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Scientific Reports

Experimental demonstration of coherent superpositions in an ultrasonic pseudospin

Lázaro Calderín, M. Arif Hasan, Neil G. Jenkins, Trevor D. Lata, Pierre Lucas, Keith Runge, Pierre Alix Deymier

Published Oct 2, 2019• 14 citations
Publisher

Abstract

We experimentally demonstrate the existence and control of coherent superpositions of elastic states in the direction of propagation of an ultrasonic pseudospin i.e., a φ-bit. The experimental realization of this mechanical pseudospin consists of an elastic aluminum rod serving as a waveguide sandwiched between two heavy steel plates. The Hertzian contact between the rod and the plates leads to restoring forces which couple the directions of propagation (forward and backward). This coupling generates the coherence of the superposition of elastic states. We also demonstrate φ-bit gate operations on the coherent superposition analogous to those used in quantum computing. In the case of a φ-bit, the coherent superposition of states in the direction of propagation are immune to wave function collapse upon measurement as they result from classical waves.

DOI
10.1038/s41598-019-50366-y

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We experimentally demonstrate the existence and control of coherent superpositions of elastic states in the direction of propagation of an ultrasonic pseudospin i.e., a φ-bit. The experimental realization of this mechanical pseudospin consists of an elastic aluminum rod serving as a waveguide sandwiched between two heavy steel plates. The Hertzian contact between the rod and the plates leads to restoring forces which couple the directions of propagation (forward and backward). This coupling generates the coherence of the superposition of elastic states. We also demonstrate φ-bit gate operations on the coherent superposition analogous to those used in quantum computing. In the case of a φ-bit, the coherent superposition of states in the direction of propagation are immune to wave function collapse upon measurement as they result from classical waves.
One of the quantum theory concepts on which quantum information processing stands is superposition. Here we provide experimental evidence for the existence of classical analogues to the coherent superposition of energy states, which is made possible by the Hertz-type nonlinearity of the granules together with the external driving field. The granules' nonlinear vibrations are projected into the linear modes of vibration, which depend on one another through the phase and form a coherent superposition. We show that the amplitudes of the coherent states form the components of a state vector that spans a two-dimensional Hilbert space, and time enables the system to span its Hilbert space parametrically. Thus, the superposition of states can be exploited in two-state quantum-like computations without decoherence and wave function collapse. Finally, we demonstrate the experimental realization of applying a reversible Hadamard gate to a pure base state that brings the state into a superposition.
Nonlinear mechanical oscillators can emulate qubit analogue algebra by leveraging multiple harmonics of large‑amplitude vibrations. We realize a logical elastic bit—a room temperature mechanical analogue of a qubit—in a two‑mass oscillator joined by a conical spring whose graded stiffness generates a robust sequence of phase‑coherent harmonics. From time‑series velocity measurements, a Fourier–projection maps the response onto the complete space of in‑phase and out‑of‑phase eigenvectors. The resulting complex coefficients define a Bloch‑sphere representation in which classical superpositions are directly controllable. Moreover, we gain more control over the coefficients by pairing the Fourier harmonics in different orders. When the paired Fourier components share a frequency, the coefficients are independent of time, producing tunable states that can serve as phase-defined memory. Pairing distinct harmonics introduces a beat frequency that drives deterministic precession of the Bloch vector, realizing single‑bit rotations (e.g., Pauli‑X and Hadamard analogues) without the need of additional external input, with time as the gate clock. By splitting the spectrum into blocks, a single resonator can host several elastic bits at once. The Hilbert space grows with the number of blocks while the hardware stays the same, allowing scalable architectures that show classical non-separable correlations. A linear mass-spring model yields closed‑form eigenfrequencies and Bloch‑angle formulas that overlay measured trajectories across resonance and provide design rules for state initialization and gate timing. All operations occur at ambient conditions and require no feedback or cryogenics, establishing a simple, reproducible route to quantum‑inspired logic in macroscopic mechanics.
The Controlled-NOT (CNOT) gate is the key to unlock the power of quantum computing as it is a fundamental component of a universal set of gates. We demonstrate the operation of a two-bit C-NOT quantum-like gate using classical qubit acoustic analogues, called herein logical phi-bits. The logical phi-bits are supported by an externally driven nonlinear acoustic metamaterial composed of a parallel array of three elastically coupled waveguides. A logical phi-bit has a two-state degree of freedom associated with the two independent relative phases of the acoustic wave in the three waveguides. A simple physical manipulation involving the detuning of the frequency of one of the external drivers is shown to operate on the complex vectors in the Hilbert space of pairs of logical phi-bits. This operation achieves a systematic and predictable C-NOT gate with unambiguously measurable input and output. The possibility of scaling the approach to more phi-bits is promising.
The creation of multilevel quantum states, qudits, has revolutionized concepts for quantum computing. Classical systems that capture behavior analogous to quantum systems have been demonstrated. In this spirit, we consider a three-level classical analogue of the qudit composed of coupled acoustic waveguides. Here, we demonstrate both the experimental realization of a three-level classical analogue of the qudit and the creation and tuning of nonseparable superpositions of two of these analogues, which are classically “entangled.” Measurements of velocities and transmission inform our assignment of these nonseparable states.
In quantum information science and technology (QIST), the superposition of states is pivotal. Current qubits, formed by superposing two pure states, are foundational in QIST but are susceptible to decoherence, causing loss of superposed states. Addressing this, we introduced an "elastic bit" using the Hertz-type nonlinearity of elastic beads under external driving. These beads exhibit phase-dependent nonlinear vibrations, leading to coherent superpositions of energy states in linear vibration modes. The resulting amplitudes span a two-dimensional Hilbert space, with time allowing parametric spanning. We showcased the Hadamard gate, translating one state superposition to another through a unitary transformation. Remarkably, these elastic bits operate at room temperature and resist decoherence, making them apt for intermediate-scale quantum analogue computing. They offer solutions for challenges in supercomputing for artificial intelligence and autonomous vehicle data processing.
Abstract We present a model of an externally driven acoustic metamaterial constituted of a nonlinear parallel array of coupled acoustic waveguides that supports logical phi-bits, classical analogues of quantum bits (qubit). Descriptions of correlated multiple phi-bit systems emphasize the importance of representations of phi-bit and multiple phi-bit vector states within the context of their corresponding Hilbert space. Experimental data are used to demonstrate the realization of the single phi-bit Hadamard gate and the phase shift gate. A three phi-bit system is also used to illustrate the development of multiple phi-bit gates as well as a simple quantum-like algorithm. These demonstrations set the stage for the implementation of a digital quantum analogue computing platform based on acoustic metamaterial that can implement quantum-like gates and may offer promise as an efficient platform for the simulation of materials.