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arXiv:2402.07672 · Physical Review Research

Photonic cellular automaton simulation of relativistic quantum fields: Observation of Zitterbewegung

Alessia Suprano, Danilo Zia, Emanuele Polino, Davide Poderini, Gonzalo Carvacho, Fabio Sciarrino, Matteo Lugli, Alessandro Bisio, Paolo Perinotti

Published Aug 5, 2024• 2 citations

Abstract

Quantum cellular automaton (QCA) is a model for universal quantum computation and a natural candidate for digital quantum simulation of relativistic quantum fields. Here we introduce the first photonic platform for implementing QCA simulation of a free relativistic Dirac quantum field in 1 + 1 dimension, through a Dirac quantum cellular automaton (DQCA). Encoding the field position degree of freedom in the orbital angular momentum (OAM) of single photons, our state-of-the-art setup experimentally realizes eight steps of a DQCA, with the possibility of having complete control over the input OAM state preparation and the output measurement making use of two spatial light modulators. Therefore, studying the distribution in the OAM space at each step, we were able to reproduce the time evolution of the free Dirac field observing the , an oscillatory movement extremely difficult to see in a real-case experimental scenario that is a signature of the interference of particle and antiparticle states. The accordance between the expected and measured oscillations certifies the simulator performances, paving the way towards the application of photonic platforms to the simulation of more complex relativistic effects. Published by the American Physical Society 2024

DOI: 10.1103/physrevresearch.6.033136

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It has been shown that certain quantum walks give rise to relativistic wave equations, such as the Dirac and Weyl equations, in their long-wavelength limits. This intriguing result raises the question of whether something similar can happen in the multiparticle case. We construct a one-dimensional quantum cellular automaton (QCA) model, which matches the quantum walk in the single particle case and which approaches the quantum field theory of free fermions in the long-wavelength limit. However, we show that this class of constructions does not generalize to higher spatial dimensions in any straightforward way and that no construction with similar properties is possible in two or more spatial dimensions. This rules out the most common approaches based on QCAs. We suggest possible methods to overcome this barrier while retaining locality.

The Thirring Quantum Cellular Automaton (QCA) describes the discrete time dynamics of local fermionic modes that evolve according to one step of the Dirac cellular automaton, followed by the most general on-site number-preserving interaction, and serves as the QCA counterpart of the Thirring model in quantum field theory. In this work, we develop perturbative techniques for the QCA path sum approach, expanding both the number of interaction vertices and the mass parameter of the Thirring QCA. By classifying paths within the regimes of very light and very heavy particles, we computed the transition amplitudes in the two- and three-particle sectors to the first few orders. Our investigation into the properties of the Thirring QCA, addressing the combinatorial complexity of the problem, yielded some useful results applicable to the many-particle sector of any on-site number-preserving interactions in one spatial dimension.

This study presents a unitary quantum cellular automaton (QCA) that, in the continuum limit, converges to the (1+1)-dimensional Generalized Dirac Equation (GDE). We outline the construction of the unitary, discrete-time evolution and derive the model's exact dispersion relation from its eigenvalue spectrum, showing it possesses the correct relativistic limit. We then explore its dynamics through numerical simulations. While the total probability density is shown to be insensitive to the chiral angle parameter

(ρ)

of the GDE, we demonstrate that this parameter has a profound and directly observable effect on the particle's local spin polarization. By measuring this quantity, we reveal the hidden internal dynamics of the spinor, providing a clear, visual confirmation of the physical consequences of the chiral mass term.

Dirac Cellular Automata (DCA) provide a framework for simulating Dirac dynamics, yet the rigid coupling between spatial and temporal resolutions can introduce artificial phase-matching symmetries on finite grids that suppress interference phenomena such as Zitterbewegung. In this work, we propose a Split-Step Dirac Cellular Automaton (SDCA) that enables continuous-time Dirac evolution at fixed spatial discretization. By employing a Trotterized fractional-step scheme in the momentum representation, SDCA breaks the phase-matching cancellation present in the standard DCA and recovers the interference dynamics of the continuous-time limit. We benchmark the SDCA through analytical and numerical studies and demonstrate its implementation on IBM Quantum processors. Despite the increased circuit depth required for finer temporal resolution, the NISQ implementation reproduces the characteristic velocity oscillations and entanglement-entropy dynamics of the continuous-time model. We further investigate hardware-topology trade-offs and dynamic circuit implementations of the Quantum Fourier Transform (QFT), highlighting the competing effects of gate errors, measurement, and feed-forward latency. These results demonstrate that SDCA provides a practical framework for improving temporal resolution while maintaining a fixed spatial quantum register, enabling the exploration of relativistic quantum dynamics on near-term quantum devices.

We study the dynamical behavior of a quantum cellular automaton which reproduces the Dirac dynamics in the limit of small wave vectors and masses. We present analytical evaluations along with computer simulations, showing that the automaton exhibits typical Dirac dynamical features, such as the Zitterbewegung and, considering the scattering from potential, the so-called Klein paradox. The motivation is to show concretely how pure processing of quantum information can lead to particle mechanics as an emergent feature, an issue that has been the focus of solid-state, optical, and atomic-physics quantum simulators.