# Duality

**Duality** – The self-interaction of a wave through its supporting medium creates acoustic wave states determined by self-interference phenomena. Analogously, the self-interaction of a sound source with its associated wave can lead to a path memory effect, which reveals the quantum mechanical concept of particle–wave duality in macroscopic systems. In the phonon representation of sound waves, these self-interference phenomena uncover the notion of duality in the quantum statistics (i.e., boson *vs*. fermion characterized by the symmetry of multiple particle states). Analogies with quantum phenomena interrogate mechanical waves in ways thought to be reserved for the microscopic realm. By advancing duality and exploiting new quantum-like states,* NewFoS* opens up **unprecedented parallelizable sound-based communication devices** that challenge the conventional transistor and integrated circuit—today’s conventional digital information processing and computational paradigms.

Critical developments in wave–particle duality in macroscopic systems, a phenomenon thought to be restricted to a peculiar microscale behavior, have been developed in our laboratories for 2-D hydrodynamic systems [1,4]. Bouncing drops of oil “walk” on an excited liquid oil surface near a Faraday instability (Fig. 4). Each droplet is accompanied by a surface wave that pilots its motion. Quantization emerges from the dynamic constraint imposed on the oil drop by its monochromatic pilot wave field. The droplet and the pilot wave interaction is but one example of classical phenomena displaying the fundamentally probabilistic nature of quantum mechanical phenomena: single-particle diffraction, tunneling, singly/doubly quantized orbits, orbital level splitting, spin and multimodal statistics.

**Figure 4 – Duality:** (A) A droplet bouncing on the surface of a vibrating fluid exhibits many quantum-like properties, including double-slit interference. (B) The droplet passes randomly through one opening while its “pilot wave” passes through both openings. After many repeat runs, a quantum-like interference pattern appears in the distribution of droplet trajectories. (C) Oscillating gas bubbles in a liquid irradiated with ultrasonic and megasonic sound waves behave as a 3-D quantum analog of the bouncing droplet. (D) High-speed imaging shows that interaction between secondary sound waves emitted by multiple oscillating gas form bound states.

In our laboratories, we have shown sound-supporting chiral media (describable within Dirac’s formalism of relativistic wave mechanics) lead to duality in the quantum-like statistics of sound waves [5,6]. Phonons are quanta of sound waves generally known to behave like bosons—there is no restriction on the number of particles per state. In some chiral media, the non-conventional spinor topology of the sound wave amplitude leads to an unusual constraint reminiscent of Pauli’s exclusion principle, i.e., no more than one “particle” in a state. The spinor degrees of freedom are quantized in the space of the propagation direction and the Eigen vectors for chiral media are a superposition of states traveling in opposite directions—referred to as quasi-standing waves. Within the context of field-theoretical approaches, a “second quantization” of these waves leads to anti-commutation rules for their creation and annihilation operators, and a multiphonon wave function with fermion-like phase characteristics [6].

- J.W.M. Bush, “Pilot-wave hydrodynamics,”
*Ann. Rev.**Fluid Mech*. 47, 269 (2014). - Y. Couder, S. Protiere, E. Fort and A. Boudaoud, “Dynamical phenomena: Walking and orbiting droplets,”
*Nature*437, 208 (2005). - A.U. Oza, D.M. Harris, R.R. Rosales and J.W.M. Bush, “Pilot-wave dynamics in a rotating frame: on the emergence of orbital quantization,”
*J. Fluid Mechanics*744, 404 (2014). - J. Molacek and J.W.M. Bush, “Drops walking on a vibrating bath: towards a hydrodynamic pilot-wave theory,”
*J. Fluid. Mechanics*727, 612 (2013). - P.A. Deymier, K. Runge, N. Swinteck and K. Muralidharan, “Torsional Topology and Fermion-like Behavior of Elastic Waves in Phononic Structures,”
*Comptes Rendus de L’académie des Sciences*(France), in press. - P.A. Deymier, K. Runge, N. Swinteck and K. Muralidharan, “Rotational modes in a phononic crystal with fermion-like behavior,”
*J. Appl. Phys*. 115, 163510 (2014).