Mimicking quantum tunnelling using programmable active acoustic metamaterials

In a recent pilot study, researchers report a peculiar tunnelling phenomenon that occurs in acoustical systems with non-reciprocal couplings, which can open new opportunities for controlling the propagation of sound waves in ducts and other acoustical waveguides.

By Felix Langfeldt and Joe Tan, Institute of Sound & Vibration Research, University of Southampton, and Lea Beilkin and Sayan Jana, School of Mechanical Engineering, Tel Aviv University

Tunnelling is a phenomenon in quantum mechanics. It allows particles to pass, with a certain probability, through a barrier that would be impenetrable for the particle according to classical mechanics. Known phenomena include, for example, tunnelling of relativistic particles through energy barriers of arbitrary heights and widths in graphene and other structures, across the event horizon of black holes, through superconducting junctions, and more. Translating this exciting property into the acoustic realm holds the potential to substantially advance waveguiding capabilities in acoustical systems.

In the project Active acoustic metamaterials for non-Hermitian sound propagation phenomena inspired by quantum mechanics, funded by the UK Acoustics Network’s third funding call, the team of scientists from the University of Southampton and Tel Aviv University derived and experimentally demonstrated a new form of a tunnelling-like phenomenon for sound waves through a non-Hermitian barrier. Non-Hermitian physics refers to non-conservative systems that interact with the environment. One of the underlying mechanisms of these systems is non-reciprocity. The wave propagation through such non-reciprocal systems is usually amplified in one direction and attenuated in the other, which is known by the non-Hermitian skin effect.

Above: Figure 1: Schematic representation of the active acoustic metamaterial setup that was derived and studied during the project. The drawing on the left shows the overall setup with several active unit cells (grey) embedded in the wall of an acoustic waveguide. Each unit cell consists of two loudspeakers and two microphones (right). A programmable controller allows to change the parameter η that determines the strength and direction of the system’s non-reciprocity

Design of the active acoustic metamaterial
For studying the behaviour of sound waves propagating through a non-Hermitian acoustic system, the team designed a one-dimensional waveguide with an array of loudspeakers, microphones, and controllers (see Figure 1), forming an active acoustic metamaterial. Metamaterials are periodic structures composed of engineered unit cells that are smaller than the wavelength and by carefully designing the properties of the unit cells and including active elements like loudspeakers and microphones, the behaviour of sound waves propagating through the metamaterial can be tailored towards a desired response. The key to realising non-reciprocity in this system lies in the control parameter η, shown in Figure 1, which determines the degree of non-reciprocity in each part of the metamaterial. In the left half (blue arrows), the weaker coupling (1-η) is in the left direction and the stronger coupling (1+η) is in the right direction. The couplings are mirrored about the centre of the metamaterial. As a standalone system, each half of the system supports the non-Hermitian skin effect, in which modes are accumulated at the boundary of the weaker coupling. However, in the project the team investigated what happens to a wave that propagates along the array and hits the interface. The skin mode accumulation, which effectively constitutes a barrier, forbids wave penetration into the interface, but surprisingly, under certain conditions the wave is transmitted to the other side, keeping the interface dark, as if the wave invisibly tunnelled through it, as illustrated in Figure 2. Remarkably, the tunnelling is independent of the interface length.

Time-domain finite element method simulations were used to verify the tunnelling effect in a continuous acoustic system as shown in Figure 1, investigate the impact of the electroacoustic properties of the loudspeakers and how they are driven on the tunnelling performance, and explore possible approaches for re-programming the control laws to achieve different tunnelling strengths or even different non-Hermitian sound propagation effects. To verify the numerical results, the team then built an experimental demonstrator (see Figure 2), which consists of an array of nine loudspeakers and microphones and a digital controller. The experimental results confirm the tunnelling phenomenon, and demonstrate the re-programmability of the system by modifying the control law on the digital controller for each unit cell, e.g. to study different non-reciprocity strengths η.

Above: Figure 2: Experimental setup for demonstrating the quantum tunnelling effect realised using the active acoustic metamaterials developed in the project. This photograph shows the rectangular duct with an array of nine loudspeakers attached to the top wall of the duct, representing the active metamaterial unit cells. The colour map along the centre of the duct is a computer-generated overlay created using a simulation model, illustrating the tunnelling of a sound wave through the metamaterial

Applications
This pilot study uncovered how an interesting tunnelling phenomenon can be realised for acoustic waves using active acoustic metamaterials. This can lead to a range of possible useful applications, e.g. to realise quiet zones in acoustical systems or tunnel sound waves through obstacles for sensing applications. The theoretical, numerical, and experimental framework that was developed in the project will enable further research on this specific phenomenon, e.g. to study the stability of the control laws, extend the concept to higher dimensions, or investigate wave tunnelling through other types of non-Hermitian interfaces, which may also include nonlinearities, time-dependence and more.

Acknowledgement
This project was supported by the UK’s Engineering and Physical Sciences Research Council (EPSRC) through the third funding call by the UK Acoustics Network Plus EP/V007866/1.