of real energy spectrum—a natural expectation for many physical systems in both classical and quantum mechanics. However, many studies in the past two decades have shown that non-Hermitian formalism can sometimes play a better role in describing open systems, which exchange energy with their environment. Systems with gain and/or loss belong to such a category. In this work, the researchers illustrate that non-Hermitian systems are distinguished from their Hermitian counterparts by a key characteristic, complex eigenvalues and eigenfunctions. This feature allows multiple interconnected eigenvalue Riemannian surfaces, on which branch singularities known as exceptional point (EP) can be found. “When you draw a closed loop around an EP, you may end up on a different surface, just like going up or down a spiral staircase. This process is accompanied by the exchange of the eigenstates. This is the key to realizing the state permutations in our work.” Ding adds.
This study is based on a three-state non-Hermitian system that forms two separate EAs, smooth trajectories of order-2 EPs, in a 3D parameter space. Researchers find that one EA is constituted by the coalescence of state-1 and 2, thus encircling any EP of this EA results in the swapping of state-1 and 2, corresponding to the operation μ2 of the non-Abelian dihedral group. In contrast, another EA is generated by the coalescence of state-2 and 3, indicating that state-2 and 3 can be exchanged (operation μ1).
Researchers further concatenate the encircling loops around two EAs in different orders, thus attaining two distinct three-state permutations, which is definitely a manifestation of the non-Abelian characteristics. These two states permutations can be described by the operations of the non-Abelian group, ρ1 and ρ2.
These permutations are experimentally observed in acoustics. The acoustic system is similar to the one reported in Science (2020). In this experiment, the researchers tune the loss and cavity volume of the acoustic cavities to the specific values defined by the encircling loop around the EP. Then they measure the eigenvalues and eigenfunctions of the acoustic system at different parameter points. The evolution of the eigenvalues and the parallel transport of the eigenfunctions are recorded, demonstrating the non-Abelian permutations in a three-state non-Hermitian system.
“Our work reveals that non-Hermitian systems can be applied to the study of non-Abelian phenomena and provides the theoretical and experimental evidence for the evolution of multiple states in non-Hermitian systems,” Ma adds. “At the same time, it also paves a new way for the development of non-Hermitian systems in the fields of acoustics, optics, and mechanics.”
The research was published in National Science Review.
Researchers find new way of gaining quantum control from loss
Weiyuan Tang et al, Experimental Realization of Non-Abelian Permutations in a Three-State Non-Hermitian System, National Science Review (2022). DOI- 10.1093/nsr/nwac010
Science China Press
Researchers experimentally realize non-Abelian permutations in an acoustic system (2022, March 18)
retrieved 20 March 2022
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