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information in conventional electronic systems necessi- tates the conversion of sound into electrical signals.
Topological acoustics enables new forms of acoustic infor- mation processing that rely on integrated circuits. The development of acoustic metamaterials has provided physi- cal platforms for the realization of acoustic Boolean logic gates. By exploiting their unique spectral, refractive, and phase properties, we can tailor the constructive or destruc- tive interference of input and control acoustic waves to achieve Boolean functions of choice (Bringuier et al., 2011). Similarly, interference has been used to demonstrate Bool- ean logic gates in acoustic metamaterials (Zhang et al., 2016), and acoustic logic elements have been demonstrated in driven chains of spherical particles (Li et al., 2014).
Until recently, all these acoustic information processing elements have made use of the spectral and refractive properties of composite materials. Topological acous- tics enables all-acoustic information processing that goes beyond the canonical attributes of sound, that is, frequency, wave vector, and dynamical phase (Deymier and Runge, 2017). The pseudospins enabling topologi- cal sound can be used as new degrees of freedom for information transport, realizing a wide range of acoustic Boolean logic elements with enhanced robustness and operating with low-energy requirements.
Interestingly, the features of topological acoustics offer avenues to go even beyond Boolean logic to pursue sound- based quantum-like information processing. In contrast to conventional computing, where a bit can be in a zero or one state, quantum computing processes a zero and a one at the same time by using a coherent superposition of states. Topological acoustic quantization, for example, based on two opposite pseudospins, coherence and cor- relations, can be harnessed to overcome stability and scalability challenges in current approaches to massive- data information processing, within the context of the second quantum revolution (Dowling and Milburn, 2003).
The pseudospin degrees of freedom of topological sound offer intriguing opportunities to achieve quantum-like phenomena like entanglement. Entanglement occurs when the state of a composite system composed of sub- systems cannot be described in terms of the states of independent subsystems. Entangled superpositions of quantum states exhibit the attributes of nonlocality and
nonseparability. Nonlocality is a unique feature of quan- tum mechanics that Einstein dubbed a “spooky action at a distance.” Nonlocality allows, for example, two photons of light to affect each other instantly, irrespective of their distance of separation. Acoustic waves, because of their nonquantum nature, that is, their “classical character,” are limited to local interactions. Nonetheless, nonseparabil- ity or classical entanglement can be realized in systems supporting classical waves, including sound.
An acoustic wave propagating in a cylindrical pipe can be represented by the product of three functions, each depen- dent on three degrees of freedom: one variable describing the pipe along its length and radial and angular variables characterizing the pipe through its cross section. In this sense, conventional guided acoustic waves are separable. A nonseparable acoustic wave, in contrast, is represented by a wave function that cannot be factored into a product of func- tions. Such waves can be created in externally driven systems composed of parallel arrays of waveguides coupled elastically and uniformly along their length (Hasan et al., 2019). These classically nonseparable states are constructed as a superpo- sition of acoustic waves, each a product of a plane wave and a spatial degree of freedom analogous to OAM (Figure 1c). The plane wave portion describes an elastic wave propagat- ing along the waveguides, and the spatial degree of freedom characterizes the amplitude and phase profile across the array of waveguides (Figure 4a). These nonseparable and therefore nonindependent degrees of freedom are the classical analogue of two correlated qubits. The amplitude of the nonseparable acoustic state is then analogous to the simplest examples of quantum entanglement of two qubits, known as Bell states in quantum mechanics. The displacement fields of the modes supported in the array of coupled waveguides are shown in Figure 4, b-d. Although Figure 4, b and d, shows separable OAM and plane wave states, characterized by symmetric pat- terns, Figure 4c shows a nonseparable and largely asymmetric linear combination of waves, with distinct momentum and OAM degrees of freedom. This type of acoustic superposition of states dramatically expands the opportunities for massive information storage and processing (Deymier et al., 2020).
Implementation of quantum-like algorithms necessitates the manipulation of nonseparable classical states, pro- viding the parallelism required to achieve the goals of quantum information science (Jozsa and Linden, 2003). The analogies between quantum mechanics and classi- cal wave physics have been recently exploited to emulate
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