Non-Hermitian quantum systems, characterized by gain and loss terms, have recently attracted great interest. The focus of researchers extends beyond the non-Hermitian degeneracies, known as exceptional points (EPs), to the nonadiabatic transitions (NATs) that occur during the dynamic encircling of the EPs. The presence of NATs around the EPs can lead to a chiral state transfer, in which the final state is determined by the encircling direction and not the initial state. This intriguing phenomenon has been extensively studied in classical systems, but there is still a lack of experimental evidence in quantum systems. To start my investigations into non-Hermitian systems, I collaborated with our partner groups to successfully demonstrate chiral state transfer, both with and without encircling an exceptional point, in a many-body spin-orbit-coupled cold-atom system.

During my investigation of chiral state transfer, we found that NATs can actually serve as an intermediate process to assist interband transitions when the initial and final states are positioned in the parameter space governed by a Hermitian Hamiltonian (Hermitian parameter space). Conversely, intraband transitions can be realized using the adiabatic theorem by slowly altering the parameters of a Hermitian system. As a result, we achieved full control of band transitions (both intraband and interband transitions) in the Hermitian parameter space.

While non-Hermitian systems exhibit NATs, these transitions can also occur in Hermitian systems due to the large interband Berry connections in a rapidly time-varying system. In Hermitian systems, NATs can be avoided if the interband Berry connections are significantly smaller than the instantaneous band gap. However, in non-Hermitian systems, this becomes impossible because the NATs arise from the competition between the interband Berry connections and the complex dynamic phase factors. When the parameters change rapidly, the interband Berry connections increase and induce NATs. Conversely, when parameters change slowly, even though the interband Berry connections are small, the complex dynamic phase factors grow exponentially over time, ultimately leading to NATs.

Recent research has proposed minimizing or shutting down the interband Berry connection as shortcuts to adiabaticity (STA) to avoid NATs in time-varying Hermitian systems with rapid changes. First, I would like to test and enhance these minimization approaches, specifically fast quasiadiabatic 2 driving (FAQUAD). With this method, by optimizing the maximum value of the interband Berry connection over the band gap (adiabaticity parameter), I can perform seemingly adiabatic evolution in a significantly shorter time than the conventional adiabatic theorem allows. I chose not to use a quantum experimental platform to demonstrate this approach, as observing the intermediate process could cause the quantum system to lose coherence. However, classical systems, which are generally robust against detection, are excellent candidates for demonstrating STA approaches. In this case, we utilized a coupled elastic waveguide system to mimic the quantum system and test the FAQUAD approach.

Second, the approach to shutting down the interband Berry connection in Hermitian systems typically requires additional non-Hermitian parameters. This original path in Hermitian systems is referred to as the reference path, while paths with added non-Hermitian parameters are known as nonHermitian shortcuts. Here, unlike previous research, I demonstrate that this approach is also useful when the reference path is within the non-Hermitian parameter space, so that the state can undergo seemingly adiabatic evolution around the Reimann surface. In testing my concept within the elastic waveguide system, I experimentally assessed the complex Young's modulus of polydimethylsiloxane (PDMS), which acts as a medium to introduce artificial loss.

In this thesis, with a comprehensive understanding of NATs, I have achieved full control of band transitions, including both intraband and interband transitions, within the parameter space governed by a Hermitian Hamiltonian. Moreover, I have accelerated the adiabatic process by minimizing the adiabaticity parameter in the Hermitian system and avoided NATs in non-Hermitian systems by shutting down the Berry connection, thus achieving seemingly adiabatic evolution near the Riemann surface. These concepts have been experimentally demonstrated not only in a quantum system (cold atoms) but also in a classical system (elastic system).