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arXiv:2207.09296 · Scientific Reports

Classical analogue to driven quantum bits based on macroscopic pendula

Heribert Lorenz, Sigmund Kohler, Anton V. Parafilo, Mikhail N. Kiselev, Stefan Ludwig

Published Oct 26, 2023• 4 citations
arXivHTMLPDF

Abstract

Quantum mechanics increasingly penetrates modern technologies but, due to its non-deterministic nature seemingly contradicting our classical everyday world, our comprehension often stays elusive. Arguing along the correspondence principle, classical mechanics is often seen as a theory for large systems where quantum coherence is completely averaged out. Surprisingly, it is still possible to reconstruct the coherent dynamics of a quantum bit (qubit) by using a classical model system. This classical-to-quantum analogue is based on wave mechanics, which applies to both, the classical and the quantum world. In this spirit we investigate the dynamics of macroscopic physical pendula with a modulated coupling. As a proof of principle, we demonstrate full control of our one-to-one analogue to a qubit by realizing Rabi oscillations, Landau-Zener transitions and Landau-Zener-Stückelberg-Majorana interferometry. Our classical qubit demonstrator can help comprehending and developing useful quantum technologies.

DOI
10.1038/s41598-023-45118-y

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Quantum mechanics increasingly penetrates modern technologies but, due to its non-deterministic nature seemingly contradicting our classical everyday world, our comprehension often stays elusive. Arguing along the correspondence principle, classical mechanics is often seen as a theory for large systems where quantum coherence is completely averaged out. Surprisingly, it is still possible to reconstruct the coherent dynamics of a quantum bit (qubit) by using a classical model system. This classical-to-quantum analogue is based on wave mechanics, which applies to both, the classical and the quantum world. In this spirit we investigate the dynamics of macroscopic physical pendula with a modulated coupling. As a proof of principle, we demonstrate full control of our one-to-one analogue to a qubit by realizing Rabi oscillations, Landau-Zener transitions and Landau-Zener-Stückelberg-Majorana interferometry. Our classical qubit demonstrator can help comprehending and developing useful quantum technologies.
A quantum system can be driven by either sinusoidal, rectangular, or noisy signals. In the literature, these regimes are referred to as Landau-Zener-Stückelberg-Majorana (LZSM) interferometry, latching modulation, and motional averaging, respectively. We demonstrate that these pronounced and interesting effects are also inherent in the dynamics of classical two-state systems. We discuss how such classical systems are realized using either mechanical, electrical, or optical resonators. In addition to the fundamental interest of such dynamical phenomena linking classical and quantum physics, we believe that these are attractive for the classical analogue simulation of quantum systems.
Coupled nanomechanical resonators have recently attracted much attention for both fundamental studies in physics and broad applications in high-precession detection or sensing. By studying the Landau-Zener transitions and Rabi oscillation of two coupled resonators, it has been shown that such a two-mode system acts as a classical two-level system bearing the analog to a quantum mechanical two-level one. Here we construct a Stückelberg interferometer with two coupled cantilevers by driving the system through the avoided crossing twice. By measuring the non-adiabatic phase acquired at the Landau-Zener transition, we unveil an in-depth analog between the two-mode and quantum two-level systems. Our study opens up new opportunities for constructing interferometers with classical devices.
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.
In a previous paper [J. S. Briggs and A. Eisfeld, Phys. Rev. A 85, 052111 (2012)] we showed that the time development of the complex amplitudes of NNN coupled quantum states can be mapped by the time development of positions and velocities of NNN coupled classical oscillators. Here we examine to what extent this mapping can be realized to simulate the ``quantum,'' properties of entanglement and qubit manipulation. By working through specific examples, e.g., of quantum gate operation, we seek to illuminate quantum and classical differences which hitherto have been treated more mathematically. In addition, we show that important quantum coupled phenomena, such as the Landau-Zener transition and the occurrence of Fano resonances can be simulated by classical oscillators.
We find that a technique analogous to hybrid mechanics previously developed for Gaussian coherent states and molecular vibrations, can be extended to SU(2) coherent states describing coupled qubits. This formulation reveals a sparse structure of the propagator, leading to a computationally efficient scheme while maintaining numerical accuracy. As a result, the method enables simulations of larger qubit systems on classical computers. In particular, we demonstrate accurate real-time propagation for systems of up to 21 qubits, which would be computationally demanding using straightforward quantum-dynamics approaches.
A conventional realization of quantum logic gates and control is based on resonant Rabi oscillations of the occupation probability of the system. This approach has certain limitations and complications, like counter-rotating terms. We study an alternative paradigm for implementing quantum logic gates based on Landau-Zener-Stückelberg-Majorana (LZSM) interferometry with nonresonant driving and the alternation of adiabatic evolution and nonadiabatic transitions. Compared to Rabi oscillations, the main differences are a nonresonant driving frequency and a small number of periods in the external driving. We explore the dynamics of a multilevel quantum system under LZSM drives and optimize the parameters for increasing the gate speed. We define the parameters of the external driving required for implementing a specific quantum logic gate using the adiabatic-impulse model. In particular, we demonstrate the implementations of single-qubit X, Y, Hadamard gates, and two-qubit i and gates using the LZSM transitions. The considered LZSM approach for implementing arbitrary quantum logic gates can be applied to a large variety of multilevel quantum systems and external driving. Published by the American Physical Society 2024